journal of naval architecture and marine engineering june, 2016 http://dx.doi.org http://dx.doi.org/10.3329/jname.v13i1.25338 http://www.banglajol.info 1813-8235 (print), 2070-8998 (online) © 2016 aname publication. all rights reserved. received on: oct. 2015 dual solutions for unsteady flow of powell-eyring fluid past an inclined stretching sheet p. mohan krishna 1 , n. sandeep 2 , j. v. ramana reddy 1 and v. sugunamma 1* 1 department of mathematics, sri venkateswara university, tirupati-517502, andhra pradesh, india. .*email: vsugunar@gmail.com 2 division of fluid dynamics, vit university, vellore-632014, tamilnadu, india abstract: this paper deals with the heat and mass transfer in unsteady flow of powell-eyring fluid past an inclined stretching sheet in the presence of radiation, non-uniform heat source/sink and chemical reaction with suction/injection effects. the governing equations are reduced into system of ordinary differential equations using similarity transformation and solved numerically using runge-kutta based shooting technique. results display the influence of governing parameters on the flow, heat and mass transfer, friction factor, local nusselt and sherwood numbers. comparisons are made with the existed studies. present results have an excellent agreement with the existed studies. results indicate that an increase in the chemical reaction parameter depreciates the friction factor, heat transfer rate and enhances the mass transfer rate. dual solutions exist only for certain range of suction/injection parameters. keywords: powell-eyring fluid, non-uniform heat source/sink, radiation, chemical reaction, suction/injection. introduction the flow of non-newtonian fluids has attained a greatest importance and increasing interest in the theory of fluid mechanics. fluids which do not obey newton’s law of motion are called non-newtonian fluids. few examples of non-newtonian fluids are tooth paste, food products, flow of blood etc. many investigators studied electrically conducting non-newtonian fluids of the two-dimensional magneto hydrodynamic boundary layer flows. it is assumed that non-newtonian behaviour described by power-law model. the powell-eyring model is mathematically more complex, and it has advantages of power-law model. firstly, in the case of power-law model, it is deduced from kinematic theory of liquid rather than the empirical relation. secondly it correctly reduces to high shear stress and newtonian behaviour for law. crane (1970) investigated an exact analytical solution for the steady two dimensional flows due to a stretching surface in a quiescent fluid. the effect of variable viscosity and viscous dissipation on the flow of a third grade fluid in a uniform pipe has been studied by massoudi and christie (1995). nadeem and ali (2009) and nadeem et al. (2010) investigated the pipe flow of non-newtonian fluid with variable viscosity by considering no slip and partial slip conditions and found that heat transfer performance of the flow is significant in the presence of slip effect. javed et al. (2013) studied the boundary layer flow over a stretching sheet for non-newtonian fluid, namely the eyring-powell model. a theory based model for evaluating the temperature and volume fraction effects on nano fluids was presented by hosseini and ghader (2010). ibrahim et al. (2008) studied the unsteady mhd free convection flow past a semi-infinite vertical plate in the presence of chemical reaction and radiation absorption. muthucumaraswamy and ganesan (2001) investigated the effect of heat transfer on the unsteady flow past an impulsively started vertical plate with first order chemical reaction. eldabe et al. (2003) and zueco and beng (2009) discussed the effects of couple stress between two parallel plates using eyring-powell model. prasad et al. (2013) presented the non-newtonian powell-eyring fluid over a non-isothermal stretching sheet. patel and timol (2009) presented mhd powell-eyring fluid flow using the method of asymptotic boundary conditions. the effect of radiation on the flow of a micropolar fluid over a nonlinearly stretching sheet was discussed by babu et al. (2015). anderson et al. (1992) investigated the non-newtonian flow of a power-law fluid past a stretching surface in steady two dimensional flows. mohankrishna et al. (2015) presented the radiation and chemical reaction effects on mhd convective flow over a permeable stretching surface with suction and heat generation by using the shooting technique. sandeep et al. (2016) analyzed the effect of heat and mass transfer in thermophoretic radiative hydro magnetic nano fluid flow over an exponentially stretching porous sheet. makinde and aziz mailto:vsugunar@gmail.com p. m. krishna, n. sandeep, j. v. r. reddy and v. sugunamma / journal of naval architecture and marine engineering 13(2016) 89-99 dual solutions for unsteady flow of powell-eyring fluid past an inclined stretching sheet 90 (2011) discussed the effect of boundary layer flow of a nano fluid past a stretching sheet. sandeep et al. (2014) analyzed aligned magnetic field effect on unsteady convection flow over a moving vertical plate in porous medium in the presence of radiation. the unsteady two-dimensional flow of a non-newtonian fluid over a stretching surface having a prescribed surface temperature is investigated by mukhopadhyay et al. (2013). nadeem et al. (2013) investigated mhd casson fluid flow in two lateral directions past a porous linear stretching sheet. raju et al. (2015) investigated the effects of cross-diffusion and radiation in steady twodimensional flow over a vertical stretching surface in the presence of aligned magnetic field. layek et al. (2007) analysed the effect of heat and mass transfer analysis for boundary layer stagnation-point flow of stretching sheet. hossain and takhar (1996) worked the effect radiation on mixed convection flow with uniform surface temperature. the influence of non uniform heat source on unsteady flow of a stretching sheet was studied bt tsai et al. (2008). peristaltic flow of non-newtonian fluid in a symmetric channel was investigated by hayat et al. (2014). to the authors knowledge no studies have been reported yet on the heat and mass transfer in unsteady flow of powell-eyring fluid past an inclined stretching sheet in the presence of radiation, non-uniform heat source/sink and chemical reaction with suction/injection effects. in this study, we analyzed the heat and mass transfer in unsteady flow of powell-eyring fluid past an inclined stretching sheet in the presence of radiation, non-uniform heat source/sink and chemical reaction with suction/injection effects numerically. 2. mathematical formulation consider an incompressible, two-dimensional unsteady flow of power-eyring fluid past an inclined stretching sheet. the sheet makes an angle α with the vertical direction as shown in fig. 1. the x-axis is taken along the sheet and y axis is normal to it. in addition, we considered the effects of thermal radiation, chemical reaction and non-uniform heat source/sink. fig. 1: physical model and coordinate system the cauchy stress tensor in power-eyring fluid is given by 11 1 sinh , i i ij j j u u x x                 where μ is the viscosity coefficient, β and γ are the material fluid parameters. the boundary layer equations comprising the balance laws of mass, linear momentum and energy can be written as hayat al. (2015) 0, u v x y       (1)     2 2 2 02 23 1 1 cos , 2 t c u u u u u u u v g t t c c t x y yy y                                          (2) 2 2 ''',r p qt t t t c u v k q t x y yy                  (3) p. m. krishna, n. sandeep, j. v. r. reddy and v. sugunamma / journal of naval architecture and marine engineering 13(2016) 89-99 dual solutions for unsteady flow of powell-eyring fluid past an inclined stretching sheet 91   2 2 , m l c c c c u v d k c c t x y y               (4) where t is the time,     is the kinematic viscosity, k is the thermal conductivity of the fluid, ρ is the fluid density, t is the fluid temperature, c is the fluid concentration, pc is the specific heat, og is the acceleration due to gravity, t  and c is the volumetric coefficient of thermal and mass exponential, md is the is the molecular diffusivity of the species concentration, lk is the chemical reaction parameter, * 3 * 16 3 r t t q k y      is the linearized radiative heat flux,  is the inclined angle, * k is the mean absorption coefficient, *  is the stefan-boltzmann constant, '''q is the non-uniform heat source ( '''q >0) or sink ( '''q <0) per unit volume. the non-uniform heat source/sink, '''q is modeled by the following expression    * * (x, t) ''' ' , s s ku q a t t f b t t x         (5) in which * a and * b are the coefficients of space and temperature dependent heat source/sink, respectively. here two cases arise. for internal heat generation * 0a  and * 0b  and for internal heat absorption, we have * 0a  and * 0.b  the surface velocity is denoted by     , 1 s bx u x t at   , whereas the surface temperature     2 3 2 , 1 2 s ref bx t x t t t at       and surface concentration     2 3 2 , 1 2 s ref bx c x t c c at       . here b (stretching rate) and a are positive constants having dimension time. also ref t , ref c are constant reference temperature and concentration respectively. the boundary conditions are taken as follows:       , , , , , , , 0, 0, , s s s s u u x t v v t t x t c c x t at y u t t c c as y             (6) by introducing the similarity transformations           0 ' , , , 1 1 , , , 1 (1 ) s l s t tbx b u f v f at at t t kc cb y k at c c at                           (7) equation (1) is identically satisfied and equations (2)-(6) become     2 2 1 1 ''' ' '' '' ''' ' '' cos cos 0, 2 f f ff f f f f gr gc                       (8)   * * 4 1 1 '' pr ' 2 ' 3 ' ' 0, 3 2 r f f a f b                          (9)   1 1 '' 3 ' 2 ' ' 0, 2 f f kr sc             (10) boundary conditions are , ' 1, 1, 1 0, ' 0, 0, 0 , f s f at f as                 (11) where prime denotes differentiation with respect to , f is the dimensionless stream function,  is the dimensionless temperature, is the dimensionless concentration and the dimensionless numbers are p. m. krishna, n. sandeep, j. v. r. reddy and v. sugunamma / journal of naval architecture and marine engineering 13(2016) 89-99 dual solutions for unsteady flow of powell-eyring fluid past an inclined stretching sheet 92     3 23* 3 0 0 * 2 2 2 2 2 3 2 0 0 2 2 2 2 41 , , , ,s , re , , pr , , , re t ss x s x pc s x ms x g t t xu gr vt r gr kk x u x vb cg c c x gc ka gc sc kr b k d bu x                               (12) where and  are the dimensionless material fluid parameters, r is the radiation parameter, gr is the thermal grashof number, gc is the mass grashof number, ε is the unsteadiness parameter, pr is the prandtl number and sc is the schmidt number, kr is the chemical reaction parameter and s is the suction/injection parameter, where 0s  for suction and 0s  for injection. for engineering interest the coefficient of skin friction, local nusselt and sherwood numbers are defined as 1/2 re ''(0), f x c f (13) 1/ 2 4 re 1 '(0), 3 x x nu r           (14) 1/ 2 re '(0), x x sh     (15) where re s x u x v  is the local reynolds number. 3. results and discussion eqs. (8) (10) with the boundary conditions (11) have been solved numerically using runge-kutta based shooting technique. the results obtained shows the influence of the non-dimensional governing parameters, namely radiation parameter r , material fluid parameter г, unsteadiness parameter ε, inclined angle α, chemical reaction parameter kr, schmidt number sc, mass grashof number gc and non-uniform heat source/sink parameters a* and b* on velocity, temperature, concentration, friction factor, nusselt and sherwood numbers. for numerical results we considered 0.2, 0.5, / 4kr gc r           pr 1,gr  3,  a*= b*=0.1, sc=0.6. these values are kept as common in entire study except the varied values as displayed in the respective figures and tables. figs. 2 and 3 depict the influence of radiation parameter on velocity and temperature profiles of the flow. it is evident from the figures that an increase in the radiation parameter enhances the velocity and temperature profiles of the flow. it is also observed that enhancements in the velocity and temperature fields are significantly high in injection case. generally, an increase in the radiation releases the heat energy to the flow field. this causes to enhance the velocity and thermal boundary layers. in view of this we can conclude that influence of radiation is more significant as 0 ( 0)r r  and it can be neglected as r   . this agrees the general physical behavior of the radiation parameter. figs. 4 and 5 illustrate the influence of inclined angle on the velocity and temperature profiles for suction and injection cases. it is observed from the figures that raise the values of inclined angle depreciates the velocity filed and enhances the temperature filed. this is due to the fact that at 0  the sheet is in vertical direction and maximum gravitational force acts on the flow. as / 2  the sheet takes horizontal direction, the strength of buoyancy forces decreases and hence reduces the velocity boundary layer and enhances the thermal boundary layer. figs. 6 and 7 depict the effect of material fluid parameter on velocity and temperature profiles of the flow for both suction and injection cases. it is clear from the figures that the increase in the value of  enhances the velocity boundary layer and depreciates the thermal boundary layer thickness. it is also noticed that the enhancement in the velocity filed is more on injection case compared with suction case. p. m. krishna, n. sandeep, j. v. r. reddy and v. sugunamma / journal of naval architecture and marine engineering 13(2016) 89-99 dual solutions for unsteady flow of powell-eyring fluid past an inclined stretching sheet 93 fig. 2: velocity field for different values of radiation parameter r fig. 3: temperature field for different values of radiation parameter r fig. 4: velocity field for different values of inclined angle α fig. 5: temperature field for different values of inclined angle α figs. 8 and 9 represent the influence of unsteadiness parameter on the velocity and temperature profiles of the flow for both suction and injection cases. we noticed an interesting result that the enhancement in the value of unsteadiness parameter depreciates the velocity profiles of the flow in injection case and increases the velocity of the flow in suction case. it is also observed the decrease in temperature profiles in both cases by increase in the value of unsteadiness parameter. this is due to the fact that increase in unsteadiness parameter reduces the 0 1 2 3 4 5 6 7 8 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1  fi (  ) r=1,3,5 blue suction red injection 0 1 2 3 4 5 6 7 8 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1   (  ) blue suction red injection r=1,3,5 0 1 2 3 4 5 6 7 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1  fi (  ) =0,/4,/3 blue suction red injection 0 1 2 3 4 5 6 7 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1   (  ) blue suction red injection =0,/4,/3 p. m. krishna, n. sandeep, j. v. r. reddy and v. sugunamma / journal of naval architecture and marine engineering 13(2016) 89-99 dual solutions for unsteady flow of powell-eyring fluid past an inclined stretching sheet 94 velocity boundary layer in injection case but in suction case it works in opposite manner. at the same time unsteadiness parameter causes to reduce the thermal boundary layer thickness and enhances the heat transfer rate. fig. 6: velocity field for different values of material fluid parameter г fig. 7: temperature field for different values of material fluid parameter г fig. 8: velocity field for different values of unsteadiness parameter ε fig. 9: temperature field for different values of unsteadiness parameter ε figs. 10-13 illustrate the effect of non-uniform heat source/sink parameters on the velocity and temperature profiles of the flow. it is evident from the figures that an increase in the values of * * anda b enhances the velocity and temperature profiles of the flow. this may happen due to the fact that the positive values of * * anda b acts like heat generators. generating the heat means releases the heat energy to the flow. this help 0 1 2 3 4 5 6 7 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1  fi (  ) blue suction red injection =0,1,2 0 1 2 3 4 5 6 7 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1   (  ) =0,1,2 blue suction red injection 0 1 2 3 4 5 6 7 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1  fi (  ) =1,3,5 =1,3,5 blue suction red injection 0 1 2 3 4 5 6 7 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1   (  ) =1,3,5 blue suction red injection p. m. krishna, n. sandeep, j. v. r. reddy and v. sugunamma / journal of naval architecture and marine engineering 13(2016) 89-99 dual solutions for unsteady flow of powell-eyring fluid past an inclined stretching sheet 95 to enhance the velocity and thermal boundary layer thickness. it is also observed that the velocity and temperature profiles shown better enhancement in injection case compared with suction case. fig. 10: velocity field for different values of nonuniform heat source/sink parameter a * fig. 11: temperature field for different values of nonuniform heat source/sink parameter a * fig. 12: velocity field for different values of nonuniform heat source/sink parameter b * fig. 13: temperature field for different values of nonuniform heat source/sink parameter b * figs. 14-16 display the influence of chemical reaction parameter, schmidt number and mass grashof number on concentration profiles of the flow. it is observed from the figures that an increase in the values of chemical reaction parameter, schmidt number and mass grashof number depreciates the concentration profiles of the flow and enhances the mass transfer rate. this is due to the fact that the increase in chemical reaction parameter, schmidt number and mass grashof number reduces the concentration boundary layer thickness. 0 1 2 3 4 5 6 7 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1  fi (  ) blue suction red injection a * =0.2,0.4,0.6 0 1 2 3 4 5 6 7 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1   (  ) a * =0.2,0.4,0.6 blue suction red injection 0 1 2 3 4 5 6 7 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1  fi (  ) blue suction red injection b * =0.2,0.4,0.6 0 1 2 3 4 5 6 7 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1   (  ) b * =0.2,0.4,0.6 blue suction red injection p. m. krishna, n. sandeep, j. v. r. reddy and v. sugunamma / journal of naval architecture and marine engineering 13(2016) 89-99 dual solutions for unsteady flow of powell-eyring fluid past an inclined stretching sheet 96 fig. 14: concentration field for different values of chemical reaction parameter kr fig. 15: concentration field for different values of schmidt number sc fig. 16: concentration field for different values of mass grashof number gc table 1 depicts the comparison of the present results with the existed results of tsai et al. (2008), hayat et al. (2015). under some special conditions present results have an excellent agreement with the existed results. this shows the validity of the present results along with the accuracy of the numerical technique we used in this study. table 2 displays the influence of non-dimensional governing parameters on friction factor, nusselt and sherwood numbers. it is evident from the table that an increase in the radiation parameter enhances the friction factor, mass transfer rate and reduces the heat transfer rate. a raise in the value of unsteadiness parameter showed opposite results to the radiation parameter. the enhancement in the value of material fluid parameter increases the friction factor, heat and mass transfer rate. but increase in inclined angle shows reverse results to the material parameter. an increase in chemical reaction parameter depreciates the skin friction coefficient and nusselt number and enhances the sherwood number. a raise in the values of non-uniform heat source/sink parameters depreciates the heat and mass transfer rate and enhances the friction factor. 0 0.5 1 1.5 2 2.5 3 3.5 4 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1   (  ) kr=1,3,5 blue suction red injection 0 1 2 3 4 5 6 7 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1   (  ) sc=0.2,0.6,1 blue suction red injection 0 1 2 3 4 5 6 7 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1   ( ) blue suction red injection gc=1,3,5 p. m. krishna, n. sandeep, j. v. r. reddy and v. sugunamma / journal of naval architecture and marine engineering 13(2016) 89-99 dual solutions for unsteady flow of powell-eyring fluid past an inclined stretching sheet 97 table 1: comparison of the present with the existed studies of tsai et.al (2008) and hayat al. (2015) for wall temperature gradient when 0gr gc s r           pr *a * b tsai et.al (2008) hayat al. (2015) present study 1 -1 0 -1.710937 -1.71094 -1.7109372 -2 -1 -2.36788 -2.3678798 2 -1 0 -2.485997 -2.25987 -2.2598771 -2 -1 -2.48600 -2.4860000 table 2: variation in ''(0), '(0) and '(0)f    at different non-dimensional parameters s r    kr * * /a b ''(0)f '(0) '(0) 1 1 -0.611964 0.825574 1.254709 3 -0.563830 0.550547 1.266303 5 -0.540636 0.439350 1.271903 -1 1 -0.368571 0.682532 1.022146 3 -0.346913 0.491671 1.027719 5 -0.336007 0.404676 1.030554 1 1 -0.631188 0.999770 1.212916 3 -0.636013 1.008084 1.132503 5 -0.643304 1.008735 1.095664 -1 1 -0.433298 0.815952 1.010578 3 -0.534345 0.887745 1.002026 5 -0.574345 0.919080 1.001011 1 0 -0.665999 0.977881 1.244200 1 -0.549080 1.006002 1.264042 2 -0.483320 1.022728 1.276059 -1 0 -0.382825 0.776950 1.018290 1 -0.360320 0.785661 1.024225 2 -0.339795 0.792168 1.028788 1 0 -0.487717 1.007818 1.266435 / 4 -0.635062 0.985020 1.249223 / 3 -0.743444 0.966682 1.235789 -1 0 -0.248542 0.798351 1.034814 / 4 -0.378544 0.778914 1.019606 / 3 -0.473163 0.763538 1.007866 1 1 -0.649569 0.985466 1.429083 3 -0.662573 0.982180 1.798841 5 -0.670821 0.980316 2.102173 -1 1 -0.387774 0.778540 1.189780 3 -0.398065 0.775834 1.545607 5 -0.404897 0.774235 1.841451 1 0.2 -0.628843 0.948003 1.250751 0.4 -0.615966 0.872625 1.253896 0.6 -0.602492 0.795400 1.257162 -1 0.2 -0.374346 0.744978 1.020743 0.4 -0.365781 0.676362 1.023045 0.6 -0.357002 0.606762 1.025384 p. m. krishna, n. sandeep, j. v. r. reddy and v. sugunamma / journal of naval architecture and marine engineering 13(2016) 89-99 dual solutions for unsteady flow of powell-eyring fluid past an inclined stretching sheet 98 4. conclusion this paper presents a numerical solution for heat and mass transfer in unsteady flow of powell-eyring fluid past an inclined stretching sheet in presence of radiation, non-uniform heat source/sink and chemical reaction with suction/injection effects. results display the influence of governing parameters on the flow, heat and mass transfer, friction factor, local nusselt and sherwood numbers. the conclusions of the present study are made as follows: • raise in the values of material fluid parameter enhances the heat and mass transfer rate. • an increase in unsteadiness parameter depreciates the friction factor, sherwood number and enhances the nusselt number. • positive values of non-uniform heat source/sink parameters acts like heat generators and negative values acts like heat observers. • dual solutions exist only for certain range of suction/injection parameter. • enhancement in chemical reaction parameter depreciates the concentration profiles and increases the mass transfer rate. • raise in the values of non-uniform heat source/sink parameters enhances the velocity and thermal boundary layers. references anderson, h. i., bech, k. h. and dandapat, b. s. 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(2004) investigated the effect of temperature dependent viscosity and density of air on the fluid flow and heat transfer from a heated cylinder for the range. it is observed that thermal conductivity of nanofluid is higher than that of the base fluids when the nanoparticles are mixed with a small amount (eastman et al., 2001, xuan and li, 2000). the application of nanofluid in convection for different industrial purpose is introduced in preceding studies (buongiorno, 2006, kakaç and pramuanjaroenkij, 2009). recently, application of nanofluid on enhancement of heat transfer has studied (daungthongsuk and wongwises, 2007, trisaksri and wongwises, 2007). mixed convection around a heated circular cylinder placed inside a vented square cavity has been studied for different diameter of the cylinder and different ri in steady flow regime (rahman et al., 2009). prasenjit dey, abhijit sarkar, ajoy. kumar das/journal of naval architecture and marine engineering, 12(2015), 57-71 prediction of unsteady mixed convection over circular cylinder in the presence of nanofluida comparative study of ann and gep 58 various numerical and experimental studies of heat transfer characteristics by utilizing nanofluids has been studied and concluded in the literature (khanafer et al., 2003, tiwari and das, 2007). khanafer et al. (2003) investigated the heat transfer performance of nanofluids in a differentially heated enclosure and concluded that there is an enhancement in heat transfer rate due to the mixing of nanoparticles in the base fluid. the authors of the present paper (dey and das, 2014) also studied the natural convection heat transfer of a heated square cylinder placed inside a square enclosure by varying volume fraction of nanofluid and the rayleigh number (ra). most recently, (sarkar et al., 2012), a study of the mixed convection of nanofluid under aiding and opposing buoyancy effect has been completed. they have taken volume fraction up to 25% and varying ri as 1 and -1. and the same authors (sarkar et al., 2011) also studied the mixed convection over a circular cylinder at high prandtl number; varying 0.7 to 100. mixed convection heat transfer of nanofluid in steady state condition in a lid driven cavity having some combination of heaters and coolers (hacs) inside is examined numerically (garoosi et al., 2015) and found that at low ri, heat transfer rate is directly related to number of heaters and coolers but at higher ri, the heat transfer rate is not changing considerably. another numerical solution of mixed convection of nanofluid in a square enclosure has been studied recently (chen et al., 2014). a numerical study of mixed convection heat transfer of nanofluid around a heated circular cylinder placed inside a backward facing step channel has been accompanied (selimefendigil and öztop, 2015) and found that heat transfer rate is directly proportional to re and volume fraction. although, all the heat transfer studies are based on experimentally or numerically and by using working fluid as air, water or nanofluid; but over the last few years, prediction of different characteristics of heat transfer and aerodynamic behavior is becoming an area of research in various engineering applications due to its less time consuming method. there are various techniques are used in production; between them artificial neural network (ann) is one of the most utilizing method. recently, (santra et al., 2009) the prediction of heat transfer in the presence of nanofluid using ann has been studied and found that ann can be used to predict heat transfer characteristics more efficiently and rapidly. where, gene expression programming (gep) is an another algorithm having the advantages of both genetic algorithm (ga) and genetic programming (gp) to evaluate more complex function to present an expression of the relation between input and output (ferreira, 2001). gep is more efficient to predict the output as compared with ann; is discussed recently (martí et al., 2013). by considering the foregoing studies, it is altogether okay to conclude that there is no prior study has been conducted on prediction of nanofluid based mixed convection over a circular cylinder. therefore, in the present study, the prediction of unsteady mixed convection by utilizing nanofluid is studied by back propagation ann and gep. the reynolds number is varied in the range of 80 to 180, ri is varying as 1 and -1 and solid volume fraction as 0 to 15%. the input parameters are partly similar to sarkar et al (2012). this present study aims to fill the gap in literature. 2. geometrical configuration and mathematical formulation the system of interest here is to predict the mixed convection heat transfer characteristics around a circular cylinder in a channel at the symmetric horizontal line, schematically shown in fig. 1. the circular cylinder of diameter „d‟ with constant wall temperature „tw‟ is held in a channel exposed to an upstream unsteady laminar flow of x-velocity, „u∞‟ (free stream velocity) and temperature, „t∞‟. the objective is to perform on an infinitely long channel; but, the computational domain has to be limited. the distance of the upstream and the downstream boundaries from the center of the cylinder are lu=10d and ld=40d. the distance between the upper and lower side-walls, h, is specified according the blockage ratio (d/h=0.05). the free-slip boundary condition is associated with the side-walls. 2.1. governing equations the dimensionless governing equations for the two dimensional, laminar, incompressible nanofluid flow and heat transfer with constant thermo-physical properties and negligible dissipation effect can be expressed in the following forms: prasenjit dey, abhijit sarkar, ajoy. kumar das/journal of naval architecture and marine engineering, 12(2015), 57-71 prediction of unsteady mixed convection over circular cylinder in the presence of nanofluida comparative study of ann and gep 59 continuity equation: 0 u v x y       (1) fig. 1: a schematic diagram of the problem description x-momentum equation: 2 2 2 2 1 re f nf nf f nf u u u p u u u v t x y x v x y                          (2) y-momentum equation:  2 2 2 2 11 ( ) re f nf p p f f nf f nf nf f v v v p v v u v ri t x y y v x y                                  (3) energy equation: 2 2 2 2 1 repr nf f u v t x y x y                      (4) where u, v are the dimensionless velocity components along x and y directions of a cartesian coordinate system respectively, p is the dimensionless pressure, re u d           is the reynolds number based on the cylinder dimension, θ is the dimensionless temperature, pr p c k            is the prandtl number and t is the dimensionless time. the fluid properties are described by the density ρ, dynamic viscosity µ, thermal diffusivity α and thermal conductivity k. the dimensionless variables are expressed as: __ _ _ _ , , , , , w t t u tu v x y u v x y t u u d d t t d               (5) where u and v are the velocity components in the x and y directions respectively, t is the temperature. 2.2. thermophysical properties of nano-fluid the different thermophysical properties of nano-fluid are defined as follows (yacob et al., 2011): 2.5 (1 ) f nf     (6) prasenjit dey, abhijit sarkar, ajoy. kumar das/journal of naval architecture and marine engineering, 12(2015), 57-71 prediction of unsteady mixed convection over circular cylinder in the presence of nanofluida comparative study of ann and gep 60 where, ϕ is the nanoparticle volume fraction and is given as: volume of nanoparticles total volume of solution   (1 ) nf f p      (7) (1 )( ) ( ) p f p p pnf nf c c c       (8) ( 2 ) 2 ( ) ( 2 ) ( ) nf p f f p f p f f p k k k k k k k k k k         (9) the thermophysical properties of fluid and nanoparticles at room temperature are given in table 1 (abu-nada and oztop, 2009). table.1: thermophysical properties of nanofluid fluid/ nanoparticle thermophysical properties ρ (kg/m 3 ) cp (j/kg k) k (w/mk) µ (kg/ms) water 997.1 4179 0.613 0.001 cu 8933 385 400 _ 2.3. boundary conditions the physical boundary condition for the above discussed problem configuration are written as follows:  the left wall of the computational domain is designed as the inlet. the “velocity inlet” boundary condition is assigned at the inlet boundary with free stream velocity, u∞, temperature t and neumann boundary condition for pressure is used 0 p x       .  the usual no-slip boundary condition is assigned for flow at the surface of the cylinder, i.e. u=0; v=0 with constant wall temperature of θ=1 and normal gradient condition for pressure  . 0,p n where n is the unit normal  .  free-slip boundary condition is assigned at the upper and lower surface of the computational domain, i.e. 0 u y    ; v=0; θ=0.  the extreme right surface of the computational domain is assigned as an outlet. the “pressure outlet” boundary condition is employed at the exit boundary with a fully developed flow situation 0, 0, 0 u v x x x             of dirichlet type pressure boundary condition (p=0). prasenjit dey, abhijit sarkar, ajoy. kumar das/journal of naval architecture and marine engineering, 12(2015), 57-71 prediction of unsteady mixed convection over circular cylinder in the presence of nanofluida comparative study of ann and gep 61 the heat transfer characteristic between the cylinder and the surrounding fluid is calculated by the nusselt number. the local nusselt number based on the cylinder dimension is given by: nf f f khd nu k k n            (10) where, h is the local heat transfer coefficient. surface average heat transfer is obtained by integrating the local nusselt number along the cylinder face. 3. cfd model 3.1. grid structure and grid independence study the grid structure of the computational domain used in the present investigation is shown in fig. 2. it is observed from the fig. 2, that the non-uniform structured and non-structured grid assembly for the whole computational domain is assigned. grids are generated by using the grid generation package gambit. the expanded view of the section of the computational domain that is having the cylinder is shown in fig. 2(b). (a) (b) fig. 2: (a) grid distribution of the computational domain (b) zoomed view of the grid distribution of the cylinder. the whole computational domain grid sizes are selected inadequate of the blockage ratio. the surface of the cylinder and the area nearer to it has the finer mesh to adequately capture the wake wall interactions in both direction and the grids becoming coarser non-uniformly towards the boundary wall. the smallest grid size of 0.07d of triangular element is incorporated with the surface of the cylinder and the grid size for the remaining surfaces is increasing linearly to 0.5d from the cylinder surface to the boundary. in this study, three different mesh sizes (grid1-15000, grid2-25000 and grid3-40000) are adopted in order to check the mesh independence. a detailed grid independence study has been performed and results are obtained for the average nusselt number at ɸ=0.0 and ri=1 but there is no considerable changes between grid2 and grid3 (the results are shown in table.2). thus, a grid size 25000 is found to meet the requirements of the both grid independence and computation time limit. table 2: study of effect of grid size for grid independence test no. of cells re-100 re-180 nuavg nuavg 15000 10.13792 15.79207 25000 10.29234 16.08154 40000 10.35409 16.15391 prasenjit dey, abhijit sarkar, ajoy. kumar das/journal of naval architecture and marine engineering, 12(2015), 57-71 prediction of unsteady mixed convection over circular cylinder in the presence of nanofluida comparative study of ann and gep 62 3.2 numerical details in the present investigation, the numerical simulation is performed by using the finite volume based commercial cfd solver fluent 6.3 (fluent, 2006). fluent is used to solve the governing equations which are the partial differential equations, using the control volume based technique in a collocated grid system. the solver used in the present work is pressure-based implicit method. semi-implicit method for pressure-linked equation (simple) is selected for the pressure-velocity coupling scheme. the pressure term is discretized under the scheme of standard whereas the convective terms are discretized by second order upwind scheme. the unsteady laminar viscous model is used for the low reynolds number consideration. the convergence criteria for the continuity and velocity are set to 10 -5 . 4. artificial neural network model artificial neural network (ann) is a computational structure inspired by a biological neural system. an ann consists of very simple and highly interconnected units called neurons. the neurons are connected to each other by links in which individual weights are passed and over which signals can pass. each neuron receives multiple inputs from other neurons in proportion to their connection weights and generates a single output, which may be propagated to several other neurons (sreekanth et al., 1999). there are abundant distinct ways of implementation of a single artificial neuron. the general mathematical formulation of a single artificial neuron could be signified as: 0 ( ) j i i i y z f w z b          (11) where, z is a neuron with j input (z0 to zn) and one output y(z) and where (wi) are weights determining how much the inputs should be weighted with b denoting the bias (kurtulus, 2009). „f’ is an activation function that weights how powerful the output should be from the neuron, based on the sum of the inputs and expressed as: 1 ( ) 1 x f x e    (12) the basic feedforward network performs a nonlinear transformation of input values in order to approximate the output values. for the present ann model, three layers are used, namely one input layer, one hidden layer and one output layer. connections in these kinds of network only go forward from one layer to the next where all the neurons in each layer are connected to all the neurons in the next layer. the designed neural network structure 3-5-1 (3 neurons in input layer, 5 neurons in hidden layer and 1 neuron in output layer) of the present study is shown in fig. 3. fig.3: schematic representation of a multilayer feed forward network consisting of three inputs, one hidden layer with five neurons and one output. prasenjit dey, abhijit sarkar, ajoy. kumar das/journal of naval architecture and marine engineering, 12(2015), 57-71 prediction of unsteady mixed convection over circular cylinder in the presence of nanofluida comparative study of ann and gep 63 4.1. training ann the back-propagation method is the most popular training algorithm. the input and output data are trained in ann so that the weights can be adjusted to give the same outputs as found in the training data. the inputs (v) into a neuron are multiplied by their corresponding connection a weight (w), summed together and bias is added to the sum. this sum is transformed through a transfer function (f) to produce the required output, which may be passed to other neurons. after propagating an input through the network, the error is calculated and the error is propagated back through the network while the weights are adjusted in order to make the error smaller. the number of iterations of predicting the output is selected as 500 for the present network. the training data have been selected 70% of the total data and the remaining data are selected for testing. neural network requires that the range of the both input and output values should between 0.1 and 0.9 due to the restriction of the sigmoid function. therefore, the numerical data evaluated in this study are normalized by the following equation: min max min i n v v v v v        (13) where vn = normalized value, vi = actual input (or output) value, vmax =maximum value of the inputs (or outputs), vmin =minimum value of the inputs (or outputs) 5. gene expression programming (gep) gep is an algorithm of developing functions through population based evolutionary technique combining the advantages of genetic algorithm (ga) and genetic programming (gp), proposed by ferreira(ferreira, 2001). it is an extension of ga in which simple or linear chromosomes are encoded to the individuals, after that transformed into an expression parse tree completely separating the genotype and phenotype which makes gep much faster (100 – 10,000 times) than the gp (ferreira, 2001, ferreira, 2002). for example, an expression tree of an algebraic expression (eq. 14) is represented in fig. 4. ( ) *( )o q r s  (14) fig.4: expression tree of eq. 14. in gep, there are multiple genes in a chromosome and several subprograms are encoded with each gene. therefore, any program can be encoded for efficient evolution the solutions by the novel structures of the genes in the gep algorithm (ferreira, 2001). the entire feasible region of the problem is used by the novel structure of the genes in the gep to have efficient genetic operators looking for the solutions. in gep, more complex scientific and technological programs can be solved with the help of linear chromosomes and expression trees (et). each linear chromosome is manipulated genetically, i.e. replication, mutation, recombination and transposition (ferreira, 2001). they are composed of genes structurally comprised of the head and tail part. the tail length (tl) is a function of head length (hl) and number of arguments of the function (m) and expressed as the following equation: ( 1) 1 l l t h m   (15) the flow chart of gep is presented in the fig. 5 (ferreira, 2001). prasenjit dey, abhijit sarkar, ajoy. kumar das/journal of naval architecture and marine engineering, 12(2015), 57-71 prediction of unsteady mixed convection over circular cylinder in the presence of nanofluida comparative study of ann and gep 64 fig. 5: flowchart of gep. 5.1. gep model of present input and output the relation between input and output data can be expressed by gep. in the present study, the training and testing data for the proposed gep are re, ri and ɸ as input and nulocal as output. as same as the ann model, 70% data have been selected for training and remaining 30% data for testing. the data are evaluated on the genexprotools (genexprotools, 2014) to develop the empirical models of input and output data. different symbols from function set and terminal set are used in mathematical expressions. various arithmetic operators and mathematical functions available in function set are used in this study (encapsulated in table. 3) to generate the relation between input and output by evolving the model. table.3: different parameters of gep model parameter values function set +,-,*,/,sqrt, pow, log, exp, not (1-x), x 2 , x 3 , cube root( 3 x ) chromosomes 30, 40 head size 5, 8, 10 number of genes 5 linking function addition mutation rate 0.044 inverse rate 0.1 one-point recombination rate 0.3 two-point recombination rate 0.3 gene recombination rate 0.1 gene transportation rate 0.1 prasenjit dey, abhijit sarkar, ajoy. kumar das/journal of naval architecture and marine engineering, 12(2015), 57-71 prediction of unsteady mixed convection over circular cylinder in the presence of nanofluida comparative study of ann and gep 65 the numbers of programs in the each linear chromosome are set by the population size i.e. number of chromosomes. the more population size makes the iteration time longer. the program is considered as converged when there are no considerable changes in the performance of the model. in the present study, the main goal of utilization of gep is to obtain the explicit expression of nuavg relating with re, ri and ɸ. the et of the explicit formula of nuavg is depicted in figs. 6-8 and the formula based on the et is expressed in the eq. 16 where d(0), d(1) and d(2) be the volume fraction (ɸ), reynolds number (re) and richardson number (ri) respectively. avg (((1.0+4.60)-(re/9.52))+( *ri))3 3 nu = ((2.45+(re+((ri+re)-( *13.07))))) + (((-4.63-(-4.38))/((-1.26-ri)-ri))+ ) + ((( +re)-(1.48+0.46))-(re-(6.48+0.46))) + (-7.93+e ) + (( * 8.83* (22.32-re)*(-8.77) ))       ; (16) fig.6: sub expression tree of genes 1-2 for nuavg. 6. results and discussions 6.1. validation of present results the present numerical data are validated with the available published data. the present data are validated with circular cylinder at re=100 with both adding and opposing buoyancy forces. numbers of trials have been performed to find quite accurate value and the time step is chosen for every case as 0.01. the parameter used for prasenjit dey, abhijit sarkar, ajoy. kumar das/journal of naval architecture and marine engineering, 12(2015), 57-71 prediction of unsteady mixed convection over circular cylinder in the presence of nanofluida comparative study of ann and gep 66 validation is nuavg. the present numerical values are in very good agreement with the published results, tabulated in table 4. fig.7: sub expression tree of genes 3-4 for nuavg. fig. 8: sub expression tree of gene 5 for nuavg. table 4: validation of present numerical results with other literature re ri volume fraction (%) study nuavg 100 1 0 present 11.52009 sarkar et. al (2012) 11.52155 -1 present 11.15485 sarkar et. al (2012) 11.1629 prasenjit dey, abhijit sarkar, ajoy. kumar das/journal of naval architecture and marine engineering, 12(2015), 57-71 prediction of unsteady mixed convection over circular cylinder in the presence of nanofluida comparative study of ann and gep 67 6.2. heat transfer prediction the heat transfer characteristic over the circular cylinder due to the presence of nanofluid is presented by means of local and average nusselt number. the present numerical outputs are in very good agreement with sarkar et al., (2012). it is found that by increasing of reynolds number and solid volume fraction, heat transfer rate is increased for both the condition of adding and opposing of buoyancy. as the reynolds number is increased, more clustering of the isotherms is seen at the front surface of the cylinder, due to which the heat transfer rate is enhanced. even also, by increasing the solid volume fraction, the volume of nano particles striking the cylinder is increased, which causes more heat transfer from cylinder surface, by means of which the heat transfer rate is also increased. the training and testing data are collected from numerical analysis for re= 80 to 180, ɸ= 0 to 15% and ri= 1 and -1. the training data are separated from the total data by keeping the particular testing data alongside. for training the network, different combinations of reynolds number and solid volume fraction are selected for ri= 1 and -1. for training the present ann model and gep model, reynolds number, ri and volume fraction have taken as input and local nusselt number is found as output. the average nusselt number (nuavg) has been calculated by time averaging the local nusselt number over the cylinder surfaces. the variation of local nusselt number about the cylinder surface is shown in fig.6 for re=100, ɸ=0.05 and ri=1. also the instantaneous vorticity and isotherm is displayed in fig.9. only one case is of predicting the local nusselt number is depicted and it shows that a very good agreement between the numerical data and the predicted data (refer fig. 10). fig 11 and 12 (a) shows the variation of numerical and predicted data after testing the model, which are clearly depicted that the predicted data are in good agreement with the numerical data for every reynolds number and volume fraction. the error between the numerical values and the ann and gep predicted values are presented as adj. r 2 (it is defined as the error measuring value which is used to measure the quantity of the discrepancy in the dependent variable accounted for by the explanatory variables in a multiple linear regression.) and mean relative error (mre) which is expressed as given below and the comparison between them is tabulated in table 5:   2 2 2 ( ) 1 . 1 1 i i s r i i s i n p n t adj r n n n          (17) 1 1 100 sn i i is i n p mre n n    (18) where, ns=sample size, tr=total number of regressors in the training model. ni= actual value. pi=predicted value and 1 1 s n i is n n n    fig.9: instantaneous vorticity (left) and isotherm (right) at re=100, ri=1 and ɸ=0.05. prasenjit dey, abhijit sarkar, ajoy. kumar das/journal of naval architecture and marine engineering, 12(2015), 57-71 prediction of unsteady mixed convection over circular cylinder in the presence of nanofluida comparative study of ann and gep 68 (a) (b) fig.10: (a) comparison of nulocal of numerical and predicted data and (b) fitting line plot of predicted data at re=100, ri=1 and ɸ=0. 05. (a) (b) (c) fig.11: (a) comparison of nuavg of numerical and predicted data (b) fitting line plot of predicted data by ann and (c) fitting line plot of predicted data by gep at ri=1. prasenjit dey, abhijit sarkar, ajoy. kumar das/journal of naval architecture and marine engineering, 12(2015), 57-71 prediction of unsteady mixed convection over circular cylinder in the presence of nanofluida comparative study of ann and gep 69 (a) (b) (c) fig.12: (a) comparison of nuavg of numerical and predicted data (b) fitting line plot of predicted data by ann and (c) fitting line plot of predicted data by gep at ri=-1. it is found that the mean relative error of ann predicted data of average nusselt number with ri=1 and -1 are 2.645%, and 2.638%, respectively, whereas for gep, the errors are 0.49% and 0.578%, respectively (refer table. 5). it is obvious for prediction models that, more values in training, more accurate will be the prediction. table 5: a comparison of gep and ann value ri model adj. r 2 mre adj. r 2 gep / adj. r 2 ann nuavg 1 gep 0.99928 0.49 1.001 ann 0.9986 2.645 -1 gep 0.99864 0.578 1.01 ann 0.99251 2.638 prasenjit dey, abhijit sarkar, ajoy. kumar das/journal of naval architecture and marine engineering, 12(2015), 57-71 prediction of unsteady mixed convection over circular cylinder in the presence of nanofluida comparative study of ann and gep 70 conclusion back propagation artificial neural network and gep are used to predict the mixed convection heat transfer characteristics of water based nanofluid flowing over a circular cylinder at low reynolds number with adding and opposing of buoyancy force. for this purpose, series of numerical data have been developed for the cylinder model with a validation which shows a very good agreement of present result with the previously available published data. for training and testing the network, several numerical cases with combinations of input variables are created and output data are generated. the validations of the applied predicted methods were checked in several cases to ensure the effectiveness to establish the results with less permissible error. it is found that the finite volume method based numerical procedure can calculate efficiently the behavior of mixed convection over a circular cylinder. it is found that the average nusselt number is directly proportional to the nanofluid volume fraction and reynolds numbers for both adding and opposing buoyancy cases. it can also be concluded by analyzing the results that the back propagation artificial neural network and gep both can predict the local and average nusselt number accurately with a minimum mean relative error where gep is a more efficient algorithm for prediction than ann; hence reducing the computational time in the cfd calculation while achieving acceptable accuracy. therefore, in different analysis of heat transfer where experimental and simulation requires more time and also more expensive; gep can be utilized to predict the heat transfer characteristics. references: abu-nada, e. and oztop, h. f. 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(2011): boundary layer flow past a stretching/shrinking surface beneath an external uniform shear flow with a convective surface boundary condition in a nanofluid, nanoscale research letters, 6, 1-7. http://dx.doi.org/10.1186/1556-276x-6-314 http://dx.doi.org/10.1016/s0017-9310(03)00156-x http://dx.doi.org/10.1007/s00521-008-0186-2 http://dx.doi.org/10.1007/s00521-008-0186-2 http://dx.doi.org/10.1016/j.ijheatfluidflow.2008.05.001 http://dx.doi.org/10.1016/j.compag.2013.08.016 http://dx.doi.org/10.1016/j.jfluidstructs.2006.09.004 http://dx.doi.org/10.3329/jname.v5i2.2504 http://dx.doi.org/10.1016/j.ijthermalsci.2008.11.009 http://dx.doi.org/10.1016/j.ijheatmasstransfer.2011.03.032 http://dx.doi.org/10.1016/j.ijheatmasstransfer.2012.04.046 http://dx.doi.org/10.1016/j.compfluid.2014.12.007 http://dx.doi.org/10.1111/j.1745-4549.1999.tb00389.x http://dx.doi.org/10.1016/j.ijheatmasstransfer.2006.09.034 http://dx.doi.org/10.1016/j.rser.2005.01.010 http://dx.doi.org/10.1017/s0022112059000829 http://dx.doi.org/10.1016/s0142-727x(99)00067-3 http://dx.doi.org/10.1186/1556-276x-6-314 separation points of magnetohydrodynamic boundary layer flow along a vertical plate with exponentially decreasing free stream velocity¬ journal of naval architecture and marine engineering december, 2016 http://dx.doi.org/10.3329/jname.v13i2.23537 http://www.banglajol.info 1813-8235 (print), 2070-8998 (online) © 2016 aname publication. all rights reserved. received on: may, 2015 mhd boundary layer slip flow of a casson fluid over an exponentially stretching surface in the presence of thermal radiation and chemical reaction p. bala anki reddy department of mathematics, school of advanced sciences, vit university, vellore, t.n.-632014. e-mail: pbarmaths@gmail.com, sireesha.siri7@gmail.com abstract: an analysis is carried out to investigate the steady two-dimensional magnetohydrodynamic boundary layer flow of a casson fluid over an exponentially stretching surface in the presence of thermal radiation and chemical reaction. velocity, thermal and solutal slips are considered instead of no-slip conditions at the boundary. stretching velocity, wall temperature and wall concentration are considered in the exponential forms. the non-linear partial differential equations are converted into a system of non-linear ordinary differential equations by similarity transformations. the resultant non-linear ordinary differential equations are solved numerically by fourth order runge-kutta method along with shooting technique. the influence of various parameters on the fluid velocity, temperature, concentration, wall skin friction coefficient, the heat transfer coefficient and the sherwood number have been computed and the results are presented graphically and discussed quantitatively. comparisons with previously published works are performed on various special cases and are found to be in excellent agreement. keywords: exponentially stretching surface, casson fluid, mhd, thermal radiation, chemical reaction, suction/blowing. 1. introduction investigations on the boundary layer flows of newtonian fluids over a stretching surface have gained considerable attention because of its wide range of applications in technology and industry. such applications include polymer extrusion from a dye, wire drawing, the boundary layer along a liquid film in condensation processes, accelerators, paper production, artificial fibers, hot rolling, glass blowing, cooling of metallic sheets or electronic chips, metal spinning, drawing plastic films and many others. crane (1970) pioneered a closed form analytical solution for an incompressible fluid flow due to a linearly stretching sheet. many researchers (bidin and nazar, 2009; magyari and keller, 200; mukhopadhyay and reddy, 2012; misra and sinha, 2013; fang et al., 2009; aziz, 2009; srinivas et al., 2014; nadeem et al., 2011; reddy and reddy, 2011 and ishak, 2011) extended the work of crane by considering several physical aspects taking newtonian fluid. but in real life, some materials like, melts muds, condensed milk, glues, printing ink, emulsions, tomato paste, paints, soaps, shampoos, sugar solution, etc. shows different characters which are not properly understandable using newtonian theory. therefore, the analysis of non-newtonian fluid flows is an essential part in the study of fluid dynamics and heat and mass transfer. constitutive equations are used for the casson fluid. casson fluid is one of the non-newtonian fluids which exhibits yield stress. however such fluids behaves like a solid when shear stress less than the yield stress is applied and it moves if applied shear stress is greater than yield stress. examples of casson fluid model jelly, soup, honey, tomato sauce, concentrated fruit juices and many others. human blood is also a casson fluid. in fact because of several substances like protein, fibrinogen and globin in aqueous base plasma, human red cells from a chain like structure, known as aggregates or rouleaux. if the rouleaux behave like a plastic solid then there exists a field stress that can be identified with the constant stress in casson fluid. this fluid can be defined as a shear thinning liquid having infinite viscosity at zero shear rate, a yield stress below which no flow occurs and a zero viscosity at an infinite shear rate was examined by dash et al (1996). several studies (nadeem et al., 2012; nadeem et al., 2014; hayat et al., 2012, mukhopadhyay et al., 2013; mustafa et al., 2012; bhattacharya et al., 2013 and mustafa. 2011) have been reported investigating the casson fluid flow over a stretching surface. bhattacharya et al. (2014) studied the boundary layer flow of casson fluid over a stretching/shrinking sheet. numerical solutions for the steady boundary layer cason fluid flow and heat transfer passing a nonlinearly stretching surface was studied by mukhopadhyay (2013). recently, the numerical solutions for steady boundary layer flow and heat transfer for a casson fluid over an exponentially permeable stretching surface in the presence of thermal radiation are analyzed by pramanik (2014). very recently, the steady two-dimensional mhd convective boundary layer flow of a casson fluid over an mailto:pbarmaths@gmail.com p. b. a. reddy / journal of naval architecture and marine engineering, 13(2016) 165-177 mhd boundary layer slip flow of a casson fluid over an exponentially stretching surface in the presence of thermal radiation… 166 exponentially inclined permeable stretching surface in the presence of thermal radiation and chemical reaction was discussed by reddy (2016). to the best of author’s knowledge, no investigation has been made yet to analyze the steady two-dimensional boundary layer flow of a casson fluid over an exponentially stretching surface in the presence of thermal radiation and chemical reaction. the present work aims to fulfill the gap in the existing literature. motivated by the above studies, a mathematical model is presented here to understand the effects of thermal radiation and chemical reaction on the steady two-dimensional boundary layer flow of a casson fluid over an exponentially stretching surface. the governing partial differential equations of the governing flow are transformed into nonlinear coupled ordinary differential equations by a similarity transformation. the resulting nonlinear coupled differential equations are solved numerically by using fourth order runge-kutta scheme together with shooting method and the flow characteristics are analysed with the help of their graphical representations. 2. mathematical formulation: consider two-dimensional flow of an incompressible viscous electrically conducting casson fluid over an exponentially stretching surface coinciding with the plane .0y the x-axis is taken along the stretching surface in the direction of the motion while the y-axis is perpendicular to the surface. the flow is confined to 0y  . two equal and opposite forces are applied along the x-axis so that the wall is stretched keeping the origin fixed. the coordinate system and flow model are shown in fig.1. fig. 1: sketch of the physical flow problem a variable magnetic field 2 0 x lb b e is applied normal to the sheet, 0b is a constant. assume that the rheological equation of state for an isotropic and incompressible flow of a casson fluid is as follows: (mukhopadhyay, 2013 and mukhopadhyay et al., 2013) 2 , 2 2 , 2 y b ij c ij y b ij c c p e p e                            u v boundary layer x y p. b. a. reddy / journal of naval architecture and marine engineering, 13(2016) 165-177 mhd boundary layer slip flow of a casson fluid over an exponentially stretching surface in the presence of thermal radiation… 167 where b  is the plastic dynamic viscosity of the non-newtonian fluid, y p is the yield stress of the fluid, , ij ij e e  ij e is the ( , ) th i j component of the deformation rate and c  is the critical value of this product based on the non-newtonian model. under these assumptions, the governing boundary layer along with the boussinesq approximation, the continuity, momentum, energy and concentration species can be written as 0      y v x u (1) 2 2 2 1 1 u u u b u v u x y y                  (2) y q cy t c k y t v x t u r pp             1 2 2 (3)             cc y c d y c v x c u 2 2 (4) subject to the boundary conditions: y u nuu     , )(xvv  , y t mtt w    y c pcc w    at 0y ,0u , tt  cc as y (5) here l x euu 0  is the stretching velocity, l x w ettt 0   is the temperature at the sheet, l x w eccc 0   is concentration at the sheet, 00 , tu and 0c are the reference velocity, temperature and concentration respectively, l x enn   1 is the velocity slip factor, l x emm   1 is the thermal slip factor and l x epp   1 is the solutal slip factor. the no-slip conditions can be recovered, by setting .0 pmn l x evxv 2 0 )(  a special case of velocity at the wall considered, 0)( xv be the velocity of suction and 0)( xv be the velocity of blowing. it is assumed that the exponential reaction rate is in the form of l x ek 0  . where u and v are the velocity components in the x and y directions respectively,  is the density of the fluid,  is the kinematic viscosity,  is the casson parameter, t is the temperature, t is the temperature of the ambient fluid, c is the concentration, c is the concentration of the ambient fluid,  is the electrical conductivity,  is the kinematic viscosity, p c is the specific heat at constant pressure, k is the thermal conductivity, r q is the radiative heat flux, d is the mass diffusion coefficient and l is the reference length. in (5), u is a constant with 0u  for stretching and 0u  for shrinking sheet. the subscript w denotes the values at the solid surface. furthermore, n, m and p represents the velocity, thermal and solutal slip factors respectively and when l=k=p=0, the slip condition is recovered. thermal radiation is simulated using the rosseland diffusion approximation and in accordance with this, the radiative heat flux rq is given by y t k q r    4 * * 3 4 (6) where *  is the stefan–boltzmann constant and * k is the rosseland mean absorption coefficient. if the temperature differences within the mass of blood flow are sufficiently small, then equation (6) can be linearized by expanding 4 t into the taylor’s series about t and neglecting higher order terms, we get p. b. a. reddy / journal of naval architecture and marine engineering, 13(2016) 165-177 mhd boundary layer slip flow of a casson fluid over an exponentially stretching surface in the presence of thermal radiation… 168 4 3 4 4 3t t t t     (7) invoking equations (6) and (7), equation (3) can be written as 3 2 * 2 16 3 p p tt t k t u v x y c c k y                (8) we introduce the similarity variables as , 2 2 2 1 0 ye l u l x           0 20 ( ), ( ) ( ) , 2 x x l l u u u e f v e f f l          ),(2 0 l x ettt   )(2 0 l x eccc   (9) where  is the similarity variable. now substituting (9) into the eqs. (2), (4) and (8), we get the following set of ordinary differential equations 21 1 2 ( ) 0f ff f hf               (10) 0)pr( 3 4 1         ffr (11) 0)(   scffsc (12) with the boundary conditions ),0(1, fsfsf f  ),0(1   ts )0(1   cs at 0 ,0f ,0 0 as  (13) where , , pr,h r sc ,  and s are non-dimensional parameters called respectively the dimensionless magnetic parameter, radiation parameter, prandtl number, schmidt number, chemical reaction parameter and suction parameter are given by 2 0 2 , b l h u  , 4 * 3* kk t r   ,pr k c p   d v sc  , 0 0 2 u lk  and l u v s 2 0 0   (14) in eq. (13), 0s and 0s correspond to injection and suction respectively. the non-dimensional velocity slip , f s thermal slip ts and solutal slip cs are defined by 0 0 1 1 , 2 2 f t u u s n s m l l      and 0 1 2 c u s p l  . (15) the quantities of physical interest in this problem are the skin-friction coefficient, heat transfer rate and mass transfer, which are defined as l x w f eu c 2 2 0 2    , )(    ttk xq nu w w x and )(    ccd xj sh w w x (16) respectively, where the surface shear stress w , surface heat flux wq and mass flux wj are given by 0 0 , w w y y u t q k y y                     and 0 w y c j d y          (17) substituting (9) and (17) into equations. (16) give re (0), (0) 2 re 2 xx f x nu c f x l     and (0) re 2 x x sh x l   (18) p. b. a. reddy / journal of naval architecture and marine engineering, 13(2016) 165-177 mhd boundary layer slip flow of a casson fluid over an exponentially stretching surface in the presence of thermal radiation… 169 where  l x x exu 0 re  is the local reynolds number. the above skin-friction coefficient, local nusselt number and sherwood number shows that its variation depends on the variation of the factors )0(),0( f and )0( respectively. 3. numerical method for solution: the set of nonlinear coupled differential equations (10)-(12) subject to the boundary conditions (13) are solved using shooting method, by converting them to an initial value problem. we set , ,f z z p   2 p (2 z ) 1 fp hz            (19) 3 pr , ( ) 4 3 q q fq z r             (20) , ( )r r sc fr z        (21) with the boundary conditions (0) , (0) 1 , (0), f f s f s f      (0) 1 s , (0),t      (0) 1 s , (0).c      (22) in order to integrate eqns.(19)-(21) as an initial value problem one requires a value for (0)p i.e., (0)f  , (0)q i.e., (0) and (0)r i.e., (0) but no such values are given at the boundary. the suitable guess values for (0), (0)f   and '(0) are chosen and the fourth order runge-kutta method with step size 0.01 is applied to obtain the solution. 4. results and discussion in the present study to gain a physical insight into the problem, velocity, temperature and concentration distributions have been discussed by assigning numerical values to various parameters obtained in the mathematical formulation of the problem and the numerical results are shown graphically. the default values of the various parameters which we considered were β = 2.0, h = 1.0, r = 0.5, pr = 0.72, sc = 0.60, ,5.0 ,5.0 f s 0.5, t s  5.0cs and s = 0.5 unless otherwise specified. in order to check the accuracy and validity of the applied numerical scheme, comparisons of the present numerical results corresponding to the heat transfer coefficient for various values of prandtl number and thermal radiation in the absence of casson fluid parameter, magnetic parameter, thermal radiation, schmidt number, suction parameter, velocity slip, thermal slip and solutal slip with the available published results of bidin and nazar (2009), magyari and keller (2000), mukhopadhyay and reddy (2012), ishak (2011), nadeem et al. (2011) and pramanik (2014) are made (see table 1 and table 2) and are found to be in excellent agreement. table 1: comparison (0) for several values of prandtl number in the absence of casson fluid parameter, magnetic parameter, thermal radiation, schmidt number, suction parameter, velocity slip, thermal slip and solutal slip. pr bidin and nazar (2009) magyari and keller (2000) el-aziz (2009) ishak (2011) pramanik (2014) present 1 0.9547 0.95478 0.9548 0.9548 0.9547 0.95477 2 1.4714 1.4715 1.4714 1.47144 3 1.8691 1.8691 1.8691 1.8691 1.8691 1.86916 5 2.5001 2.5001 2.5001 2.5001 2.50016 10 3.6604 3.6604 3.6604 3.6604 3.66038 p. b. a. reddy / journal of naval architecture and marine engineering, 13(2016) 165-177 mhd boundary layer slip flow of a casson fluid over an exponentially stretching surface in the presence of thermal radiation… 170 table 2: comparison (0) for several values of prandtl number and thermal radiation parameter in the absence of casson fluid parameter, magnetic parameter, schmidt number, suction parameter, velocity slip, thermal slip and solutal slip. bidin and nazar (2009) nadeem et al. (2011) pramanik (2014) present pr 0.5 1 0.5 1 0.5 1 0.5 1 1 0.6765 0.5315 0.680 0.534 0.6765 0.5315 0.6765 0.5315 2 1.0735 0.8627 1.073 0.863 1.0734 0.8626 1.0734 0.8627 3 1.3807 1.1214 1.381 1.121 1.3807 1.1213 1.3807 1.1213 the velocity, temperature and concentration for different values of casson fluid parameter, magnetic parameter, velocity slip, thermal radiation parameter, prandtl number, thermal slip, schmidt number, solutal slip and chemical reaction for both cases of suction and blowing are shown graphically in figs. 2-11. the effects of the casson fluid parameter on the velocity, velocity gradient, temperature, temperature gradient, concentration and concentration gradient profiles in the presence of suction/blowing are exhibited in figs. 2(a)-(f) respectively. from fig. 2(a), we observed that, the velocity and the momentum boundary layer thickness decreases with the increase of casson fluid parameter for both the cases of suction and blowing. it is found that the magnitude of shear stress decreases initially with the casson fluid parameter but increases significantly after a certain distance η normal to the sheet (fig. 2(b)). fig. 2(c), it is very clear that, the temperature and the thermal boundary layer thickness increases as the casson fluid parameter increases for both the cases of suction and blowing. the variations in the temperature gradient for changes in the casson fluid parameter are presented in fig. 2(d). it is noticed that the temperature gradient increases initially but it decreases after a certain distance η normal to the sheet. the effect of the casson fluid parameter on the concentration is presented in fig. 2(e). it can be observed that an increase in the casson fluid parameter enhances the concentration profiles. from fig. 2(f), it is noticed that the concentration gradient increases initially but it decreases after a certain distance η normal to the sheet. figs. 3(a)-(d) focus on the velocity, velocity gradient, temperature and concentration distributions for various values of the magnetic parameter. fig. 3(a) represents the velocity profiles for various values of magnetic parameter in the boundary layer for suction and blowing cases. from this figure it can be seen that the velocity and the momentum boundary layer thickness reduces with increasing values of magnetic parameter. this is due to the fact that an increase in magnetic parameter signifies an enhancement of lorentz force, thereby reducing the magnitude of the velocity. fig. 3(b) is aimed to shed light on the effect of magnetic parameter on the velocity gradient distribution. it can be noticed from this figure that the velocity gradient distribution of the flow field reduces as the magnetic parameter increases. the variations in the temperature profiles for changes in the magnetic parameter are presented in fig. 3(c) for suction and blowing cases. it is observed that the temperature profiles and the thermal boundary layer thickness increases with an increase in the magnetic parameter. fig. 3(d) describes the variation of concentration distribution for different values of magnetic parameter for suction and blowing cases. it is found that an increase in the magnetic parameter enhances concentration gradient for the suction and blowing cases. figs. 4(a)–(b) depict the influence of velocity and the velocity gradient for different values of velocity slip. the variations in the velocity profiles for changes in the velocity slip are presented in fig. 4(a). it can be noticed that the velocity of the boundary layer reduces with increasing values of velocity slip parameter for suction and blowing cases. fig. 4(b) displays the variations of velocity gradient profile with changes in the dimensionless velocity slip for suction and blowing cases. results indicate that the increase in the velocity slip enhances the velocity. figs. 5(a)-(b) focus on the temperature and temperature gradient for various values of the thermal radiation parameter. fig. 5(a) represents the temperature profiles for various values of thermal radiation parameter in the boundary layer for suction and blowing. from this it can be inferred that an increase in the thermal radiation enhances the heat transfer. fig. 5(b) is aimed to shed light on the effect of thermal radiation parameter on the temperature gradient distribution. it is noticed that the temperature gradient increases initially but it decreases after a certain distance η normal to the sheet. fig. 6(a) describes the variation of temperature distribution for different values of prandtl number for the suction and blowing. from this plot, it is evident that the thickness of the boundary layer as well as the temperature profiles decreases with an increase in the value of the prandtl number. the changes in the temperature gradient profiles for various values of prandtl number is plotted in fig. 6(b). it is noticed that the temperature gradient decrease initially but it increases after a certain distance η normal to the sheet. the variations in the temperature profiles for changes in the thermal slip are presented in fig. 7(a) for the suction and blowing. from this figure it can be seen that the temperature of the boundary layer p. b. a. reddy / journal of naval architecture and marine engineering, 13(2016) 165-177 mhd boundary layer slip flow of a casson fluid over an exponentially stretching surface in the presence of thermal radiation… 171 reduces with increasing values of thermal slip parameter. fig. 7(b) represents the temperature gradient profiles for various values of thermal slip. it can be inferred that an increase in the thermal slip parameter enhances temperature gradient for the suction and blowing cases. fig. 8(a) demonstrates the effect of the schmidt number on the concentration distribution. one can observe that the concentration distribution of the flow field reduces as the schmidt number increases for the suction and blowing cases. fig. 8(b) is a plot of concentration gradient distribution for various values of schmidt number for the suction and blowing. it is observed that the concentration gradient decreases initially but it increases after a certain distance η normal to the sheet. fig. 9(a) shows that the concentration decreases with the increase of solutal slip for suction and blowing. we noticed from the fig. 9(b) that the concentration gradient profiles increase with an increase in the solutal slip. figs. 10(a)-(b) shows the influence of the chemical reaction on the concentration and the concentration gradient profiles in the boundary layer for the case of suction. it reveals that the concentration decreases with an increase in the destructive ( )0 chemical reaction, whereas the reverse trend is observed in the case of generative ( )0 chemical reaction (fig. 10(a)). from the fig. 10(b), it is observed that the concentration gradient decreases with an increase in the destructive ( )0 chemical reaction, whereas the reverse trend is observed in the case of generative ( )0 chemical reaction. figs. 11(a)-(b) focus the influence of the chemical reaction on the concentration and the concentration gradient profiles in the boundary layer for the case of blowing. it can be seen that the concentration decreases with an increase in the destructive ( )0 chemical reaction, whereas the reverse trend is observed in the case of generative ( )0 chemical reaction (fig. 11(a)). from the fig. 11(b), it is noticed that the concentration gradient decreases with an increase in the destructive ( )0 chemical reaction, whereas the reverse trend is observed in the case of generative ( )0 chemical reaction. fig. 2: (a) velocity profiles for several values of casson parameter of suction/blowing (b) velocity gradient profiles for several values of casson parameter of suction/blowing fig. 2: (c) temperature profiles for several values of casson parameter of suction/blowing (d) temperature gradient profiles for several values of casson parameter of suction/blowing s = 0.5 s = -0.5 s = 0.5 s = -0.5 s = 0.5 s = -0.5 s = 0.5 s = -0.5 β = 0.5, 1.0, 2.0 β = 0.5, 1.0, 2.0 β = 0.5, 1.0, 2.0 β = 0.5, 1.0, 2.0 p. b. a. reddy / journal of naval architecture and marine engineering, 13(2016) 165-177 mhd boundary layer slip flow of a casson fluid over an exponentially stretching surface in the presence of thermal radiation… 172 fig. 2: (e) concentration profiles for several values of casson parameter of suction/blowing (f) concentration gradient profiles for several values of casson parameter of suction/blowing fig. 3: (a) velocity profiles for several values of magnetic parameter of suction/blowing (b) velocity gradient profiles for several values of magnetic parameter of suction/blowing fig. 3: (c) temperature profiles for several values of magnetic parameter of suction/blowing (d) concentration profiles for several values of magnetic parameter of suction/blowing s = 0.5 s = -0.5 s = 0.5 s = -0.5 s = 0.5 s = -0.5 s = 0.5 s = -0.5 s = 0.5 s = -0.5 s = 0.5 s = -0.5 β = 0.5, 1.0, 2.0 β = 0.5, 1.0, 2.0 h = 0.5, 1.0, 1.5 h = 0.5, 1.0, 1.5 h = 0.5, 1.0, 1.5 h = 0.5, 1.0, 1.5 p. b. a. reddy / journal of naval architecture and marine engineering, 13(2016) 165-177 mhd boundary layer slip flow of a casson fluid over an exponentially stretching surface in the presence of thermal radiation… 173 fig. 4: (a) velocity profiles for several values of velocity slip of suction/blowing (b) velocity gradient profiles for several values of velocity slip of suction/blowing fig. 5: (a) temperature profiles for several values of radiation parameter of suction/blowing (b) temperature gradient profiles for several values of radiation parameter of suction/blowing fig. 6: (a) temperature profiles for several values of prandtl number of suction/blowing (b) temperature gradient profiles for several values of prandtl number of suction/blowing s = 0.5 s = -0.5 s = 0.5 s = -0.5 s = 0.5 s = -0.5 s = 0.5 s = -0.5 pr = 0.72, 1.0, 1.5 s = 0.5 s = -0.5 s = 0.5 s = -0.5 sv = 0.0, 0.5, 1.0 sv = 0.0, 0.5, 1.0 r = 0.5, 1.0, 1.5 r = 0.5, 1.0, 1.5 pr = 0.72, 1.0, 1.5 p. b. a. reddy / journal of naval architecture and marine engineering, 13(2016) 165-177 mhd boundary layer slip flow of a casson fluid over an exponentially stretching surface in the presence of thermal radiation… 174 fig. 7: (a) temperature profiles for several values of thermal slip of suction/blowing (b) temperature gradient profiles for several values of thermal slip of suction/blowing fig. 8: (a) concentration profiles for several values of schmidt number of suction/blowing (b) concentration gradient profiles for several values of schmidt number of suction/blowing fig. 9: (a) concentration profiles for several values of solutal slip of suction/blowing (b) concentration gradient profiles for several values of solutal slip of suction/blowing s = 0.5 s = -0.5 s = 0.5 s = -0.5 s = 0.5 s = -0.5 s = 0.5 s = -0.5 s = 0.5 s = -0.5 s = 0.5 s = -0.5 st = 0, 0.5, 1.0 st = 0, 0.5, 1.0 sc = 0.22, 0.6, 0.78 sc = 0.22, 0.6, 0.78 sc = 0, 0.5, 1.0 sc = 0, 0.5, 1.0 p. b. a. reddy / journal of naval architecture and marine engineering, 13(2016) 165-177 mhd boundary layer slip flow of a casson fluid over an exponentially stretching surface in the presence of thermal radiation… 175 fig. 10: (a) concentration profiles for several values of chemical reaction parameter of suction (b) concentration gradient profiles for several values of chemical reaction parameter of suction fig. 11: (a) concentration profiles for several values of chemical reaction parameter of blowing (b) concentration gradient profiles for several values of chemical reaction parameter of blowing table 3: the values of skin friction coefficient, sherwood number and the nusselt number for various values of h, sf , r, pr, st, sc, , sc and s.  h sf r pr st sc γ sc s )0( )0( 2.0 0.5 0.5 0.5 0.7 0.5 0.6 0.5 0.5 0.5 -0.84779 0.43524 0.63787 3.0 0.5 0.5 0.5 0.7 0.5 0.6 0.5 0.5 0.5 -0.90234 0.42431 0.61722 2.0 1.0 0.5 0.5 0.7 0.5 0.6 0.5 0.5 0.5 -0.90317 0.42195 0.62803 2.0 0.5 1.0 0.5 0.7 0.5 0.6 0.5 0.5 0.5 -0.57730 0.40871 0.61514 2.0 0.5 0.5 1.0 0.7 0.5 0.6 0.5 0.5 0.5 -0.84779 0.37775 0.63787 2.0 0.5 0.5 0.5 1.0 0.5 0.6 0.5 0.5 0.5 -0.84779 0.51543 0.63787 2.0 0.5 0.5 0.5 0.7 1.0 0.6 0.5 0.5 0.5 -0.84779 0.35745 0.63787 2.0 0.5 0.5 0.5 0.7 0.5 1.0 0.5 0.5 0.5 -0.84779 0.43524 0.48362 2.0 0.5 0.5 0.5 0.7 0.5 0.6 1.0 0.5 0.5 -0.84779 0.43524 0.71746 2.0 0.5 0.5 0.5 0.7 0.5 0.6 0.5 1.0 0.5 -0.84779 0.43524 0.48362 2.0 0.5 0.5 0.5 0.7 0.5 0.6 0.5 0.5 1.0 -0.93642 0.50393 0.71383 )0(f  γ = -0.5, 0, 0.5, 1.0, 1.5, 2.0 γ = -0.5, 0, 0.5, 1.0, 1.5, 2.0 γ = -0.5, 0, 0.5, 1.0, 1.5, 2.0 γ = -0.5, 0, 0.5, 1.0, 1.5, 2.0 p. b. a. reddy / journal of naval architecture and marine engineering, 13(2016) 165-177 mhd boundary layer slip flow of a casson fluid over an exponentially stretching surface in the presence of thermal radiation… 176 5. conclusions a numerical study is performed to analyze the boundary layer casson fluid flow over an exponentially stretching sheet in the presence of slips, thermal radiation and chemical reaction. the governing partial differential equations have been transformed by a similarity transformation into a system of ordinary differential equations, which are solved numerically using the runge–kutta fourth order along with a shooting technique. the main numerical results of the present analysis can be listed as follows: (i) momentum boundary layer thickness decreases with an increasing the casson fluid parameter, whereas the reverse trend for the thermal boundary layer thickness. also the same trend is observed in the case of magnetic parameter. (ii) the temperature increases with an increasing the radiation parameter, whereas the reverse trend is observed in the case of prandtl number. (iii) the concentration decreases with an increase in the destructive chemical reaction, whereas the reverse trend is observed in the case of generative chemical reaction. (iv) the skin friction coefficient decreased with increasing values of casson fluid parameter, magnetic parameter and suction parameter, whereas the reverse trend is observed in the case of velocity slip. (v) the nusselt number decreases with an increasing the casson fluid parameter, radiation parameter and thermal slip, where as it increases with an increase in the prandtl number and the suction parameter. (vi) the sherwood number decreases as the casson fluid parameter, schmidt number and the solutal slip. it is hopeful that the present investigation will contribute a better understanding of the flow dynamics and heat and mass transfer of casson fluid flow over an exponential stretching surface as well as real application. references aziz, e.l. (2009): viscous dissipation effect on mixed convection flow of a micropolar fluid over an exponentially stretching sheet, can. j. phys., vol. 87, pp. 359-368. https://doi.org/10.1139/p09-047 bhattacharya, k., hayat, t. and alsaedi, a. (2013): analytic solution for magnetohydrodynamic boundary layer flow of casson fluid over a stretching/shrinking sheet with mass transfer, chin. phys. b, vol. 22, pp. 024702. https://doi.org/10.1088/1674-1056/22/2/024702 bhattacharya, k.., hayat, t. and alsaedi, a. 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(2013): casson fluid flow and heat transfer over a nonlinearly stretching surface, chin. phys. b, vol. 22, no. 7, pp. 074701-5. https://doi.org/10.1088/1674-1056/22/7/074701 mukhopadhyay, s., prativa ranjan de, bhattacharya, k. and layek, g.c. (2013): casson fluid flow over an unsteady stretching surface, ain shams eng. j., vol. 4, pp. 933-938. https://doi.org/10.1016/j.asej.2013.04.004 mukhopadhyay, s. and reddy, g.r.s. (2012): effects of partial slip on boundary layer flow past a permeable exponential stretching sheet in presence of thermal radiation, heat mass transf., vol. 48, pp. 1773-1781. https://doi.org/10.1007/s00231-012-1024-8 mustafa, m., hayat, t., pop, i. and aziz, a. (2011): unsteady boundary layer flow of a casson fluid due to an impulsively started moving flat plate, heat transf. asian res., vol. 40, pp. 563-576. https://doi.org/10.1002/htj.20358 mustafa, m., hayat, t., pop, i. and hendi, a. 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(2011): effects of thermal radiation on the boundary layer flow of a jeffery fluid over an exponentially stretching surface, numer. algor., vol. 57, pp. 187-205. https://doi.org/10.1007/s11075-010-9423-8 pramanik, s. (2014): casson fluid flow and heat transfer past an exponentially porous stretching surface in presence of thermal radiation, ain shams eng. j., vol. 5, pp. 205-212. https://doi.org/10.1016/j.asej.2013.05.003 reddy, p. b. a. (2016): magnetohydrodynamic flow of a casson fluid over an exponentially inclined permeable stretching surface with thermal radiation and chemical reaction, ain shams engineering journal, vol. 7, pp.593602. reddy, p. b. a. and reddy, n. b. (2011): thermal radiation effects on hydromagnetic flow due to an exponentially stretching sheet, int. j. appl. math. comp., vol. 3, pp. 300-306. srinivas, s., reddy, p.b.a. and prasad, b.s.r.v. (2014): effects of chemical reaction and thermal radiation on mhd flow over an inclined permeable stretching surface with non-uniform heat source/sink: an application to the dynamics of blood flow, j. mech. med. biol., vol. 14, pp. 1450067 1-24. https://doi.org/10.1088/1674-1056/22/7/074701 https://doi.org/10.1016/j.asej.2013.04.004 https://doi.org/10.1007/s00231-012-1024-8 https://doi.org/10.1002/htj.20358 https://doi.org/10.5560/zna.2011-0057 https://doi.org/10.1016/j.aej.2013.08.005 https://doi.org/10.1016/j.ijthermalsci.2013.12.001 https://doi.org/10.1007/s11075-010-9423-8 https://doi.org/10.1016/j.asej.2013.05.003 separation points of magnetohydrodynamic boundary layer flow along a vertical plate with exponentially decreasing free stream velocity¬ journal of naval architecture and marine engineering december, 2014 http://dx.doi.org//10.3329/jname.v11i2.6477 http://www.banglajol.info 1813-8235 (print), 2070-8998 (online) © 2014 aname publication. all rights reserved. received on: november, 2010 unsteady mhd mixed convection flow past an oscillating plate with heat source/sink a. m. okedoye department of mathematics and computer science, federal university of petroleum resources, p. m. b. 1221, effurun, delta state nigeria, email: okedoye.akindele@fupre.edu.ng, dele.mikeoke@gmail.com abstract: this paper study unsteady mhd mixed convection flow past an infinite vertical oscillating plate through porous medium, taking account of the presence of free/forced convection and mass transfer. using similarity transformation, the coupled non – linear governing equations are solved numerically by applying the combination of the base scheme sub-methods – midpoint, and a method enhancement scheme richardson extrapolation technique together with fehlberg fourth-fifth order runge-kutta shooting iteration method with degree four interpolant. the results are obtained for velocity, temperature, concentration. the effects of various material parameters are discussed on flow variables and presented by graphs. keywords: mixed convection, magnetohydrodynamic flows, porous medium, mass transfer, oscillating plate generative reactions,convergence, runge – kutta, dufour effect, soret effect, isotope separation nomenclature thermal conductivity dimensionless group u, v velocity components along xand yaxes, respectively mass grashof number concentration of the fluid thermal grashof number diffusion coefficient prandtl number fluid temperature schmidt number free steam velocity hartmann number free stream concentration dufour number free stream temperature soret number ha hartmann number greek symbols heat generation coefficient non dimensional fluid temperature specific heat at constant pressure ratio of free stream velocity parameter to stretching sheet parameter surface concentration heat source/sink coefficient surface temperature coefficient of concentration expansion magnetic induction coefficient of thermal expansion time kinematic viscousity coefficient of heat transfer electrical conductivity acceleration due to gravity density is the darcy permeability subscripts is the empirical constant condition on the wall is the suction/blowing parameter ambient condition 1. introduction the study of magneto-hydrodynamic (mhd) flows have stimulated considerable interest due to its important applications in cosmic fluid dynamics, meteorology, solar physics and in the motion of earth’s core (cramer and pai, 1973)]. in a broader sense, mhd has applications in three different subject areas, such as astrophysical, geophysical and engineering problems. convection flow driven by temperature and concentration differences  grc c grt dm pr t sc 0 u m  c du  t sr q  p c  w c  w t c  0 b   t  q  g  'k b w w v  a. m. okedoye/ journal of naval architecture and marine engineering 11(2014) 167-176 unsteady mhd mixed convection flow past an oscillating plate with heat source/sink. 168 has been the objective of extensive research because such processes exists in nature and has engineering applications. the process occurring in nature includes photo-synthetic mechanism, calm-day evaporation and vaporization of mist and fog, while the engineering application includes the chemical reaction in a reactor chamber consisting of rectangular ducts, chemical vapour deposition on surfaces and cooling of electronic equipment. sharma and singh (2009) report the effects of variable thermal conductivity and heat source/sink on mhd flow near a stagnation point on a linearly stretching sheet. the analysis of propagation of thermal energy through mercury and electrolytic solution in the presence of external magnetic field and heat absorbing sinks has wide range of applications in chemical and aeronautical engineering, atomic propulsion, space science etc. recently, ahmed and ahmed (2004) analyzed two-dimensional mhd oscillatory flow along a uniformly moving infinite vertical porous plate bounded by porous medium. further, ogulu and prakash (2006) investigated the effects magnetic field on heat transfer unsteady flow past an infinite moving vertical plate with variable suction. later on, the effects of unsteady free convective mhd flow through a porous medium bounded by an infinite vertical porous plate were investigated by ahmed (2007). ahmed (2010) studied mixed convection hydro-magnetic oscillatory flow and periodic heat transfer of a viscous incompressible and electrically conducting fluid past an infinite vertical porous plate. it was reported that the mean and transient velocity decreases with the increase in prandtl number. physically this is true because the increase in the prandtl number is due to increase in the viscosity of the fluid, which makes the fluid thick and hence a decrease in the velocity of the fluid. zueco and ahmed (2010) studied combined heat and mass transfer by mixed convection mhd flow along a porous plate with chemical reaction in presence of heat source. it was reported that as the chemical reaction parameter kincreases, the temperature profiles decrease. sharma et al. (2011) investigated the influence of chemical reaction and radiation on an unsteady mhd free convective flow and mass transfer through a viscous incompressible, electrically conducting fluid past an infinite vertical heated porous plate with suction, embedded in porous medium in the presence of a uniform transverse magnetic field, oscillating free stream and heat source by taking into account the viscous dissipation. they discovered that the magnitude of fluid temperature decreases with the increase of chemical reaction parameter, viscous dissipation effect and molecular diffusivity; while it increases with an increase of intensity of magnetic field and heat source. also an increase in the grashof number, leads to a rise in the magnitude of fluid velocity due to enhancement in buoyancy force. the peak value of the velocity increases rapidly near the porous plate as buoyancy force for heat transfer increases and then decays the free stream velocity. sharma, chand and chaudhary (2011) report an approximate analysis of unsteady mixed convection flow of an electrically conducting fluid past an infinite vertical porous plate embedded in porous medium under constant transversely applied magnetic field, where the transient velocity increases with increase in distance from plate until it attains its maximum value (nearly y = 1), after which it decreases. osman et al. (2011) studied thermal radiation and chemical reaction effects on the unsteady mhd convection through a porous medium bounded by an infinite vertical plate with heat source/sink. they observed that the presence of porous media increases the resistance flow resulting in a decrease in the flow velocity. several other researcher have worked on mhd convection flow, okedoye et al (2008) has good review of it. due to the importance of soret (thermal-diffusion) and dufour (diffusionthermo) effects for the fluids with very light molecular weight as well as medium molecular weight many investigators have studied and reported results for these flows. for the problem of coupled heat and mass transfer in mhd mixed convective flow of a conducting fluid through a porous medium in the presence of chemical reaction, the effect of both dufour and soret effects were neglected, on the basis that they are of a smaller order of magnitude than the effects described by fourier’s and fick’s laws. there are, however, exceptions the soret effect, for instance, has been utilized for isotope separation and in mixture between gases and with very light molecular weight (h 2 , he), and for medium molecular weight (h 2 , air) the dufour effect was found to be of considerable magnitude such that it cannot be neglected. hence, as a complementary study to that of postelnicu (2004) and alam and rahman (2005), we propose to study the above-mentioned unsteady free forced convection flow past an oscillating plate in a porous medium under the influence of transversely applied magnetic field.the aim of this paper is to present the numerical analysis of unsteady mhd mixed convective flow of a conducting fluid with variable properties through a porous medium in the presence of chemical reaction and heat source or sink when the plate is made to oscillate in time about a non-zero constant mean with a specified velocity. a. m. okedoye/ journal of naval architecture and marine engineering 11(2014) 167-176 unsteady mhd mixed convection flow past an oscillating plate with heat source/sink. 169 2. mathematical formulation we consider the mixed convection flow of an incompressible and electrically conducting viscous fluid along an infinite non conducting vertical flat plate through a porous medium. the x axis is taken along the plate in the vertically upward direction and y axis is taken normal to the plate. a magnetic field of uniform strength b0 is applied in the direction of flow and the induced magnetic field is neglected. initially, the plate and the fluid are at same temperature t∞in a stationary condition with concentration level c at all points. at time 0t the plate starts oscillating in its own plane with a velocity tu cos0 . its temperature is raised to wt and the concentration level at the plate is raised to wc . the coordinates system and the configuration are shown in fig. 1 in view of these, we consider that: (i) all the fluid properties except density in the buoyancy force term are constant; (ii) the influence of the density variations in other terms of the momentum and energy equations and the variation of the expansion coefficient with temperature is negligible; (iii) the eckert number and the magnetic reynolds number are small so that the induced magnetic field can be neglected. (iv) all the physical variables are independent of x, except possibly the pressure. (v) the plate is subjected to a constant suction velocity. (vi) there exists a first-order homogeneous chemical reaction with a constant rate k between the diffusing species and the fluid. with foregoing assumptions using the boussinesq approximation, the governing equations for the flow are given by: 0   y v (1)     2 2 0 2 2 u k b u k u b ccgttg y u y u v t u c                  (2)                     ttq y c cc dmk y t y t v t t c ps t p 2 2 2 2  (3)                     cca y t t dmk y c dm y c v t c m t 02 2 2 2  (4) the boundary conditions are given by       0,0,,0,,00,   tycyctytyu (5)             0,,,0 0,0,0,,0,cos,0 0 tyasccttu tyatctcttttutu ww  (6) t∞, c∞ g u0 b0 fig. 1: flow configuration a. m. okedoye/ journal of naval architecture and marine engineering 11(2014) 167-176 unsteady mhd mixed convection flow past an oscillating plate with heat source/sink. 170 now integrating (1) we have,   consttyv , . then an appropriate value of 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡for the problem under consideration is taking to be, h vv w   , (7) where wv > 0 is the suction parameter and wv < 0 is the injection parameter and h is the scale parameter. let us introduce the non-dimensional variables               cc cc tt tt fuu y ww w  ,,)(, h (8) with the help of (7) and (8), the governing equations (2) – (4) reduce to                grcgrtff k hffvf dt dhh aw        22 1 (9)            0 prpr 1  du v dt dhh w (10)            1 1  sc sr sc v dt dhh w (11) equations (9) to (11) are similar except for the term dt dhh  where t appears explicitly. thus the similarity condition required that dt dhh  must be a constant. thus c dt dhh   that is tch  without loss of generality, we take 2c , and so th 2 , which define the well – established scaling parameter for unsteady boundary layer problem. hence equations (9) – (11) together with the boundary and initial conditions (5) and (6) becomes               0 1 2 22         grcgrtff k hfcf a (12)           pr2pr 0  duc (13)           scsrcsc 1 2  (14)                    0,0,0 010,10,cos0 f f (15) where                        4 ,,, ,,,, ,, 222 02 2 2 0 1 2 0 2 22 hhb h k hbu tt cc cc dmk du dm sc ha cc tt tm k sr c qhc p h k k u tchg grc u tthg grt a w w w ps t w wt p p r w wc w w                                    the physical variables have their usual meanings as defined in nomenclature. a. m. okedoye/ journal of naval architecture and marine engineering 11(2014) 167-176 unsteady mhd mixed convection flow past an oscillating plate with heat source/sink. 171 3. numerical computation the set of equations (12) – (14) under the boundary conditions (15) have been solved numerically by applying the combination of the base scheme sub methods – midpoint (hairer and wanner (1996)), and a method enhancement scheme richardson extrapolation (for detail discussion of the method see ascher, mattheij, and russell (1995) and ascher, and petzold (1998)) technique together with fehlberg fourth-fifth order rungekutta shooting iteration method with degree four interpolant. the results are presented in figs. (2) – (17) 3.1. convergence. in this paper, we considered rk methods with 2s and coefficients satisfying the hypotheses 0:1  ji ah for ;,,1 sj  :1h thesubmatrix 2,)(: ~   jiji aa is invertible; isi abh :2 for ,,,1 sj  i.e., the method is stiffly accurate. in order to show the convergence criteria however, for convenience, we present here the theorems the detail proof is analogous to the one in ascher and petzold (1991) theorem 1.0.let ,iy iz be the solution of (12, 13, 14) subject to (15) and consider perturbed values , ~ i y i z ~ satisfying (1.1) si yg hzyfahy ii s j ijjjii ,,1 ) ~ (0 ) ~ , ~ (~ ~ 1             with .: ~ 1 z in addition to the assumptions of theorem 4.1, suppose that (1.2) ).(),(),( ~ ),(~ 2 hohohozho iii   then we have for 0hh  the estimates (1.3)  ,~~ 2   hhcyy ii (1.4)    hhh h c zz ii ~~ where ,),,( 1 t s   ii  max and similarly for  . remarks. 1) the conditions (1.2) ensure that all terms )(o in the proof below are small. 2) we introduce the notation ,~   ,  ,),,( 1 t s yyy  , ~ yyy  ii yy  max and similarly for the z-component. over a multiple-vector a tilde '~' indicates the removal of its first subvector, e.g. .),,( ~ 2 t s yyy  theorem 2.0.in addition to the assumptions of theorem 1.0, suppose that the conditions c(q), d(r) and the hypothesis h3 hold, and that )(),)(( k y hofg  with 1k . then we have (2.1)  ,~)~)(,(~ 22    hhhhopyy m ss (2.2)  hhorzz ii   ~))(( ~ where ,0),1,1min(  rqkm r is the stability function, and p is the projectordefined under the condition (1.3) by (2.3) ,: qip n  .)(: 1 yzyz gfgfq   remarks. the important result consists in the factor 2m h in front of   in (2.1) (2.4). a. m. okedoye/ journal of naval architecture and marine engineering 11(2014) 167-176 unsteady mhd mixed convection flow past an oscillating plate with heat source/sink. 172 theorem 2.0 yields the main component for the convergence proof of rk methods with singular rk matrix a. 4. results and discussion here we have investigated numerically mhd mixed convection flow past an oscillating plate with heat source or sink. in order to point out the effects of various parameters on flow characteristic, the following discussion is set out. in simulation, the values of the prandtl number are considered to be 0.70, 1.70, 2.97 and 4.34 that corresponds to helium, sulfur dioxide, methyl chloride and water, respectively. the values of the schmidt number are chosen to represent the presence of species water vapour (0.60). numerical results have been obtained for different values of flow condition and are presented graphically. special cases (i) in the absence of magnetic field i.em= 0, the results of the present paper are reduced to those obtained by pop, grosan, pop (2004) and mahapatra and gupta (2002). (ii)in the absence of magnetic field, heat source/sink and variable thermal conductivity, the results of the present paper are reduced to those obtained by pop, grosan, pop (2004) inthe absence of radiation effect with constant thermal conductivity and mahapatra and gupta (2002) in absence of viscous dissipation and constant thermal conductivity. (iii) in the absence of chemical reaction, the result of the present paper are reduced to those obtained by sharma and singh (2008) (iv) in the absence of dufour and soret effect, the results of the present paper are reduced to those obtained by osman, abo-dahab, and r. a. mohamed1 (2011) fig. 2: velocity distribution for various  fig. 3: velocity distribution for various c fig. 4: temperature distribution for various c fig. 5: concentration distribution for various c in fig. 2 represents the velocity profiles due to the variations in ωt. it is evident from the figure, under the chosen condition that the maximum velocity is at the plate. furthermore, the magnitude of the velocity decreases with increasing phase angle (ωt). figs. 3, 4 and 5 reveal the velocity, temperature and species concentration variations with suction/injection parameter. it is observed that in case of cooling of surface (an a. m. okedoye/ journal of naval architecture and marine engineering 11(2014) 167-176 unsteady mhd mixed convection flow past an oscillating plate with heat source/sink. 173 increase in gr), decrease in injection rate decreases velocity, temperature and species concentration distribution, while increase in suction rate increases the fields. it is found that the velocity decreases with increase in magnetic parameter. it is because that the application of transverse magnetic field will result a resistive type force (lorentz force) similar to drag force which tends to resist the fluid flow and thus reducing its velocity. the presence of a porous medium increases the resistance to flow resulting in decrease in the flow velocity, since 1/kis an addition to ha 2 . this behaviour is depicted by the decrease in the velocity as k decreases and when k = ∞ (i.e. the porous medium effect is vanished) the velocity is greater in the flow field as shown in fig. 6. in figs. 7 and 8, it is observed that greater cooling of surface (an increase in grt) and increase in grc results in an increase in the velocity, respectively. it is due to the fact increase in the values of thermal grashof number and mass grashof number has the tendency to increase the thermal and mass buoyancy effect. this gives rise to an increase in the induced flow. furthermore, the velocity near the plate is greater than at the plate. the maximum velocity attains near the plate and is in the neighbourhood of point η = 0.4. after η > 0.4, the velocity decreases and tends to zero as η → ∞. figs. 9 and 10 display the effects of du (dufour number) on the velocity and temperature fields for the case grt> 0 and grc> 0. it is seen that increase in dufour number brings about increase in both the velocity and temperature distribution in the flow field, while in fig.11, it is sown tat concentration distribution increases with an increase in sr (soret number) with maximum concentration near the plate for higher value of sr. fig. 6: velocity distribution for various hartmann number fig. 7: velocity distribution for various thermal grashof number fig. 8: velocity distribution for various mass grashof number fig. 9: velocity distribution for various values of dufor number figs. 12, 13 and 14 depict the effect of internal heat generation/absorption on velocity, temperature and concentration fields respectively. it is observed that internal heat generation increases the velocity, temperature and concentration distribution wile international eat absorption reduces the velocity and temperature distribution. furthermore the magnitude of temperature is maximum at the plate whereas for internal eat generation the maximum velocity and concentration is near the wall and then decays to zero asymptotically. effect of chemical reactivity on velocity, temperature and concentration fields are sown in figs. 15, 16 and 17 respectively. it should be noted here that, 0 1  correspond to destructive chemical reaction and 0 1  a. m. okedoye/ journal of naval architecture and marine engineering 11(2014) 167-176 unsteady mhd mixed convection flow past an oscillating plate with heat source/sink. 174 correspond to generative chemical reaction. it is observed that increase in generative chemical reaction increases the velocity and concentration distribution but it reduces the flow temperature distribution. while the reverse is the case for destructive chemical reaction. furthermore the magnitude of temperature is maximum at the plate whereas for generative chemical reaction, the maximum velocity and concentration is near the wall and then decays to zero asymptotically. fig. 10: temperature distribution for various values of dufor number fig. 11: concentration distribution for various values of soret number fig. 12: velocity distribution for various values of reaction parameter fig. 13: temperature distribution for various values of reaction parameter fig. 14: concentration distribution for various values of reaction parameter fig. 15: velocity distribution for various values of heat parameter a. m. okedoye/ journal of naval architecture and marine engineering 11(2014) 167-176 unsteady mhd mixed convection flow past an oscillating plate with heat source/sink. 175 fig. 16: temperature distribution for various values of heat parameter fig. 17: concentration distribution for various values of heat parameter 5. conclusion in this paper effect of temperature dependent thermal conductivity on mhd free convection flow along a vertical flat plate have been studied numerically. implicit finite difference method together with keller box scheme is employed to integrate the equations governing the flow. comparison with previously published work is performed and excellent argument has been observed. from the present numerical investigation, following conclusions may be drawn:  for increased value of magnetic parameter, the velocity profile decreases but the temperature profile increases slightly.  in case of cooling of the plate (grt> 0), the velocity decreases with an increase in phase angle, injection and magnetic parameter. on the other hand, it increases with an increase in the value of thermal grashof number and mass grashof number, suction parameter, dufour number, internal heat generation and generative chemical reaction.  the local skin friction coefficient decreases as well as the surface temperature distribution increase with the increase in values of the magnetic parameter.  the concentration decreases with an increase in injection and increases with increase in soret number, internal heat generation and generative chemical reaction.  the temperature decreases with an increase in injection and increases with increase in dufour number, suction, internal heat generation and generative chemical reaction. references ahmed, s. (2010): free and forced convective mhd oscillatory flow over an infinite porous surface in an oscillating free stream,latin american applied research,vol. 40, pp. 167-173 ahmed, s. (2007): effects of unsteady free convective mhd flow through a porous medium bounded by an infinite vertical porous plate, bulleting of calcuttamathematical society, vol. 99, pp. 511-522. ahmed, s. and ahmed, n. (2004): two-dimensional mhd oscillatory flow along a uniformly moving infinite vertical porous plate bounded by porous medium, international journal of pure and applied mathematics,vol.35, pp. 1309-1319 alam, m. s. and rahman, m. m. (2005): dufour and soret effects on mhd free convective heat and mass transfer flow past a vertical flat plate embedded in a porous medium, journal of naval architecture and marine engineering, vol. 2, no. 1, pp. 55–65. ascher, u. m. and petzold, l. r. (1991): projected implicit runge-kuttamethods for differential-algebraic equations,society for industrial and applied mathematics, journal of numerical analysis, vol. 28, pp. 10971120.http://dx.doi.org/10.1137/0728059 http://dx.doi.org/10.1137/0728059 a. m. okedoye/ journal of naval architecture and marine engineering 11(2014) 167-176 unsteady mhd mixed convection flow past an oscillating plate with heat source/sink. 176 ascher, u., mattheij. r. and russell, r. (1995): numerical solution of boundary value problems for ordinary differential equations, society for industrial and applied mathematics, classics in applied mathematics, vol. 13. ascher, u. m. and petzold, l. (1998): computer methods for ordinary differential equations and differential algebraic equations,society for industrial and applied mathematics, philadelphia.http://dx.doi.org/10.1137/1.9781611971392 cramer, k. r. and pai, s. i. (1973): magneto-fluid dynamics for engineers and applied physicists, mcgrawhill co., new york. hairer, e. and wanner, g. (1996): solving ordinary differential equations ii. 2nd ed. new york: springer.http://dx.doi.org/10.1007/978-3-642-05221-7 mahapatra, t. r. and gupta, a. s. (2002): heat transfer in a stagnation point flow toward stretching sheet, journal of heat and mass transfer,vol. 38, pp. 517 – 521.http://dx.doi.org/10.1007/s002310100215 ogulu, a. and prakash, j. (2006): heat transfer to unsteady magneto-hydrodynamic flow past an infinite moving vertical plate with variable suction, royal swedish academy of society, vol. 74, pp. 232-239. okedoye, a. m. faryol,a, p. i. and bello, o. a. (2008): mhd flow of a uniformly stretched vertical permeable surface under oscillatory suction velocity, j. nigeria association of mathematical physics, vol. 13,pp. 117 – 130. osman, a. a., abo-dahab, s. m. and mohamed, r. a. (2011): analytical solution of thermal radiation and chemical reaction effects on unsteady mhd convection through porous media with heat source/sink, mathematical problems in engineering, vol. 2011, pp.1-18.http://dx.doi.org/10.1155/2011/205181 pop, s. r., grosan, t. and pop, i. (2004):.radiation effect on the flow near the stagnation point of a stretching sheet, technischemechanik, vol. 25, pp. 100-106. postelnicu, a. (2004): influence of a magnetic field on heat and mass transfer ay natural convection from vertical surfaces in porous media considering soret and dufour effects, international journal of heatand mass transfer, vol. 47, no. (6–7), pp. 1467–1472. sharma, b. k., chand, t. and chaudhary, r. c. (2011): hydro-magnetic unsteady mixed convection flow past an infinite vertical porous plate,applied mathematics,vol. 1, no.1,pp.39-45. http://dx.doi.org/1. 5923/j.am.20110101.05 sharma, p. r. and singh, g. (2009): effects of variable thermal conductivity and heat source/sink on mhd flow near a stagnation point on a linearly stretching sheet, journal of applied fluid mechanics,vol. 2, no. 1, pp. 13-21. sharma, p. r., kumar, n. and sharma, p. (2011): influence of chemical reaction and radiation on unsteady mhd free convective flow and mass transfer through viscous incompressible fluid past a heated vertical plate immersed in porous medium in the presence of heat source, applied mathematical sciences, vol. 5, no. 46, 2249 – 2260. zueco,j. and ahmed, s. (2010): combined heat and mass transfer by mixed convection mhd flow along a porous plate with chemical reaction in presence of heat source,applied mathematics and mechanics,english edition, vol. 31, no. 10, 217 – 230, doi 10.1007/s10483-010-1355-6.http://dx.doi.org/10.1007/s10483-0101355-6 http://dx.doi.org/10.1137/1.9781611971392 http://dx.doi.org/10.1007/978-3-642-05221-7 http://dx.doi.org/10.1007/s002310100215 http://dx.doi.org/10.1155/2011/205181 http://dx.doi.org/1.%205923/j.am.20110101.05 http://dx.doi.org/1.%205923/j.am.20110101.05 http://dx.doi.org/10.1007/s10483-010-1355-6 http://dx.doi.org/10.1007/s10483-010-1355-6 flow behavior of hydromagnetic mixed convection through an octagonal channel with a heat generating hollow cylinder journal of naval architecture and marine engineering december, 2016 http://dx.doi.org/10.3329/jname.v13i2.27774 http://www.banglajol.info 1813-8235 (print), 2070-8998 (online) © 2016 aname publication. all rights reserved. received on: may, 2015 assited convective heat transfer and entropy generation in a solar collector filled with nanofluid r. nasrin* , a, b , m. a. alim a , m. hasanuzzaman b a department of mathematics, bangladesh university of engineering and technology, dhaka-1000, bangladesh. e-mail: rehena@math.buet.ac.bd b um power energy dedicated advanced centre (umpedac), level 4, wisma r&d, university of malaya, 59990 kuala lumpur, malaysia. e-mail: hasan@um.edu.my *e-mail: rehena@um.edu.my abstract: heat transfer phenomena of flat plate solar collector filled with different nanofluids has been investigated numerically. galerkin’s finite element method is used to solve the problem. heat transfer rate, average bulk temperature, average sub-domain velocity, outlet temperature, thermal efficiency, mean entropy generation and bejan number has been investigated by varying the solid nanoparticle volume fraction of water/cu, water/ag and water/cu/ag nanofluids from 0% to 3%. it is found that the solid nanoparticle volume fraction has great effect on heat transfer phenomena. it is observed that the increase in the solid volume fraction (up to 2%) enhances the heat transfer rate and collector efficiency where after 2% the rate of change almost constant. higher heat transfer rate and collector efficiency has been obtained 19% and 13% for water/ag nanofluid respectively. keywords: assisted convection, flat plate collector, finite element method, nanofluids, entropy generation, bejan number. nomenclature greek symbols a surface area of the collector (m 2 ) α fluid thermal diffusivity (m 2 s -1 ) ae area base on the perimeter of collector (m 2 ) β tilt angle ( 0 ) c constant defined in subsection 2.4 λ transmitivity cp specific heat at constant pressure (j kg -1 k -1 )  nanoparticles volume fraction h convective heat transfer coefficient (wm -2 k -1 ) ν kinematic viscosity (m 2 s -1 ) ha convective heat transfer coefficient between glass and ambient air (wm -2 k -1 ) η collector efficiency i intensity of solar radiation (wm -2 ) θ dimensionless temperature ae area base on the perimeter of collector (m 2 ) ρ density (kgm -3 ) k thermal conductivity (w m -1 k -1 ) μ dynamic viscosity (nsm -2 ) kb back insulation conductivity (wm -1 k -1 ) κ absorption coefficient ke edge insulation conductivity (wm -1 k -1 ) subscripts m mass flow rate (kg s -1 ) a absorber n number of glass amb ambient r. nasrin, m.a. alim and m. hasanuzzaman/journal of naval architecture and marine engineering, 13(2016) 135-150 assisted convective heat transfer and entropy generation in a solar collector filled with nanofluid 136 nu nusselt number av average p pressure (kg m s -2 ) b bottom pr prandtl number col collector q heat flux (wm -2 ) e edge re reynolds number f fluid t dimensional temperature (k) in input u, v dimensional x and y components of velocity (m s -1 ) loss lost u, v dimensionless velocities nf nanofluid ul overall heat transfer coefficient (w m -2 k -1 ) out output v magnitude of velocity (m s -1 ) recv received x, y dimensionless coordinates s solid particle x, y cartesian coordinates (m) t top xb back insulation thickness (m) usfl useful xe edge insulation thickness (m) 1 cu nanoparticle 2 ag nanoparticle 1. introduction today, renewable energy based power generation is broadly encouraged all over the world because of the limited non-renewable energy resources, environmental awareness, and abundant renewable resources schroeder (2009), hasanuzzaman et al. (2015) and hosenuzzaman et al. (2015). among the renewable energy, solar energy is one of the most potential resources ahmed et al. (2013). solar collector is one the key elements in many applications (i.e. building heating systems, solar drying devices, etc). flat-plate solar collector is commonly used for low temperature solar thermal energy (i.e. solar water-heating systems, solar space heating etc.) where performance of the solar collector depends on the various parameters. heat transfer process and performance of the solar collector has been investigated numerically and experimentally. saleh (2012) simulated one-dimensional mathematical model for transient processes which occurs in liquid flat-plate solar collectors. flat plate and heat pipe solar collectors with and without color absorbers are also numerically studied kazeminejad (2002) and azad (2009). struckmann (2008) analyzed flat-plate solar collector where efforts had been made to combine a number of important factors into a single equation. martín et al. (2011) experimentally studied heat transfer in enhanced flat-plate solar collectors. to test the enhanced solar collector and compare with a standard one, an experimental side-by-side solar collector test bed was designed and constructed. they concluded that the pressure loss ratio remained constant at about 1.8 for reynolds numbers higher than 500 inside the raisers and increasing pumping power improved thermal efficiency. sandhu (2013) experimentally studied temperature field in flat-plate collector and heat transfer enhancement with the use of insert devices. various new configurations of the conventional insert devices were tested over a wide range of reynolds number (200-8000). testing of thermal efficiency, heat transfer system, and optimization of solar thermal collectors were addressed and discussed by zambolin (2011). karuppa et al. (2012) experimentally investigated a new solar flat plate collector. experiments had been carried out to test the performance of both the water heaters under water circulation with a small pump. their results showed that rising values of temperature difference between the outlet water and the ambient air decreased thermal efficiency due to heat loss. iordanou (2009) investigated flat-plate solar collectors for water heating with improved heat transfer for application in climatic conditions of the mediterranean region. dara et al. (2013) conducted evaluation of a passive flat-plate solar collector and investigated the variations of top loss heat transfer coefficient with absorber plate emittance and air gap spacing between the absorber plate and the cover plate. enhanced heat transfer using oscillatory flows in solar collectors, different geometric absorber configuration were analyzed by lambert (2006) and amrutkar (2012). karanth et al. (2011) performed numerical simulation of a solar flat plate collector using discrete transfer radiation model (dtrm)–a cfd approach. they concluded that temperature difference between absorber plate and fluid increased with increasing flow velocity and absorber plate temperature was almost linear at all considered values of r. nasrin, m.a. alim and m. hasanuzzaman/journal of naval architecture and marine engineering, 13(2016) 135-150 assisted convective heat transfer and entropy generation in a solar collector filled with nanofluid 137 flow velocities. álvarez et al. (2010) studied finite element modelling of a solar collector and presented a mathematical model of a serpentine flat-plate solar collector using finite elements. all thermofluidic processes involve irreversibilities and therefore incur an efficiency loss. in practice, the extent of these irreversibilities can be measured by the entropy generation rate. in designing practical systems, it is desirable to minimize the rate of entropy generation and to maximize the available energy hooman et al. (2008), delavar, and hedayatpour (2012). the entropy generation induced by natural convection heat transfer in a square cavity was studied by shahi et al. (2011) and found that the nusselt number increased and the entropy generation reduced as the nanoparticle volume fraction was increased. enhancement of collector thermal performance with different nanofluids was conducted polvongsri and kiatsiriroat (2011), natarajan and sathish (2009). mahian et al. (2013-2014) performed review of the applications of nanofluids and entropy generation in solar energy. the effect of nanofluids on the performance of solar collectors was shown. modeling of solar water collector of different shapes with water based nanofluids using various nanoparticles was conducted by nasrin et al. (2013, 2014a-b, 2015). hussein et al. (2012, 2014, 2015a-b, 2016a-c) studied computational analysis in heat transfer and entropy generation considering different types of geometry as well as solar collector in both two dimensional and three dimensional forms. from the above discussion, it is clear that the application of nanofluid is one most potential options to enhance the heat transfer of the solar collector systems. as this is a potential field, a good number of research works have been conducted in the field and still have many scopes to work with fluid flow, heat transfer, enhancement of collector efficiency and entropy generation using different nanofluids. the aim of the paper is to investigate the heat transfer phenomena in term of heat transfer rate, velocity, collect efficiency, entropy generation and bejan number by varying the solid nanoparticle volume fraction of water/cu, water/ag and water/cu/ag nanofluids. 2. problem formulatıon figure 1 shows the schematic diagram of the collector. the numerical computation is carried on taking single riser pipe of fpsc. fpsc with single riser pipe gives the average heat transfer and fluid flow phenomena. the glass cover is at the top of the fpsc. it is highly transparent and anti-reflected. it is also called as glazing. the glass top surface is exposed to solar irradiation. the glass cover is made of borosilicate. it has thermal conductivity of 1.14 w/mk, specific heat of 750 j/kgk and coefficient of sunlight transmission of 95%. thickness of glass cover is 0.005m. air gap of 0.005m lies between glass cover and absorber plate. air density = 1.269 kg/m 3 , specific heat = 287.058 j/kgk and thermal conductivity = 0.0243 w/mk. all these properties of air domain represent air of temperature at 298k. length, width and thickness of the absorber plate are 1m, 0.15m and 0.0005m respectively. coefficients of heat absorption and emission of absorber plate are 95% and 5% respectively. the riser pipe has height 1m, inner diameter 0.01 m and thickness 0.0005m. a trapezium shaped bonding conductance is attached to the absorber and riser pipe. it is located from middle one-third part of width of the absorber plate. figure 1: longitudinal cross-section of a fpsc housing insulation glass cover air gap absorber plate solar irradiation inflow out flow y x r. nasrin, m.a. alim and m. hasanuzzaman/journal of naval architecture and marine engineering, 13(2016) 135-150 assisted convective heat transfer and entropy generation in a solar collector filled with nanofluid 138 it covers the three-fourth part of the riser pipe. it is as long as the absorber plate and tube. the absorber plate is modeled to provide for conduction, convection and radiation in the analysis. the absorber, riser and bonding conductance are made of copper metal. the computation domain is a fluid passing riser pipe which is attached ultrasonically to the absorber plate. let i be the intensity of solar radiation and a be any surface area, then the amount of energy received by any surface is: aiqi . (1) this equation is modified for solar collector surface as it is the product of the rate of transmission of the cover (λ) and the absorption rate of the absorber (к). thus,  aiqrecv  (2) then the temperature of absorber becomes higher than that of the surrounding. some heat is lost to the atmosphere by convection and radiation. the rate of heat loss (qloss) depends on the collector overall heat transfer coefficient (ul), the collector temperature (tcol) and the surrounding ambient temperature (tamb).  loss l col ambq u a t t  (3) the rate of useful energy (qusfl) is:    u recv loss l col ambsflq q q i a u a t t     (4) it is also known that the rate of extraction of heat from the collector may be measured by means of the amount of heat carried away in the fluid passed through it. thus  inoutpusfl ttmcq  (5) where m, cp, tin and tout are the mass flow rate per unit area, the specific heat at constant pressure, inlet and outlet fluid temperatures respectively. to calculate the collector average temperature is difficult. the equation (4) may not be convenient. it is suitable to describe a quantity that communicates the actual useful energy gain of a collector to the useful gain if the whole collector surface is at the fluid inlet temperature. this quantity is known as “the collector heat removal factor (fr)” and is stated as:       p out i n r l in amb mc t t f a i u t t       (6) the expression of actual useful energy gain (qusfl) can be written as:    u r l in ambsflq f a i u t t     (7) the heat flux q per unit area of absorber plate is  am binl sflu ttuiq a q   . the two dimensional governing equations are given as: 0      y v x u (8) 2 2 2 2nf nf u u p u u u v x y x x y                          (9) 2 2 2 2nf nf v v p v v u v x y y x y                          (10) 2 2 2 2nf t t t t u v x y x y                 (11) 2 2 2 2 0 a at t x y           (12) where,  1nf f s       is the density, (13)       1p p p nf f s c c c       is the heat capacitance, (14)   2.5 1 f nf      is the viscosity of brinkman model (1952) (15) r. nasrin, m.a. alim and m. hasanuzzaman/journal of naval architecture and marine engineering, 13(2016) 135-150 assisted convective heat transfer and entropy generation in a solar collector filled with nanofluid 139     2 2 2 s f f s nf f s f f s k k k k k k k k k k          is thermal conductivity of maxwell garnett model (1904) (16)  nf nf p nf k c  is the thermal diffusivity, f f ν pr α  is the prandtl number and in f u h re   is the reynolds number. the boundary conditions are: at all solid boundaries: u = v = 0 at the solid-fluid interface: a nf solid nf solid tt k k y y              at the inlet boundary: int t , u = uin at the outlet boundary: convective boundary condition p = 0 at the top surface of absorber: heat flux  aa l in amb t k q i u t t y         at outer surface of riser pipe: 0 t y    2.1 collector efficiency the instantaneous collector efficiency is:         usfl r l in amb in amb r r l f a i u t t t tq f f u ai ai i             (17) 2.2 average nusselt number the average nusselt number (nu) is expected to depend on a number of factors such as thermal conductivity, heat capacitance, viscosity, flow structure of nanofluids and volume fraction, dimensions and fractal distributions of nanoparticles. equation of local nusselt number for flow through the absorber tube of solar collector can be written as l f f f f u l q l q l nu k t k kql k                        where l is the length of riser pipe, t is the difference between riser pipe surface temperature and ambient temperature, q is the energy received or lost by the absorber pipe surface. due to assigned constant heat flux at absorber top surface and water based nanofluid flow this equation becomes nf nf f f f f t k k ky l t nu k k q yql k                                     dimensionless form of local nusselt number at top of riser pipe is nf f k nu k y     . the above equations are non-dimensionalized by using the following dimensionless dependent and independent variables:   , , , , in f in in t t kx y u v x y u v l l u u ql        r. nasrin, m.a. alim and m. hasanuzzaman/journal of naval architecture and marine engineering, 13(2016) 135-150 assisted convective heat transfer and entropy generation in a solar collector filled with nanofluid 140 thus rate of heat transfer at top surface of riser pipe according to nasrin and alim (2015) is as 1 0 nu nu dx  (18) normalized nusselt number is defined as the ratio of average nusselt number at any volume fraction of nanoparticles to that of pure water, is     * 0 nu nu nu     2.3 mean bulk temperature and velocity the mean bulk temperature and average sub domain velocity of the fluid inside the collector may be written as 1 a av a a tda t tda hlda      and v 1 v va av a a da da hlda      . 2.4 overall heat transfer coefficient the overall heat transfer loss from the collector is the summation of three separate components, the top loss coefficient, the bottom loss coefficient and the edge loss coefficient. the empirical relations for these coefficients are mentioned by duffie and beckman (1991) as follows: l t b e u u u u   where      1 2 2 1 1 2 1 0.13 0.00591 a amb a amb t c aamb a amb a amb c a t t t tn u n fht tc nh n t n f                               , b b b k u x  ,  2 e e e e e l w hu a k u a x lw         and          2 520 * 1 0.000051 , 1 0.089 0.1166 * 1 0.07866 and 0.43* 1 100 / a amb a am c f h h n e t           2.5 entropy generation the entropy generation in the flow field is caused by the non-equilibrium flow imposed by boundary conditions. in the convection process, the entropy generation is due to the irreversibility caused by the heat transfer phenomena and fluid flow friction. according to bejan (1996), the dimensional local entropy generation, sgen, can be expressed by: 2 2 22 2 2 00 2 2 nf nf gen k t t u v u v s x y t x y x yt                                                       (19) where 0 2 col int t t   . in equation (19), the first term represents the dimensional entropy generation due to heat transfer ( sgen,h), while the second term represents the dimensional entropy generation due to viscous dissipation (sgen,v). by using r. nasrin, m.a. alim and m. hasanuzzaman/journal of naval architecture and marine engineering, 13(2016) 135-150 assisted convective heat transfer and entropy generation in a solar collector filled with nanofluid 141 dimensionless parameters presented in eq. (18), the expression of the non-dimensional entropy generation, sgen can be written by:   2 0 2 2 2 2 2 2 , , 2 2 gen gen f col in nf nf f f gen h gen v t l s s k t t k u v u v k x y x y x y s s                                                                  (20) here sgen, h and sgen, v are the dimensionless entropy generation for heat transfer and viscous effect respectively. in eq. (20), χ is the irreversibility factor which represents the ratio of the viscous entropy generation to thermal entropy generation. it is given as:   2 0 2 if f col in ut k t t     the dimensionless average entropy generation, s for the entire computational domain is as follows: , , , , 1 gen gen h m gen v ms s dv s s v    (21) where v is the volume occupied by the nanofluid and sgen, h, m and sgen, v, m are the average entropy generation for heat transfer and viscous effect respectively. normalized entropy generation can be written as     * 0 s s s     2.6 bejan number the bejan number, be, defined as the ratio between the entropy generation due to heat transfer by the total entropy generation, is expressed as , ,gen h ms be s  (22) it is known that the heat transfer irreversibility is dominant when be approaches to 1. when be becomes much smaller than 1/2 the irreversibility due to the viscous effects dominates the processes and if be = 1/2 the entropy generation due to the viscous effects and the heat transfer effects are equal khorasanizadeh et al. (2013). 3. numerical implementation the galerkin finite element method of reddy and gartling (1994) is used to solve the governing equations (8) (12) along with convective boundary condition for the considered problem. conservation equations are solved for the finite element method to yield the velocity and temperature fields for the water flow in the absorber tube and the temperature field for the absorber plate. the equation of continuity has been used as a constraint due to mass conservation and this restriction may be used to find the pressure distribution. then the velocity components (u, v) and temperatures (t, ta) of governing equations (8) (12) are expanded using a basis set. the galerkin finite element technique yields the subsequent nonlinear residual equations. gaussian quadrature technique is used to evaluate the integrals in these equations. the non-linear residual equations are solved using newton–raphson method to determine the coefficients of the expansions. the convergence of solutions is assumed when the relative error for each variable between consecutive iterations is recorded below the convergence criterion such that 1 -6 1.0e n n      , where n is the number of iteration and ψ is a function of any one of u, v, t and ta. 3.1. mesh generation the finite element meshing of the computational domain is displayed by the figure 2. extra fine meshing is chosen for this geometry. r. nasrin, m.a. alim and m. hasanuzzaman/journal of naval architecture and marine engineering, 13(2016) 135-150 assisted convective heat transfer and entropy generation in a solar collector filled with nanofluid 142 3.2 grid check a grid-independent check is performed at pr = 5.8, i = 215w/m 2 and re = 480 for a fpsc. five different nonuniform grid systems are checked with the number of elements: 42,010, 99,832, 1,50,472, 1,68,040 and 1,92,548. heat transfer rate for water-copper nanofluid ( = 2%) as well as water ( = 0%) is considered as supervising parameter. it is noticed from the fifth and sixth column of table 1 that there is no considerable alteration in the value of mean nusselt number but time unbearable. thus 1,68,040 elements are considered for numerical analysis. table 1: grid sensitivity check at pr = 5.8, i = 215w/m 2 and re = 480 elements 42,010 99,832 1,40,472 1,68,040 1,92,548 nu (nanofluid) 1.87872 1.99127 2.10934 2.14351 2.14378 nu (water) 1.59326 1.69225 1.81524 1.84333 1.84341 time (s) 127.52 308.75 581.11 897.23 1295.31 3.3 thermo-physical properties the thermo-physical properties of the nanoparticle are taken from nasrin and alim (2014) and given in table 2. table 2: thermo-physical properties of fluid and nanoparticles physical properties fluid phase (water) ag cu cp (j/kgk) 4179 235 385  (kg/m 3 ) 997.1 10500 8933 k (w/mk) 0.613 429 400 α10 7 (m 2 /s) 10 6 (ns/ m 2 ) 1.47 855 1738.6 -- 1163.1 ---- 3.4 nanofluid with double nanoparticles water based nanofluid with copper and silver nanoparticles is also used as heat transfer medium in this numerical computation. for this the effective properties of nanofluid that is equations (13-16) can be modified as: the effective thermal diffusivity  nf nf p nf k c  (23) effective density  1 2 1 1 2 21nf f s s            (24) the effective heat capacitance figure 2: mesh generation of the 2d domain r. nasrin, m.a. alim and m. hasanuzzaman/journal of naval architecture and marine engineering, 13(2016) 135-150 assisted convective heat transfer and entropy generation in a solar collector filled with nanofluid 143         1 2 1 2 1 2 1p p p p nf f s s c c c c            (25) the effective dynamic viscosity (modified form) of brinkman model (1952)   2.5 1 21nf f        (26) and the effective thermal conductivity (modified form) of maxwell garnett model (1904)             1 2 1 1 2 2 1 2 1 1 2 2 2 2 2 2 s s f f s f s nf f s s f f s f s k k k k k k k k k k k k k k k k                  (27) where suffixes 1 and 2 represent two types of nanoparticles such as cu and ag respectively. 4. results and discussion rate of heat transfer, average bulk temperature, mean sub domain velocity, mean output temperature, percentage of collector efficiency, mean entropy generation, bejan number, normalized nusselt number, normalized entropy generation are shown graphically for various values of solid volume fraction () of the water/cu, water/ag and water/cu/ag nanofluids. the considered values of  are 0%, 1%, 2% and 3%. the reynolds number (re) = 480, surface area of collector (a) = 1.8m 2 , mass flow rate per unit area (m) = 0.0248kg/s, overall heat transfer coefficient (ul) = 8 w/m 2 k, solar irradiation (i) = 215 w/m 2 and prandtl number (pr) = 5.8 are chosen fixed. 4.1 rate of heat transfer the nu- profile for water-cu, water-ag and water-cu/ag nanofluids as well as base fluid are depicted in the figure 3. the distribution of  (= 1 + 2) for water based nanofluid having copper and silver nanoparticles is as  = 1 + 2 = 0.05% + 0.05% = 1% and so on. from the plot of the average nusselt number (nu)-solid volume fraction () shows that rate of heat transfer rises monotonically upto 2% of solid volume fraction of all nanofluids. and then there is almost no change in nu- profile for extra variation of  from 2% to 3%. it is happened due to lower value of heat capacitance of nanofluid. here rate of heat transfer remains constant for clear water ( = 0%) with the variation of  . heat transfer rate increases by 16%, 18% and 17% with the variation of  from 0% to 2% of water-cu, water-ag and water-cu/ag figure 4: average bulk temperature for effect of solid concentration figure 3: mean nusselt number for effect of solid volume fraction r. nasrin, m.a. alim and m. hasanuzzaman/journal of naval architecture and marine engineering, 13(2016) 135-150 assisted convective heat transfer and entropy generation in a solar collector filled with nanofluid 144 nanofluids respectively. thus, adding more nanoparticles is not beneficial because of increasing nanoparticles concentration increases viscosity. on the other hand heat capacitance of nanofluid is lower than base fluid. 4.2 average bulk temperature average bulk temperature (θav) for the effect of the solid volume fraction is shown in the figure 4. θav grows successively for  upto 2%. mean temperature remains constant for further increasing of solid volume fraction. it is well known that higher concentration of solid particle enhances thermal conductivity as well as temperature of the working nanofluids. figure 4 expresses that mean bulk temperature of water/silver nanofluid becomes greater than water/ag/cu and water/cu nanofluid. 4.3 mean velocity mean sub-domain velocity (vav) against solid volume fraction of different nanofluids is expressed in the figure 5. vav has notable changes with different values of solid concentration. growing  devalues mean velocity of the nanofluid through the riser pipe of the flat plate solar collector. this happens because more solid concentrated nanofluid can’t move freely like base fluid water. greater  represents higher thermal conductivity simultaneously higher density properties of the nanofluid. thus motion of the nanofluid diminishes with enhancing . velocity of copper/water nanofluid is higher than other nanofluids. this is due to the fact that density of water/cu nanofluid is lower than other nanofluids. 4.4 average outlet temperature figure 6 displays the mean output temperature (tout) of different nanofluids with the influences of volume fraction. from this figure it is shown that the inlet temperature of fluid is maintained at 300k and then it increases gradually with the contact of heated solid riser pipe. and finally the mean output temperature is obtained as 310k, 312k, 314k and 314k for water/cu, 310k, 314k, 316k and 316k for water/ag, 310k, 313k, 315k and 315k for water/cu/ag nanofluid at  = 0%, 1%, 2%, and 3% respectively. tout remains unchanged for  = 3%. 4.5 collector efficiency the variation of percentage of collector efficiency as a function of the solid volume fraction varies from 0%-3% that is shown in figure 7. it is observed that increasing solid volume fraction (upto 2%) enhances the collector efficiency. figure 6: mean outlet temperature for the effect of solid concentration figure 5: mean velocity for effect of solid concentration r. nasrin, m.a. alim and m. hasanuzzaman/journal of naval architecture and marine engineering, 13(2016) 135-150 assisted convective heat transfer and entropy generation in a solar collector filled with nanofluid 145 the enhancement of collector efficiency is found from 40% to 48% for cu/water, 40% to 53% for ag/water and 40% to 51% for cu/ag/water nanofluids in the case with the variation of  from 0%-2% respectively. mean output temperature of nanofluids has not been increased at 3% concentration. consequently thermal efficiency has been decreased at  = 3% for all nanofluids. this result agrees the formula of thermal efficiency. so thermal efficiency has not been improved for further mixing nanoparticles with clear water. 4.6 mean entropy generation the variations of average entropy generation (s) against solid volume fraction () is displayed in the figure 8. the entropy generation increases by  upto 2%. the increment of nanoparticles, in terms of enhancing heat transfer rate, observed in figure 3 is also obtained in terms of increasing entropy production. this result is to be expected since the addition of a greater concentration of nanoparticles increases the thermal conductivity and viscosity of the working fluid. the higher thermal conductivity results in a smaller temperature gradient within the riser pipe of the flat plate solar collector. thus the average entropy generation caused by heat transfer irreversibility increases. the greater viscosity of the working fluid increases the local entropy generation due to fluid friction irreversibility. after the level of  = 2%, there is no change in mean entropy generation for all types of nanofluids. 4.7 bejan number figure 9 depicts be- profile for the considered values of  from 0% to 3%. increasing be is observed for increasing solid volume fraction of nanoparticles within the level 0% to 2%. for further increment of  no variation is found in the bejan number for different nanofluids. if the bejan number (be) approaches unity, the fluid friction irreversibility effect can be ignored. in other words, mean entropy generation is dominated by the heat transfer irreversibility effect. note that lower value of be is observed for base fluid. 4.8 normalized nusselt number the variation of normalized nusselt number against  is displayed in the figure 10. nu* is the ratio of mean nusselt number for nanofluid and base fluid. from the figure 3, it is seen that nu- profile increases, so nu*- profile also grows up with the variation of  from 0% to 2% for all nanofluids. values of normalized nusselt number are found higher for the variation of water/ag nanofluid than other considered nanofluids. figure 7: collector efficiency for the effect of solid concentration figure 8: mean entropy generation for the effect of solid concentration r. nasrin, m.a. alim and m. hasanuzzaman/journal of naval architecture and marine engineering, 13(2016) 135-150 assisted convective heat transfer and entropy generation in a solar collector filled with nanofluid 146 4.9 normalized entropy generation figure 11 displays the variations of normalized entropy generation against solid volume fraction. s* is high for higher solid volume fraction upto 2% which is similar to the figure 8 for water/cu, water/ag and water/cu/ag nanofluids. 5. correlation from the current two dimensional numerical study the calculated average nusselt number (nu) and bejan number (be) are correlated with solid volume fraction ( ) of water/ag nanofluid in the range of 0% ≤  ≤ 2% through the fpsc. these correlations can be written as: figure 10: normalized nusselt number for the effect of solid concentration figure 9: bejan number for the effect of solid concentration figure 12: comparison between present code and kalogirou (2004) figure 11: normalized entropy generation for the effect of solid concentration r. nasrin, m.a. alim and m. hasanuzzaman/journal of naval architecture and marine engineering, 13(2016) 135-150 assisted convective heat transfer and entropy generation in a solar collector filled with nanofluid 147 nu = 2.102*() 0.16 , where the confidence coefficient is r 2 = 99.8%. and be = 0.899*() 0.042 , where the confidence coefficient is r 2 = 99.9%. 6. tilt angle for dhaka city the sun’s position on the celestial sphere is usually specified in terms of the solar azimuth angle ψ and the solar altitude angle α. the solar zenith angle φ is the sun’s angular distance from the zenith, which is the point directly overhead on the celestial sphere. thus φ and α are complementary angles. the tilt angle is just the angle at which the surface is inclined from horizontal and is taken positive for south facing surfaces. the solar altitude and azimuth angles are computed for any time, date and location using the formula: 0 90     , where φ is the latitude taken north of the equator and δ is declination angle. another form of δ is  0 360 23.45 sin 284 365 d        , where d is the day of the year like from 1 to 365. for the incident ray to be perpendicular to the surface is: α + β = 90 0 , where α is elevation angle and β is tilt angle of the solar collector measured from the horizontal surface. then from above equation it is clear that β = φ δ for dhaka city latitude and longitude are 23.7 0 n and 90.41 0 e. the tilt angle will be a) for may 25, 2015 (summer), d = 31+28+31+30+25 =145. so, δ = 20.916 0 now β = φ δ = 23.7 0 20.916 0 = 2.78 0 b) for october 3, 2015 (autumn), d = 276. so, δ = -5.007 0 now β = φ δ = 23.7 0 – (-5.007 0 ) = 28.707 0 similarly, the tilt angle of solar collector for 365 days in dhaka city is calculated and average value is obtained as 23.8 0 . as β is positive, collector will be facing south. maximum yearly solar radiation can be achieved using a tilt angle approximately equal to a site's latitude. to optimize performance in winter, collector can be tilted 15° greater than the latitude. to optimize performance in summer, collector can be tilted 15° less than the latitude. 7. comparison the current numerical result is compared for collector efficiency temperature difference [ti ta] profile of water with the graphical representation of kalogirou (2004) of flat plate solar thermal collector. ti and ta are temperatures of input fluid and ambient air respectively. the computation is done taking heat transfer medium as water with irradiation level 1000 w/m 2 and the mass flow rate per unit area 0.015 kg/s. a survey report on various types of solar thermal collectors and applications was presented by kalogirou (2004). author presented that generally the collector efficiency linearly depended on temperature difference for a fpsc. figure 12 demonstrates the above stated comparison. 8. conclusion the influences of solid volume fraction of different nanofluids on forced convection boundary layer flow inside the riser pipe of a flat plate solar collector is accounted. various  have been considered for the heat transfer rate, mean bulk temperature, average velocity, mean output temperature and collector efficiency through the solar collector. the results of the numerical analysis lead to the following conclusions: r. nasrin, m.a. alim and m. hasanuzzaman/journal of naval architecture and marine engineering, 13(2016) 135-150 assisted convective heat transfer and entropy generation in a solar collector filled with nanofluid 148  rate of heat transfer enhancement about 19% for ag/water nanofluid with  = 2%.  collector efficiency increases about 13% for 2% solid volume 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(2013): experimental study of temperature field in flat-plate collector and heat transfer enhancement with the use of insert devices, m. of engg. sci. theseis, the school of graduate and postdoctoral studies, the university of western ontario london, ontario, canada. schroeder, m. (2009): utilizing the clean development mechanism for the deployment of renewable energies in china, appl. energ. 86, 2, 237-242. https://doi.org/10.1016/j.apenergy.2008.04.019 shahi, m., mahmoudi, ah., raouf, ah. (2011): entropy generation due to natural convection cooling of a nanofluid, int. commun. in heat and mass transf., 38, 972-983. https://doi.org/10.1016/j.icheatmasstransfer.2011.04.008 struckmann, f. (2008): analysis of a flat-plate solar collector, project report 2008 mvk160 heat and mass transport, lund, sweden. zambolin, e. (2011): theoretical and experimental study of solar thermal collector systems and components, ph. d. thesis, scuola di dottorato di ricerca in ingegneria industriale, indirizzo fisica tecnica. https://doi.org/10.1016/j.ijheatmasstransfer.2013.12.043 https://doi.org/10.1016/j.apenergy.2008.04.019 https://doi.org/10.1016/j.icheatmasstransfer.2011.04.008 microsoft word 5954.docx journal of naval architecture and marine engineering june, 2012 doi: 10.3329/jname.v8i1.5954 http://www.banglajol.info 1813-8235 (print), 2070-8998 (online) © 2012 aname publication. all rights reserved. received: august 2010 joule heating effect on magnetohydrodynamic natural convection flow along a vertical wavy surface nazma parveen* and m. a. alim department of mathematics, bangladesh university of engineering and technology, dhaka-1000, bangladesh *email: nazma@math.buet.ac.bd abstract: in this paper, the effect of joule heating on magnetohydrodynamic natural convection flow of viscous incompressible fluid along a uniformly heated vertical wavy surface has been investigated. the governing boundary layer equations with associated boundary conditions for this phenomenon are converted to nondimensional form using a suitable transformation. the equations are mapped into the domain of a vertical flat plate and then solved numerically employing the implicit finite difference method, known as the keller-box scheme. effects of pertinent parameters, such as the joule heating parameter (j), prandtl number (pr), magnetic parameter (m) and the amplitude of the wavy surface α on the surface shear stress in terms of the skin friction coefficient (cfx), the rate of heat transfer in terms of local nusselt number (nux), the streamlines and the isotherms are discussed. a comparison with previously published work is performed and the results show excellent agreement. keywords: magnetohydrodynamics, joule heating, natural convection, uniform surface temperature, keller-box method, wavy surface nomenclature greek symbols cfx local skin friction coefficient α amplitude of the wavy surface cp specific heat at constant pressure (jkg -1k-1) β volumetric coefficient of thermal expansion(k-1) f dimensionless stream function β0 applied magnetic field strength g acceleration due to gravity (ms-2) η dimensionless similarity variable gr grashof number θ dimensionless temperature function j joule heating parameter ψ stream function (m2s-1) k thermal conductivity of fluid (wm-1k-1) µ dynamic viscosity of the fluid (kg m-1s-1) l wavelength associated with the wavy ν kinematic viscosity of the fluid (m2s-1) surface (m) ρ density of the fluid (kg m-3) m magnetic parameter σ0 electrical conductivity of the fluid (ω -1m-1) nux local nusselt number τw shearing stress p pressure of the fluid (nm-2) σ(x) surface profile function defined in equation (1) pr prandtl number t temperature of the fluid in the boundary subscripts layer (k) w wall conditions tw temperature at the surface (k) ∞ ambient conditions t∞ temperature of the ambient fluid (k) u, v dimensionless velocity components along superscripts the (x, y) axes (ms-1) ' differentiation with respect to η x, y axis in the direction along and normal to the tangent of the surface nazma parveen and m. a. alim / journal of naval architecture and marine engineering 9(2012) 11-24 joule heating on magnetohydrodynamic natural convection flow …... 12 1. introduction laminar natural convection boundary layer flow and heat transfer problem from a vertical wavy surface get a great deal of attention in various branches of engineering. along with the natural convection flow the phenomenon of the boundary layer flow of an electrically conducting fluid in the presence of joule heating and magnetic field are also very common because of their applications in nuclear engineering in connection with the cooling of reactors. if the surface is roughened the flow is disturbed by the surface and this alters the rate of heat transfer. these types of roughened surface are taken into account in several heat transfer collectors, flat plate condensers in refrigerators and heat exchanger. one common example of a heat exchanger is the radiator used in car, in which the heat generated from engine transferred to air flowing through the radiator. the interface between concurrent or countercurrent two-phase flow is another example remotely related to this problem. such an interface is always wavy and momentum transfer across it is by no means similar to that across a smooth, flat surface and neither is the heat transfer. also a wavy interface can have an important effect on the condensation process. the effects of nonuniformities of surface waviness on the natural convection boundary layer flow of a newtonian fluid have studied by yao (1983) and moulic and yao (1989). yao used an extended prantdl’s transposition theorem and a finite-difference scheme. he proposed a simple transformation to study the natural convection heat transfer for an isothermal vertical sinusoidal surface. these simple coordinate transformations method to change the wavy surface into a flat plate. hossain (1992) analyzed the viscous and joule heating effects on mhd free convection flow with variable plate temperature. rees and pop (1994) investigated the natural convection boundary layer induced by vertical and horizontal wavy surface exhibiting small amplitude waves embedded in a porous medium. the magnetohydrodynamic boundary layer flow and heat transfer from a continuous moving wavy surface have been investigated by hossain and pop (1996). alam et al. (1997) studied the problem of free convection from a wavy vertical surface in presence of a transverse magnetic field. on the other hand, the combined effects of thermal and mass diffusion on the natural convection flow of a viscous incompressible fluid along a vertical wavy surface have been investigated by hossain and rees (1999). in this paper the effect of waviness of the surface on the heat and mass flux is investigated in combination with the species concentration for a fluid having prandtl number equal to 0.7. cheng (2000) investigated the natural convection heat and mass transfer near a vertical wavy surface with constant wall temperature and concentration in a porous medium. hossain et al. (2002) considered the problem of natural convection of fluid with temperature dependent viscosity along a heated vertical wavy surface. amin (2003) analyzed combined effect of viscous dissipation and joule heating on mhd forced convection over a non isothermal horizontal cylinder embedded in a fluid saturated porous medium. molla et al. (2004) investigated natural convection flow along a vertical wavy surface with uniform surface temperature in presence of heat generation/absorption. they found the effect of varying the heat generation/absorption on the heat transfer rate in terms of local nusselt number as well as on the streamlines and isotherm patterns for very small prandtl number pr ranging from 0.001 to 1.0. they concluded that the velocity and temperature distributions for the case of heat generation higher than that of the heat absorption case. yao (2006) considered natural convection flow along a vertical complex wavy surface. alim et al. (2007) investigated joule heating effect on the coupling of conduction with mhd free convection flow from a vertical flat plate. combined effects of viscous dissipation and joule heating on the coupling of conduction and free convection along a vertical flat plate have also been studied by alim et al. (2008). nasrin and alim (2009) studied combined effects of viscous dissipation and temperature dependent thermal conductivity on mhd free convection flow with conduction and joule heating along a vertical flat plate. very recently, parveen and alim (2011) analyzed the effect of temperature dependent thermal conductivity on magnetohydrodynamic natural convection flow along a vertical wavy surface. at the same time parveen and alim (2011) also studied the effect of temperature dependent variable viscosity on magnetohydrodynamic natural convection flow along a vertical wavy surface. the above literature survey shows that the joule heating effect on magnetic field is an interesting macroscopic physical phenomenon in fluid dynamics. none of the above investigations considered the effect of joule heating on mhd natural convection flow along wavy surface. joule heating in electronics and in physics refers to the increase in temperature of a conductor as a result of resistance to an electrical current flowing through it. in this paper, attention has been given to study the joule heating effect in presence of magnetic field of electrically conducting fluid with free convection boundary layer flow along a vertical wavy surface. it is assumed that the wavy surface is electrically insulated and is maintained at a uniform temperature tw. far above the wavy plate, the fluid is stationary and is kept at a temperature t∞, where tw > t∞. using the appropriate transformations, the boundary layer equations are reduced to non-similar partial differential forms. the transformed boundary layer equations are solved numerically using implicit finite difference method known as nazma parveen and m. a. alim / journal of naval architecture and marine engineering 9(2012) 11-24 joule heating on magnetohydrodynamic natural convection flow …... 13 the keller box technique (1978). consideration is given to the situation where the buoyancy force assist the natural convection flow for various values of the joule heating parameter j, prandtl number pr, magnetic parameter m and the amplitude of the wavy surface α. 2. formulation of the problem the boundary layer analysis outlined below allows )(xσ being arbitrary, but our detailed numerical work assumed that the surface exhibits sinusoidal deformations. the wavy surface may be described by ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ == l xn xy w π ασ sin)( (1) where l is the wavelength associated with the wavy surface. the geometry of the wavy surface and the two-dimensional cartesian coordinate system are shown in fig. 1. fig. 1: the coordinate system and the physical model under the usual boussinesq approximation, the equations governing the flow can be written as: 0= ∂ ∂ + ∂ ∂ y v x u (2) ( ) uttgu x p y u v x u u ρ βσ βν ρ 2 0021 −−+∇+ ∂ ∂ −= ∂ ∂ + ∂ ∂ ∞ (3) 21v v pu v v x y y ν ρ ∂ ∂ ∂ + = − + ∇ ∂ ∂ ∂ (4) 2 2 002 u c t c k y t v x t u pp ρ βσ ρ +∇= ∂ ∂ + ∂ ∂ (5) where ),( yx are the dimensional coordinates along and normal to the tangent of the surface and ),( vu are the velocity components parallel to ),( yx , )//( 22222 yx ∂∂+∂∂=∇ is the laplacian operator, g is the acceleration due to gravity, p is the dimensional pressure of the fluid, ρ is the density, β0 is the strength of magnetic field, σ0 is the electrical conduction, k is the thermal conductivity, β is the coefficient of thermal y x t∞ g l v u α tw 0β nazma parveen and m. a. alim / journal of naval architecture and marine engineering 9(2012) 11-24 joule heating on magnetohydrodynamic natural convection flow …... 14 expansion, ν ( = µ/ρ) is the kinematic viscosity, µ is the dynamic viscosity of the fluid in the boundary layer and cp is the specific heat due to constant pressure. the boundary conditions relevant to the present problem are )(,0,0 xyyatttvu ww σ===== (6a) ∞→=== ∞∞ yaspp,tt,0u (6b) where tw is the surface temperature, t∞ is the ambient temperature of the fluid and p∞ is the pressure of fluid outside the boundary layer. following yao (1983), we now introduce the following nondimensional variables: 4 1 , gr l y y l x x σ− == pgr l p 1 2 2 −= ρν )(, 4 1 2 1 uvgr l vugr l u xσνν −== −− , ∞ ∞ − − = tt tt w θ (7) 3 2 ( ) , wx g t td d gr l dx dx βσ σ σ ν ∞−= = = where θ is the dimensionless temperature function and (u, v) are the dimensionless velocity components parallel to (x, y). here (x, y) are not orthogonal, but a regular rectangular computational grid can be easily fitted in the transformed coordinates. it is also worthwhile to point out that (u, v) are the velocity components parallel to (x, y) which are not parallel to the wavy surface. introducing the above dimensionless dependent and independent variables into equations (2)–(5), the following dimensionless form of the governing equations are obtained after ignoring terms of smaller orders of magnitude in gr, the grashof number defined in (7). 0= ∂ ∂ + ∂ ∂ y v x u (8) ( ) θσσ +− ∂ ∂ ++ ∂ ∂ + ∂ ∂ −= ∂ ∂ + ∂ ∂ mu y u y p gr x p y u v x u u xx 2 2 24 1 1 (9) ( ) 2 2 2 24 1 1 u y u y p gr y u v x u u xxxxx σσσσ −∂ ∂ ++ ∂ ∂ −=⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ ∂ ∂ + ∂ ∂ (10) ( ) 2 2 2 21 pr 1 ju yy v x u x +∂ ∂ += ∂ ∂ + ∂ ∂ θ σ θθ (11) in the above equations pr, j and m are respectively known as the prandtl number, the joule heating parameter and magnetic parameter, which are defined as α ν =pr , ( )∞− = ttc gr j wpρ νβσ 2 12 00 and 2 1 22 00 gr l m µ βσ = (12) it can easily be seen that the convection induced by the wavy surface is described by equations (8)–(11). we further notice that, equation (10) indicates that the pressure gradient along the y-direction is )( 4 1− gro , which implies that lowest order pressure gradient along x -direction can be determined from the inviscid flow solution. for the present problem this pressure gradient ( 0=∂∂ xp ) is zero. equation (10) further shows that ypgr ∂∂ − /4 1 is )1(o and is determined by the left-hand side of this equation. thus, the elimination of yp ∂∂ / from equations (9) and (10) leads to nazma parveen and m. a. alim / journal of naval architecture and marine engineering 9(2012) 11-24 joule heating on magnetohydrodynamic natural convection flow …... 15 ( ) θ σσσ σσ σ 22 2 22 2 2 1 1 11 1 xxx xxx x u m u y u y u v x u u + + + − + − ∂ ∂ += ∂ ∂ + ∂ ∂ (13) the corresponding boundary conditions for the present problem then turn into ⎭ ⎬ ⎫ ∞→=== ==== yaspu yatvu 0,0 01,0 θ θ (14) now we introduce the following transformations to reduce the governing equations to a convenient form: ),(,),,( 4 1 4 3 ηθθηηψ xyxxfx === − (15) where η is the pseudo similarity variable and ψ is the stream function that satisfies the equation (8) and is defined by x v y u ∂ ∂ −= ∂ ∂ = ψψ , . (16) introducing the transformations given in equation (15) into equations (13) and (11) the momentum and energy equations transformed into the new co-ordinate system. thus the resulting equations are obtained ( ) ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ ∂ ∂ ′′− ∂ ′∂ ′= ′ + − + +′⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ + +−′′+′′′+ x f f x f fx f mx f x fff xxx xxx x 2 2 1 2 2 2 2 11 1 12 1 4 3 1 σ θ σσ σσ σ (17) ( ) ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ ∂ ∂ ′− ∂ ∂ ′=′+′+′′+ x f x fxfxjfx θ θ θθσ 22 32 4 3 1 pr 1 (18) the boundary conditions (14) now take the following form: ⎭ ⎬ ⎫ =∞=∞′ ==′= 0),(,0),( 1),(,0),(),( xxf oxoxfoxf θ θ (19) in the above equations prime denote the differentiation with respect to η. however, it is important to calculate the values of the shearing stress τw and the rate of heat transfer in terms of the skin friction coefficients cfx and nusselt number nux respectively, which can be written as 2 2 u c wfx ρ τ = and )( ∞− = ttk xq nu w w x (20) 00 ).( and ).(where == ∇−=∇= ywyw tnkqunµτ (21) using the transformations (15) into equation (20), the local skin friction coefficient, cfx and the rate of heat transfer in terms of the local nusselt number, nux takes the following form: ),(12/)/( 24 1 oxfxgrc xfx ′′+= σ (22) ),(1)/( 24 1 oxxgrnu xx θσ ′+−= − (23) nazma parveen and m. a. alim / journal of naval architecture and marine engineering 9(2012) 11-24 joule heating on magnetohydrodynamic natural convection flow …... 16 3. method of solution this paper concerns the natural convection flow of viscous incompressible fluid along a uniformly heated vertical wavy surface in presence of joule heating and magnetic field has been investigated using the very efficient implicit finite difference method known as the keller box scheme developed by keller (1978), which is well documented by cebeci and bradshaw (1984). to apply the aforementioned method, equations (17) and (18) their boundary condition (19) are first converted into the following system of first order equations. for this purpose we introduce new dependent variables ,),(),,( ηξηξ vu ),( ηξp and ),( ηξg so that the transformed momentum and energy equations can be written as uf =′ (24) vu =′ (25) pg =′ (26) )(54 2 321 ξξ ξ ∂ ∂ − ∂ ∂ =−+−+′ f v u uupgpupvfpvp (27) )( pr 1 2 621 ξξ ξ ∂ ∂ − ∂ ∂ =++′ f p g uuppfppp (28) where x = ξ, θ = g and ( ) 1 2 2 1 2 3 4 52 2 2 3 1 1 1 , , , , 4 2 1 1 1 x xx x x x x x mx p p p p p σ σ σ σ σ σ = + = = + = = + + + and 2 3 6 xjp = and the boundary conditions (19) are ( ) 0),(,0),( 10,,0)0,(,0)0,( =∞=∞ === ξξ ξξξ gu guf (29) now consider the net rectangle on the (ξ,η) plane shown in the fig. 2 and denote the net points by jjh nnk jjj n nn ,,2,1,0 ,,2,1,,0 ,10 10 llll llll =+== =+== − − ηηη ξξξ (30) here n and j are just sequence of numbers on the (ξ,η) plane, kn and hj are the variable mesh widths. approximate the quantities f, u, v and p at the points (ξn,ηj) of the net by n j n j n j n j pvuf ,,, which call net function. it is also employed that the notation njp for the quantities midway between net points shown in fig. 2 and for any net function as )( 2 1 121 −− += nnn ξξξ (31) )( 2 1 121 −− += jjj ηηη (32) )( 2 1 121 −− += nj n j n j ggg (33) )( 2 1 121 n j n j n j ggg −− += (34) nazma parveen and m. a. alim / journal of naval architecture and marine engineering 9(2012) 11-24 joule heating on magnetohydrodynamic natural convection flow …... 17 fig. 2: net rectangle of difference approximations for the box scheme. the finite difference approximations according to box method to the three first order ordinary differential equations (24) – (26) are written for the mid point (ξn,ηj-1/2 ) of the segment p1p2 shown in the fig. 2. 2 1 21 1 n j n jn j j n j n j uuu h ff + == − − − − (35) 2 1 21 1 n j n jn j j n j n j vvv h uu + == − − − − (36) 2 pp p h gg nj n 1jn 21j j n 1j n j +== − − − − (37) the finite difference approximations to the two first order differential equations (27) and (28) are written for the mid point (ξn-1/2,ηj-1/2 ) of the rectangle p1p2p3p4. this procedure yields ( ) ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ − − − =−+ −+⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ − +⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ − − −−− − − −−− − − − − − − − − − − − − − − − − − − n n j n jn j n n j n jn j n j n j n j n j n j j n j n jn j j n j n jn j k ff v k uu uupgp upfvp h vv p h vv p 1 2/12/12/1 2/1 1 2/12/12/1 2/1 2/1 2/1 1 2/15 1 2/14 1 2/1 2 3 2/1 2/12 1 1 1 1 2/11 1 2/11 )()( )()( 2 1 )( 2 1 ξ (38) { } { } ( ) ( ) ⎟⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ − − − =+ +⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ − +⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ − − −−− − − −−− − − − − − − − − − − − − − − n n j n jn j n n j n jn j n j n j n j j n j n jn j j n j n jn j k ff p k gg uup fpp h pp p h pp p 1 2/12/12/1 2/1 1 2/12/12/1 2/1 2/1 2/1 2/1 2/1 2 6 2/1 2/12 1 1 1 1 2/11 1 2/11 )(pr2 1 )( pr2 1 ξ (39) the above equations are to be linearized by using newton’s quasi-linearization method. then linear algebraic equations can be written in block matrix which form a coefficient matrix. the whole procedure, namely reduction to first order followed by central difference approximations, newton’s quasi-linearization method and the block thomas algorithm, is well known as the keller-box method. hj kn ηj-1/2 ηj ηj-1 ξn-1 ξn-1/2 ξ n p1 p4 p3 p2 nazma parveen and m. a. alim / journal of naval architecture and marine engineering 9(2012) 11-24 joule heating on magnetohydrodynamic natural convection flow …... 18 4. results and discussion here we have discussed the numerical results obtained from parabolic differential equations (17)-(18) using the method mentioned above. it can be seen that the solutions are affected by four parameters, namely the joule heating parameter j, prandtl number pr, magnetic parameter m and the amplitude of the wavy surface α. numerical values of local shearing stress and the rate of heat transfer are calculated from equations (22) and (23) in terms of the skin friction coefficient cfx and nusselt number nux respectively for a wide range of the axial distance x starting from the leading edge. these are shown graphically in figs. 3-7 for different values of the aforementioned parameters j, m, pr and α. the effect of magnetic parameter m on the surface shear stress in terms of the local skin friction coefficient cfx and the rate of heat transfer in terms of the local nusselt number nux are depicted graphically in fig. 3 and fig. 4 for j = 0 and j > 0, respectively while α = 0.2 and pr = 1.0. the skin friction coefficient cfx and local rate of heat transfer nux varies according to the slope of the wavy surface. this is due to the alignment of the buoyancy force 1/(1+σx 2), as shown in equation (17), which drives the flow tangentially to the wavy surface. it is observed from figs. 3 and 4 that with and without effects of joule heating parameter the skin friction coefficient, the rate of heat transfer and their amplitude reduce at a great extent for increasing values of the magnetic parameter m. the magnetic field acts against the direction of fluid flow and reduce the skin friction and the rate of heat transfer. again from fig. 4 considering the joule heating parameter j the skin friction coefficient is higher and the rate of heat transfer becomes slower than that of not considering j, which is expected. fig. 3: variation of (a) skin friction coefficient cfx and (b) rate of heat transfer nux for varying of magnetic parameter m against x while j = 0.0, pr = 1.0 and α = 0.2. the effect of joule heating parameter j the local skin friction coefficient cfx and the rate of heat transfer in terms of the local nusselt number nux from the wavy surface while α = 0.2, m = 0.01 and pr = 0.5 is illustrated in fig. 5. from fig. 5 it is noted that the skin friction coefficient increases slowly along the upstream direction of the surface and to decrease of the heat transfer rates. the maximum values of the skin friction coefficient cfx are 1.01264 and 1.01382 for j = 0.00 and 0.05 respectively which occurs at the same point x = 0.5. furthermore, the maximum values of the rate of heat transfer nux are 0.30649 and 0.30379 for j = 0.00 and 0.05 respectively which occurs at the different position of x. it is observed that the skin friction coefficient increases by 0.12% and the rate of heat transfer decreases by 0.88% when j increases from 0.00 to 0.05. the variation of the local skin friction coefficient cfx and local rate of heat transfer nux for different values of prandtl number pr for j = 0.001, m = 1.0 and α = 0.2 are depicted graphically in fig. 6(a) and fig. 6(b) respectively. the skin friction coefficient decreases and the rate of heat transfer increases for increasing value of the prandtl number pr. increasing values of prandtl number pr, speed up the decay of the temperature field away from the heated surface with a consequent increase in the rate of heat transfer and reduces the skin friction coefficient. the maximum values of the local skin friction coefficient cfx are 0.78581, 0.75212, 0.64204 and 0.55692 for pr = 0.70, 1.0, 3.0 and 7.0 respectively and each of which occurs at x = 0.45. it is noted that the skin friction coefficient decreases by 29.13% when pr increases from 0.70 to 7.0. furthermore, the maximum values of the rate of heat transfer nux are 0.32504, 0.36889, 0.53037 and 0.68546 for pr = 0.70, 1.0, 3.0 and 7.0 0 2 4 6 8 x 0 0.3 0.6 0.9 c fx m = 0.00 m = 0.50 m = 1.00 m = 2.00 pr = 1.0 , j = 0.0 , (a) α = 0.2 0 2 4 6 8 x 0 0.2 0.4 n ux m = 0.00 m = 0.50 m = 1.00 m = 2.00 α = 0.2 , pr = 1.0 , j = 0.0(b) nazma parveen and m. a. alim / journal of naval architecture and marine engineering 9(2012) 11-24 joule heating on magnetohydrodynamic natural convection flow …... 19 respectively and each of which occurs at the surface. it is observed that the rate of heat transfer increases by 52.58% when pr increases from 0.70 to 7.0. fig. 4: variation of (a) skin friction coefficient cfx and (b) rate of heat transfer nux for varying of magnetic parameter m against x while j > 0 (j = 0.001), pr = 1.0 and α = 0.2. fig. 5: variation of (a) skin friction coefficient cfx and (b) rate of heat transfer nux for varying of joule heating parameter j against x while α = 0.2, m = 0.01 and pr = 0.5. fig. 6: variation of (a) skin friction coefficient cfx and (b) rate of heat transfer nux for varying of prandtl number pr against x while m = 1.0, j = 0.001 and α = 0.2. 0 2 4 6 8 x 0 0.3 0.6 0.9 c fx m = 0.00 m = 0.50 m = 1.00 m = 1.50 pr = 1.0 , j = 0.001 , (a) α = 0.2 0 2 4 6 8 x 0 0.2 0.4 n ux m = 0.00 m = 0.50 m = 1.00 m = 1.50 α = 0.2 , pr = 1.0 , j = 0.001(b) 0 2 4 6 8 10 x 0 0.3 0.6 0.9 1.2 c fx j = 0.000 j = 0.025 j = 0.035 j = 0.050 (a) m = 0.01 , pr = 0.5α = 0.2 , 0 2 4 6 8 10 x 0 0.1 0.2 0.3 n ux j = 0.000 j = 0.025 j = 0.035 j = 0.050 (b) m = 0.01, α = 0.2 ,pr = 0.5 0 2 4 6 8 x 0 0.2 0.4 0.6 0.8 c fx pr = 0.70 pr = 1.00 pr = 3.00 pr = 7.00 (a) m = 1.0 , j = 0.001α = 0.2 , 0 2 4 6 8 x 0 0.2 0.4 0.6 0.8 n ux pr = 0.70 pr = 1.00 pr = 3.00 pr = 7.00 (b) m = 1.0 , α = 0.2 , j = 0.001 nazma parveen and m. a. alim / journal of naval architecture and marine engineering 9(2012) 11-24 joule heating on magnetohydrodynamic natural convection flow …... 20 figs. 7(a) and 7(b) show that increase in the value of the amplitude of wavy surface (α = 0.0, 0.1, 0.2, 0.3) leads to decrease the value of the skin friction coefficient and the rate of heat transfer in terms of the local nusselt number while prandtl number pr = 1.0, magnetic parameter m = 1.0 and joule heating parameter j = 0.001. frictional force depends on the smoothness of the surface, temperature and nature of fluid. surface becomes more roughened for increasing values of amplitude of the wavy surface. velocity force decreases at the local points. it is seen that the skin friction coefficient and the heat transfer rate decrease by 16.84% and 18.86% respectively as α increases from 0.0 to 0.3. fig. 7: variation of (a) skin friction coefficient cfx and (b) rate of heat transfer nux for varying of amplitude of the wavy surface α against x while m = 1.0, j = 0.001 and pr = 1.0. fig. 8: streamlines for (a) j = 0.00 (b) j = 0.025 (c) j = 0.035 (d) j = 0.05 while pr = 0.5, α = 0.2 and m = 0.01. the influence of the joule heating parameter j on the development of streamlines and isotherms profile which are plotted for the amplitude of the wavy surface α = 0.2, prandtl number pr = 0.5 and m = 0.01 are shown in figs. 8 and 9 respectively. it is observed that as the value of j increases, the maximum value of ψ increase steadily. when j = 0.0, maxψ = 10.44 and j = 0.05, maxψ = 11.28. from fig. 9, it is noted that the thermal boundary layer becomes thicker for increasing value of j. 0 2 4 6 8 x 0 0.2 0.4 0.6 0.8 1 c fx α = 0.00 α = 0.10 α = 0.20 α = 0.30 (a) m = 1.0 , pr = 1.0 , j = 0.001 0 2 4 6 8 x 0 0.1 0.2 0.3 0.4 n ux α = 0.00 α = 0.10 α = 0.20 α = 0.30 (b) m = 1.0 , pr = 1.0 , j = 0.001 0 2 4 6 8 10x 0 5 10 15 20 y 10.44 0.70 9.74 9.048.35 7.656.96 6.26 5.57 2.09 3.48 4.17 (a) 0 2 4 6 8 10x 0 5 10 15 20 y 10.87 0.72 10.15 9.42 8.70 2.17 3.62 5.07 6.52 7.97 (b) 0 2 4 6 8 10x 0 5 10 15 20 y 11.04 0.74 10.30 9.57 8.83 8.10 7.36 6.62 5.89 5.15 4.423.68 2.21 (c) 0 2 4 6 8 10x 0 5 10 15 20 y 11.28 10.53 9.78 9.02 8.27 7.526.77 6.02 5.26 4.51 3.762.26 0.75 (d) nazma parveen and m. a. alim / journal of naval architecture and marine engineering 9(2012) 11-24 joule heating on magnetohydrodynamic natural convection flow …... 21 fig. 9: isotherms for (a) j = 0.00 (b) j = 0.025 (c) j = 0.035 (d) j = 0.05 while pr = 0.5, α = 0.2 and m = 0.01. fig. 10: streamlines for (a) m = 0.0 (b) m = 0.5 (c) m = 1.0 (d) m = 1.5 while pr = 1.0, α = 0.2 and j = 0.001. figs. 10 and 11 illustrate the effect of magnetic parameter m on the streamlines and isotherms profile for α = 0.2, j = 0.001 and pr = 1.0. for increasing values of the magnetic parameter m, the flow rate within the boundary layer decreases and the thermal boundary layer becomes thicker. fig. 10 depicts that the maximum values of ψ decreases steadily while the values of m increases. the maximum values of ψ, that is, maxψ are 8.08, 5.33, 4.03 and 3.26 for m = 0.0, 0.5, 1.0 and 1.5 respectively. the magnetic field acting along the horizontal direction retards the fluid velocity. for this there creates a lorentz force by the interaction between the applied magnetic field and flow field. this force acts against the direction of fluid flow and reduces the velocity. the magnetic field decreases the temperature gradient at the surface and increases the temperature in the flow region due to the interaction. so the thermal boundary layer becomes higher. 0 2 4 6 8 10x 0 5 10 15 20 y 0.06 0.13 0.38 0.63 (a) 0 2 4 6 8 10x 0 5 10 15 20 y 0.06 0.19 0.3 1 0.69 (b) 0 2 4 6 8 10x 0 5 10 15 20 y 0.06 0.13 0.38 (c) 0 2 4 6 8 10x 0 5 10 15 20 y 0.06 0.13 0.31 0.75 0.94 (d) 0 2 4 6 8 10x 0 5 10 15 20 y 4.03 3.763.493.232.96 0.27 0.81 1.34 1.88 2.15 2.42 2.69 (c) 0 2 4 6 8 10x 0 5 10 15 20 y 3.05 2.83 2.612.39 2.18 1.96 1.74 0.65 1.09 1.52 0.22 3.26 (d) 0 2 4 6 8 10x 0 5 10 15 20 y 5.33 4.984.624.27 3.91 3.56 3.202.84 0.36 1.07 1.78 2.13 (b) 0 2 4 6 8 10x 0 5 10 15 20 y 8.08 0.54 7.54 7.00 6.46 5.92 5.38 4.854.31 3.77 1.62 2.69 (a) nazma parveen and m. a. alim / journal of naval architecture and marine engineering 9(2012) 11-24 joule heating on magnetohydrodynamic natural convection flow …... 22 the influence of the magnetic parameter m, on the local nusselt number are illustrated in figs. 12 and 13 respectively with pr = 1.0, α = 0.0 and j = 0.0. the results for without joule heating (j = 0.0) and a fluid having pr = 1.0 are compared with those of alam et al. (1997) and a very good agreement is found. fig. 11. isotherms for (a) m = 0.0 (b) m = 0.5 (c) m = 1.0 (d) m = 1.5 while pr = 1.0, α = 0.2 and j = 0.001. fig. 12. local nusselt number for different values of magnetic parameter m with pr = 1.0, α = 0.0 and j = 0.0 (alam et al., 1997). 0 2 4 6 8 10x 0 5 10 15 20 y 0.06 0.13 0.38 (a) 0 2 4 6 8 10x 0 5 10 15 20 y 0.06 0.13 0.38 0.75 (b) 0 2 4 6 8 10x 0 5 10 15 20 y 0.06 0.13 0.3 1 0.94 0.44 (c) 0 2 4 6 8 10x 0 5 10 15 20 y 0.06 0.13 0.19 0.88 0.380.25 (d) nazma parveen and m. a. alim / journal of naval architecture and marine engineering 9(2012) 11-24 joule heating on magnetohydrodynamic natural convection flow …... 23 fig. 13. local nusselt number for different values of magnetic parameter m with pr = 1.0, α = 0.0 and j = 0.0 (present work). 5. conclusion the effect of joule heating on natural convection flow of viscous incompressible fluid including the magnetic field along a uniformly heated vertical wavy surface has been studied. new variables to transform the complex geometry into a simple shape and were used a very efficient implicit finite difference method known as the keller-box scheme was employed to solve the boundary layer equations. from the present investigation the following conclusions may be drawn: • the skin friction coefficient decreases for increasing values of prandtl number pr, over the whole boundary layer but the significantly increase the rate of heat transfer. • the effect of increasing joule heating parameter j results in decreasing the local rate of heat transfer nux and increasing the local skin friction coefficient cfx. • an increase in the values of m and α leads to decrease the local skin friction coefficient cfx and the local rate of heat transfer nux. • the velocity and the thermal boundary layer become thicker when joule heating parameter j increases. • the flow rate decreases and the thermal boundary layer grows thick when the effect of magnetic field is considered. references alam, k. c. a., hossain, m. a. and rees, d. a. s. (1997): magnetohydrodynamic free convection along a vertical wavy surface, int. j. appl. mech. engrg 1, 555–566. alim, m. a., alam, m. and mamun, a. a. (2007): joule heating effect on the coupling of conduction with mhd free convection flow from a vertical flat plate, nonlinear analysis: modell. and cont. 12, 307-316. alim, m. a., alam, m. m., mamun, a. a. and hossain, m. b. (2008): combined effect of viscous dissipation & joule heating on the coupling of conduction & free convection along a vertical flat plate, int. commun. in heat and mass transfer 35, 338-346. doi:10.1016/j.icheatmasstransfer.2007.06.003 cebeci, t. and bradshaw, p. (1984): physical and computational aspects of convective heat transfer, springer, new york. cheng, c. y., (2000): natural convection heat and mass transfer near a vertical wavy surface with constant wall temperature and concentration in a porous medium, int. comm. heat and mass transfer 27, 1143–1154. el-amin, m. f. (2003): combined effect of viscous dissipation and joule heating on mhd forced convection over a non isothermal horizontal cylinder embedded in a fluid saturated porous medium, journal of magnetism and magnetic materials 263, 337-343. doi:10.1016/s0304-8853(03)00109-4 0 1 2 3 40.2 0.3 0.4 0.5 m = 0.0 m = 0.5 m = 1.0 (a) pr = 1.0 , j = 0.0 , α = 0.0 n u x nazma parveen and m. a. alim / journal of naval architecture and marine engineering 9(2012) 11-24 joule heating on magnetohydrodynamic natural convection flow …... 24 hossain, m. a. (1992): the viscous and joule heating effects on mhd free convection flow with variable plate temperature, int. j. heat and mass transfer 35, 3485-3487. doi:10.1016/0017-9310(92)90234-j hossain, m. a. and pop, i. (1996): magnetohydrodynamic boundary layer flow and heat transfer on a continuous moving wavy surface, arch. mech. 48, 813–823. hossain, m. a. and rees, d. a. s. (1999): combined heat and mass transfer in natural convection flow from a vertical wavy surface, acta mechanica 136, 133–141. hossain, m. a., kabir, s. and rees, d. a. s. (2002): natural convection of fluid with variable viscosity from a heated vertical wavy surface, z. angew. math. phys. 53, 48–52. keller, h. b. (1978): numerical methods in boundary layer theory, annual rev. fluid mechanics, 10, 417433. doi:10.1146/ annurev.fl.10.010178.002221 molla, m. m., hossain, m. a. and yao, l. s. (2004): natural convection flow along a vertical wavy surface with uniform surface temperature in presence of heat generation/absorption, int. j. thermal. sciences 43, 157-163. doi:10.1016/j.ijthermalsci.2003.04.001 moulic, s. g. and yao, l. s. (1989): mixed convection along wavy surface, asme j. heat transfer 111, 974– 979. moulic, s. g. and yao, l. s. (1989): natural convection along a wavy surface with uniform heat flux, asme j. heat transfer 111, 1106–1108. nasrin, r. and alim, m. a. (2009): combined effects of viscous dissipation and temperature dependent thermal conductivity on mhd free convection flow with conduction and joule heating along a vertical flat plate, j. naval arch. marine eng., 6, 30-40. doi:10.3329/jname.v6i1.2648 parveen, n. and alim, m. a., (2011): effect of temperature dependent thermal conductivity on magnetohydrodynamic natural convection flow along a vertical wavy surface, international journal of energy & technology 3 (9), 1–9. parveen, n. and alim, m. a., (2011): effect of temperature-dependent variable viscosity on magnetohydrodynamic natural convection flow along a vertical wavy surface, international scholarly research network mechanical engineering vol. 2011, article id 505673, 1-10. rees, d. a. s and pop, i. (1994): a note on free convection along a vertical wavy surface in a porous medium, j. heat transfer 116, 505–508. rees, d. a. s and pop, i. (1994): free convection induced by a horizontal wavy surface in a porous medium, fluid dyn. res. 14, 151–166. yao, l. s. (1983): natural convection along a vertical wavy surface, asme j. heat transfer 105, 465–468. yao, l. s. (2006): natural convection along a vertical complex wavy surface, int. j. heat and mass transfer 49, 281–286. microsoft word 2789-18238-2-ed.doc journal of naval architecture and marine engineering december, 2009 doi: 10.3329/jname.v6i2.2789 http://www.banglajol.info 1813-8235 (print), 2070-8998 (online) © 2009 aname publication. all rights reserved. received on: 12 july 2009 coupled dynamic response of a three-column mini tlp anitha joseph, lalu mangal and precy sara george department of civil engineering, thangal kunju musaliar college of engineering, kollam-691005, kerala, india. e-mail: anithareebu@yahoo.co.uk abstract: for the development of deepwater marginal fields, many new platform concepts and designs are on the anvil. the mini tlp is a proven concept in this regard wherein an optimized conventional tlp system economically and efficiently serves in developing small marginal deepwater reserves. various new geometric configurations and designs of mini tlps are reported in the literature. this paper presents a new geometric configuration which could be a better alternative to an existing configuration. a 3-column mini tlp is designed and its platform-mooring coupled dynamic behaviour is investigated and compared with an existing 4-column mini tlp. the numerical investigation is carried out for the 1:56 scaled model using a finite element computer program suitable for compliant offshore platforms. a combination wave force model with diffraction-radiation loading on large members and morison loading on slender members is adopted for computing the non-linear dynamic response of the structure. the effects of parameters such as pretension in tethers and wave approach angle have been studied. the results obtained are compared with published results of the 4-column mini tlp. it is found that the dynamic responses of the 3-column mini tlp are close to the 4-column mini tlp with relatively higher surge and tether tension. accounting for this in the design stage, the newly designed structure could be a promising candidate which can be used as an alternative to the 4-column mini tlp. reducing the number of columns from four to three has added advantages in terms of cost and time during fabrication, installation and maintenance of the platform. keywords: deepwater structures; coupled dynamics; finite element method; mini tlp; nonlinear dynamic analysis. 1. introduction numerous small oil fields have been discovered in very deep seabeds. new concepts of platform construction, exploration, drilling and production are necessary for economic development of these minimal oil fields situated in deep, remote locations in hostile environment. mini tension leg platform (mini tlp) is a proven concept towards this objective. it is evident from the literature that the study of structural behaviour and dynamic response of these platform concepts are presently being actively pursued for design optimisation. several mini tlp designs have been discussed in the literature. these are: a downsized four column mini tlp for 1000 m water depth (hudson et al. 1996, hudson and vasseur 1996); a family of concrete mini tlps for generic applications (logan 1996); a three column tlp for 800 m water depth, easily extendable to 1500 m under shorter development schedule with significant reduction in cost (muren 1996); a mini tlp with vertical cylindrical hull connected to three radially tapered rectangular pontoons called seastar (kibbee 1996 and kibbee et al. 1999). the importance of coupled response analysis of offshore compliant structural systems has also been discussed in the literature. analysis of deepwater compliant structures must address significant hull-tether coupling because the uncoupled method ignores the interaction effects between the platform hull and its tethers. the coupled nonlinear dynamic analysis of the` seastar mini tlp in the time domain using finite element method has been reported (sreekumar et al. 2001). the results were compared with the similar calculations based on morison equation for wave loading as well as with experimental results using a 1:50 scale model. kim and sclavounos (2001) described the fully coupled response simulations of theme offshore structures in water depths of up to 10,000 feet. bhattacharya et al. (2003) presented a detailed finite element methodology for a. joseph, l.mangal and p.s. george / journal of naval architecture and marine engineering 6(2009)52-61 coupled dynamic response of a three-column mini tlp 53 the analysis of coupled dynamics of mini tlps and the nature of hull-tether coupling and its physical modeling principles have been brought out. joseph et al. (2004) reported detailed experimental and numerical investigations on the coupled dynamic behaviour of a 4-column mini tlp with special attention to hull-tether coupling. numerical investigation using several wave force models and validation with experiments identified the best suited wave force model, which is a combination of potential theory based wave loading on the hullcolumn entity and morison-type wave loading on the slender cantilevering arms and tethers. liagre and niedzwecki (2006) presented the non-linear coupled dynamic response of a deepwater mini tlp considering non-linear stiffness, quadratic damping, surge-pitch and sway-roll coupling. in addition, behaviour of hydrodynamic added-mass and damping coefficient was simulated using an industry standard diffractionradiation software package. the results obtained were used for modelling the local offshore environment, numerical simulation and model test verification of the platform response characteristics. chen et al. (2006) compared numerical results with measurements for a mini tlp. they considered only morison wave loading. joseph et al. (2007) presented the design details of a 3-column mini tlp based on an existing 4-column mini tlp and carried out its numerical analysis with morison equation alone for the wave force model thus the importance of coupled analysis and combined wave force modelling is underscored in the literature. several innovative designs and interesting geometrical configurations also are reported aiming towards savings and advantages in terms of material, time and cost, applicable to marginal deepwater sites. numerous marginal oil fields have been discovered in very deepwater all around the world. for the economic development of these deepwater minimal fields, optimized new platform concepts is necessary. based on the critical assessment of the literature and the motivation outlined above, the scope of the paper is a new mini tlp design which could have favourable dynamic response along with advantages in terms of cost, material and time. 2. design details of the model the new geometrical configuration is arrived at as a part of a screening of alternative mini tlp concepts. the geometric parameters are designed with reference to the 4-column mini tlp model. the design is done by keeping the draft, the total weight, the weight displacement and the vertical centre of buoyancy same as that of the 4-column model. the model geometry and construction details of the 1:56 scaled model (froude modeling) of the 3-column mini tlp are given in fig. 1. table 1 shows detailed data of both models. the principal parts of the newly designed mini tlp model are the large pontoon, three slender columns, the upper and lower decks and three cantilevering trusses to which the tendons are attached. the pontoon consists of a 400 mm diameter pvc cylinder of height 232 mm with closed bottom and top ends and provides most of the buoyancy. three 75 mm diameter columns extend vertically upwards from the top surface of the pontoon to support the lower and upper decks. the still water level is at the level of the columns (fig. 1) such that the draft in tethered condition is 598 mm (which is same as that of 4-column tlp model). therefore the water plane area consists of the three columns (awp= 3×4418 = 13253 mm 2). the three tendon-supporting cantilevered trusses, each of length 246 mm consisting of two 40 mm diameter horizontal pvc tubes and two 32 mm diameter inclined pvc tubes, extend radially from the pontoon’s circumference, at an angle of 120º from each other. pvc flats of size 80 mm  50 mm  20 mm (with a 10 mm diameter hole drilled centrally for receiving the tendon assembly) connect the ends of each pair of horizontal truss members. for model tether: young's modulus is 0.457  1011 n/m2, crosssectional area is 4.04 mm2, and mass density is 0.0317 kg/m. the weight of model (w) without ballast (additional weight) is 17.882 kg (= wmin). at design draft of 598 mm, the weight displacement (∆) of the model is 37.84 kg. two tether pretension levels were used in experiments, tt/∆ = 0.26 (tt = w = 9.84 kg; w = 28 kg, ballast = 10.06 kg, w/tt = 2.85) and tt/ = 0.17 (tt = w = 6.43 kg; w = 31.41 kg, ballast = 13.47 kg, w/tt = 4.88). steel ballast weights could be used to achieve desired model weight. the condition with w/tt ratio of 2.86 is in the range of typical tlp designs. the two tether pretension levels can be achieved by varying the weight of the model, by inserting the ballast in the three columns. the percentage difference of weight displacement of the present model from that of 4-column model is 1.96. the percentage difference of the water plane area is 5.9 and that of the total weight of the model is 0.65. all these values are sufficiently low so that results of the present model could be compared with published results of the 4-column mini tlp. a. joseph, l.mangal and p.s. george / journal of naval architecture and marine engineering 6(2009)52-61 coupled dynamic response of a three-column mini tlp 54 fig. 1. three-column mini tlp model (scale 1:56) table 1: model data parameter 3-column mini tlp 4-column mini tlp displacement () 37.84 kg 38.6 kg weight (w) tt/=0.26 28.6 kg 28 kg tt/=0.17 32 kg 31.41 kg total tether pretension (tt) tt/=0.26 10 kg 9.84 kg tt/=0.17 6.56 kg 6.43 kg water depth 4.3 m 4.3 m draft 598 mm 598 mm height of pontoon 232 mm 232 mm diameter of cylindrical pontoon 400 mm 400 mm water plane area of hull (awp) 13253 mm 2 12469 mm2 moment of area of water plane (iwp) 53.37×106 mm 4 140.5×106 mm4 diameter of columns 75 mm 63 mm length of tether (lt) 3.6 m 3.6 m vertical centre of gravity (vcg) from keel w = 18 kg 429 mm 450 mm w = 28.6 kg 295 mm 310 mm w = 32 kg 280 mm 292 mm vertical centre of buoyancy 159 mm 157 mm the tethers of the present model is taken as the same as that for the 4-column model. they are strand type twisted steel wire ropes of 3 mm outer diameter comprising of six strands of seven threads each (i.e. a total of 42 threads). diameter of each thread is 0.35 mm and hence its c/s area is 0.0962 mm2. on this basis, the cross section area (a) of the steel wire is 420.0962 = 4.04 mm2. mass density of the tether (t) is 0.0317 kg/m, so that the total mass of one tether is 0.03173.6 = 0.114 kg, which is negligible compared to the platform mass. the value of young's modulus is e = 0.4571011 n/m2. tether 400  pontoon (d1) 75  column (d2) deck 610610 32  (d3) 5 9 8 (d ) 48 2 3 57 3 66 2 32 1 2 5 z x 246 z y y x 1 2 3 (a) elevation x-z (b) elevation y-z (c) plan a. joseph, l.mangal and p.s. george / journal of naval architecture and marine engineering 6(2009)52-61 coupled dynamic response of a three-column mini tlp 55 the damped natural periods of vibration of the mini tlp model (table 2) reveal that the surge, sway and yaw (soft modes) frequencies lie much below while the heave, roll and pitch (stiff modes) frequencies lie above the practical range of wave frequencies. table 2: comparison natural periods of vibration and damping ratios of the mini tlp model (tt/δ = 0.26,  = 0) degree of freedom natural period (s) damping ratio surge, sway 10.600 0.040 heave 0.115 0.129 roll, pitch 0.130 0.118 yaw 3.630 0.274 3. coupled response analysis the dynamic response of the model is investigated using nonlinear finite element method in the time domain. the underwater hull of the mini tlp comprises of three sets of slender columns and cantilevering arms as well as a hydro-dynamically compact pontoon with a slender tether system providing the compliance. so a wave force model with diffraction-radiation loading on the pontoon-column unit and morison loading on slender members was adopted for computing the non-linear dynamic response of the structure. for evaluating wave forces on hydrodynamically transparent (slender) members, morison equation is generally used. depending on the keulegan-carpenter (kc) numbers of various members, drag coefficient (cd) values ranging from 0.5 to 0.76 and inertia coefficient (cm) values ranging from 2.2 to 2.3 were selected from experimental values given in chakrabarti (1987). the hydrostatic parameters used in the numerical analysis are heave stiffness = 43.34 n/m, pitch stiffness (= roll stiffness) = 0.01524 nm/rad. to carry out linear diffraction-radiation analysis of the scaled model a cylindrical fluid domain of diameter approximately five times that of the larger diameter (400mm) hull is selected. the finite element mesh of the fluid domain with horizontal seabed boundary has 12702 nodes and 11442 eight-noded brick elements. the diffraction-radiation analysis is carried out using a finite element code which has been extensively used in a variety of problems yielding frequency-dependent added mass and radiation damping coefficients and first order wave forces (sathyapal, 2001). second order dampers have been used at the radiation boundary. in the context of the nonlinear dynamic analysis, the added mass matrix is modeled by a 3d global coupled mass element and the radiation damping matrix is modeled by a 3d global coupled damper element, locating both at the cg of the hull. to carry out the nonlinear dynamic analysis, a finite element code for the nonlinear time domain simulation of dynamic response based on updated lagrangian formulation considering both hull-tether coupling and coupling between six platform degrees of freedom is used. the dynamic equilibrium equation is solved in the time domain using the incremental-iterative newmark-beta algorithm (sreekumar, 2001). the finite element model is presented in fig. 2. it comprises of 39 nodes, 62 beam elements, 3 spring elements, 7 mass elements and 180 equations. there are 4 beam elements per tether, 16 in pontoon, 2 each per cantilevering truss, 3 in columns below swl, and 15 above the swl representing the columns and the upper and lower decks. table 3 shows the diameters for hydrodynamic calculations for the different sets of elements (1 to 7 marked in the basic fe model in fig. 2). the three dimensional beam elements are modelled such that (i) the mass distribution should match with the overall mass of 17.882 kg of the mini tlp model (ii) the vcg should match with 450 mm, as in the case of the 4-column mini tlp model. six 3d coupled mass elements (at the top and bottom nodes of the pontoon column elements) model the ballast (lumped mass) for varying the pretension values. effect of two tether pretensions of 26 % and 17 % and two wave approach angles 0º and 60º are studied. the tether nodes at the seabed are given fixed boundary conditions. being a compliant structure, the large displacements are of interest than the structural deformation of the platform. so the elastic tethers are modelled as beam elements with their true stiffness while arbitrarily high values are used for the rigid platform. a. joseph, l.mangal and p.s. george / journal of naval architecture and marine engineering 6(2009)52-61 coupled dynamic response of a three-column mini tlp 56 fig. 2. finite element model of mini tlp table 3: diameters of beam elements for hydrodynamic calculation element set description of member diameter (m) structural area of cross section (m2) 1 tether 0.003 4.040910-06 2 horizontal cantilevering arm 0.0566 8.503810-04 3 sloping cantilevering arm 0.0453 3.445710-04 4 pontoon central member 0 4.036210-03 5 submerged column 0 2.07710-03 6 column above swl 0.075 6.785810-04 7 deck and other pontoon members 0 1.2509610-03 4. results and discussion the results of the diffraction-radiation analysis are added mass [], radiation damping coefficients [], diffraction force components {f} and their phases for unit wave amplitude. the analysis is carried out for frequencies ranging from 0.36 hz to 2.5 hz. figure 3 shows that the surge added mass for the 3-column mini tlp is 12.5 % lower than that of 4-column mini tlp (for a wave frequency of 1.2 hz). the sway added mass for the 3-column mini tlp is 11 % lower than that of 4-column mini tlp (for a wave frequency of 1.2 hz). the heave added mass for the 3-column mini tlp is 10 % lower than that of 4-column mini tlp (for a wave period of 1.6 hz). roll added inertia of the 3column mini tlp is 10 % higher and pitch added inertia of 3-column mini tlp is 9 % higher than that of the 4column mini tlp (for a wave frequency of 1.2hz). 4#3d-beam elements in one tether node with constraints beam element y x z p b o s s s 1 2 3 4 6 7 7 7 7 7 7 7 7 7 5 39 nodes 62 elements a. joseph, l.mangal and p.s. george / journal of naval architecture and marine engineering 6(2009)52-61 coupled dynamic response of a three-column mini tlp 57 fig. 3. comparison of added mass of mini tlp models. figure 4 shows that all damping coefficients, except heave, are higher for the 3-column mini tlp model. surge and sway damping coefficients are 60 % higher, heave damping coefficient is 55 % lesser, and roll as well as pitch damping coefficients are 45 % higher for the 3-column mini tlp (for a wave frequency of 1.2 hz) compared to the 4-column mini tlp. the reduction in the case of 4-column mini tlp could be attributed to the cancellation by interference from the reflected waves from the four columns and supporting frames, whereas for a 3-column mini tlp, the interference is only from three columns and supporting frames resulting in an increase of waves passing out of the radiation boundary thus increasing the radiation damping. in fig. 5 the diffraction force components as well as their phases for unit wave amplitude (for  = 0) are compared with that of 4-column mini tlp. surge force is 13 % and heave force is 21 % higher for 3-column mini tlp (for a wave frequency of 0.8 hz). pitch moment is almost same for both the models for both the models, sway force, roll moment and yaw moment vanish for  equal to 0 and hence not shown.. the plots 12 16 20 24 28 0.4 0.8 1.2 1.6 2 f (hz) 3 column mini tlp 4 column mini tlp μ 1 1 (k g ) 12 16 20 24 28 0.4 0.8 1.2 1.6 2 f (hz) μ 2 2 (k g ) 12 16 20 24 28 0.4 0.8 1.2 1.6 2 f (hz) μ 3 3 (k g ) 0 1 2 3 4 5 0.4 0.8 1.2 1.6 2 f (hz) μ 4 4 (k g m 2 ) 0 1 2 3 4 5 0.4 0.8 1.2 1.6 2 f (hz) μ 5 5 ( k g m 2 ) 0 1 2 3 4 5 0.4 0.8 1.2 1.6 2 f (hz) μ 6 6 (k g m 2 ) -10 -5 0 5 0.4 0.8 1.2 1.6 2 f (hz) μ 1 5 (k g m ) -10 -5 0 5 10 0.4 0.8 1.2 1.6 2 f (hz) μ 2 4 (k g m ) surge added mass sway added mass heave added mass pitch added mass roll added mass yaw added mass coupled added mass coupled added mass a. joseph, l.mangal and p.s. george / journal of naval architecture and marine engineering 6(2009)52-61 coupled dynamic response of a three-column mini tlp 58 show that the phase angles in surge and sway are independent of wave approach angle. also, the phases of surge and sway are identical for both 3-column and 4-column mini tlps. fig. 4. radiation damping coefficients of mini tlp models. the results of the non-linear dynamic analysis are presented in fig. 6 and fig. 7 in the form of response amplitude operators (rao). from fig. 6 it is seen that the surge response, which is the only major motion response is more for the 3-column mini tlp i.e., a maximum of 20 % increase (for a wave period of 1.5 s, tt/δ = 0.17). the heave response is small and is almost same for both the structures. the maximum value is less than 2 mm for wave amplitude of 1 m. the pitch response also is small but higher for 3-column model. the maximum value is less than 0.17o against 0.11o per unit wave amplitude for the 4-column mini tlp. it is evident from fig. 7 that the dynamic tether tension is higher for the 3-column model, the maximum being 94 n per unit wave amplitude (for a wave period of 1.5 s, tt /δ = 0.17) against 60 n for the 4-column mini tlp. 0 5 10 15 20 0.4 0.8 1.2 1.6 2 f (hz) β 2 2 (n s/ m ) 0 5 10 15 20 0.4 0.8 1.2 1.6 2 f (hz) β 3 3 (n s/ m ) -0.5 0 0.5 1 1.5 2 0.4 0.8 1.2 1.6 2 f (hz) β 4 4 ( n s m ) 0 5 10 15 20 0.4 0.8 1.2 1.6 2 f (hz) 3 column mini tlp 4 column mini tlp β 1 1 (n s/ m ) -0.5 0 0.5 1 1.5 2 0.4 0.8 1.2 1.6 2 f (hz) β 5 5 (n s m ) -0.5 0 0.5 1 1.5 0.4 0.8 1.2 1.6 2 f (hz) β 6 6 (n s m ) -8 -4 0 4 8 0.4 0.8 1.2 1.6 2 f (hz) β 1 5 ( n s ) -8 -4 0 4 8 0.4 0.8 1.2 1.6 2 f (hz) β 2 4 (n s ) surge sway heave roll pitch yaw coupled (surge-pitch) coupled (sway-roll) a. joseph, l.mangal and p.s. george / journal of naval architecture and marine engineering 6(2009)52-61 coupled dynamic response of a three-column mini tlp 59 fig. 5. diffraction force components and their phases. parametric studies reveal almost similar observations for tether pretension values of tt /δ = 0.17 and 0.26 which show that the chosen pretension levels which are in the practical range do not affect the dynamic response appreciably. for a change of wave approach angle from 0º to 60º, it is observed that there is considerable decrease in responses for surge and pitch while, the heave and tether responses shows little change. but the resultant horizontal displacement in the direction of the wave is the same as that for θ = 0º. 5. conclusion the study thus shows that there is no drastic change in the dynamic response of the proposed 3-column mini tlp compared to the 4-column mini tlp. the increase in the case of surge and tether tensions is expected as the number of tethers is reduced to three from four, and can be taken into account while designing the structure. as its dynamic responses are found to be close, the 3-column structure can be considered as an alternative to the 4-column mini tlp. reducing the number of columns from four to three has added advantages in terms of cost and time during fabrication, installation and maintenance of the platform. 0 100 200 300 400 500 0.4 0.8 1.2 1.6 2 f (hz) 3 column mini tlp 4 column mini tlp fd 1/ a ( n /m ) 0 100 200 300 400 500 0.4 0.8 1.2 1.6 2 f (hz) fd 3/ a ( n /m ) 0 100 200 300 400 500 0.4 0.8 1.2 1.6 2 f (hz) fd 5/ a ( n m /m ) -200 -100 0 100 200 0.4 0.8 1.2 1.6 2 f (hz) p ha se ( de gr ee s) -200 -100 0 100 200 0.4 0.8 1.2 1.6 2 f (hz) p h as e (d eg re e) -200 -100 0 100 200 0.4 0.8 1.2 1.6 2 f (hz) p h as e (d eg re es ) surge surge heave heave pitch pitch a. joseph, l.mangal and p.s. george / journal of naval architecture and marine engineering 6(2009)52-61 coupled dynamic response of a three-column mini tlp 60 fig. 6. surge, heave and pitch responses. fig. 7. dynamic tether tension responses. acknowledgement the authors thankfully acknowledge the support from department of ocean engineering, indian institute of technology madras, and all india council for technical education (research promotion scheme-2006). references anitha joseph, idichandy,v.g. and bhattacharyya, s.k. (2004): experimental and numerical study of coupled dynamic response of a mini tension leg platform, asme transactions, journal of offshore mechanics and arctic engineering, vol. 126, pp. 18-33. doi:10.1115/1.1833358 anitha joseph, lalu mangal and anima, v. (2007): numerical study of nonlinear dynamic response of a three column mini tension leg platform, proc. fourth indian national conf. harbour and ocean engineering, nitk, surathkal, india, pp. 307314. bhattacharyya, s.k., sreekumar, s. and idichandy, v.g. (2003): coupled dynamics of seastar mini tension leg platform, ocean engineering, vol. 30, pp. 709-737. doi:10.1016/s0029-8018(02)00061-6 h ea ve r a o (m /m ) 0 0.002 0.004 0.006 0 0.5 1 1.5 2 2.5 3 wave period (s) 0 0.002 0.004 0.006 0 0.5 1 1.5 2 2.5 3 wave period (s) p it ch r a o (r ad /m ) s u r g e r a o ( m /m ) 0 0.3 0.6 0.9 1.2 0 0.5 1 1.5 2 2.5 3 wave period (s) 3 column mini tlp 4 column mini tlp t et h er 1 t en si on r a o (n /m ) t et h er 2 t en si o n r a o (n /m ) 0 50 100 150 0 0.5 1 1.5 2 2.5 3 wave period (s) 3 column mini tlp 4 column mini tlp 0 50 100 150 0 0.5 1 1.5 2 2.5 3 wave period (s) 0 50 100 150 0 0.5 1 1.5 2 2.5 3 wave period (s) t et h er 3 t en si o n r a o (n /m ) a. joseph, l.mangal and p.s. george / journal of naval architecture and marine engineering 6(2009)52-61 coupled dynamic response of a three-column mini tlp 61 chakrabarti, s.k. (1987): hydrodynamics of offshore structures, computational mechanics limited and springer-verlag, new york, pp. 199-200. chen, x., yu ding, jun zhang, pierre liagre, niedzwecki, j. and teigen, p. (2006): coupled dynamic analysis of a mini tlp: comparison with measurements, ocean engineering, vol. 33, no. 1, pp. 93-117. doi:10.1016/j.oceaneng.2005.02.013 hudson, w.l., meurant, o. and vasseur, j. (1996): mini tlp for deep but mild waters, journal of offshore technology, vol. 4, 4, pp. 16-19. hudson, w.l. and vasseur, j.c. (1996): small tension leg platform for marginal deepwater fields, proc. offshore technology conf., otc 8046. kibbee, s.e., john chianis., davies, k.b., and sarvano, b.a. (1994): the sea star tension leg platform, proc. offshore technology conf., otc 7535. kibbee, s., leverette, s.j., davies, k.b.and matten, r.b. (1999): morpeth seastar minitlp, proc. offshore technology conf., otc 10855. kim, s. and p.d. sclavounos (2001): fully coupled response simulations of theme offshore structures in water depths of up to 10,000 feet, proc. eleventh int. offshore and polar engineering conf. liagre p.f.and niedzwecki, j. m. (2006): estimating non-linear coupled frequency-dependent parameters in offshore engineering, applied ocean research, vol. 25, 1-19. doi:10.1016/s0141-1187(03)00029-4 logan, b.l., s. naylor, t. munkejord and nygaard, c. (1996): ‘atlantic alliance: the next generation tension leg platform’, proc. offshore technology conf., otc 8264. muren, j., p. flugstad, b. greiner, r. d'souza and solberg, i.c. (1996): the 3 column tlp a cost efficient deepwater production and drilling platform’, proc. offshore technology conf., otc 8045. sathyapal, s. (2001): finite element procedures for second order steady and low frequency wave forces on elongated bodies with forward speed, ph.d. thesis, indian institute of technology, madras. sreekumar, s. (2001): analytical and experimental investigations on the dynamics of deepwater mini tension leg platforms, ph.d. thesis, indian institute of technology, madras. sreekumar, s., idichandy,v.g. and bhattacharyya, s.k. (2001): coupled dynamics of seastar mini tension leg platform using linear diffraction-radiation theory, proc. offshore mechanics and arctic engineering conf., 1074. local similar solutions for steady mhd free convection and mass transfer flow past journal of naval architecture and marine engineering june, 2005 http://jname.8m.net 1813-8535 © 2005 aname publication. all rights reserved. dufour and soret effects on mhd free convective heat and mass transfer flow past a vertical porous flat plate embedded in a porous medium md. shariful alam1 and mohammad mansur rahman2 1department of mathematics, dhaka university of engineering and technology (duet), gazipur-1700, bangladesh, email: msalam631@yahoo.com 2department of mathematics, university of dhaka, dhaka-1000, bangladesh, email: mansurdu@yahoo.com abstract a two-dimensional steady mhd free convection and mass transfer flow past a semi-infinite vertical porous plate in a porous medium has been studied numerically including the dufour and soret effects. the resulting momentum, energy and concentration equations are then made similar by introducing the usual similarity transformations. these similar equations are then solved numerically by using the nachtsheim-swigert shooting method along with runge-kutta sixth order integration scheme. the numerical results are displayed graphically showing the effects of various parameters entering into the problem. finally, the local values of the skin-friction coefficient (cf), nusselt number (nu) and sherwood number (sh) are also shown in tabular form. keywords: mhd, free convection, vertical plate, steady flow, porous medium, dufour effect, soret effect. nomenclature: b empirical constant sc schmidt number b0 magnetic field intensity sh sherwood number c concentration sr soret number cp specific heat at constant pressure t temperature cs concentration susceptibility tm mean fluid temperature da local darcy number u0 uniform velocity dm mass diffusivity u, v darcian velocities in the x and y-direction respectively du dufour number x, y cartesian coordinates along the plate and normal to it, respectively fw dimensionless suction velocity α thermal diffusivity fs1 local forchheimer number β coefficient of thermal expansion g acceleration due to gravity β∗ coefficient of concentration expansion gr local grashof number σ electrical conductivity gm local modified grashof number ρ density of the fluid k darcy permeability ν kinematic viscosity kt thermal diffusion ratio θ dimensionless temperature m magnetic field parameter φ dimensionless concentration nu nusselt number w condition at wall pr prandtl number ∞ condition at infinity re1 local reynods number 1. introduction the study of magnetohydrodynamic (mhd) flows have stimulated considerable interest due to its important applications in cosmic fluid dynamics, meteorology, solar physics and in the motion of earth’s core [cramer & pai (1973)]. in a broader sense, mhd has applications in three different mailto:msalam631@yahoo.com mailto:mansurdu@yahoo.com m. s. alam and m. m. rahman / journal of naval architecture and marine engineering 1(2005) 55-65 subject areas, such as astrophysical, geophysical and engineering problems. in light of these applications, steady mhd free convective flow past a heated vertical flat plate has been studied by many researchers such as gupta (1961), lykoudis (1962), and nanda and mohanty (1970). raptis and kafoussias (1982) studied free convection and mass transfer flow through a porous medium in the presence of transverse magnetic field, due to the importance of mass transfer and that of applied magnetic field in the study of star and planets. recently sattar et al. (2001) obtained similar solutions of a steady mhd free convection and mass transfer flow with viscous dissipation. they have used the perturbation method to solve the problem. however, in all the above studies, dufour and soret effects were neglected, on the basis that they are of a smaller order of magnitude than the effects described by fourier’s and fick’s laws. there are, however, exceptions. the soret effect, for instance, has been utilized for isotope separation and in mixture between gases and with very light molecular weight (h2, he), and for medium molecular weight (h2, air) the dufour effect was found to be of considerable magnitude such that it can not be neglected [eckert and drake (1972)]. dursunkaya and worek (1992) studied the diffusion-thermo and thermal-diffusion effects in transient and steady natural convection from a vertical surface. recently, anghel et al. (2000) included the dufour and soret effects on free convection boundary layer over a vertical surface embedded in a porous medium. very recently, postelnicu (2004) studied the influence of a magnetic field on heat and mass transfer by natural convection from vertical surfaces in porous media considering soret and dufour effects. hence the objective of the present paper is to study the above-mentioned dufour and soret effects on steady free convection and mass transfer flow past a continuously moving semi-infinite vertical porous flat plate embedded in a porous medium under the influence of a transversely applied magnetic field. 2. mathematical formulation: let us consider the steady free convection and mass transfer flow of a viscous, incompressible and electrically conducting fluid past a continuously moving semi-infinite vertical porous plate embedded in a porous medium under the influence of a transversely applied magnetic field. the flow is assumed to be in the x-direction, which is taken along the plate in the upward direction and y-axis is normal to it. initially it is assumed that the plate and the fluid are at the same temperature t and the concentration level everywhere in the fluid is same. at time >0, the plate temperature and concentration are instantly raised to ( > ) and ( > ), which are thereafter maintained constant, where and are the temperature and concentration respectively outside the boundary layer. the induced magnetic field is assumed to be negligible, such that b = (0, b t wt ∞t wc ∞c ∞t ∞c 0, 0). the equation of conservation of electric charge ∇.j = 0 gives jy = constant, where j = (jx, jy, jz). since the plate is electrically nonconducting, this constant is zero and hence jy = 0 everywhere in the flow. assuming that the boussinesq and boundary-layer approximations hold and using the darcy-forchheimer model, the governing equations relevant to the problem are given by: continuity equation 0= ∂ ∂ + ∂ ∂ y v x u , (1) momentum equation ( ) ( ) , 22 0 2 2 k ub k uub ccgttg y u y u v x u u −−−−+−+ ∂ ∂ = ∂ ∂ + ∂ ∂ ∞ ∗ ∞ υ ρ σ ββυ (2) energy equation 2 2 2 2 y c cc kd y t y t v x t u ps tm ∂ ∂ + ∂ ∂ = ∂ ∂ + ∂ ∂ α , (3) concentration equation 2 2 2 2 y t t kd y c d y c v x c u m tm m ∂ ∂ + ∂ ∂ = ∂ ∂ + ∂ ∂ , (4) 56 m. s. alam and m. m. rahman / journal of naval architecture and marine engineering 1(2005) 55-65 where the variables and related quantities are defined in the nomenclature. the boundary conditions for the model are given by: ( ) ⎭ ⎬ ⎫ ∞→==== ===== ∞∞ ,,,0,0 ,0,,, 00 yasccttvu yatccttxvvuu ww (5) where u0 is the uniform velocity and v0(x) is the velocity of suction at the plate. we now introduce the following dimensionless variables: ( ) ( ) ( ) ⎪ ⎪ ⎪ ⎪ ⎭ ⎪ ⎪ ⎪ ⎪ ⎬ ⎫ − − = − − = ′= = ∞ ∞ ∞ ∞ . , , , 2 0 0 cc cc tt tt fuu x u y w w ηφ ηθ η υ η (6) also by introducing the relation (6) into equation (1) we obtain ( ff x u v −′= η υ 2 0 ) . (7) introducing equations (6) and (7) into the equations (2)-(4) we obtain the following local similarity equations: 0 re 1 21 1 =′−′−′−++′′+′′′ f da fs f da fmgmgrfff φθ , (8) 0prpr =′′+′+′′ φθθ duf , (9) 0=′′+′+′′ θφφ scsrscf , (10) where ( ) 2 0 2 u xttg gr w ∞ − = β is the local grashof number, ( ) 0 2 u xccg gm w υ β ∞ ∗ − = is the local modified grashof number, 0 2 0 2 u xb m ρ σ = is the magnetic field parameter, 22x k da = is the local darcy number, υ xu 0 1re = is the local reynolds number, x b fs =1 is the local forchheimer number, α υ =pr is the prandtl number, md sc υ = is the schmidt number, ( ) ( )∞ ∞ − − = cct ttkd sr wm wtm υ is the soret number, ( ) ( )∞ ∞ − − = ttcc cckd du wps wtm υ is the dufour number. the boundary conditions are now transformed to: ⎭ ⎬ ⎫ ∞→===′ ====′= ,0,0,0 ,01,1,1, ηφθ ηφθ asf atfff w (11) where 0 0 2 u x vf w υ −= is the dimensionless suction velocity and prime denotes differentiation with respect to the variable η. 57 m. s. alam and m. m. rahman / journal of naval architecture and marine engineering 1(2005) 55-65 the parameters of engineering interest for the present problem are the local skin-friction coefficient (cf), the local nusselt number (nu) and the local sherwood number (sh), which are given respectively by the following expressions: ( )0re 2 1 2 1 1 fc f ′′= , (12) fig. 1: velocity profiles for different values of gr and m. fig. 2: velocity profiles for different values of gm and fw. 0 1 2 3 4 5 6 0 0.25 0.5 0.75 1 1.25 1.5 1.75 2 η f ' gm = 5, fw = 0.5, pr = 0.71, sc = 0.22, sr = 0.5, du = 0.12, da = 0.5, re1= 200 and fs1= 1.0 curve gr m 1 4 0.4 2 4 1.2 3 10 0.4 3 1 2 0 1 2 3 4 5 6 0 0.25 0.5 0.75 1 1.25 1.5 1.75 2 η f ' gr = 10, m = 0.5, pr = 0.71, sc = 0.22, sr = 0.5, du = 0.12, da = 0.5, re1= 200 and fs1= 1.0 curve gm fw 1 3 0.5 2 3 1.5 3 7 0.5 58 m. s. alam and m. m. rahman / journal of naval architecture and marine engineering 1(2005) 55-65 ( )0)(re 2 1 1 θ ′−= − nu , (13) ( )0)(re 2 1 1 φ′−= − sh . 14) fig. 3: velocity profiles for different values of sr and du. fig. 4: velocity profiles for different values of da. 0 1 2 3 4 5 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 curve sr du 1 2.0 0.03 2 0.5 0.12 3 0.1 0.60 η f ' gr = 10, gm = 4, pr = 0.71, sc = 0.22, m = 0.5, fw = 0.5, da = 0.5, re1 = 200 and fs1 = 1.0 0 1 2 3 4 5 0 0.25 0.5 0.75 1 1.25 1.5 1.75 2 2.25 2.5 gr = 10, gm = 4, m = 0.5, pr = 0.71, sc = 0.22, sr = 0.5, du = 0.12, fw = 0.5, re1= 200 and fs1= 1.0 curve da 1 0.5 2 1.0 3 1.5 1 2 3 η f ' 59 m. s. alam and m. m. rahman / journal of naval architecture and marine engineering 1(2005) 55-65 60 fig. 5: temperature profiles for different values of gr and m. fig. 6: temperature profiles for different values of gm and fw. 0 1 2 3 4 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 η θ gm = 5, fw = 0.5, pr = 0.71, sc = 0.22, sr = 0.5, du = 0.12, da = 0.5, re1= 200 and fs1= 1.0 curve gr m 1 4 0.4 2 4 1.2 3 10 0.4 0 1 2 3 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 η θ gr = 10, m = 0.5, pr = 0.71, sc = 0.22, sr = 0.5, du = 0.12, da = 0.5, re1= 200 and fs1= 1.0 curve gm fw 1 3 0.5 2 3 1.5 3 7 0.5 the set of equations (8) (10) under the boundary conditions (11) have been solved numerically by applying the nachtsheim-swigert (1965) shooting iteration technique together with runge-kutta sixthorder integration scheme. from the process of numerical computation, the skin-friction coefficient, the m. s. alam and m. m. rahman / journal of naval architecture and marine engineering 1(2005) 55-65 local nusselt number and the local sherwood number, which are respectively proportional to , ( )0f ′′ ( )0θ ′− and ( )0φ′− , are also worked out and their numerical values are presented in a tabular form. fig. 7: temperature profiles for different values of sr and du. fig. 8: temperature profiles for different values of da. 0 1 2 3 4 0 0.2 0.4 0.6 0.8 1 gr = 10, gm = 4, pr = 0.71, sc = 0.22, m = 0.5, fw = 0.5, da = 0.5, re1 = 200 and fs1 = 1.0 curve sr du 1 2.0 0.03 2 0.5 0.12 3 0.1 0.60 η θ 0 1 2 3 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 gr = 10, gm = 4, m = 0.5, pr = 0.71, sc = 0.22, sr = 0.5, du = 0.12, fw = 0.5, re1= 200 and fs1= 1.0 η θ curve da 1 0.5 2 1.0 3 1.5 3. results and discussion: during the course of the discussion of the effects of various parameters on the flow field the following considerations are made: (i) the value of prandtl number pr is taken equal to 0.71, which corresponds, physically to air. 61 m. s. alam and m. m. rahman / journal of naval architecture and marine engineering 1(2005) 55-65 (ii) the value of schmidt number sc is chosen at 0.22, which represents hydrogen at approx. 250c and 1 atm. (iii) the values of dufour number du and soret number sr are chosen in such a way that their product is constant provided that the mean temperature tm is kept constant as well. fig. 9: concentration profiles for different values of gr and m. fig. 10: concemtration profiles for different values of gm and fw. 0 2 4 6 8 0 0.25 0.5 0.75 1 gm = 5, fw = 0.5, pr = 0.71, sc = 0.22, sr = 0.5, du = 0.12, da = 0.5, re1= 200 and fs1= 1.0 η φ curve gr m 1 4 0.4 2 4 1.2 3 10 0.4 0 1 2 3 4 5 6 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 η φ gr = 10, m = 0.5, pr = 0.71, sc = 0.22, sr = 0.5, du = 0.12, da = 0.5, re1= 200 and fs1= 1.0 curve gm fw 1 3 0.5 2 3 1.5 3 7 0.5 62 m. s. alam and m. m. rahman / journal of naval architecture and marine engineering 1(2005) 55-65 (iv) finally, the values of the local grashof number gr, local modified grashof number gm, suction parameter fw, magnetic field parameter m, local reynolds number re1, local darcy number da and local forchheimer number fs1 are chosen arbitrarily. fig. 11: concentration profiles for different values of sr and du. fig. 12: concentration profiles for different values of da. 0 1 2 3 4 5 0 0.2 0.4 0.6 0.8 1 gr = 10, gm = 4, pr = 0.71, sc = 0.22, m = 0.5, fw = 0.5, da = 0.5, re1 = 200 and fs1 = 1.0 η φ curve sr du 1 2.0 0.03 2 0.5 0.12 3 0.1 0.60 0 1 2 3 4 5 6 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 η φ gr = 10, gm = 4, m = 0.5, pr = 0.71, sc = 0.22, sr = 0.5, du = 0.12, fw = 0.5, re1= 200 and fs1= 1.0 curve da 1 0.5 2 1.0 3 1.5 63 m. s. alam and m. m. rahman / journal of naval architecture and marine engineering 1(2005) 55-65 under the above assumptions, results are shown in figs. 1-12 and in table 1. the effects of grashof number and magnetic field parameter on the velocity field are shown in fig.1. it is seen from this figure that the velocity decreases with the increase of magnetic field parameter while it increases with the increase of grashof number (or increase of free convection current). in fig.2 the effects of modified grashof number and suction parameter on the velocity field are shown. fig. 2 shows that the velocity increases when the concentration difference between the mean and free stream values increases, whereas it decreases with an increase of suction parameter indicating suction stabilizes the boundary layer growth. the influence of soret number sr and dufour number du on the velocity field are shown in fig. 3. quantitatively, when η = 1.0 and sr decreases from 2 to 0.5 (or du increases from 0.03 to 0.12), there is 5.48% decrease in the velocity. on the other hand, when sr decreases from 0.5 to 0.1, there is 3.42% increase in the velocity. the effect of darcy number da on the velocity field is shown in fig. 4. from this figure we observe that velocity increases with the increase of darcy number. for large darcy number porosity of the medium increases, hence fluid flows quickly. the effects of grashof number and magnetic field parameter on the temperature field are shown in fig. 5. from this figure we observe that the temperature increases with an increase of magnetic field parameter and decreases with an increase of free convection current. the effects of modified grashof number and suction parameter on the temperature profiles are shown in fig.6. this figure shows that the temperature decreases with the increase of both suction parameter and modified grashof number. from fig. 7 when 0.1=η and sr decreases from 2 to 0.5 (or du increases from 0.03 to 0.12), there is 6.19% increase in the temperature, whereas the corresponding increase is 16.97% when sr decreases from 0.5 to 0.1. the effect of darcy number da on the temperature field is shown in fig. 8. from this figure we observe that temperature decreases with the increase of darcy number. as the fluid velocity increases with the increase of the darcy number, consequently temperature surrounding the plate decreases. table 1: numerical values of skin-friction coefficient (cf), nusselt number (nu) and sherwood number (sh) for gr = 10, gm = 4, pr = 0.71, sc = 0.22, fw = 0.5, m = 0.5, da = 0.5, re1 = 200 and fs1 = 1.0 sr du cf nu sh 2.0 0.03 3.4231141 1.0283189 0.1296854 1.0 0.06 3.3457474 1.0155338 0.2992750 0.5 0.12 3.3162482 1.0019868 0.3844602 0.4 0.15 3.3141130 0.9965224 0.4017999 0.2 0.30 3.3287043 0.9718535 0.4381199 0.1 0.60 3.3828661 0.9248360 0.4602605 the effects of grashof number and magnetic field parameter on the concentration field are shown in fig. 9. from this figure we observe that the concentration increases with an increase of magnetic field parameter and decreases with an increase of free convection current. the effects of modified grashof number and suction parameter on the concentration profiles are shown in fig. 10. this figure shows that the concentration decreases with the increase of suction parameter as well as the modified grashof number. in fig. 11 when η = 1.0 and sr decreases from 2 to 0.5 (or du increases from 0.03 to 0.12), there is 23.11% decrease in the concentration, whereas the corresponding decrease is 7.91% when sr decreases from 0.5 to 0.1. the effect of darcy number da on the concentration field is shown in fig. 12. this figure reveals that concentration of the fluid within the boundary layer decreases with the increase of darcy number. 64 m. s. alam and m. m. rahman / journal of naval architecture and marine engineering 1(2005) 55-65 finally, the effects of soret and dufour numbers on the skin-friction coefficient, nusselt number and sherwood number are shown in table 1. the behaviour of these parameters is self-evident from the table 1 and hence they will not discuss any further due to brevity. 4. conclusions in this paper we have studied numerically the dufour and soret effects on a steady mhd free convention and mass transfer flow past a semi-infinite vertical plate embedded in a porous medium. from the present study the following conclusions can be drawn: • the velocity profiles decrease whereas temperature and concentration profiles increase with an increase of magnetic field parameter. • the velocity profiles increase whereas temperature and concentration profiles decrease with an increase of free convection currents. • the suction stabilizes the boundary layer growth. • large darcy number (large porosity of the medium) leads to the increase of the velocity and decrease of the temperature as well as concentration of the fluid within the boundary layer. • for fluids with medium molecular weight (h2, air), dufour and soret effects should not be neglected. references anghel, m., takhar, h. s. and pop, i. (2000): dufour and soret effects on free-convection boundary layer over a vertical surface embedded in a porous medium, studia universitatis babes-bolyai, mathematica vol. xlv, pp. 11-21. cramer, k. r. and pai, s. i. (1973): magnetofluid dynamics for engineers and applied physicists, mcgraw-hill co., new york. dursunkaya. z. and worek, w. m. (1992): diffusion-thermo and thermal-diffusion effects in transient and steady natural convection from vertical surface, int. j. heat mass transfer, vol. 35, pp. 20602065. eckert, e. r. g. and drake, r. m. (1972): analysis of heat and mass transfer, mcgraw-hill, new york. gupta, a. s. (1961): steady and transient free convection of an electrically conducting fluid from a vertical plate in the presence of magnetic field, appl. sci. res., vol. 9a, pp. 319-333. lykoudis, p. s. (1962): natural convection of an electrically conducting fluid in the presence of a magnetic field, int. j. heat mass transfer, vol. 5, pp. 23-34. nanda, r. s. and mohanty, h. k. (1970): hydromagnetic free convection for high and low prandtl numbers, j. phys. soc. japan, vol. 29, pp. 1608-1618. nachtsheim, p. r. and swigert, p. (1965): satisfaction of the asymptotic boundary conditions in numerical solution of the system of nonlinear equations of boundary layer type, nasa tnd-3004. postelnicu, a. (2004): influence of a magnetic field on heat and mass transfer by natural convection from vertical surfaces in porous media considering soret and dufour effects, int. j. heat mass transfer, vol. 47, pp. 1467-1472. raptis, a. and kafoussias, n. g. (1982): magnetohydrodynamic free convection flow and mass transfer through porous medium bounded by an infinite vertical porous plate with constant heat flux, can. j. phys. vol. 60, pp. 1725-1729. sattar, m. a., rahman, m. m. and samad, m. a. (2001): similar solutions of an mhd free convection and mass transfer flow with viscous dissipation, dhaka univ. j. sci., vol. 49, pp. 67-78. 65 sc sh sr t du gr gm k kt m nu pr re1 results and discussion: sr du cf nu sh tunnel effect on the resistance of high-speed journal of naval architecture and marine engineering june, 2005 http://jname.8m.net 1813-8535 © 2005 aname publication. all rights reserved. effect of tunnel on the resistance of high-speed planing craft v. anantha subramanian1 and p.v.v. subramanyam2 1associate professor, department of ocean engineering, indian institute of technology madras, chennai60036, india. ph. +91-44-22574812, email: subru@iitm.ac.in 2research scholar, department of ocean engineering, indian institute of technology madras, chennai60036, india. email: subbupvv@rediffmail.com abstract tunnels are provided in ship hulls to accommodate propellers under reduced draught conditions, thereby avoiding reduction of propeller diameter and consequent loss of efficiency. in this work the hydrodynamic effect of propeller tunnels in high speed craft, by way of modified resistance and pressure distribution, are studied both numerically using cfd, and experimentally using geometrically scaled models. the experimental study has been conducted on the model of a single hard chine hull form designed for specific length to displacement ratio. the parametric study considers two other draught conditions. in order to bring out pressure influences, cfd based simulations have been carried out. it is noted that the pressure distributions are altered around the tunnel region and, for an investigated case of tunnel area ratio, there is consistent reduction of resistance for all the three draught conditions tested. the study also compares the merits of the modified froude extrapolation method, after correction for flow velocity due to pressure development in the hull zone, with the classical savitsky’s method for planing hulls. the qualitative aspects with respect to pressure distribution are brought out in the cfd based studies and the pressure predictions do show consistency with obtained experimental data. keywords: high speed planing craft, tunnels, resistance, extrapolation methods, pressure distribution, cfd studies. nomenclature: k turbulent kinetic energy ε turbulent energy dissipation rate l length on waterline t draught lcg longitudinal centre of gravity cg centre of gravity ap projected area of the planing hull at projected area of the tunnels cf schoenherr turbulent friction coefficient τ trim angle v1 average bottom velocity λ wetted length to beam ratio ∇ volume of displacement cv coefficient of velocity fn froude number fn∇ volume based froude number ρ mass density of water t propeller thrust ∆ displacement vcg vertical cenre of gravity β deadrise angle bt beam at transom v forward speed d draft of keel at transom b maximum beam at chine cp centre of pressure df viscous component of drag d total drag a distance between df and cg f distance between t and cg c distance between n and cg lbp length between perpendiculars n resultant of pressure forces acting normal to bottom rtm total model resistance b buoyancy of the model dm drag of the model lm lift force acting on the mode pd maximum bottom pressure p1 p 14 pressure transducer locations from 1 to 14 wm displacement of the model x distance of pressure tapping location measured from transom rn reynolds number mailto:subru@iitm.ac.in mailto:subbupvv@rediffmail.com v. a. subramanian, p.v.v. subramanyam / journal of naval architecture and marine engineering 1(2005) 1-14 2 1. introduction planing crafts are high-speed marine vehicles, with applications ranging from small pleasure boats to large military crafts. generally, in a properly configured planing hull form, the deadrise angle diminishes from bow towards stern. high-speed planing crafts have hard chine, and may have both longitudinal and transverse steps at intermediate positions over the wetted region. the planing craft is typically run with a small bow-up trim or attack angle. because of the constant deadrise angle at the aft, planing crafts often have constraint of space for accommodating propellers. a solution to this problem is to provide propellers on inclined shafts. another alternative is to provide tunnels (also called "propeller pockets") at the bottom of planing hulls. the enhancement achieved by using a partial tunnel includes reducing the shaft angle, decreasing navigational draft and allowing the propulsion machinery to be moved aft for an appropriate longitudinal centre of gravity location with improved arrangement of machinery space. by using tunnels, reduction of propeller diameter can be avoided. therefore the provision of tunnels gives the designer freedom not to reduce propeller diameter and therefore efficiency. the question is how beneficial are tunnels, and if so, is there any trade offs in terms of other characteristics. in more recent years computational fluid dynamics (cfd) codes have been applied to modelling ship flows. the increase is due to advances in computational methods together with the increase in performance and affordability of computers. the increased use of cfd has established commercial cfd as credible design tools for solving practical flow problems such as the highly complex problem of flow past ship hulls. today cfd does give qualitative information to decide the relative merits such as flow alteration in and around the ship hull due to the geometry of the tunnel. 2. motivation since the early 1960’s several different planing hull forms have been systematically investigated for obtaining total resistance. blount and clement (1963) presented a simplified prediction method for the estimation of planing hull resistance. savitsky (1964) presented a performance prediction method using the empirical equations for lift, drag, wetted area and centre of pressure. the method is still used as a first estimate method for planing hull resistance. harbaugh and blount (1973) presented model resistance and self propulsion data from experiments modified for shallow and deep tunnels and with propellers of different diameters. they observed that the deep tunnelled hull in combination with propellers of large diameter and the smallest permissible tip clearance compare well performance-wise to the hull with no tunnels. koelbel (1979) studied the effect of tunnels and observed the changes in drag and propeller performance. blount (1997) has given guidelines for the design of partial propeller tunnels and relative placement of propellers to achieve exceptional vessel performance. experimental and cfd studies have been carried out by thornhill et al (2003) to measure the drag as well as pressure distribution on the planing vessel at steady speed through calm water. the lack of rigorous qualitative analysis of flow, pressure and resistance effects due to the presence of tunnels is the major motivation for the present study. 3. methodology the major difference in the method of extrapolation of resistance from model to prototype for planing hulls and displacement hulls, is essentially due to the different velocity conditions near the hull. in a planing hull, due to the dynamic lift condition, there is alteration of trim as well as draught in running condition. savitsky (1964) proposed the use of average bottom velocity instead of the free stream velocity in the calculations for frictional resistance component. the scheme predicted the performance of a prismatic planing hull based on the empirical equations for lift, drag, wetted area, centre of pressure, and porpoising limits as a function of speed, trim angle, deadrise angle and loading. it is an iterative method based on choosing trim angles, which are then used in empirical equations to obtain v. a. subramanian, p.v.v. subramanyam / journal of naval architecture and marine engineering 1(2005) 1-14 3 values for lift and equilibrium moment in trim. the method progresses till equilibrium of body forces are obtained. the forces acing on the planing hull is shown in fig. 1. the planing hull is said to be in equilibrium when it satisfies the following equation tan cos fdd τ τ = ∆ + 1.1 2 12 2 0.0120 cos 2 cos d v p b τ ρ λ τ λ τ ∆ = = 1 2 1 2 2 1 d p v v vρ ⎛ ⎞ = −⎜ ⎟ ⎝ ⎠ ( )10 0.242 log f f rn c× (1) ( ) ( ) 1 sin sin sin 0 co s f c f d a f τ τ ε τ τ − +⎡ ⎤⎪ ⎪⎣ ⎦⎧ ⎫∆ − + − =⎨ ⎬ ⎪ ⎪⎩ ⎭ c = (2) fig. 1 forces acting on the planing hull where (3) (4) it may be noted that the above scheme can only be iteratively used, ensuring that the first equation is satisfied by iterative choice of values of trim angle. once the correct trim angle is obtained, the hydrodynamic drag is obtained from 3.1 the modified froude extrapolation (mfe) method this method is used for planing craft for extrapolating experimental data from model to obtain prototype values. the lift lm, is calculated on the basis of the following formula based on resolution of lift, drag, weight and buoyancy forces sin tan tan m m t m w b l r τ τ τ ⎛ ⎞ = + −⎜ ⎟ ⎝ ⎠ the drag dm, is given by f df n v c τ ε a ∆ lcg t tan 2 b β 2 2 1 0.75 5.21 / 2.39p v c c λ = − + ( )4 tana vcg b β= −pc lcg c bλ= − ( )2 21 2 cos f f c v b d ρ λ τ = v. a. subramanian, p.v.v. subramanyam / journal of naval architecture and marine engineering 1(2005) 1-14 4 ( )sin co s t m m m r l d τ τ − = the resultant normal force n, is calculated as ( )c o s s i nm mn l dτ τ= − the maximum bottom pressure pd, is co s 2 d n p l b τ = × × the resultant velocity v1, using bernoulli’s equation, is obtained as 21 2 dpv v ρ = − the frictional resistance is calculated from the ittc ’57 correlation line using the above modified velocity and appropriate wetted length. hence in the modified extrapolation scheme, the modified velocity is associated with the calculation of ctm and the subsequent extrapolation to prototype values. 4. numerical studies 4.1 computational method for pressure and resistance the objective of the cfd simulations was to obtain pressure distributions. these were later validated by comparison with experimental data. the study was also extended to obtaining the modified pressures due to the existence of the tunnels in the aft region. cfd simulations can overcome the shortcomings of the experiments to measure the pressure distribution on the bottom of the hull. hence the simulations of the flow field around the high speed planing hull model with and without tunnels for l/∇1/3 = 6.5 were performed in order to find out the hull pressure distribution. for this purpose, the flow problem was solved by continuity and momentum equation around the body in the domain of interest. the velocity components are governed by momentum equations and this requires determination of pressure. the procedure is based on simple (semi-implicit method for pressure linked equations) algorithm. in principle, a pressure field is initially guessed and the corresponding velocity field is computed using momentum equation. on substituting the velocity field in the continuity equation, and based on conservation of mass, pressure correction is obtained by solving the continuity equation written in terms of pressure correction. using the corrected pressure, the above steps are repeated till the converged solution is obtained. the cfd software fluent version 6.0.20 is used for the present computations. the finite volume method is used for discretising the governing equations and employing the cartesian coordinate system for mapping the physical plane to the computational domain. the computational models chosen are shown in fig. 2. all cfd studies performed here are based on model scale in order to directly compare with model based results. the computations are carried out over a model speed range from 2.58 to 4.24m/s (froude no. 0.55 to 1.03). using a modelling pre-processor gambit 2.0, the geometry is generated with 3-d unstructured mesh with tetrahedral elements. unstructured grids have flexibility to match with the surface of the complex geometries of the hull surface. the fluid domain is shown in fig. 3 and the domain including the mesh with and without tunnel is shown in fig. 4. grid independence studies were carried out by increasing the degree of fineness of the mesh to obtain optimum computational grid for accuracy. assuming steady flow field, the computations apply measured velocity at the inlet and outlet as boundary conditions. symmetry v. a. subramanian, p.v.v. subramanyam / journal of naval architecture and marine engineering 1(2005) 1-14 5 boundary condition is applied at the central surface and a solid boundary condition with slip is enforced to the top of the domain and a solid wall with no slip condition is prescribed for the hull surface, side and bottom. (see fig. 5) a segregated solver is used with the standard k-ε turbulence model. convergence is obtained when the sum of the residual errors for the pressure, velocity and dissipation of kinetic energy is less than 10-4. fig. 2 planing hull models (a) without tunnel (b) with tunnel using pre-processor gambit 2.0 0.18m 0.11m 1.875 0.75m 1.5 1.4 1.0m model fig. 3 planing hull models flow domain 5. experimental investigation 5.1 choice of models and experiments a high speed planing hull form of single hard chine type and designed for speed of 35 knots (corresponding to froude number, fn =1.0 where fn = v/ √(gl) was selected. the form is characterized by a fairly constant deadrise angle over the after half of the vessel. the model was v. a. subramanian, p.v.v. subramanyam / journal of naval architecture and marine engineering 1(2005) 1-14 fabricated in glass-reinforced plastic (grp) to a scale of 1:20. the model was modified with special bottom inserts which could be removed or filled to represent tunnel shape or ‘no tunnel’ condition respectively. the details of the hull and tunnels are shown in tables1 and 2. 4c 4b 4a fig. 4 (a) domain with mesh and zoomed view near the aft region of the planing hull (b) without tunnel (c) with tunnel inlet – velocity inlet outlet – outflow sides & bottom wall (no slip) top – wall (allows slip) hull – wall (no slip) fig. 5 boundary conditions 6 v. a. subramanian, p.v.v. subramanyam / journal of naval architecture and marine engineering 1(2005) 1-14 7 table 1 main particulars of the prototype particulars prototype length overall, (loa), m 37.8 beam at transom (bt), m 7 beam max. at chine (b), m 7.1 depth (d), m 5.36 b/t 4.7 l/b 4.68 deadrise at transom (βt), deg. 14 deadrise at midsection (β), deg. 20 design speed , knots 35 model scale 20 volume of displacement (∇), m3 150/142* length on waterline, (l), m 34.4/34.4 wetted surface area, (s), m2 192/193* lcg from transom, m 12.88/13.5* * values with and without tunnels at 1.5m draught the body plan view is shown in fig. 6. fig. 7 shows the photograph of the model with tunnels. the tests were performed at three different draught conditions (l/∇1/3 ratios of 6.8, 6.5 and 6.0 corresponding to draught of 1.3m, 1.5m and 1.7m) to firmly establish the trend of drag component with and without tunnel influence. the model was designed with a full featured tunnel with at/ap = 0.12 (i.e., tunnel area ratio which is defined as projected area of tunnels to projected water plane area of hull). the towing carriage fixture permitted the natural heave and trim of the model during steady speed runs. fig. 7 photograph of planing hull model fig. 6 bodyplan of planing hull model table 2 particulars of the tunnel particulars of tunnel prototype propeller immersion, % 33 at/ap 0.12 lt/lp 0.319 projected area, m2 24.4 length, m 11 width at transom, m 1.3 width near propeller region, m 1.46 depth near propeller region, m 0.48 v. a. subramanian, p.v.v. subramanyam / journal of naval architecture and marine engineering 1(2005) 1-14 8 5.2 test facility and set up the towing experiments were carried out in a tank of dimensions 82m × 3.2m × 2.8m at iit madras, india. the maximum carriage speed is 5m/s. drag measurements were made using an electronic dynamometer and the trim and c.g. rise were recorded using fore and aft trim indicators. for pressure measurements, 0.2 bar capacity strain gauge based underwater pressure transducers were used. carrier frequency amplifiers combined with data logger (hp bench link) were used for data acquisition. the test setup for the resistance measurement and pressure measurements is shown in fig. 8. for the pressure measurement, the tapping locations are shown in fig. 9. planing hull model fig. 8 experimental set-up of planing hull model for resistance and pressure measurement tests fig. 9 location of pressure tappings in planing hull model 5.3 experimental procedure and extrapolation to prototype the planing hull model was towed with constant forward speed in calm water. the model was run at different speeds ranging from model speed of 2.30m/s (prototype speed = 20knots; fn = 0.55) to 4.24m/s (prototype speed = 36.84 knots; fn = 1.03). the tests were conducted for different l/ ∇1/3 of each model with and without tunnels and for the above speed range. the model is free to heave and trim naturally during the steady speed measurement phase. while conducting resistance tests on the planing hull, the dynamic condition water line reading at forward and aft was noted by means of a calibration grid on the side of the model. afterwards, the wetted surface corresponding to each speed in the entire range of speeds was interpolated between the low speed and the high speed dynamic trim condition waterlines in the tests. model speed for all runs was selected on the basis of equivalent v. a. subramanian, p.v.v. subramanyam / journal of naval architecture and marine engineering 1(2005) 1-14 9 froude number identity with prototype. for the total pressure acting on the bottom of the flat bottom hull model, pressures were measured at a point of 0.25b (where b is the breadth of the model) from the centre line of the model. typical sampling interval of 10 milliseconds was used for acquiring data from pressure transducers. the modified froude’s extrapolation method has been used for extrapolating the model planing hull resistance to prototype values. the main difference arises due to the fact that the velocity for frictional resistance assessment is different from the free stream velocity because of the associated pressure build up in the vicinity of the hull. also in the case of planing hulls, the dynamic wetted surface during planing must be recorded carefully. therefore the component of viscous resistance is obtained using the modified velocity and the residuary resistance term is then extrapolated from model scale to prototype values. these values have been compared with savitsky’s scheme. 6. results and discussion 6.1 resistance results for a particular draught condition (t=1.5m, l/∇1/3 = 6.5) the measured trim and c.g. changes of the model due to the effect of the tunnel are brought out in figs. 10 and 11. at full planing speed (fn∇ > 1.5) the trim is more for the case of the vessel without tunnel. it is obvious that with the present tunnel (at/ap = 0.12) the flow for the aft is favourably modified to give the ship a reduced (favourable) trim condition. similarly the centre of gravity rise is reduced in the case of the vessel with tunnel. there is a characteristic drop of centre of gravity in both the cases (i.e. with and without tunnel) at pre-planing speeds. -0.04 -0.03 -0.02 -0.01 0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.5 1.0 1.5 2.0 2.5 3.0 with tunnel without tunnel l/∇1/3 = 6.5 (t = 1.5m) fig. 11 cg rise with respect to speed c g r is e/ ∇ 1/ 3 fn∇ the non-dimensionalized resistance plots are shown in figs. 12 to 14. both the methods of analysis have been used viz., the modified froude’s method and the savitsky’s method. irrespective of the method used, the resistance with tunnel has always shown reduced value. this trend is in conformity with the beneficial effects as seen in both favourable trim as well as c.g. changes in the cases with and without tunnels. the comparison of resistance due to the influence of the tunnel is brought out in fig. 12. to rule out the possibility of erroneous conclusion, the experiments were repeated for the cases of without tunnel and with tunnels for three different draughts (l/∇1/3 = 6.8, 6.5, 6.0; t = 1.3m, 1.5m, 1.7m) and the curves presented consistently show improved resistance due to the presence of tunnel at all 3 draught conditions. the results in figs. 13 and 14 shows that the resistance prediction by savitsky scheme gives very close values with the experimental results. comparisons are reasonable up to fn∇ = 1.9. 0.0 5 0 5 0 5 0 5 0 0.5 1.0 1.5 2.0 2.5 3.0 1. 1. 2. 2. 3. 3. 4. t ri m b y st er n (d eg .) l/∇1/3 = 6.5 (t = 1.5m) with tunnel without tunnel 0. fn∇ fig. 10 trim changes with respect to speed v. a. subramanian, p.v.v. subramanyam / journal of naval architecture and marine engineering 1(2005) 1-14 10 0.00 0.02 0.04 06 08 0.10 0.12 0.14 1.0 1.2 1.4 1.6 1.8 2.0 2.2 2.4 2.6 2.8 3.0 0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14 1.0 1.2 1.4 1.6 1.8 2.0 2.2 2.4 2.6 2.8 3.0 with tunnel without tunnel fn∇ l/∇1/3 = 6.5 (t = 1.5m) l/∇1/3 = 6.8 (t = 1.3m) fn∇ with tunnel without tunnel 0. 0. r /∆ 0.00 0.02 0.04 06 08 0.10 0.12 0.14 1.0 1.2 1.4 1.6 1.8 2.0 2.2 2.4 2.6 2.8 3.0 l/∇1/3 = 6.0 (t = 1.7m) fn∇ with tunnel without tunnel 0. 0. r /∆ fig. 12 resistance with and without tunnels at different draughts (modified froude extrapolation) l/∇1/3 = 6.5 (t = 1.5m) l/∇1/3 = 6.5 (t = 1.5m) 0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14 l/∇1/3 = 6.5 (t = 1.5m) 1.0 1.2 1.4 1.6 1.8 2.0 2.2 2.4 2.6 2.8 3.0 mfe by exp. savitsky (1964) 0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14 l/∇1/3 = 6.5 (t = 1.5m) mfe by exp. savitsky (1964) 1.0 1.2 1.4 1.6 1.8 2.0 2.2 2.4 2.6 2.8 3.0 r /∆ fn∇ fn∇ it is very evident that the tunnel helps improve resistance. in this particular case the tunnel has been designed with smooth streamlined forward entry. the tunnel depth increases gradually and linearly towards the aft. hence discontinuity due to tunnel is minimal. fig. 14 resistance with tunnel fig. 13 resistance without tunnel 6.2 pressure measurements and comparison the objective of pressure measurements was to obtain the distribution of pressure in the planing condition and to validate cfd based measurements by comparison. for this purpose the pressure measurements were confined to the case without tunnel. pressure plots are shown in figs. 15 and 16. each curve represents the pressure measurement at a particular location. the pressures measured at the forward most point of contact with water i.e., the spray root region, shows the highest growth of pressure with speed. this pressure vs. velocity curve has the highest gradient. pressure at mid aft region is consistently high and the pressure vs. velocity gradient is mild. pressures measured at other points lie between the bandwidth of these points. after the transition to full planing mode, the v. a. subramanian, p.v.v. subramanyam / journal of naval architecture and marine engineering 1(2005) 1-14 11 pressures are concentrated maximum at the spray root region and are high at the mid-aft region. further towards aft, the pressures diminish. fig. 16 non-dimensional pressure along the length of the planing hull model without tunnel 0.0 2.0 4.0 6.0 8.0 10.0 12.0 14.0 -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 fn = 0.56 fn = 0.63 fn = 0.68 fn = 0.74 fn = 0.80 fn = 0.84 fn = 0.91 fn = 0.96 fn = 1.03 t ot al p re ss ur e/ st at ic p re ss ur e x/lbp fig. 15 total pressure for various model speeds at different locations t ot al p re ss ur e (p a) model speed (m/s) 0 200 400 600 800 1000 1200 1400 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 p1 p2 p3 p4 p5 p6 p13 p14 mid-aft forward most pressure the spatial pressure distribution is shown in fig. 16. this brings out the consistent concentration of pressure at the spray root region and immediately near it. further behind this point, the pressures are nearly constant and much less. the cfd based contours of bottom pressure distributions are shown in figs. 17 and 18. it may be noted that by the choice of coarser pressure range, the contour plots would depict a simpler pressure distribution pattern in the projected hull area with the characteristic fall of pressure from the spray root region at the bow towards the aft. the pressure re-distribution due to the tunnel effect is shown in fig. 19. the pressure patterns are thus brought out. the cfd based assessment of resistance in the two cases viz., with and without tunnel are shown in fig. 20. from the experimental study it is established that the re-distribution of pressure results in a favourable trim of the hull resulting in reduced resistance for the case of, with tunnel. the experimentally measured pressures are compared with corresponding values obtained from cfd in fig. 21. v. a. subramanian, p.v.v. subramanyam / journal of naval architecture and marine engineering 1(2005) 1-14 12 model speed = 4.24m/s fn = 1.03 fn∇ = 2.60 fig. 17 contours of total pressure (pa) of planing hull model without tunnel model speed = 4.24m/s fn = 1.03 fn∇ = 2.60 fig. 18 contours of total pressure (pa) of planing hull model with tunnel 7. conclusion the numerical and experimental investigation on resistance and pressure measurement on planing hulls establish that modified froude extrapolation results show reasonable match with savitsky based scheme at the speed range up to fn∇ = 1.9. at higher speeds, the divergence of values is higher. there may be a limitation due to the fact that the savitsky method is generally applicable for prismatic hulls. the present hull has a degree of warp. the conventional modified velocity froude extrapolation based results show closer match with the savitsky based scheme at lower speeds. the full featured tunnel v. a. subramanian, p.v.v. subramanyam / journal of naval architecture and marine engineering 1(2005) 1-14 13 offers beneficial effect in term of reduced resistance. the cfd simulation of flow with tunnel shows that there is a re-distribution of pressure besides reduction, due to the presence of tunnels. in the tunnel area, there are locations with reduced pressures. pressure integration in the horizontal direction gives the total drag and the reduction in drag due to the presence of tunnels is re-confirmed in the cfd based studies. the pressure measurements bring out the nature of pressure variation in the underwater hull, with a marked peak at the spray root region. the study directly contributes to the preliminary planing hull design process. without tunnel 302 760 394 302 302 211 120 485-245 28 577 -62.8 66839428.6 302 211 760 211 485 851 with tunnel 668 760577 485 485 577211 -1800 302 -245 577 668 394394 302 851 760 485 952 302 fig. 19 contours of total pressure (pa) in the tunnel region of planing hull model for a model speed of 4.24m/s (ship speed = 36.84knots) references blount d.l. (1997) design of propeller tunnels of high-speed craft. fast’97, sydney, australia. blount d.l. and e.p. clement (1963) resistance tests of a systematic series of planing hull forms. sname transactions, 491-579. harbaugh, k. h. and d. l. blount (1973) an experimental study of a high performance tunnel hull craft, sname spring meeting, lake buena vista, florida. koelbel jr., j.g. (1979) tunnel hull designs for u.s. navy small craft, combatant craft engineering, naval ship engineering center, norfolk division. savitsky, d. (1964) hydrodynamic design of planing hulls, marine technology, 1(1), 71-95. thornhill, e., n. bose, b. veitch, and p. liu (2003) planing hull performance evaluation using a general purpose cfd code, proceedings of 24th symposium on naval hydrodynamics. v. a. subramanian, p.v.v. subramanyam / journal of naval architecture and marine engineering 1(2005) 1-14 14 0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14 0.16 0.18 0.50 0.55 0.60 0.65 0.70 0.75 0.80 without tunnel with tunnel l/∇1/3 = 6.5 r /∆ fn fig. 20 cfd based comparison of resistance for planing hull with and without tunnel 0 200 400 600 800 1000 1200 1400 -1 -0.75 -0.5 -0.25 0 0.25 0.5 0.75 experimental k-epsilon model (cfd) v = 4.24m/s fn =1. 03 fn∇ = 2.60 t ot al p re ss ur e (p a) x/lbp fig. 21 comparison of total pressure along the length of the planing hull model for without tunnel case for l/∇1/3=6.50 (25% breadth from centre line) journal of naval architecture and marine engineering june, 2016 http://dx.doi.org/10.3329/jname.v13i1.20773 http://www.banglajol.info 1813-8235 (print), 2070-8998 (online) © 2016 aname publication. all rights reserved. received on: oct. 2014 thermal diffusion and chemical reaction effects on unsteady flow past a vertical porous plate with heat source dependent in slip flow regime siva kumar narasu 1 , b. rushi kumar 2* , a. g. vijaya kumar 3 1,2,3 fluid dynamics division, school of advanced sciences, vit university, vellore, t.n, india – 632014, * email: rushibkumar@gmail.com abstract: the present study investigates an analytical solution of free convective unsteady fluid flow in the presence of thermal diffusion and chemical reaction past a vertical porous plate with heat source dependent in slip flow regime. the plate is assumed to move with a constant velocity in the direction of fluid flow while free stream velocity is assumed to follow exponentially increasing small perturbation law. the velocity, temperature, and concentration profiles are presented graphically for different values of the parameters entering into the problem. finally, the effects of pertinent parameters on the skin friction coefficient, nusselt number, and sherwood numbers are studied and are shown through graphs and tables by using perturbation technique. it is observed in the case of air that the temperature decreases very rapidly whereas, in the case of liquid, the temperature decreases steadily. keywords:free convection, heat and mass transfer, slip flow regime, thermal diffusion, heat source, chemical reaction. nomenclature u  the component of velocity along x-axis (m/s) v  the component of velocity along y-axis (m/s) t time (s) g the acceleration due to gravity (ms -2 ) t  thermal expansion coefficient c  concentration expansion coefficient  kinematic viscosity(m 2 s -1 ) k  permeability of the porous medium non dimensional form s0 soret effect  density of the dusty fluid (kgm -3 ) q heat source cp specific heat at constant pressure (kj/kg.k) t  temperature in nondimensional form c  concentration in nondimensional form u0 suction velocity(m/s) d molecular diffusivity(m 2 /s) d1 thermal diffusivity (m 2 /s) c k  chemical reaction parameter in nondimension form gt thermal grashof number gm mass grashof number c k chemical reaction parameter k permeability of porous medium (m 2 ) pr prandtl number h heat source parameter(w/m 2 k) sc schmidt number x m maxwell’s reflection coefficient t   temperature of the fluid far away from the plate c   concentration of the fluid far away from the plate w t  temperature of the fluid at the plate w c  concentration of the fluid at the plate 1. introduction the problem of free convection flow involving the combined mechanism of heat and mass transfer are encountered in many natural processes, in many chemical processing systems, and in many industrial applications. the study of free convective heat and mass transfer flow has become the object of extensive research as the effects of heat transfer along with mass transfer are dominant features in many engineering mailto:rushibkumar@gmail.com s. k. narasu, b. r. kumar, a. g. v. kumar/journal of naval architecture and marine engineering 13(2016), 51-62 thermal diffusion and chemical reaction effects on unsteady flow past a vertical porous plate... 52 applications such as cooling of nuclear reactors, rocket nozzles, high-speedaircraft and their atmospheric reentry, chemical devices process equipment. the phenomenon of free convection arises in the fluid when the temperature changes cause density variation leading to buoyancy forces acting on the fluid elements. this can be seen in our everyday life in the atmospheric flow, which is driven by temperature differences. there are many transport process occurring in nature due to temperature and chemical differences. the process of heat and mass transfer is encountered in aeronautics, fluid fuel nuclear reactor, chemical process industries and many engineering applications in which the fluid is a working medium and chemical reaction can be codified either heterogeneous or homogeneous processes. its effect depends on the nature of the reaction whether the reaction is heterogeneous or homogeneous. a reaction is of the firstorder if the rate of reaction is directly proportional to concentration itself. in nature, the presence of pure air or water is not possible. some foreign mass may be present naturally mixed with air or water. the presence of a foreign mass in air or water causes some kind of chemical reaction. the study of such type of chemical reaction processes is useful for improving a number of chemical technologies, such as food processing, polymer production, manufacturing of ceramics and glassware. several authors have analyzed physical problems in this field. chambre and young (1958)analyzed the effect of homogeneous first order chemical reactions in the neighborhood of a flat plate for destructive and generative reactions. das et al. (1994) studied the effect of first order reaction on the flow past an impulsively started infinite vertical plate with uniform heat flux and mass transfer. alam and rahman (2005) analyzed dufour and soret effects on mhd free convective heat and mass transfer flow past a vertical porous plate embedded in a porous medium.anjali devi and kandasamy (2000) investigated the effect of chemical reaction on the flow in the presence of heat transfer and magnetic field. muthucumaraswamy and ganesan (2001) studied the effect of chemical reaction on flow past an impulsively started vertical plate with uniform heat and mass flux.a study on steady laminar free convection flow of an electrically conducting fluid along a porous vertical plate in the presence of heat source was carried out by sharma and mathur (1995). the effect of thermal diffusion on steady laminar free convective flow along a moving porous hot vertical plate in the presence of heat source with the mass transfer was studied by varshney and kumar (2004). heat transfer in mhd free convection flow over an infinite vertical plate with time-dependent suction was investigated in detail by mishra (2005). sharma and singh (2008) examined unsteady mhd free convective flow and heat transfer along a vertical porous plate with variable suction and internal heat generation. in many practical applications, the particle adjacent to a solid surface no longer takes the velocity of the surface. the particle at the surface has a finite tangential velocity; it slips along the surface. the flow regime is called the slip flow regime and this effect cannot be neglected. using these assumptions, sharma and chaudary (2003) discussed the free convection flow past a vertical plate in the slip-flow regime and also discussed its various applications for engineering purpose. also, sharma (2005) investigated the effect of periodic heat and mass transfer on the unsteady free convection flow past a vertical flat plate in the slip-flow regime when suction velocity oscillates in time. coupled non-linear partial differential equations governing free convection flow, heat, and mass transfer have been obtained analytically using the perturbation technique. the fluids considered in this investigation are air (pr = 0.71) and water (pr = 7) in the presence of hydrogen (sc = 0.22). magnetohydrodynamic convective heat and mass transfer in a boundary layer slip flow past a vertical permeable plate with thermal radiation and chemical reaction were investigated by pal and talukdar (2010). kumar and gangadhar (2012) have studied the heat generation effects on mhd boundary layer flow of a moving vertical plate with suction. das and mitra (2009) discussed the unsteady mixed convective mhd flow and mass transfer past an accelerated infinite vertical plate with suction. recently, das et al. (2009) analyzed the effect of mass transfer on mhd flow and heat transfer past a vertical porous plate through a porous medium under oscillatory suction and heat source. das et al. (2007) investigated numerically the unsteady free convective mhd flow past an accelerated vertical plate with suction and heat flux.dufour and soret effects on steady mhd free convection and transfer fluid flow through a porous medium was analyzed by mahmud and nazmul (2007). das et al. (2006) estimated the mass transfer effects on unsteady flow past an accelerated vertical porous plate with suction employing finite difference analysis. kumar et al.(2009) investigated effects of chemical reaction and mass transfer on mhd unsteady free convection flow past an infinite vertical plate with constant suction and s. k. narasu, b. r. kumar, a. g. v. kumar/journal of naval architecture and marine engineering 13(2016), 51-62 thermal diffusion and chemical reaction effects on unsteady flow past a vertical porous plate... 53 heat sink. kumar et al. (2015), have examined the effects of thermal diffusion and radiation effects on unsteady free convection flow in the presence of magnetic field fixed relative to the fluid or to the plate. hayat et al. (2010) analyzed a mathematical model in order to study the heat and mass transfer characteristics in mixed convection boundary layer flow over a linearly stretching vertical surface in a porous medium filled with a viscous – elastic fluid, by taking into account the diffusion thermal and thermal diffusion effects. recently, poonia and chaudhary (2010) studied an unsteady, two-dimensional, hydro-magnetic, laminar mixed convective boundary layer flow of an incompressible and electrically conducting fluid along an infinite vertical plate embedded in the porous medium with heat and mass transfer, by taking into account the effects of viscous dissipation.in further studies by saxena et al. (2009) investigates the unsteady two-dimensional magnetohydrodynamic heat and mass transfer free convection flow of an incompressible, viscous, electrically conducting polar fluid through a porous medium past a semi-infinite vertical porous moving plate in the presence of a transverse magnetic field with thermal diffusion and heat generation. the plate moves with a constant velocity in the longitudinal direction, and the free stream velocity follows an exponentially increasing or decreasing small perturbation law. kumar (2013) analyzed the mhd boundary layer flow of heat and mass transfer over a stretching sheet with slip effect. the objective of the present paper is to study an analytical solution of free convective thermal diffusion and chemical reaction effects on unsteady flow past a vertical porous plate with heat source dependent in slip flow regime. in obtaining the solution, the heat source term is taken into account in the energy equation and chemical reaction parameter, thermal diffusion parameter is taken into account in the concentration equation. to the best of author’s knowledge, this model has not been discussed in the literature. the permeability of the porous medium and suction velocity are considered to be as exponentially decreasing the function of time. 2. mathematical formulation we consider a two-dimensional unsteady free convection flow of an incompressible viscous fluid past an infinite vertical porous plate. in rectangular cartesian coordinate system, we take x-axis along the plate in the direction of flow and y-axis normal to it. further, the flow is considered in the presence of thermal diffusion and chemical reaction effect past a vertical porous plate with heat source dependent in slip flow regime. in the analysis for small velocity the viscous dissipation and darcy’s dissipation are neglected. it is assumed that radioactive heat transfer is negligible in the energy equation, the fluid has small electric conductivity and the electromagnetic force is very small. fig. a: physical model and coordinate system x * 0, , w w u t t c c         u g boundary layer v porous medium y * 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 , w w u u l y t t c c                 s. k. narasu, b. r. kumar, a. g. v. kumar/journal of naval architecture and marine engineering 13(2016), 51-62 thermal diffusion and chemical reaction effects on unsteady flow past a vertical porous plate... 54 the flow in the medium is entirely due to buoyancy force caused by temperature difference between the porous plate and the fluid. under the above assumptions, the conservation of mass, momentum, energy, concentration is given bythe following relations, respectively (rao et al., 2013). (1)     2 2 ( ) t c u u u v g t t g c c u t y k ty                               (2) (3) (4) the boundary conditions relevant to the problem are (5) where the equation of continuity (1) yields that v * is either a constant or some function of time, hence we assume that (6) the negative sign indicates that the suction velocity acts towards the plate. the permeability of the porous medium in non-dimensional form is considered as (7) now, we introduce the following non-dimensional quantities introducing the equations (6), (7) in the equations (2), (3), (4), we obtain (8) (9) (10) 0 v y      2 2 t p t t t q t v k t y c yy                     2 2 2 2 1c c c c t v d k c c d t y y y                          1 , , 0 0, , w w w w u u l t t c c at y y u t t c c as y                             x x m m l   2 1 0 (1 ) n t v u e        0 ( ) (1 ) n t k t k e        2 2 2 0 0 0 0 2 2 0 0 4 , , , t , , , 4 w w y u u t k ut t c cu n u y n t k c u u t t c c                               2 12 1 (1 ) 4 nt t m u u u e g t g c m u t y y              2 2 1 1 (1 ) 4 nt r t t t t e h t y p t y              2 2 02 2 1 1 (1 ) 4 nt c c c c c t e k c s t y s t y               h0 o l g t interface tb insulated b v u  , 0t x y h0 x s. k. narasu, b. r. kumar, a. g. v. kumar/journal of naval architecture and marine engineering 13(2016), 51-62 thermal diffusion and chemical reaction effects on unsteady flow past a vertical porous plate... 55 and the boundary conditions (5) reduce to (11) where 3. solution of the problem in order to solve the nonlinear partial differential equations, the above systems of partial differential equations (8), (9) and (10) are reduced to a system of ordinary differential equations in a dimensionless form. to obtain the solution, the following perturbation method, which is given by singh et al. (2006), is used for ‘ε’ << 1. the velocity, temperature and concentration are assumed in the following form: (12) substituting above expressions (12) in to the equations (8), (9), (10) and equating the coefficient of , (neglecting terms etc.,), we obtain the following set of ordinary differential equations (13) (14) (15) (16) (17) (18) here and the boundary conditions (11) reduce to (19) , 1 , 1 0 0 , 0 , 0 u u h t c at y y u t c as y               2 0 13 3 0 0 1 02 2 0 0 ( ) (c ) 1 , , , k(1 ) ( ) , , , , ( ) (c ) t w c w t m nt p c w r c c t p w w lu g t t g c h g t g c m u u u e c k d t tq p h s k s k c u t t d u c                                                           0 1 u( , ) ( ) ( ) e nt y t f y f y    0 1 t( , ) ( ) ( ) e nt y t g y g y    0 1 c( , ) ( ) ( ) e nt y t h y h y    0  1  2  0 0 1 0 0 0t m f f m f g g g h      1 1 2 1 1 1 0t m f f m f g g g h f        0 0 (1 ) 0 r g p h g    1 1 1 0 (1 ) 4 r r r np g p h g g p g       0 0 0 0 0c c c c h s h k s h s s g      1 1 1 0 1 0 4 c c c c c n h s h s k h s s g s h             12 4 m n m  0 0 1 1 0 1 0 1 0 1 0 1 0 1 , , 1 , 0 , 1 , 0 0 0, 0 , 0, 0, 0, 0 f hf f hf g g h h at y f f g g h h as y                 s. k. narasu, b. r. kumar, a. g. v. kumar/journal of naval architecture and marine engineering 13(2016), 51-62 thermal diffusion and chemical reaction effects on unsteady flow past a vertical porous plate... 56 the equations from (13) to (18) are 2 nd order linear differential equations with constant coefficients. the solutions of these paired equations under the corresponding boundary conditions (19) are (20) (21) (22) (23) (24) (25) introducing the equations (20) to (25) in the equations (12), we get the expressions for velocity, temperature and concentration as (27) (28) (29) skin-friction: the expression for the skin-friction ( ) at the plate is (30) nusselt number: the expression for nusselt number (nu) in terms of the rate of heat transfer is (31) sherwood number: the expression for sherwood number in terms of the rate of mass transfer is (32) 4. results and discussion the problem of thermal diffusion and chemical reaction effects on unsteady fluid flow past a vertical porous plate with heat source dependent in slip flow regime has been investigated and the analytical solution for velocity, temperature and concentration have been presented in the previous section. the skin friction 1m y o g e   1 2 1 1 ( ) m y m y g a e e     3 1 3 0 2 ( ) m y m y m y h e a e e       4 1 2 3 1 3 4 5 3 4 5 ( ) m y m y m y m y h a a a e a e a e a e            5 1 3 0 8 6 7 m y m y m y f a e a e a e       6 1 2 3 4 5 1 14 9 10 11 12 13 m y m y m y m y m y m y f a e a e a e a e a e a e                 5 31 6 3 51 2 4 8 6 7 14 9 10 11 12 13 u(y, t) m y m ym y m y m y m ym y m y m y nt a e a e a e a e a e a e a e a e a e e                  1 1 21t( , ) ( ) m y m y m y nt y t e a e e e            3 1 3 4 1 2 3 2 3 4 5 3 4 5 c( , ) ( ) ( ) m y m y m y m y m y m y m y nt y t e a e e a a a e a e a e a e e                    0 1 15 16 nt nt y o y o y o f fu e a a e y y y                                  0 1 1 17 nt nt u y o y o y o g gt n e m a e y y y                                  0 1 18 19 nt nt h y o y o y o h hc s e a a e y y y                                 s. k. narasu, b. r. kumar, a. g. v. kumar/journal of naval architecture and marine engineering 13(2016), 51-62 thermal diffusion and chemical reaction effects on unsteady flow past a vertical porous plate... 57 coefficient, heat transfer coefficient, and mass transfer coefficient are also found. from the available analytical solutions, the numerical values for the distributions of velocity, temperature, concentration, skin friction coefficient, nusselt number and sherwood number are calculated by fixing various values of the nondimensional parameters involved in the problem. in the present work, we have chosen t = 1, ε = 0.1, k = 1 and n=1 while the other non-dimensional parameters take various values. fig. 1: the effect of slip parameter h on velocity fig. 2: the effect of thermal grashoff number gt on velocity fig. 3: the effect of mass grashoff number gm on velocity fig. 4: the effect of porous medium k on velocity to assess the physical depth of the problem, the effects of various parameters like slip parameter h, thermal grashof number gt, permeability of porous medium k, heat source parameter h, chemical reaction parameter kc, modified grashof number gm, schmidt number sc, soret number s0 and prandtl number pron velocity distribution, temperature distribution, and concentration distribution are studied in figs. (1) to (9), while keeping the other parameters as constants. fig. (1) depicts the velocity profiles with the variations in h, it is observed that the significance of the velocity is high near the plate and thereafter it decreases and reaches to the stationary position at the other side of the plate. as expected, velocity increases with an increase in h. the effects of gt on velocity distribution are presented in fig. (2). from this figure, it is noticed that velocity increases as an increase in gt. in fig. (3) the effects of gm on velocity are shown. from this figure, it is noticed that velocity increases as gm increases. from fig. (4), it is observed that the velocity increases as permeability of porous medium k increases. in figs. (5) & (6) the temperature distribution increases as heat source parameter handprandtl number prdecreases respectively. in fig. (7), concentration increases with an increase in s0. in figs. (8) and (9) the concentration decreases as sc and kc increase respectively. to be realistic, the numerical values of prandtl 0 1 2 3 4 5 6 7 8 9 10 0 1 2 3 4 5 6 7 8 y u h=1,2,3 sc=0.22;s0=1;pr=0.71;kc=1;h=1;n=1;e=0.1; gt=5;gm=5;k=1;t=1; 0 1 2 3 4 5 6 7 8 9 10 0 1 2 3 4 5 6 7 8 9 10 y u gt=5,10,15 sc=0.22,s0=1,pr=0.71,kc=1,h=1,n=1,e=0.1, gm=5,k=3,h=1,t=1. 0 1 2 3 4 5 6 7 8 9 10 0 2 4 6 8 10 12 14 16 18 sc=0.16,s0=1,pr=0.71,kc=1,h=1,n=1, e=0.1,gt=5,k=5,h=1,t=1. gm=5,10,15 0 1 2 3 4 5 6 7 8 9 10 0 2 4 6 8 10 12 14 y u k=1,2,3 sc=0.22,s0=1,pr=0.71,kc=1,h=1,n=1, e=0.1,gt=5,gm=5,h=1,t=1. s. k. narasu, b. r. kumar, a. g. v. kumar/journal of naval architecture and marine engineering 13(2016), 51-62 thermal diffusion and chemical reaction effects on unsteady flow past a vertical porous plate... 58 number pr are chosen as pr=0.71 and pr=7, which corresponds to air and water at 20°c respectively.in heat transfer problems, the prandtl number controls the relative thickness of the momentum and thermal boundary layers. when pr is small, it means that the heat diffuses quickly compared to the velocity. this means that for liquid metals the thickness of the thermal boundary layer is much bigger than the velocity boundary layer. the numerical values of the remaining parameters are chosen arbitrarily. fig. 5: the effects of heat source parameter h on temperature fig. 6: the effects of prandtl number pr on temperature fig. 7: the effects of s0 on concentration fig. 8: the effects of chemical reaction parameter kc on concentration fig. 9: the effects of schmidt number sc on concentration 0 1 2 3 4 5 6 7 8 9 10 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 y t h=1,2,3 sc=0.22,s0=1,pr=0.71,kc=1,n=1, e=0.1,gt=5,gm=5,k=1,h=1,t=1. 0 1 2 3 4 5 6 7 8 9 10 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 y t pr=0.71,7.0 sc=0.22,s0=1,kc=1;h=1,n=1,e=0.1, gt=5,gm=5,k=1,h=1,t=1. 0 1 2 3 4 5 6 7 8 9 10 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 y c s0=1,2,3 sc=0.22,pr=0.71,kc=1,h=1,n=1, e=0.1,gt=5,gm=5,k=3,h=1,t=1. 0 1 2 3 4 5 6 7 8 9 10 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 y c kc =1,2,3 sc=0.22,s0=1,pr=0.71,h=1,n=1,e=0.1, gt=5,gm=5,k=1,h=1,t=1. 0 1 2 3 4 5 6 7 8 9 10 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 y c sc=0.22,0.30,0.60. s0=1,pr=0.71,kc=1,h=1,n=1,e=0.1, gt=5,gm=5,k=1,h=1,t=1. s. k. narasu, b. r. kumar, a. g. v. kumar/journal of naval architecture and marine engineering 13(2016), 51-62 thermal diffusion and chemical reaction effects on unsteady flow past a vertical porous plate... 59 the variations in skin friction, nusselt number, and sherwood number are studied at n=1, t=1 and 0.1  through the tables (1) to (3). table 1: skin friction at the plate whenpr=0.71, n=1, t=1 and ε=0.1 gt gm h k  5 5 1 1 3.1537 10 5 1 1 1.0803 15 5 1 1 5.0068 5 10 1 1 5.3809 5 15 1 1 7.6080 5 5 3 1 1.4073 5 5 1 2 4.3995 5 5 1 3 5.1290 table 2: rate of heat transfer at the plate pr h nu 0.71 1 -1.4489 7.0 1 -14.2599 0.71 2 -2.1572 0.71 3 -2.8667 table 3: rate of mass transfer at the plate sc kc s0 sh 0.22 1 1 -0.3552 0.30 1 1 -0.4038 0.60 1 1 -0.5407 0.22 2 1 -0.5690 0.22 3 1 -0.7315 0.22 1 2 -0.1134 0.22 1 3 0.1284 5. conclusions analytical solutions are obtained for the unsteady flow past a vertical porous plate in slip flow regime in the presence of heat source, thermal diffusion, and chemical reaction. the dimensionless governing equations are solved by using perturbation technique. the conclusions are as follows:  the velocity profiles enhance for increasing the h, gt, gm and k.  the temperature profiles decrease with increasing h.  prandtl number strongly influences the temperature profiles.  the concentration profiles increase with increasing s0 while the trend reversed for kc and sc.  the results would be useful in many areas related to the diffusive operations which involve the molecular diffusion of species with chemical reaction. s. k. narasu, b. r. kumar, a. g. v. kumar/journal of naval architecture and marine engineering 13(2016), 51-62 thermal diffusion and chemical reaction effects on unsteady flow past a vertical porous plate... 60 acknowledgments the authors are thankful to the reviewers for their suggestions that significantly improved our paper. references alam, m.s.,and rahman, m. m. 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(2012):heat generation effects on mhd boundary layer flow of a moving vertical plate with suction, journal of naval architecture and marine engineering, vol. 9, no. 2, pp. 153-167. http://dx.doi.org/10.3329/jname.v9i2.8550 kumar, b. r. (2013): mhd boundary layer flow on heat and mass transfer over a stretching sheet with slip effect, journal of naval architecture and marine engineering,vol.10, no2, pp.16-26. http://dx.doi.org/10.3329/jname.v10i2.16400 kumar, b. r., kumar, t. s. and kumar, a. g. v. (2015): thermal diffusion and radiation effects on unsteady free convection flow in the presence of magnetic field fixed relative to the fluid or to the plate, frontiers in heat and mass transfer (fhmt), vol. 6, no. 1. http://dx.doi.org/10.5098/hmt.6.12 mishra, b. k. (2005): heat transfer in mhd free convection flow over an infinite vertical plate with timedependent suction,actacienciaindica, xxxi m no.2, pp.371. http://dx.doi.org/10.3329/jname.v4i1.915 http://dx.doi.org/10.1063/1.1724336 http://dx.doi.org/10.1007/bf02601318 http://dx.doi.org/10.1016/j.cnsns.2009.05.062 http://dx.doi.org/10.3329/jname.v9i2.8550 http://dx.doi.org/10.5098/hmt.6.12 s. k. narasu, b. r. kumar, a. g. v. kumar/journal of naval architecture and marine engineering 13(2016), 51-62 thermal diffusion and chemical reaction effects on unsteady flow past a vertical porous plate... 61 muthucumaraswamy, r and ganesan, p (2001): first order chemical reaction on flow past an impulsively started vertical plate with uniform heat and mass flux, acta mechanica, vol. 147, pp. 4557.http://dx.doi.org/10.1007/bf01182351 pal, d. and talukdar, b. 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(2009): unsteady mhd heat and mass transfer free convection flow of polar fluids past a vertical moving porous plate in a porous medium with heat generation and thermal diffusion, advances in applied science research, vol. 2, no. 4, pp.259-278. sharma, p. r. and mathur, p. (1995): steady laminar free convection flow of an electrically conducting fluid along a porous hot vertical plate in the presence of heat source/sink. indian j. pure appl. math., vol. 26, no.11, pp. 1125-1134. sharma p. r. and singh, g. (2008): unsteady mhd free convective flow and heat transfer along a vertical porous plate with variable suction and internal heat generation. int. j. of.appl. math and mech., vol. 4, no. 5, pp.1-8. sharma, p. k. and chaudary, r. c. (2003): effect of variable suction on transient free convection viscous incompressible flow past a vertical plate with periodic temperature variations in slip-flow regime, emirates j. engineering research, vol. 8, pp. 33-38. sharma, p. k. (2005): fluctuating thermal and mass diffusion on unsteady free convective flow past a vertical plate in slip-flow regime, latin american applied research, vol.35, pp. 313-319. singh, n. p., kumar, a., singh a. k. and singh, atul k.(2006): mhd free convection flow of viscous fluid past a porous vertical plate through non homogeneous porous medium with radiation and temperature gradient dependent heat source in slip flow regime, ultra science, vol.18, no. 1, m, pp.39-46. varshney, n. k. and kumar, s. (2004): effect of thermal diffusion on steady laminar free convective flow along a moving porous hot vertical plate in the presence of heat source with mass transfer, acta ciencia indica, xxx m no.2, pp.489-495. http://dx.doi.org/10.1007/bf01182351 http://dx.doi.org/10.1063/1.1724336 http://dx.doi.org/10.9790/4961-0362232 s. k. narasu, b. r. kumar, a. g. v. kumar/journal of naval architecture and marine engineering 13(2016), 51-62 thermal diffusion and chemical reaction effects on unsteady flow past a vertical porous plate... 62 appendix the constants involved in the solution of the problem are:       2 1 2 2 3 4 1 2 5 6 2 0 11 1 2 2 2 1 1 1 1 2 2 0 1 1 2 1 0 1 2 3 4 2 2 1 1 2 2 1 2 4 ( 4 ) 2 2 1 1 4 1 1 4 2 2 ( ) 4 4 r r r r r r c c c c c c c c cr r c c c r r c c c c c c c p p h p p h np m p h m s s k s s s s n k m m m m m m s s mp m a a np m s m k s m p p h m s s a m s a m s s a m a a n m s m s k m s m                                           2 3 2 5 6 2 2 1 1 1 3 3 6 1 7 32 7 82 3 3 1 5 1 3 6 1 4 9 102 2 1 1 2 2 2 2 7 3 5 3 4 11 122 3 3 2 4 (1 ) 4 (1 ) (1 )(1 ) (1 ) ( c c c t m c c c m t m t m m m n s k s a m g g a a a n m m m m s m s k a hm a hmg a a a m m m hm g a g a a g a g a a a m m m m m m a m g a g a a a a m m m                                               5 2 4 4 2 8 5 13 2 5 5 2 9 10 11 12 13 1 9 2 10 3 11 4 12 5 13 14 6 15 1 6 3 7 5 8 16 1 9 2 10 3 11 4 12 5 13 6 14 17 1 2 1 18 3 2 3 1 19 4 3 4 5 ) (1 ) ( ) ( ) ( ) a m m m a m a m m m a a a a a hm a hm a hm a hm a hm a a hm a m a m a m a a m a m a m a m a m a m a a a m m a m a m m a m a a a m                                        1 3 2 4 3 5 a m a m a  403 forbidden forbidden you don't have permission to access this resource. apache/2.4.54 (ubuntu) server at www.banglajol.info port 443 microsoft word 2654.docx journal of naval architecture and marine engineering june, 2009 doi: 10.3329/jname.v6i1.2654 http://www.banglajol.info 1813-8235 (print), 2070-8998 (online) © 2009 aname publication. all rights reserved. received on: june 2009 numerical study of magnetohydrodynamic free convective heat transfer flow along a vertical flat plate with temperature dependent thermal conductivity m. m. rahman1 and m. a. alim2 1department of computer science and engineering, dhaka international university, banani, dhaka -1213, bangladesh, email: mrrahmandiu@yahoo.com 2department of mathematics, bangladesh university of engineering and technology, dhaka-1000, bangladesh, email: maalim@math.buet.ac.bd abstract: the present numerical work describes the effect of the magnetohydrodynamic (mhd) free convective heat transfer flow along a vertical flat plate with temperature dependent thermal conductivity and heat conduction. the governing equations reduce to local non-similarity boundary layer equations using suitable transformation have been integrated by employing an implicit finite difference method together with the keller box technique. comparison with previously published work is performed and excellent agreement is observed. profiles of the dimensionless velocity and temperature distributions as well as the local skin friction coefficient and surface temperature distribution are shown graphically for various values of the magnetic parameter m, thermal conductivity variation parameter  and prandtl number pr. keywords: implicit finite difference method, free convection flow, vertical flat plate, temperature dependent thermal conductivity. nomenclature greek symbols b plate thickness vu , velocity components cx local skin friction coefficient u, v dimensionless velocity components cp specific heat at constant pressure yx, cartesian coordinates f dimensionless stream function x,y dimensionless cartesian coordinates g acceleration due to gravity  co-efficient of thermal expansion gr grashof number  vector differential operator h dimensionless temperature  dimensionless similarity variable 0 strength of the magnetic field  (x,0) surface temperature distribution l length of the plate f thermal conductivity of the fluid m magnetic parameter s thermal conductivity of the solid  thermal conductivity variation parameter  thermal conductivity of the ambient fluid p conjugate conduction parameter  ,  dynamic and kinematic viscosities pr prandtl number  density of the fluid t temperature of the interface  electrical conductivity tb temperature at outside surface of the plate w shearing stress tf temperature of the fluid  stream function  fluid asymptotic temperature m. m. rahman and m. a.alim/ journal of naval architecture and marine engineering 6(2009) 16-29 numerical study of magnetohydrodynamic free convective heat transfer flow along a vertical flat plate... 17 1. introduction the most common type of body force, which acts on a fluid is due to gravity so that the body force can be defined as in magnitude and direction by the acceleration due to gravity. the electric and magnetic fields themselves must obey a set of physical laws, which are expressed by maxwell’s equations. the solution of such problems requires the simultaneous solution of the equations of fluid mechanics and of electromagnetism. one special case of this type of coupling is the field known as mhd. the interaction of the magnetic field and the moving electric charge carried by the flowing fluid induces a force, which tends to oppose the fluid motion. and near the leading edge the velocity is very small so that the magnetic force, which is proportional to the magnitude of the longitudinal velocity and acts in the opposite direction, is also very small. consequently, the influence of the magnetic field on the boundary layer is exerted only through induced forces within the boundary layer itself, with no additional effects arising from the free stream pressure gradient. magnetohydrodynamic flow is an important research area due to its potential applications in engineering and industrial fields. magnetohydrodynamic power generators and accelerators, cooling of nuclear reactors and crystal growth are included in this area. accordingly, a considerable amount of research has been accomplished on the effects of electricallyconducting fluids such as liquid metals water mixed with a little acid and others in the presence of transverse magnetic field on the flow and heat transfer characteristics over various geometries. the effect of magnetic field on free convection heat transfer has studied by sparrow and cess (1961). kuiken (1970) studied the problem of mhd free convection in a strong cross field. hossain et al. (1990, 1997, and 1998) discussed the both forced and free convection boundary layer flow of an electrically conducting fluid in the presence of magnetic field. moreover, mhd free convection flow of visco-elastic fluid past an infinite porous plate was investigated by chowdhury and islam (2000). elbashbeshy (2000) also discussed the effect of free convection flow with variable viscosity and thermal diffusivity along a vertical plate in the presence of magnetic field. in all the above studies the effects of temperature dependent thermal conductivity has not been considered. but, thermal boundary layer in liquid metals with variable thermal conductivity studied by arunachalam and rajappa (1978). the combined convection from a vertical flat plate with temperature dependent viscosity and thermal conductivity was investigated by hossain et al. (2002). hossain et al. (2004) have considered the problem of the natural convection laminar flow with temperature dependent viscosity and thermal conductivity along a vertical wavy surface. moreover, the natural convection flow from an isothermal sphere with temperature dependent thermal conductivity has studied by molla et al. (2005). therefore the objective of the present work is to investigate the numerical study on mhd free convection flow along a vertical flat plate with temperature dependent thermal conductivity and heat conduction. in our study, we have considered the conductivity of the fluid to be proportional to a linear function of temperature as considered by charraudeau (1975) .the governing partial differential equations are reduced to locally nonsimilar partial differential forms by adopting appropriate transformations. the transformed boundary layer equations are solved numerically using very efficient finite difference scheme known as keller box technique (1978). numerical results of the velocity, temperature, local skin friction coefficient and the surface temperature distribution for the thermal conductivity variation parameter, the magnetic parameter and prandtl number are presented graphically. 2. mathematical formulation we consider a steady two-dimensional laminar free convection flow of an electrically conducting, viscous and incompressible fluid along a vertical flat plate of length l and thickness b. it is assumed that the temperature at the outer surface of the plate is maintained at a constant temperature tb, where tb > t. here t the temperature of the fluid outside the boundary layer. the coordinates system and the configuration are shown in fig. 1 the governing equations for continuity, momentum and energy take the following form 0      y v x u (1)    uh ttg y u y u v x u u f 2 0 2 2 )(           (2) m. m. rahman and m. a.alim/ journal of naval architecture and marine engineering 6(2009) 16-29 numerical study of magnetohydrodynamic free convective heat transfer flow along a vertical flat plate... 18 fig.1: physical model and coordinate system )( 1 y t ycy t v x t u f f p ff             (3) here we will consider the form of the temperature dependent thermal conductivity, which is proposed by charraudeau [1975] )](1[   tt ff  (4) where  is the thermal conductivity of the ambient fluid and  is a constant. the appropriate boundary conditions to be satisfied by the above equations are (luikov 1974, merkin & pop 1996, pop & ingham 2001, cheng 2006) 0,,0 0,0)(),0,( 0,0             xyasttu xyattt by t xtt vu f bf f sf f   (5) the non-dimensional governing equations and boundary conditions can be obtained from equations (1) (5) using the following non-dimensional quantities 2 3 4 1 2 1 4 1 )( ,, ,,,                ttlg gr tt tt gr lv v gr lu ugr l y y l x x b b f (6) where l is the length of the plate, gr is the grashof number,  is the non-dimensional temperature. equations (1) to (3) we get the following non -dimensional equations 0      y v x u (7) h0 o l g t interface tb insulated b v u  , 0t x y h0 x m. m. rahman and m. a.alim/ journal of naval architecture and marine engineering 6(2009) 16-29 numerical study of magnetohydrodynamic free convective heat transfer flow along a vertical flat plate... 19          2 2 y u mu y u v x u u (8) 2 2 2 pr )1( pr 1                  yyy v x u    (9) where pr    kc p / is the prandtl number,  )(  ttb , is the dimensionless thermal conductivity variation parameter and 2122 0 grlhm  is the dimensionless magnetic parameter. the corresponding boundary conditions (5) then take the following form 0,0,0 0,0)1(1,0,0         xyasu xyat y pvu    (10) where     4/1/ grlkbkp s is the conjugate conduction parameter. this coupling parameter determines the significance of the conduction resistance within the wall. in the present investigation we have considered p = 1. to solve the equations (8) and (9) subject to the boundary conditions (10) the flowing transformations are then introduced ),()1( ,)1( ),,()1( 5 1 5 1 20 1 5 1 20 1 5 4    xhxx xxy xfxx       (11) here  is the similarity variable and  is the non-dimensional stream function which satisfies the continuity equation and is related to the velocity components in the usual way as yu   and xv   . moreover h (x,) represents the dimensionless temperature. the momentum and energy equations (equation (8) and (9), respectively) are transformed for the new coordinate system. thus we get                   x f f x f fxhfxxmf x x ff x x f 10 1 5 2 2 )1( )1(10 56 )1(20 1516 (12)                                 x f h x h fx hf x hf x x h x x hh x x h )1(5 1 )1(20 1516 1pr1prpr 1 25 1 5 1  (13) where prime denotes partial differentiation with respect to . the boundary conditions as mentioned in equation (10) then take the following form 0),(,0),( )0,()1()1( 1)0,()1( )0,( ,0)0,()0,( 20 9 5 1 4 1 5 1 5 1        xhxf xhxxx xhxx xh xfxf  (14) m. m. rahman and m. a.alim/ journal of naval architecture and marine engineering 6(2009) 16-29 numerical study of magnetohydrodynamic free convective heat transfer flow along a vertical flat plate... 20 3. numerical method of solution in the present investigation implicit finite difference method has been used to integrate equations (12) to (13). 3.1 implicit finite difference method we employed implicit finite difference method together with keller box elimination technique, which was first introduced by keller (1978) and widely used by hossain et al. (1992, 1999). to apply the aforementioned method, we first convert equations (12) and (13) into the following system of first order equations with dependent variables ),(),,(),,(  pvu and g ),(  as pgvuuf  ,, (15) )(4 2 21         f v u ugupupvfpv (16) )( prprpr 1 255 31         f p g up p pg p guppfpp (17) where  = x, h = g and  5 1 5 10 1 5 2 4321 1 ,)1(, )1(5 1 , )1(10 56 , )1(20 1516                 x x pxxmp x p x x p x x p and the boundary conditions are 0)0,(,0)0,( )0,()1()1( 1)0,()1( )0,( 0)0,(,0)0,( 20 9 5 1 4 1 5 1 5 1             gu g g p uf (18) now we consider the net rectangle on the (,) plane shown in the fig. 2 and denote the net points by jjh nnk jjj n nn ,,2,1,0 ,,2,1,,0 ,10 10         here n and j are just sequence of numbers on the ( ,) plane, nk and jh are the variable mesh widths. approximate the quantities f, u, v, p at the points ( jn  , ) of the net by n j n j n j n j pvuf ,,, . which we call net function. we also employ the notation jnp for the quantities midway between net points shown in fig. 2 and for any net function as )( 2 1 121   nnn  (19) )( 2 1 121   jjj  (20) m. m. rahman and m. a.alim/ journal of naval architecture and marine engineering 6(2009) 16-29 numerical study of magnetohydrodynamic free convective heat transfer flow along a vertical flat plate... 21 fig. 2: net rectangle for difference approximations for the box scheme. )( 2 1 121   nj n j n j ggg (21) )( 2 1 121 n j n j n j ggg   (22) the finite difference approximations according to box method to the three first order ordinary differential equations (15) are written for the mid point (n,j-1/2) of the segment p1p2 shown in the fig. 2 and the finite difference approximations to the two first order differential equations (16) and (17) are written for the mid point (n-1/2,j-1/2) of the rectangle p1p2 p3p4. this procedure yields 2 1 21 1 n j n jn j j n j n j uu u h ff       (23) 2 1 21 1 n j n jn j j n j n j vv v h uu       (24) 2 1 21 1 n j n jn j j n j n j pp p h gg       (25) the difference approximation to equations (16)-(17) become     1n 21j 1n 2/1j n 2/1j 1n 2/1j n 2/1jn n 2/1j n 2/1j4 n 2/1j 2 n2 n 2/1jn1 n 1j n j 1 j r)fvvf(g)u(p )u(p)fv(p)vv(h             (26)   1 211 2/121 2/11 21 )()(   njnjnjnnj lufvrwhere  1 2/1 1 2/14 1 2/1 2 2 1 2/11 1 1 111 21 )()()()(              n j n j n j n j n j n jj n j gupupfvpvvhlwhere where 21 21 1   n j n n k  hj kn ηj-1/2 ηj ηj-1 ξn-1 ξn-1/2 ξ n p1 p4 p3 p2 m. m. rahman and m. a.alim/ journal of naval architecture and marine engineering 6(2009) 16-29 numerical study of magnetohydrodynamic free convective heat transfer flow along a vertical flat plate... 22   1n 2/1j 1n 2/1j n 2/1j 1n 2/1j n 2/1j 1n 2/1j n 2/1j 1n 2/1j n 2/1jn n 2 1 j 2 r 5 n 1j n j 1n 2/1j 1 j r 51n 1j 1n j n 1j n j n 2/1j 1 j r 5 n 2/1jn3 n 2/1jn1 n 1j n j 1 j r t ]pffpuggu[p p p )pp(gh p2 p )pppp(gh p2 p )ug()p()fp()p()pp(h p 1                            (27) where  1 2/11 2/11 2/11 2/1 )()(   njnjnnjnj ugfpmt  the corresponding boundary conditions (18) become   0g,0u g)1(1/1g)1()0,(p,0u,0f n j n j n 0 20 9 5/14 1 n 0 5 1 5 1 n 0 n 0 n 0     (28) if assume ,,,,, 11111  nj n j n j n j n j pgvuf to be known for jj0  , equations (19) to (27) from a system of 5j+5 non-linear equations for the solutions of the 5j+5 unknowns ),,,,( nj n j n j n j n j pgvuf , j = 0,1,2,3,……j. these non-linear systems of algebraic equations are to be non-linearized by newton’s quasi-linearization method. we define the iterates ),,,,( nj n j n j n j n j pgvuf , i = 0, 1, 2, 3…imax with initial values equal those at the previous x-station. for the higher iterates thus the following form   i j i j i j fff  1 (29)   i j i j i j ufuu  1 (30)   i j i j i j vvv  1 (31)   i j i j i j ggg  1 (32)   i j i j i j ppp  1 (33) now by substituting the right hand sides of the above equations in place of n j n j n j vuf ,, and gj n dropping the terms that are quadratic in i j i j i j vuf  ,, and pj i then the equations (23), (24) and (26) in the following form  )( 1)( 1)()()( 1)( 1)()( 2 i j i j i j i j ji j i j i j i j uuuu h ffff    j i j i j ji j i j ruu h ff )()( 2 1 )( 1 )()( 1 )(    (34) j i j i j ji j i j rvv h uu )()( 2 4 )( 1 )()( 1 )(    (35) m. m. rahman and m. a.alim/ journal of naval architecture and marine engineering 6(2009) 16-29 numerical study of magnetohydrodynamic free convective heat transfer flow along a vertical flat plate... 23 j i j i j ji j i j rpp h gg )()( 2 5 )( 1 )()( 1 )(    (36) jjj i jj i jj i jj i jj i jj i jj i jj i jj rssgsgsus usfsfsvsvs )(0.)(0.)()()()( )()()()()( 219 )( 18 )( 7 )( 16 )( 5 )( 14 )( 3 )( 12 )( 1       (37) j i jj i jj i jj i jj i jj i jj i jj i jj i jj i jj rvtvtgtgtut utftftptpt )()()()()()( )()()()()( 3 )( 110 )( 9 )( 18 )( 7 )( 16 )( 5 )( 14 )( 3 )( 12 )( 1       (38) where, )( 21 )()( 11 )( i jj i j i jj uhffr   )( 21 )()( 14 )( i jj i j i jj vhuur   )( 21 )()( 15 )( i jj i j i jj phggr                                          11 1 2 1 / 2 1 1 1 2 2 4 1 1 21 1 ( ) ( ) 1 1 1 2 1 1 2 ( ) 2 1 2 2 2 2 2 i ii i nn j j j j j j j i i i i i in j j j jj j i i n i i nn n j j j j j j p r r h v v fv fv p p u u u u g g f f v v v f                                   (39)                                              1 211 1 211 1 211 1 211 )( 1 2)(251 211 15 1 1 1 11 15)( 1 )(3 )( 1 )(1)( 1 )(11 2/13 )( 2 )( 2 )( 2 )( 2pr4 )( pr4 }){( pr22 2 1 )(                             n j i j i j nn j i j i j nn j i j i j n n j i j i j ni j i j n j i j i jj n j n j n j n j i j i jj i j i j n i j i j ni j i jj r n jj pfffppugg guupp p gpph p ppppggh p ugug p fpfp p pph p tr     (40) the coefficients of momentum equation are   ,f 2 f 2 p h)s( 1n 21j ni j n11 jj1          1 21 11 2 22 )(      nj ni j n jj ff p hs      ,v 2 v 2 p )s( 1n 21j ni j n1 j3          1 211 1 4 22 )(     nj ni j n j vv p s      , 2 p up)s( 4ijn1j5       46 1 1( ) 2 i j n j p s p u      ,21)s( j7   ,21)s( j8  ,0)s( j9  0)( 10 js the coefficients of energy equation are 1 21 51 21 15)(11 1 2prpr22 )( pr 1 )(        nj nn jj i j n jj f p gh p f p ht  m. m. rahman and m. a.alim/ journal of naval architecture and marine engineering 6(2009) 16-29 numerical study of magnetohydrodynamic free convective heat transfer flow along a vertical flat plate... 24 1 21 51 21 15)( 1 11 2 2prpr22 )( pr 1 )(          nj nn jj i j n jj f p gh p f p ht  ,p 2 p 2 )p( )t( 1n 21j n)i( j n1 j3      1 21 )( 1 1 4 22 )( )(     nj ni j n j pp p t  ,g 2 g 2 )p( )t( 1n 21j n)i( j n3 j5      1 21 )( 1 3 6 22 )( )(     nj ni j n j gg p t  1 21 1 1 1 1 15)(3 7 2 }{ pr42 )( )(          nj nn j n j n j n jj i j n j upppph p u p t  1 21 1 1 1 1 15)( 1 3 8 2 }{ pr42 )( )(           nj nn j n j n j n jj i j n j upppph p u p t  ,0)t( j9  0)( 10 jt the boundary conditions (28) become   0,0 )1(1/1)1()0,(,0,0 0 20 9 5/14 1 0 5 1 5 1 000          n j n j nnnnn gu ggpuf   (41) which just express the requirement for the boundary conditions to remain during the iteration process. now the system of linear equations (34), (35), (36), (37) and (38) together with the boundary conditions (41) can be written in a black matrix form a coefficient matrix. the whole procedure, namely reduction to first order followed by central difference approximations, newton’s quasi-linearization method and the block thomas algorithm, is well known as keller-box method. 4. local skin friction coefficient and surface temperature distribution from the process of numerical computation, in practical point of view, it is important to calculate the values of the surface shear stress in terms of the skin friction coefficient. this can be written in the non-dimensional form as     wf lgrc /24/3 (42) where ])([ 0 yw yu is the shearing stress. using the new variables described in equation (6), the local skin friction co-efficient can be written as )0,()1( 20/35/2 xfxxc xf   43) the numerical values of the surface temperature distribution are obtained from the relation )0,()1()0,( 5/15/1 xhxxx  (44) we have also discussed the velocity profiles and the temperature distributions for different values of the thermal conductivity variation parameter, the magnetic parameter and the prandtl number. 5. comparison with previous work and programme validation a comparison of the surface temperature and local skin friction coefficient obtained in the present work with 5 1 x , m=0 and  = 0 and obtained by merkin and pop (1996) and pozzi and lupo (1988) have been shown in table 1 and table 2, respectively. it is clearly seen that there is an excellent agreement among the respective results. m. m. rahman and m. a.alim/ journal of naval architecture and marine engineering 6(2009) 16-29 numerical study of magnetohydrodynamic free convective heat transfer flow along a vertical flat plate... 25 table 1: comparison of the present numerical results of surface temperature with prandtl number pr = 0.733, m = 0 and  = 0  (x,0) 5 1 x pozzi and lupo (1988) merkin and pop (1996) present work 0.7 0.651 0.651 0.651 0.8 0.684 0.686 0.687 0.9 0.708 0.715 0.716 1.0 0.717 0.741 0.741 1.1 0.699 0.762 0.763 1.2 0.640 0.781 0.781 table 2: comparison of the present numerical results of local skin friction coefficient with prandtl number pr = 0.733, m = 0 and  = 0 cfx 5 1 x pozzi and lupo (1988) merkin and pop (1996) present work 0.7 0.430 0.430 0.424 0.8 0.530 0.530 0.529 0.9 0.635 0.635 0.635 1.0 0.741 0.745 0.744 1.1 0.829 0.859 0.860 1.2 0.817 0.972 0.975 6. results and discussion here we have investigated numerically mhd free convection flow along a vertical flat plate with temperature dependent thermal conductivity and heat conduction. in simulation, the values of the prandtl number are considered to be 0.70, 1.70, 2.97 and 4.34 that corresponds to helium, sulfur dioxide, methyl chloride and water, respectively. numerical results have been obtained for different values of the magnetic parameter, thermal conductivity variation parameter and prandtl number are presented graphically. (a) (b) fig: 3 (a) variation of velocity profiles and (b) temperature profiles against  for varying of m with  = 0.12 and pr = 4.34 0.0 2.0 4.0 6.0  0.0 0.1 0.2 v el o ci ty , m = 0.12 m = 0.55 m = 0.85 m = 1.25 f ' 0.0 1.0 2.0 3.0 4.0  0.0 0.2 0.4 0.6 0.8 t em p er at u re , m = 0.12 m = 0.55 m = 0.85 m = 1.25  m. m. rahman and m. a.alim/ journal of naval architecture and marine engineering 6(2009) 16-29 numerical study of magnetohydrodynamic free convective heat transfer flow along a vertical flat plate... 26 (a) (b) fig. 4 (a) variation of velocity profiles and (b) temperature profiles against  for varying of  with m = 0.12 and pr = 4.34 (a) (b) fig. 5 (a) variation of velocity profiles and (b) temperature profiles against  for varying of pr with  = 0.12 and m = 0.12 (a) (b) fig. 6 (a) variation of local skin friction coefficients and (b) surface temperature distribution profiles against x for varying of m with  = 0.12 and pr = 4.34 0.0 2.0 4.0 6.0  0.0 0.1 0.2 0.3 0.4 0.5 v el o ci ty , pr = 0.70 pr = 1.70 pr = 2.97 pr = 4.34 f ' 0.0 1.0 2.0 3.0 4.0 x 0.0 0.2 0.4 0.6 0.8 1.0 s u rf ac e te m pe ra tu re , m = 0.12 m = 0.55 m = 0.85 m = 1.25  (x ,0 ) 0.0 1.0 2.0 3.0 4.0  0.0 0.2 0.4 0.6 0.8 1.0 t em p er at u re , pr = 0.70 pr = 1.70 pr = 2.97 pr = 4.34  0.0 2.0 4.0 6.0 8.0 10.0 x 0.0 0.2 0.4 0.6 0.8 1.0 l o ca l sk in fr ic ti o n , m = 0.12 m = 0.55 m = 0.85 m = 1.25 c f x 0.0 2.0 4.0 6.0  0.0 0.1 0.2 0.3 v el o ci ty ,             f ' 0.0 1.0 2.0 3.0 4.0  0.0 0.2 0.4 0.6 0.8 t em pe ra tu re ,              m. m. rahman and m. a.alim/ journal of naval architecture and marine engineering 6(2009) 16-29 numerical study of magnetohydrodynamic free convective heat transfer flow along a vertical flat plate... 27 (a) (b) fig. 7 (a) variation of local skin friction coefficients and (b) surface temperature distribution profiles against x for varying of  with m = 0.12 and pr = 4.34 (a) (b) fig. 8 (a) variation of local skin friction coefficients and (b) surface temperature distribution profiles against x for varying of pr with  = 0.12 and m = 0.12 from fig. 3 (a) it can be observed that the magnetic field normal to the flow in an electrically conducting fluid introduces a lorentz force, which acts against the flow. the peak velocity decreases with the increase in magnetic parameter m due to this retarding effect. from fig. 3 (b), it can be seen that the temperature within the boundary layer increases with the increase in m. temperature at the interface also varies since the conduction is considered with in the plate. fig. 4 (a) and 4(b) display the numerical results of the velocity and the temperature, respectively obtained from the solution of the equations (12) and (13) subject to the boundary condition (14) for different values of thermal conductivity variation parameter  plotted against  with m = 0.12 and pr = 4.34. it is seen from fig. 4(a) and fig.4 (b) that the velocity and temperature increase within the boundary layer with the increasing value of . it means that the velocity boundary layer and the thermal boundary layer thickness increase for increasing values of . fig. 5 (a) and fig. 5 (b) illustrate the velocity and temperature profiles for different values of prandtl number pr with m = 0.12 and  = 0.12. from fig. 5 (a), it can be observed that the velocity decreases as well as its position moves toward the interface with the increase in pr. from fig. 5 (b), it is seen that the temperature profiles shift downward with the increasing values of pr. the variation of the local skin friction coefficient cfx and surface temperature  (x, 0) for different values of m with  = 0.12, and pr = 4.34 at different positions are illustrated in fig. 6. (a) and fig.6 (b), respectively. it is observed from fig. 6. (a) that the increase in value of the magnetic parameter m leads to a decrease in the skin friction factor. again fig. 6 (b) shows that the surface temperature  (x, 0) increases due to the increased value of the magnetic parameter m. 0.0 1.0 2.0 3.0 4.0 x 0.0 0.2 0.4 0.6 0.8 1.0 s u rf ac e te m pe ra tu re , pr = 0.70 pr = 1.70 pr = 2.97 pr = 4.34  (x ,0 ) 0.0 2.0 4.0 6.0 8.0 10.0 x 0.0 0.4 0.8 1.2 l o ca l sk in fr ic ti o n , pr = 0.70 pr = 1.70 pr = 2.97 pr = 4.34 c f x 0.0 2.0 4.0 6.0 8.0 10.0 x 0.0 0.2 0.4 0.6 0.8 1.0 l o ca l sk in fr ic ti o n ,             c f x 0.0 1.0 2.0 3.0 4.0 x 0.0 0.2 0.4 0.6 0.8 s u rf ac e te m pe ra tu re ,             ( x ,0 ) m. m. rahman and m. a.alim/ journal of naval architecture and marine engineering 6(2009) 16-29 numerical study of magnetohydrodynamic free convective heat transfer flow along a vertical flat plate... 28 fig. 7 (a) and fig. 7 (b) illustrate the effect of the thermal conductivity variation parameter on the local skin friction coefficient and surface temperature distribution against x with m = 0.12 and pr = 4.34. it is also seen that the local skin friction coefficient increases with the increase in . from fig. 7 (b), it is seen that the surface temperature increases with the increase in . this is to be expected because the higher value for the thermal conductivity variation parameter accelerates the fluid flow and increases the temperature as mentioned in fig. 4 (a) and fig. 4 (b), respectively. fig. 8 (a) and fig. 8 (b) plot the effect of prandtl number on the local skin friction coefficient and surface temperature distribution against x with m = 0.12 and  = 0.12. it can be observed from fig. 8 (a) that the local skin friction coefficient decreases with the increase in pr. this is expected behavior because the fluid velocity is decreased due to the increase in pr. from fig. 8 (b), it can be conclude that the surface temperature distribution decreases for the increase in values of pr. 7. conclusion in this paper effect of temperature dependent thermal conductivity on mhd free convection flow along a vertical flat plate have been studied numerically. implicit finite difference method together with keller box scheme is employed to integrate the equations governing the flow. comparison with previously published work is performed and excellent argument has been observed. from the present numerical investigation, following conclusions may be drawn:  for increased value of magnetic parameter, the velocity profile decreases but the temperature profile increases slightly.  the local skin friction coefficient decreases as well as the surface temperature distribution increase with the increase in values of the magnetic parameter.  the velocity and the temperature within the boundary layer increases for increasing values of the thermal conductivity variation parameter.  increasing values of the thermal conductivity variation parameter leads to increase the local skin friction coefficient and also the surface temperature distribution.  the velocity, temperature, local skin friction coefficient and surface temperature distribution within the boundary layer decreases with the increase in values of the prandtl number. reference arunachalam, m. and rajappa, n.r., (1978): thermal boundary layer in liquid metals with variable thermal conductivity, appl. sci. res., vol. 34, pp.179-187. doi:10.1007/bf00418866 charraudeau, j., (1975): influence de gradients de properties physiques en convection force application au cas du tube, int. j. heat mass trans., vol.18, pp.87-95. doi:10.1016/0017-9310(75)90011-3 cheng, l. c. a., (2006): numerical simulation of micro polar fluid flows along a flat plate with wall conduction and buoyancy effects, j. appl. phys. d, vol.39, pp.1132–1140. doi:10.1088/0022-3727/39/6/019 chowdhury, m. k. and islam, m. n., (2000): mhd free convection flow of visco-elastic fluid past an infinite 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flow, (communicating) keller, h.b., (1978): numerical methods in boundary layer theory, annual rev. fluid mechanics, vol.10, pp.417-433. doi:10.1146/annurev.fl.10.010178.002221 kuiken, h. k., (1970): magnetohydrodynamic free convection in strong cross flow field, j. fluid mech., vol.40, pp.21-38. doi:10.1017/s0022112070000022 luikov, a. k., (1974): conjugate convective heat transfer problems. int. j. heat mass trans., vol.16, pp. 57– 265. molla, m.m., rahman, a. and lineeya, l.t., (2005): natural convection flow from an isothermal sphere with temperature dependent thermal conductivity, j. arc. marine eng., vol.2, pp.53-64. merkin, j. h. and pop, i., (1996): conjugate free convection on a vertical surface, int. j. heat mass trans., vol.39, pp.1527-1534. doi:10.1016/0017-9310(95)00238-3 pop, i. and ingham, d. b., (2001): convective heat transfer, (pergamon, oxford). p. 179. doi:10.1016/b978008043878-8/50009-2 pozzi, a. and lupo, m., (1988): the coupling of conduction with laminar natural convection along a flat plate, int. j. heat mass trans., vol.31, no. 9, pp.1807-1814. doi:10.1016/0017-9310(88)90195-0 sparrow, e. m. and cess, r. d., (1961): effect of magnetic field on free convection heat transfer, int. j. heat mass trans., vol.3, p.267. doi:10.1016/0017-9310(61)90042-4 separation points of magnetohydrodynamic boundary layer flow along a vertical plate with exponentially decreasing free stream velocity¬ journal of naval architecture and marine engineering june, 2016 http://dx.doi.org/10.3329/jname.v13i1.20703 http://www.banglajol.info 1813-8235 (print), 2070-8998 (online) © 2016 aname publication. all rights reserved. received on: oct. 2014 natural convection on heat transfer flow of nonnewtonian second grade fluid over horizontal circular cylinder with thermal radiation v. ramachandrprasad 1 , r. bhuvanavijaya 2 and b. mallikarjuna 3* 1 department of mathematics, madanapalle institute of technology and science, madanapalle, india-517325, india, email: rcpmaths@gmail.com 2 department of mathematics, jnt university college of engineering (autonomous) anantapur, ananthapuramu, andhrapradesh -515002, india, email: bhuvanarachamalla@gmail.com 3 department of mathematics, bms college of engineering, bangalore, karnataka-560019, india, email:mallikarjuna.jntua@gmail.com, mallikarjunab.maths@bmsce.ac.in * corresponding author abstract: this article numerically studies for multi-physical transport of an optically-dense, free convective incompressible non-newtonian second grade fluid past an isothermal, impermeable horizontal circular cylinder. the governing boundary layer equations for momentum and energy transport, which are parabolic in nature, have been reduced to non-similarity non-linear conservation equations using appropriate transformations and then solved numerically by employing with most validated, efficient implicit finite difference method with keller box scheme. the numerical code is validated with previously existing results and found to be very good agreement. numerical results have been carried out for various values of the physical parameters; deborah number (0 ≤ de ≤ 1.5), prandtl number (0 ≤ pr ≤ 100) and thermal radiation (0 ≤r ≤ 5) on flow velocity and temperature profiles. furthermore, the effects of these parameters on non-dimensional wall shear stress (skin friction) and surface heat transfer rate (nusselt number) are also investigated. increasing the deborah number reduces velocity profile, skin friction where as it enhances the temperature profile. increasing prandtl number decelerates the flow velocity, temperature and skin friction. increase in radiation parameter retards the flow velocity, temperature profiles and skin friction. the rate of heat transfer (nusselt number) enhances markedly with increase in radiation parameter and prandtl number but depreciated for larger values of deborah number. increasing stream wise coordinate retards the velocity gradient whereas enhances rate of heat transfer. applications of the model arise in polymer processing in chemical engineering, metallurgical material processing. keywords: non-newtonian fluid; second grade fluid; thermal radiation; prandtl number; skin friction; nusselt number; keller box scheme nomenclature greek symbols u velocity component in x-direction [m/s] 1 2,  material fluid parameters v velocity component in y-direction [m/s]  fluid density [kg/m 3 ] g acceleration due to gravity [m/s 2 ]  the dimensionless tangential coordinate f non-dimensional stream function  the dimensionless radial coordinate a radius of the cylinder [m]  thermal expansion coefficient [1/k] tw surface temperature [k]  stefan-bolzmann constant t∞ ambient fluid temperature [k] xx normal stress in x-direction x stream wise coordinate [m] yy normal stress in y-direction y transverse coordinate [m] ,xy yx  shear stresses cf skin friction coefficient  dimensionless stream function nu nusselt number  non-dimensional temperature mailto:rcpmaths@gmail.com mailto:bhuvanarachamalla@gmail.com mailto:mallikarjuna.jntua@gmail.com mailto:mallikarjunab.maths@bmsce.ac.in r. bhuvanavijaya and b. mallikarjuna/ journal of naval architecture and marine engineering, 13(2016) 63-78 natural convection on heat transfer flow of non-newtonian second grade fluid over horizontal circular cylinder … 64 gr grashof number  stream function de deborah number  angular velocity about the y-axis a1, a2 rivlin-ericksen tensors k thermal conductivity of the second grade fluid [m 2 /s] 1. introduction the study of flow and heat transfer generated by different geometries in non-newtonian fluid plays a significant role in many engineering and industrial areas. examples of such fluids are foods, fossils, fuels, pulps and molten polymers, several fluids in pharmaceutical formulations, synthetic lubricants, salvia, synovial fluid, jams, marmalades, sewage sludge, clay and coal in water and paints. during the last five years, the boundary layer flows of non-newtonian fluids over different geometries with the influence of thermal radiation have been studied. this is in view of much importance in industrial, engineering, geo-physics and bio-science such as slurries, suspensions, dispersions, pharmaceutical formulations, paints, biological fluids, synthetic polymers, melts and solutions of naturally occurring high molecular weight, synthetic lubricants and food stuffs, naturally occurring fluids such as animal blood, behavior of exotic lubricants, liquid crystals and colloidal suspensions. tai and char (2010) studied the effects of soret and dufour on free convection flow of non-newtonian fluids along a vertical plate embedded in a porous medium with thermal radiation. hayat and qasim (2011) analyzed the effects and radiation and magnetic fields on the unsteady mixed convection flow of a second grade fluid over a vertical stretching sheet. mukhopadhyay et al (2012) investigated thermal radiation on forced convective flow and heat transfer past a permeable vertical surface in darcy-forchheimer porous medium. his results indicate that due to suction the skin-frinction increases while the rate of heat transfer increases due to suction. due to thermal radiation, temperature is found to decrease. the combined effects of suction and thermal radiation can be used as means of cooling. gireesha et al (2013) studied radiation effect on mixed convective heat and mass transfer of a dusty fluid past a stretching sheet with non-uniform heat source/sink. the radiation parameter is predicted to enhance the temperature profiles of both the fluid and dust phases for the vwt and vhf cases for larger values of radiation parameter. rashad (2014) investigated the influence of radiation and variable viscosity on unsteady mhd flow of a rotating fluid over a stretching surface in a porous medium. srinivasacharya et. al (2015) studied radiation effect on heat and mass transfer flow of a viscous fluid over a vertical wavy surface in a porous medium with variable properties and reported results that increasing radiation parameter tends to increase flow velocity and temperature profiles whereas concentration profile is reduced. rashad et.al (2016) investigated thermal radiation effects on convective boundary layer flow over a rotating cone in porous medium with thermophoresis. previous studies indicate that not much has been presented yet regarding the flow over horizontal cylinder. free convection around heated, horizontal cylinder for various fluids is of great importance due to its extensive industrial process and geological formulations applications. such as the coating of wires or polymer-fiber spinning, in the assessment of aquifers, in the exploration and thermal recovery of oil, underground nuclear waste storage sites and geothermal reservoirs. vasu et.al (2010) studied computational analysis of free convection heat and mass transfer from an isothermal horizontal circular cylinder in a micropolar fluid with soret/dufour effects. they reported the results that with increasing soret number the rate of heat transfer is boosted considerably, with the converse response with increase in dufour number. hussain and hussain (2010) numerically investigated natural convection in a uniformly heated circular cylinder embedded in square enclosure filled with air at different vertical locations. nadeem et.al (2011) investigated boundary layer flow of second grade fluid in a cylinder with heat transfer. hussain and hussain (2011) studied natural convection heat transfer flow in a differentially heated square enclosure with a heat generating-conducting circular cylinder at different diagonal locations. hussain (2013) analyzed natural convection in a parallelogrammic cavity with a hot concentric circular cylinder moving at different vertical locations. prasad et.al (2013) studied flow and heat transfer of casson fluid from a horizontal circular cylinder with partial slip in non-darcy porous medium. they reported that velocity increases for increasing values of casson fluid parameter and is found to reduce the temperature profile. tham et al (2014) studied mixed convective heat transfer flow of a nanofluid from a horizontal circular cylinder in a porous medium using buongiorno-darcy model. prasenjit et al (2015) investigated unsteady mixed convection over circular cylinder in the presence of nanofluid. they found and reported results that mixed convection heat transfer of water based nanofluid can be analyzed correctly by both the artificial neural network and gene expression programming (gep), but gep is found more efficient. r. bhuvanavijaya and b. mallikarjuna/ journal of naval architecture and marine engineering, 13(2016) 63-78 natural convection on heat transfer flow of non-newtonian second grade fluid over horizontal circular cylinder … 65 to the best of our knowledge, no one has studied the influence of thermal radiation on convective heat transfer flow of a non-newtonian second grade fluid over a horizontal circular cylinder. the governing boundary layer equations are transformed into non-dimensional equations by using non-similarity transformation and then found the numerical solutions by employing implicit finite difference method with keller box scheme. the numerical results are reported graphically and in tabular form for various physical parameters; deborah number, prandtl number and thermal radiation on flow velocity and temperature profile, as well as skin friction and nusselt number. the obtained results are validated by comparing the results with previously published work. excellent agreement has been obtained. 2. mathematical formulation we consider the steady, incompressible, laminar free convective flow of non-newtonian second grade fluid over an impermeable isothermal horizontal cylinder. fig. 1 shows the configuration of the system and co-ordinate system. where a is the radius of the cylinder. the cylinder is heated to temperature of tw which is surrounded by the free stream temperature t , far away from the surface. the fluid is assumed to be a gray, absorbingemitting radiation but non-scattering medium. fig. 1: physical configuration and coordinate system the constitutive equation for the cauchy stress tensor related to the deformation field in a second grade fluid is 2 1 1 2 2 1pi a a a        (1) where -pi is indeterminate spherical stress due to constraint of incompressibility, 1 2,  are the material moduli, and 1 2,a a are the first two rivlin-ericksen (1995) tensors such that 1 t a l l  , 12 1 1 tda a a l l a dt    , (2) where l v  , d/dt is the material derivative, fosdick and rajagopal (1979) argue that the second grade fluid, modelled by eq. (1), is to be thermodynamically compatible, in the sense that all motions of the fluid meet the clausius – duhem inequality together with helmoltz free energy being at its minimum whenever the fluid is locally at rest. the thermodynamical constrains put some restriction on the sign and magnitude of the material moduli: 1 1 20, 0, 0       . (3) where v is the velocity field in two-dimensional i.e.    , , ,v u x y v x y    , u and v are its components in x and y – directions, v denotes the velocity gradient tensor. in our model we use the two-dimensional deformation rate tensor, in place of the a1, a2 and write the part of the stress tensor, i j as beard and walters (1964) 1 22 i j i j i j dd d dt     , where i j i j k i k j k j i k dd d l d l d dt t      (4) second grade fluid cylinder a y x t r. bhuvanavijaya and b. mallikarjuna/ journal of naval architecture and marine engineering, 13(2016) 63-78 natural convection on heat transfer flow of non-newtonian second grade fluid over horizontal circular cylinder … 66 where ii j j u l x    . now x and y-momentum equations are written as: xyxxu u pu v x y x x y                    (5) yx yyv v p u v x y y x y                     (6) where ,xx yy  express the stresses in the perpendicular direction and the shear stresses are represented by ,xy yx  . under the boundary layer approximations and boussinesq approximation, we consider the governing equations for the mass, momentum and energy equations 0 u v x y       (7)   2 2 2 3 1 2 2 2 3 sin u u u u u v u x u v u v g t t x y x y ay y y y                                         (8) 2 2 1 r p t t t q u v x y c yy             (9) the associated boundary conditions are 0, 0, 0 0, wu v t t at y u t t as y         (10) where ,u v are velocity components in x and y-directions. 1 is the material constant associated with second grade fluid,  is the fluid density,  is the kinematic viscosity, g is the acceleration due to gravity,  is the thermal expansion coefficient and cp is the specific heat at constant pressure. second term in right hand side of eq. (8) represents second grade fluid which consists of mixed derivates of third order derivates. hence, momentum boundary layer equations attains higher order than the viscous (newtonian) navier-stokes flow model. the impact of non-newtonian effect takes place only on shear term of momentum equation but not on convection term. the last term on eq. (8) represents thermal buoyancy force term which couples with temperature field of eq. (9). the penultimate term in eq. (9) represents thermal radiative heat flux (qr), which is approximated for an optically thick boundary layer according to rosseland approximation. the rosseland diffusion flux model is an algebraic approximation and defined as follows: 4 * 4 3 rq t k     (11) where * , k are respectively the stefan-bolzmann constant and mean absorption coefficient. the term 4 t expressed as a linear function of the temperature and it is accomplished by expanding in a taylor series about t and neglecting the higher order terms of t to yield 4 3 4 4 3t t t t   (12) therefore, eq. (9) becomes 2 3 2 2 2 16 3 t t t t t u v x y ky y               (13) in order to transform the governing equations into non-dimensional form we introduce the following nondimensional variables  1/4 1/4 , , ( , ) , , w x y t t gr f a a t tgr                  , (14) where  is the stream function which satisfy eq. (7) such that  , ,u v y x           . by using eq. (14), eqs. (8) and (13) becomes r. bhuvanavijaya and b. mallikarjuna/ journal of naval architecture and marine engineering, 13(2016) 63-78 natural convection on heat transfer flow of non-newtonian second grade fluid over horizontal circular cylinder … 67     2 2 sin( ) 2 iv iv f ff f de f f f ff f f f f f f f f de f f f f                                                         (15) 4 1 pr pr 3 f f f r                           (16) the corresponding boundary conditions are 0, 0, 1 0 0, 0 f f at f as               (17) where primes denotes the differentiation with respect to  , the dimensionless radial coordinate,  is the dimensionless tangential coordinate, 1 r u de a    is the deborah number, 1/2 r gr u a   is the reference velocity,   3 2 wg t t a gr     is the free convection (grashof) number, pr    is the prandtl number, p k c    is the thermal conductivity and * 3 4 kk r t   is the radiation parameter. the physical interest of engineering design quantities is skin friction and nusselt number, which are defined as 0 0 w w y y u t and q k y y                    (18) in non-dimensional, skin friction cf and nusselt number nu can be written as   3/4 , 0 fc f gr  (19)   1/4 , 0 nu gr    (20) the location, 0 , corresponds to the vicinity of the lower stagnation point on the cylinder. since sin 0 0    i.e. 1. for this case the model defined by equations (15) and (16) reduce to ordinary differential equations.     2 2 sin( ) 2 0 iv f ff f de f f f ff                    4 1 pr 0 3 f r            3. numerical method of solution the set of differential equations (15) and (16) associated with boundary conditions (17) is highly nonlinear and therefore, it cannot be solved analytically, hence implicit finite difference method with keller box scheme has been used for solving it. the main ability of this implicit scheme is to solve systems of differential equations of any order as well as featuring second-order accuracy and attractive extrapolation features. for the implicit finite difference method with keller box scheme one can refer to cebeci and bradshaw (1984) and bhuvanavijaya et al (2014). we first converted the partial differential equations (15) and (16) into a system of first order differential equations. the resulting equations are written in finite difference forms by considering the functions and their derivatives in terms of center difference. the resulting central finite difference equations are to be linearized by using newton’s method. the obtained linear algebraic equations can be expressed in block diagonal matrix. the solution procedure is given as: r. bhuvanavijaya and b. mallikarjuna/ journal of naval architecture and marine engineering, 13(2016) 63-78 natural convection on heat transfer flow of non-newtonian second grade fluid over horizontal circular cylinder … 68 fig. 2: grid meshing and a keller box computation cell 3.1 convert the partial differential equations into system of first order equations to obtain a set of first order differential equation from (15) and (16) we introduce the new variables          , , , , , , , , ,u f v f w f s t s                  f u  (21a) u v  (21b) v w  (21c) s t  (21d)  2 22 f u v f w u w fv u de uw v fw sb u u de v w u w                                        (21e) 4 1 pr pr 3 s f t ft u t r                    (21f) the corresponding boundary conditions are 0, 0, 1 0 0, 0, 0 u f s at u v s as            (21g) where prime denotes differentiation with respect to  . 3.2 write the difference equation using central differences as shown in fig. 2 the mesh points are represented in ,  -plane by , n j  where n=1, 2, …..n and j=1,2,….j and j  . the derivates are approximated by centered-difference gradients and averages centered at midpoints of the net, defined by          0 1 1/2 1 1/2 1 0 1 0, , 1, 2, 3, ........... , , 1 1 , 2 2 0, , 1, 2, 3, ........., j j j j j j j j j n n n h j j k n n                            (22) where ,n jk h are respectively ,   spacing. therefore, 1 1 2 n n n n j j j j j f f u u h     (23a) 1 1 2 n n n n j j j j j u u v v h     (23b) 1 1 2 n n n n j j j j j v v w w h     (23c) r. bhuvanavijaya and b. mallikarjuna/ journal of naval architecture and marine engineering, 13(2016) 63-78 natural convection on heat transfer flow of non-newtonian second grade fluid over horizontal circular cylinder … 69 1 1 2 n n n n j j j j j s s t t h     (23d)            2 1 1 1 1 1 11 1 1 4 4 2 j j j j j j j j j j j j j j j h h h v v f f v v u u de u u q q                                     2 1 1 1 1 1 1/2 1 11 1 1 1/2 1 1/2 1 11/2 1/2 1 1 1 4 2 2 2 2 j j n j j j j j j j j j n nj jn n j j j j j j j j j j h h de v v de f f q q f f v h h v v f de q q f f f q r                                      (23e)                    1 1 1 1 1 1 1 1 1 1/2 1 1/2 1 1/2 11 1 1/2 2 1/2 4 1 1 3 4 4 2 2 2 2 j j j j j j j j j j j j j j jn n n j j j j j j j j j nj n j j j j h h t t f f t t u u s s r h h h s s u u u s t t f h f f t r                                               (23f) where                  1 2 2 1/2 1/2 1/21 1/2 1/2 1 1/2 1/21/2 1 1 2 1 1 1 n j j jn j j jj jj v fv u de uq de v r h de fq bs                                          1 1 1 1/2 2 1/2 1/2 1/2 4 1 3 n j j n j jjj j j t t ft r hr h us ft                              the boundary conditions are 0 0 00, 0, 1, 0, 0, 0 n n n n n n j j jf u s u v s      (23g) 3.3 linearize the resulting algebraic equations by newton’s method to linearize the system of nonlinear equations (23) using newton’s method, we introduce the following iterates 1 1 1 1 1 1 , , , , i i i j j j i i i j j j i i i j j j i i i j j j i i i j j j i i i j j j f f f u u u v v v w w w u s s t t t                         (24) by using eq. (24), eq. (23) becomes  1 1 1 1/2 1 ( ) 2 j j j j j jf f h u u r         (25a)  1 1 2 1/2 1 ( ) 2 j j j j j ju u h v v r         (25b)  1 1 3 1/2 1 ( ) 2 j j j j j jv v h w w r         (25c)  1 1 4 1/2 1 ( ) 2 j j j j j js s h t t r         (25d) r. bhuvanavijaya and b. mallikarjuna/ journal of naval architecture and marine engineering, 13(2016) 63-78 natural convection on heat transfer flow of non-newtonian second grade fluid over horizontal circular cylinder … 70                       1 2 1 3 4 1 5 6 1 7 8 1 9 10 1 5 1/2 j j j j j jj j j j j j j j j jj j j j j a w a w a f a f a u a u a v a v a s a s r                           (25e)                   1 2 1 3 4 1 5 6 1 7 8 1 6 1/2 j j j j j jj j j j j j j jj j j b t b t b f b f b u b u b s b s r                      (25f) where      11 1/2 1/2 1/21 2 2 2 j n j j j jj h de a de h u f f           ,      12 1/2 1/2 1/21 2 2 2 j n j j j jj h de a de h u f f               1 1 11 3 1/2 1 1/2 1/2 (1 ) 2 2 2 n n j j j jn j j j j jj j h h w wde a v w w u u de h                                  15 1/2 1/2 1/2 1/21 1 2 2 j j n j j j j jj h h a u de h w f f                    4 3 6 5, , ,j j j ja a a a   7 1/2 1/2 1 (1 ) , 2 j j jj a h f de v                   8 7 9 10, 2 j j j j j bh a a a a       11 1/2 1/2 4 pr 1 1 pr 3 2 n j j jj b h f f r                      ,     12 1/2 1/2 4 pr 1 1 pr 3 2 n j j jj b h f f r                       ,     13 4 1/2 1/2 1 pr 2 n j j jj j b b h t t                 1/2 1 5 6 1/2pr 2 j n j jj j s b b h s             ,     17 8 1/2 1/2 pr 2 j n j jj j h b b u u                    1 1 1/2 2 1 1/21/2 1/2 3 1 1/2 4 1 1/21/2 1/2 , , , j j j j j j j jj j j j j j j j j jj j r f f h u r u u h v r v v h w r s s h t                                     2 5 1 1/2 1/2 1/2 1/21/2 1/2 2 1 1/2 1/2 1/2 1/2 1 1 1/2 1/2 1/2 1/2 1/2 1/2 1 1 1 1 1/2 1/2 1 1 2 j j j j jj j j j j j j j j j n n j j j j j j j n n j j j jn j j j j r r h w f v u w w u w h de v f h h f u u f f u w w w w h de f f h                                                       1 1/2j j bs h               1/2 1/2 1 1 1/2 1/2 1/2 1/2 1/2 1/26 2 11/2 1/2 1 1 1/2 1/2 1/2 1/2 pr(1 ) 4 1 3 pr j j n n j j j j j jj j jj j n n j j j j f t u s u s u sr r t t h r t f f t                                              where 1/2 1/2 1/2 ( ) , n n n n sin b k          to complete the system (25) we recall the boundary conditions (24g) , which can be satisfied exactly with no iteration. so, to maintain these correct values in all the iterates, we take r. bhuvanavijaya and b. mallikarjuna/ journal of naval architecture and marine engineering, 13(2016) 63-78 natural convection on heat transfer flow of non-newtonian second grade fluid over horizontal circular cylinder … 71 0 0 00, 0, 0 0, 0, 0.j j j u f s u v s             (25g) 3.4 block elimination method the linear system (25) can now be solved by the block-elimination method, as it is explained in na (1979). the linearized difference equations of the system (25) have a block tri-diagonal structure. in a matrix vector form, this can be written as                                     1 1 1 1 2 2 2 2 2 1 1 1 1 1 ... ... ... ... ... j j j j j j j j j a c r b a c r b a c r b a r                                                          that is: a r  (26) where the elements are defined by                 1 1 1 1 1 1 1 1 8 2 3 11 1 1 1 2 3 11 1 1 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 , 0 0 0 0 2 0 0 0 0 0 d d d h a d d d a a a a b b b                                       6 8 10 3 1 6 8 3 1 0 0 1 0 0 1 0 0 0 0 0 1 0 0 0 , , 2 0 0 1 0 0 2 0 0 0 j j j j j j j j j j j j j j j j d d d h a d j j d a a a a a b b b b                                     4 2 4 2 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 , , 2 0 0 0 0 0 2 0 0 0 0 0 0 0 0 j j j j j j j j j d h b d j j d a a b b                                      5 7 9 5 7 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 , , 1 0 0 1 0 0 0 2 0 0 0 0 0 0 0 j j j j j j j j j j d d h c d j j a a a b b                            r. bhuvanavijaya and b. mallikarjuna/ journal of naval architecture and marine engineering, 13(2016) 63-78 natural convection on heat transfer flow of non-newtonian second grade fluid over horizontal circular cylinder … 72               1 1.2 1 0 2 1.21 0 31 1.20 1 1 4 1.2 1 5 1.2 1 6 1.2 , , 2 , , 1 j j jj j j j j j j j j j j r uu rvv rss j j r j j ff r w w r t t r                                                                                   , the coefficient matrix a in (26) has a block tri-diagonal structure and the difference equations are solved using a block matrix version of the thomas algorithm. the numerical results are strongly influenced by the number of mesh points in ,  -directions. after some trials in the η-direction a larger number of mesh points are selected whereas in the ξ-direction significantly less mesh points are necessary. ηmax has been set at appropriate position and this constitutes an adequately large value at which the prescribed boundary conditions are satisfied. ξmax is set at 2.5 for the simulations. mesh independence has been comfortably attained in the simulations. the numerical algorithm is executed in matlab®. 3.5 grid independency test in order to check the effects of step size   we found the nusselt number and sherwood number with four different step sizes as 0.1, 0.01     and 0.001  . we observe from table 1 that the results are independent with the step size   and exist with increasing cpu time (sec). hence a step size 0.01  is selected to be satisfactory for a convergence criterion of 10 -5 in all cases. 3.6 code validation in order to assess the present numerical method we compare present results with previously published work by merkin (1977) and yih (2000) and good agreement is obtained as shown in table 2. table 1: grid-independence for pr=10, de=1 and r=1  (step size) ( , 0)f  0),(  cpu time (sec) 0.1 0.6711096013 0.52420031235 1.0140 0.01 0.67110538432 0.52420013563 10.4833 0.001 0.6711032946 0.52420009563 25.8026 table 2: values of nusselt number for different values of stream wise coordinate with de = 0, pr = 10 and r = 0  merkin (1977) yih (2000) present results 0.0 0.4212 0.4214 0.4213 1.0 0.4025 0.4030 0.4031 2.0 0.3443 0.3457 0.3458 3.0 0.2252 0.2267 0.2261 4. results and discussion comprehensive solutions have been obtained and are presented in figs. 3–14 and in tables 2–4. numerical calculations have been carried out for different values of deborah number (de), prandtl number (pr) and r. bhuvanavijaya and b. mallikarjuna/ journal of naval architecture and marine engineering, 13(2016) 63-78 natural convection on heat transfer flow of non-newtonian second grade fluid over horizontal circular cylinder … 73 radiation parameter (r) on flow velocity and temperature profiles, skin friction and nusselt number. throughout the calculation we have fixed the values of de = 1, pr = 10 and r = 1. 4.1 variation of deborah number fig. 3: velocity profile for different values of de fig. 4: temperature profile for different values of de fig. 5: skin friction coefficient for different values of de. fig. 6: nusselt number for different values of de table 3: values of ( , 0)f  and ( , 0)  for different values of de for pr = 10 and r = 1.0 0  0.5  1  1.5  de ( , 0)f  ( , 0)  ( , 0)f  ( , 0)  ( , 0)f  ( , 0)  ( , 0)f  ( , 0)  0.1 0 0.6734 0.2992 0.6645 0.5479 0.6407 0.7039 0.6007 0.5 0 0.6444 0.2672 0.6373 0.4963 0.6179 0.6539 0.5853 0.75 0 0.6309 0.2526 0.6243 0.4715 0.6065 0.6269 0.5766 1.0 0 0.6195 0.2405 0.6133 0.4506 0.5967 0.6302 0.5687 1.5 0 0.6010 0.2215 0.5954 0.4170 0.5803 0.5633 0.5551 2.0 0 0.5863 0.2069 0.5811 0.3908 0.5671 0.5312 0.5436 figs. 3 & 4 represent velocity and temperature profiles for different values of deborah number. we observe that increase in de tends to decrease maximum velocity profile near the surface, while it increases far away from the cylinder surface. it is because of increase in de means increase in elasticity and decrease in viscosity of the fluid reduces the velocity boundary layer thickness. increase in deborah number results an enhancement in r. bhuvanavijaya and b. mallikarjuna/ journal of naval architecture and marine engineering, 13(2016) 63-78 natural convection on heat transfer flow of non-newtonian second grade fluid over horizontal circular cylinder … 74 temperature profile as shown in fig. 4. further it is clear that thermal boundary layer thickness increases with increase in de. for different values of deborah number de, the skin friction coefficient and nusselt number (rate of heat transfer) are shown in figs. 5 & 6 and table 3. both skin friction coefficient and nusselt number are decreased with increase in deborah number. as explained above, increase in de results depreciation in velocity gradient. the numerical values of average nusselt number for various values of deborah number are shown in table 4. this shows that increase in deborah number leads to depreciation in the average nusselt number. table 4: average nusselt number for different values of de for r=2 and pr=10 de 0.1  0.5  1.0  1.5  2.0  0.1 0.0560 0.2792 0.6627 1.0026 1.3182 0.5 0.0538 0.2681 0.6373 0.9669 1.2763 0.75 0.0527 0.2628 0.6250 0.9492 1.2549 1.0 0.0518 0.2582 0.6145 0.9339 1.2360 4.2 variation of prandtl number figs. 7 & 8 illustrate velocity and temperature profile along the radial direction, which is normal to the circular cylinder, for different values of prandtl number. from these figures it is observed that velocity and temperature profiles are reduced with increase in pr, and therefore it turns to reduce velocity and thermal boundary layer thickness. the reason is that for higher values of pr, heat is able to diffuse far away from the heated surface more rapidly and thermal boundary layer is thicker for smaller values of pr. the numerical results of skin friction coefficient and nusselt number are displayed graphically in figs. 9 & 10 and table 5 for different values of pr. it is noticed from the figures that increase in prandtl number accelerates, i.e. increase in skin friction coefficient and decelerates the nusselt number. the results of average nusselt number for different values of prandtl number pr presented in table 6. from these values we observed that there is an enhancement in average nusselt number with increase in pr. fig. 7: velocity profile for different values of pr fig. 8: temperature profile for different values of pr fig. 9: skin friction coefficient for different values of pr fig. 10: nusselt number for different values of pr r. bhuvanavijaya and b. mallikarjuna/ journal of naval architecture and marine engineering, 13(2016) 63-78 natural convection on heat transfer flow of non-newtonian second grade fluid over horizontal circular cylinder … 75 table 5: values of ( , 0)f  and ( , 0)  for different values of pr for de = 1.0 and r = 1.0 0 5.0 1 5.1 pr ( , 0)f  ( , 0)  ( , 0)f  ( , 0)  ( , 0)f  ( , 0)  ( , 0)f  ( , 0)  0.01 0 0.0945 0.3447 0.0943 0.6550 0.0940 0.9005 0.0934 0.71 0 0.2428 0.3111 0.2406 0.5889 0.2346 0.8040 0.2247 7.0 0 0.5509 0.2513 0.5456 0.4716 0.5311 0.6333 0.5068 10 0 0.6195 0.2405 0.6133 0.4506 0.5967 0.6032 0.5687 50 0 1.0218 0.1905 1.0105 0.3539 0.9798 0.4663 0.9278 100 0 1.2522 0.1692 1.2376 0.3132 1.1980 0.4099 1.1309 table 6: average nusselt number for different values of pr for de=1 and r=1 pr 0.1  0.5  1.0  1.5  2.0  0.71 0.0182 0.0908 0.2164 0.3293 0.4369 7 0.0413 0.2061 0.4907 0.7461 0.9885 10 0.0465 0.2317 0.5516 0.8385 1.1103 50 0.0766 0.3821 0.9086 1.3789 1.8210 4.3 variation of radiation parameter figs. 11& 12 show the behavior of velocity and temperature for different values of radiation parameter. it is noticed that increase in radiation (r) tends to depreciation in velocity and temperature profiles within the boundary layer, as well as reduced velocity and thermal boundary layer thickness. figs. 13 and 14 and table 7 shows the results of skin friction coefficient and nusselt number for different values of radiation parameter r. it is clear that the skin friction coefficient decreases, while nusselt number increases with increasing values of radiation parameter r. fig. 11: velocity profile for different values of r fig. 12: temperature profile for different values of r 4.4 variation of stream wise coordinate (ξ) figures 15 and 16 represent variation of transverse (stream wise) coordinate (ξ), on velocity and temperature distributions respectively. generally velocity is noticeably lowered with increasing migration from the leading edge i.e. larger  values (fig. 15). the maximum velocity is computed at the lower stagnation point (~0) for low values of radial coordinate (). the transverse coordinate clearly exerts a significant influence on momentum development. a very strong increase in temperature (), as observed in figure 16, is generated r. bhuvanavijaya and b. mallikarjuna/ journal of naval architecture and marine engineering, 13(2016) 63-78 natural convection on heat transfer flow of non-newtonian second grade fluid over horizontal circular cylinder … 76 throughout the boundary layer with increasing  values. the temperature field decays monotonically. temperature is maximized at the surface of the cylindrical body (= 0, for all ) and minimized in the free stream (= 5). although the behavior at the upper stagnation point (~2.0) is not computed, the pattern in figure 8b suggests that temperature will continue to progressively grow here compared with previous locations on the cylinder surface (lower values of ). it is an important to note that all the numerical results presented here should be claimed and are accurate and acceptable. these methods can easily be extended to more complicated heat transfer problems. fig. 13: skin friction coefficient for different values of r fig. 14: nusselt number for different values of r table 7: values of ( , 0)f  and ( , 0)  for different values of r for de = 1.0 and pr = 10 0  0.5  1  1.5  r ( , 0)f  ( , 0)  ( , 0)f  ( , 0)  ( , 0)f  ( , 0)  ( , 0)f  ( , 0)  0.5 0 0.5337 0.2541 0.5286 0.4771 0.5146 0.6412 0.4912 1.0 0 0.6195 0.2405 0.6133 0.4506 0.5967 0.6032 0.5687 1.5 0 0.6633 0.2341 0.6566 0.4380 0.6385 0.5852 0.6081 2.0 0 0.6904 0.2302 0.6834 0.4305 0.6644 0.5745 0.6325 2.5 0 0.7089 0.2276 0.7017 0.4255 0.6821 0.5673 0.6491 fig. 15: velocity profile for different values of ξ fig. 16: temperature profile for different values of ξ r. bhuvanavijaya and b. mallikarjuna/ journal of naval architecture and marine engineering, 13(2016) 63-78 natural convection on heat transfer flow of non-newtonian second grade fluid over horizontal circular cylinder … 77 5. conclusion a mathematical model has been investigated for free convection boundary layer flow of non-newtonian second grade fluid past an impermeable isothermal horizontal cylinder. the governing boundary layer equations for mass, momentum and energy are transformed into 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(1976) on two-dimensional, laminar natural convection cooling of a single, isothermal flush-mounted heater on a vertical wall inside an air-filled rectangular enclosure, the heat transfer problem of natural convection in a discretely heated enclosure is of great research interest as indicated by the considerable research activities on this subject. a natural convection heat transfer experiment in a tall vertical rectangular enclosure (aspect ratio 16.5) with an array of eleven discrete flush-heaters has been performed by keyhani et al. (1988). it was found that the discrete heating in the enclosure results in a significantly augmented local heat transfer rate over that for an enclosure with the uniformly heated vertical wall. a follow-up study (keyhani et al., 1988) for a vertical enclosure aspect ratio 4.5 with three flush heaters further revealed that the temperature of the heaters is strongly affected by the stratification of fluid inside the enclosure. moreover, the effects of enclosure width and prandtl number on natural convection liquid cooling of discrete flush heaters in a tall enclosure cooled from the top has been investigated experimentally and numerically (carmona and keyhani, 1989 and prasad et al., 1990). refai ahmed and yovanovich (1991) performed a numerical study to examine the influence of discrete heat source location on natural convection heat transfer in a vertical square enclosure. furthermore, the temperature field of natural convection within a discretely heated vertical enclosure with single and dual heaters configuration has been visualized using mach-zehnder interferometry (chadwick et al., 1991). the problem of convective heat transfer in an enclosure has been studied extensively because of the wide application of such process. ostrach (1988) provided a comprehensive review article and extensive bibliography on natural convection in cavities. other articles on the topic published are valencia and frederick (1989), selamet et al. (1992), hasnaoui et al. (1992), papanicolaou and gopalakrishna (1995), sundstrom and kimura (1996), hsu and chen (1996), elsherbiny et al. (1982), and nguyen and prudhomme (2001), among others, who investigated natural convection in rectangular enclosures under various configurations and orientations. anderson and lauiat (1986) studied the natural convection in a vertical square cavity heated from bottom and cooled from one side. convection in a similar configuration where the bottom wall of the rectangular cavity was partially heated with cooling from one side was studied by november and nansteel (1986). it was reported that the heated fluid layer near the bottom wall remains attached up to the turning corner. ganzarolli and milanez (1995) performed numerical study of steady natural convection in rectangular enclosures heated from below and symmetrically cooled from the sides. the size of the cavity was varied from square to shallow where the cavity width was varied from 1-10 times of the height. the heat source, which spanned the entire bottom wall, was either isothermal or at constant heat flux condition. aydin and yang (2000) numerically investigated the natural convection of air in a vertical square cavity with localized isothermal heating from below and symmetrical cooling from sidewalls. the top wall as well as non-heated parts of the bottom wall was considered adiabatic. the length of the symmetrically placed isothermal heat source at the bottom was varied. two counter rotating vortices were formed in the flow domain due to natural convection. the average nusselt number at the heated part of the bottom wall was shown to increase with increasing rayleigh number as well as with increasing length of the heat source. the geometry and coordinate system of the problem under consideration is depicted in fig. 1. it consists of a rectangular enclosure of dimension, w×h, whose sidewalls are kept at a constant low temperature, tc. the aspect ratio of the enclosure is defined as a = h/w. the bottom wall is maintained g. saha, s. saha, m.q. islam, m.a.r. akhanda / journal of naval architecture and marine engineering 4(2007) 1-13 natural convection in enclosure with discrete isothermal heating from below 3 at constant high temperature, th and length l. the remaining parts of the bottom wall and the entire upper wall are adiabatic. the present study reports the computations for enclosures at various aspect ratios, ranging from 0.5 to 1, and inclination angles from 0º to 30º. the natural convection parameter, grashof number, gr is varied from 103 to 106. also the ratio of the heating element to the enclosure width, ε = l/w and is varied from 0.2 to 0.8. fig. 1: schematic diagram of the physical system. 2. mathematical model natural convection is governed by the differential equations expressing conservation of mass, momentum and energy. the present flow is considered steady, laminar, incompressible and twodimensional. the viscous dissipation term in the energy equation is neglected. the boussinesq approximation is invoked for the fluid properties to relate density changes to temperature changes, and to couple in this way the temperature field to the flow field. then the governing equations for steady natural convection can be expressed in the dimensionless form as: u v 0 x y ∂ ∂ + = ∂ ∂ (1) ( ) 2 2u u p u uu v gr sin2 2x y x x y ⎛ ⎞ ⎜ ⎟ ⎜ ⎟ ⎝ ⎠ ∂ ∂ ∂ ∂ ∂ + = − + + + φ θ ∂ ∂ ∂ ∂ ∂ (2) ( ) 2 2v v p v vu v gr cos2 2x y y x y ⎛ ⎞ ⎜ ⎟ ⎜ ⎟ ⎝ ⎠ ∂ ∂ ∂ ∂ ∂ + = − + + + φ θ ∂ ∂ ∂ ∂ ∂ (3) 2 21u v 2 2x y pr x y ⎛ ⎞ ⎜ ⎟ ⎜ ⎟ ⎝ ⎠ ∂θ ∂θ ∂ θ ∂ θ + = + ∂ ∂ ∂ ∂ (4) where x and y are the coordinates varying along horizontal and vertical directions, respectively, u and v are the velocity components in the x and y directions, respectively, θ is the temperature, p is the pressure, and φ is the inclination angle of the enclosure with the horizontal direction, gr and pr, are the grashof number and prandtl number, respectively, and they defined as 3 2 g t w gr υ β∆ = and υ pr = α (5) the dimensionless parameters in the equations above are defined as follow: , w x x = , w y y = υ , uw u = υ , vw v = 2 2υ , pw p = ρ t tc t − θ = ∆ , t t th c∆ = − (6) g. saha, s. saha, m.q. islam, m.a.r. akhanda / journal of naval architecture and marine engineering 4(2007) 1-13 natural convection in enclosure with discrete isothermal heating from below 4 where ρ, β, υ, α and g are the fluid density, coefficient of volumetric expansion, kinematic viscosity, thermal diffusivity, and gravitational acceleration, respectively. the boundary conditions for the present problem are specified as follows: top wall: u v 0, 0 y ∂θ = = = ∂ right and left wall: u v 0, 0= = θ = bottom wall: u v 0= = 0 for 0 x 0.5 and 0.5 x 1 y 2 2 1 for 0.5 x 0.5 2 2 ∂θ ε ε = < < − + < < ∂ ε ε θ = − ≤ ≤ + (7) the average nusselt number (luo and yang, 2007) can be written as 0 1 nu dx y ε ∂θ = − ε ∂∫ (8) 3. finite element formulation the basic idea of the solution algorithm proposed in this paper is to use the two momentum equations for solving both of the velocity components, use the continuity equation for solving the pressure, and use the energy equation for solving the temperature. the element assumes linear interpolation for the velocity components, the pressure, and the temperature as ( ) i iu x, y n u= (9a) ( ) i iv x, y n v= (9b) ( ) i ip x, y n p= (9c) ( ) i ix, y nθ = θ (9d) where i = 1, 2, 3, 4, 5, 6; and ni is the element interpolation functions. the two momentum equations, eqs. (2), (3), are discretized using the conventional bubnov-galerkin’s method. however, a special treatment of the convection terms is incorporated. using the standard galerkin approach, each momentum equation is multiplied by weighting functions, ni, and then the diffusion terms are integrated by parts using the gauss theorem to yield the finite element equations in the form px u aau r r r= + + (10a) py v bav r r r= + + (10b) where the coefficient matrix a contains the known contributions from the convection term. the load vectors on the right-hand side of eqs. (10a), (10b) are defined by px i p r n d xω ∂ = − ω ∂∫ (11a) py i p r n d yω ∂ = − ω ∂∫ (11b) u i x y u u r n n n d x yγ ∂ ∂⎛ ⎞ = + γ⎜ ⎟∂ ∂⎝ ⎠ ∫ (11c) v i x y v v r n n n d x yγ ∂ ∂⎛ ⎞ = + γ⎜ ⎟∂ ∂⎝ ⎠ ∫ (11d) g. saha, s. saha, m.q. islam, m.a.r. akhanda / journal of naval architecture and marine engineering 4(2007) 1-13 natural convection in enclosure with discrete isothermal heating from below 5 a ir n (gr sin ) d ω = θ ω∫ (11e) b ir n (gr cos ) d ω = θ ω∫ (11f) where ω is the element area and γ is the element boundary. to derive discretized pressure equation, the method of weighted residuals is applied to the continuity equation, eq. (1), ( )i ii i x y n nu v n d u v d n u n v n d 0 x y x yω ω γ ∂ ∂∂ ∂ ⎛ ⎞⎛ ⎞ + ω = − + ω + + γ =⎜ ⎟ ⎜ ⎟∂ ∂ ∂ ∂⎝ ⎠ ⎝ ⎠ ∫ ∫ ∫ (12) where the integrations are performed over the element domain ω and along the element boundary γ; nx and ny are the direction cosines of the unit normal to the element boundary with respect to x and y directions, respectively. now we consider u ii i ij j i i j i p a u a u f n d x≠ ω ∂ = − + − ω ∂ ∑ ∫ (13a) v ii i ij j i i j i p a v a v f n d y≠ ω ∂ = − + − ω ∂ ∑ ∫ (13b) where uif and v if are the surface integral terms and the source term due to buoyancy. by assuming constant pressure gradient on an element, we get p ii i p u u k x ∂ = − ∂ (14a) p ii i p v v k y ∂ = − ∂ (14b) where u i ij j i j iii 1 u a u f a ≠ ⎛ ⎞ = − +⎜ ⎟ ⎝ ⎠ ∑ (15a) v i ij j i j iii 1 v a v f a ≠ ⎛ ⎞ = − +⎜ ⎟ ⎝ ⎠ ∑ (15b) p i i ii 1 k n d a ω ⎛ ⎞ = ω⎜ ⎟ ⎝ ⎠ ∫ (15c) by applying the element velocity interpolation functions, eqs. (9a), (9b), into the continuity equation, eq. (10), we have ( ) ( ) ( )i ij j j j i x y n n n u d n v d n u n v n d 0 x yω ω γ ∂ ∂ ω − ω + + γ = ∂ ∂∫ ∫ ∫ (16a) and introducing the nodal velocities uj and vj from eqs. (14a), (14b), then eq. (14) becomes, ( ) ( ) ( ) ( ) ( ) p pi i j j j j i i j jj j i x y n np p n k d n k d x x y y n n n u d n v d n u n v n d x y ω ω ω ω γ ∂ ∂∂ ∂ ω + ω = ∂ ∂ ∂ ∂ ∂ ∂ ω + ω − + γ ∂ ∂ ∫ ∫ ∫ ∫ ∫ (16b) finally, by applying the element pressure interpolation functions, eq. (9c), the above element equations can be written in matrix form with unknowns of the nodal pressures as ( )x y u v ck k p f f f+ = + + (17) where ( )px j jn nk n k dx xω ∂ ∂ = ω ∂ ∂∫ (18a) g. saha, s. saha, m.q. islam, m.a.r. akhanda / journal of naval architecture and marine engineering 4(2007) 1-13 natural convection in enclosure with discrete isothermal heating from below 6 ( )py j jn nk n k dy yω ∂ ∂ = ω ∂ ∂∫ (18b) ju j n f n u d xω ∂ = ω ∂∫ (18c) jv j n f n v d yω ∂ = ω ∂∫ (18d) ( )c x yf n u n v n d γ = − + γ∫ (18e) the above element pressure equations are assembled to form the global equations; boundary conditions for the specified nodal pressures are imposed prior to solving for the updated nodal pressures. the finite element equations corresponding to the energy equation are derived using an approach similar to that used in deriving element momentum equations. the standard galerkin method is applied to yield the element equations which can be written in matrix form as k rθ = (19) where i x y 1 r n n n d pr x yγ ∂θ ∂θ⎛ ⎞ = + γ⎜ ⎟∂ ∂⎝ ⎠ ∫ (20) these elements equations are again assembled to yield the global temperature equations. appropriate boundary conditions are applied prior to solving for the new temperature values. 4. numerical procedure the numerical procedure used to solve the governing equations for the present work is the combined finite element method. the application of this technique is well documented zienkiewicz and taylor (2000). it provides the smooth solutions at the interior domain including the corner regions. the nonlinear parametric solution method is chosen to solve the governing algebraic equations. this approach will result in substantially fast convergence assurance. a non-uniform triangular mesh arrangement is implemented in the present investigation especially near the heated wall to capture the rapid changes in the dependent variables. also six noded triangular elements are used in this paper since the six noded elements smoothly capture the non-linear variations of the field variables. all six nodes are associated with velocities as well as temperature, only the corner nodes are associated with pressure. solutions were assumed to converge when the following convergence criteria was satisfied for every dependent variables at every point in the solution domain 6new old old 10− ψ − ψ ≤ ψ (21) where ψ represents a dependent variable u, v, p, and θ. 5. results and discussion the working fluid is chosen as air with prandtl number, pr = 0.71. the normalized length of the constant heat source at the bottom wall, ε, is varied from 0.2 to 0.8. for each value of ε, the grashof number, gr, is varied from 103 to 106, the aspect ratio, a, is varied from 0.5 to 1 while the inclination angle, φ, is varied from 0º to 30º. to test and assess grid independence of the present solution scheme, many numerical runs are performed for higher grashof number as shown in table 1. these experiments reveal that a non-uniform spaced grid of 6394 elements for the solution domain is adequate to describe correctly the flow and heat transfer processes inside the enclosure. in order to validate the numerical model, the results are compared with those reported by sharif and mohammad (2005), for a = 1.0, gr = 103 to 106, ε = 0.2 and φ = 0°. in table 2, a comparison of the average nusselt number of the square enclosure is presented. the agreement is found to be excellent with a maximum discrepancy of about 1.01%, which validates the present computations indirectly. g. saha, s. saha, m.q. islam, m.a.r. akhanda / journal of naval architecture and marine engineering 4(2007) 1-13 natural convection in enclosure with discrete isothermal heating from below 7 table 1: comparison of the results for various grid dimensions (a = 1.0, gr = 106, ε = 0.2, φ = 0°). elements 1970 2902 3540 4608 4828 6394 12606 nu 16.229 16.854 16.379 16.487 16.534 16.581 16.581 table 2: comparison of the average nusselt number of the square enclosure for ε = 0.2 and φ = 0°. average nusselt number, nu gr sharif and mohammad (2005) present work error (%) 103 5.927 5.939 0.2 104 5.946 5.954 0.13 105 7.124 7.117 0.1 106 11.342 11.226 1.02 st re am lin es is ot he rm s gr = 10 3 gr = 106 fig. 2: evolution of the flow in the enclosure with the variation of gr for a = 1.0, ε = 0.4 and φ = 0º. 5.1 effect of grashof numbers: the evolution of the flow and thermal fields with grashof number for an enclosure of aspect ratio, a = 1 for a representative case with ε = 0.4 and φ = 0º is presented in fig. 2. for various gr = 103-106, the flow pattern is characterized by two symmetrical rolls with clockwise and anti-clockwise rotations inside the enclosure. the hot fluid rises in the central region as a result of buoyancy forces, and then it descends downwards along the vertical walls and turns horizontally to the central region after hitting the bottom wall. the flow then rises along the vertical symmetry axis and gets blocked at the adiabatic g. saha, s. saha, m.q. islam, m.a.r. akhanda / journal of naval architecture and marine engineering 4(2007) 1-13 natural convection in enclosure with discrete isothermal heating from below 8 top wall, which turns the flow horizontally towards the cold vertical walls. thus a pair of counterrotating rolls is formed in the flow domain. at gr = 103, as can be expected, heat transfer from the discrete heat source is essentially dissipated via a conduction-dominated mechanism as indicated by the isotherm pattern shown in fig. 2. for gr > 103, the buoyant convection flow in the central region between the rolls distorts the isotherms field. the distortion of the isotherm field increases with enhanced buoyancy as gr increases, where the heat transfer becomes increasingly advection dominated. with increase of gr to 106 a transformation from a primarily two symmetrical rolls pattern to a structure characterized by two large vortices near the central regions, moving towards upper wall. therefore, the prevailing conductive heat transfer for gr = 103 and the mushroom profile of the isotherms for gr = 106 are presented in fig. 2. also viscous forces are more dominant than the buoyancy forces at lower gr. at higher gr when the intensity of convection increases significantly, the core of the circulating rolls moves up and the isotherm patterns changes significantly indicating that the convection is the dominating heat transfer mechanism in the enclosure. streamlines isotherms a = 1 .0 a = 0 .5 fig. 3: streamlines and isotherms profiles for different aspect ratios with ε = 0.2, φ = 0º and gr = 104. 5.2 effect of aspect ratio: the buoyancy-driven flow and temperature fields inside the discretely heated enclosure of various aspect ratios are illustrated by means of contour maps of streamlines and isotherms, as exemplified in figs. 3 for two different aspect ratios of 0.5 and 1.0 with φ = 0º, ε = 0.2, and gr = 104. as expected, due to the cold vertical walls, fluids rise up from middle portion of the bottom wall and flow down along the two vertical walls forming two symmetric rolls with clockwise and anti-clockwise rotations inside the cavity for all aspect ratios. however, in the convection region adjacent to the heat source, the isotherms become thinner and denser producing higher temperature gradients (increasing the overall nusselt number) with increasing a, specially until the cavity changes from thin rectangle to square. this is due to the fact that the cavity volume increases with aspect ratio and more volume of cooling air is involved in cooling the heat source leading to better cooling effect. for a = 0.5, the two convection rolls appeared in the rectangular cavity, each half filled up with clockwise or anti-clockwise circulation in a square area. at gr = 104, the circulation intensity is not g. saha, s. saha, m.q. islam, m.a.r. akhanda / journal of naval architecture and marine engineering 4(2007) 1-13 natural convection in enclosure with discrete isothermal heating from below 9 much higher and the heat transfer is almost due to conduction, as evident from the isotherm plots (fig. 3). during conduction dominant heat transfer, the temperature contours with θ = 0.35 occur symmetrically near the side walls of the enclosure. the other temperature contours with θ ≥ 0.4 are nearly smooth curves which span from the middle-bottom of the enclosure and they are generally symmetric with respect to the vertical center line. with increase of the aspect ratio of the enclosure, the buoyant convection flow is increasingly strengthened, exhibiting a transformation from two square size recirculation rolls into a structure characterized by two rectangular high strength vortices. at a = 1.0, the circulation on each half of the cavity becomes stronger as they expand vertically and consequently, the temperature contour with θ = 0.15 starts getting shifted towards the side wall and they break into two symmetric contour lines. the presence of significant convection is also exhibited in other temperature contour lines which start getting deformed and pushed towards the top plate. streamlines isotherms ε = 0. 2 ε = 0. 6 fig. 4: streamlines and isotherms for different heat source ratios ε with a = 0.5, gr = 106, and φ = 0º. 5.3 effect of discrete heat source length: the flow and temperature fields in terms of computed streamlines and isotherms for two representative values of the dimensionless source length ε = 0.2 and 0.6 are shown in fig. 4. in each case, the flow descends downwards along the moving sidewalls and turns horizontally to the central region hitting the bottom wall. the circulation in each half of the cavity follows a progressive wrapping around the centers of rotation, and a more pronounced compression of the isotherms toward the boundary surfaces of the enclosure occur. visual examination of the streamlines does not reveal any significant difference among the different cases. however, noticeable difference is observed in the isotherm plots. for gr = 106, the temperature gradients near bottom and side walls tend to be significant leading to the development of a thermal boundary layer. due to greater circulations near the central core at the top half of the enclosure, there are small gradients in temperature whereas a large stratification zone of temperature is observed at the vertical symmetry line due to stagnation of flow. the convection region adjacent to the heat source becomes thinner and denser producing higher temperature gradients with increasing gr. the heat transfer rate affect significantly with the increasing ε because the energy transport increases due to the increased area of the heated port. since the isotherm plots change with gr, it is the parameter of focus in the analysis for all cases. for gr ≥ 104, the buoyancy becomes dominant the heat transport and the isotherms with high values tend to concentrate near the heat source surface. it is noticed that the temperature decreases from the bottom to the top along the centerline of the cavity for a particular value of ε. at a fixed height, the temperature increases as the heat source length ε grows. the temperature profiles clearly express the heat transfer behavior expected from the isotherms given in fig. 4, where the most intensive heat transfer region is located near the heat source surface due to the presence of large temperature gradients. the temperature gradients of the cooling air decrease as it ascends from the bottom. when the heat source length ε g. saha, s. saha, m.q. islam, m.a.r. akhanda / journal of naval architecture and marine engineering 4(2007) 1-13 natural convection in enclosure with discrete isothermal heating from below 10 increases, more heat is transferred into the system, thus the whole temperature level in the cavity is upgraded. st re am lin es is ot he rm s φ = 0º φ = 30º fig. 5: evolution of the flow in the enclosure with inclination angles for a = 1, gr = 105, and ε = 0.4. 5.4 effect of inclination angles: the evolution of the flow and thermal fields in the enclosure with increasing inclination are shown in fig. 5 for a representative case of aspect ratio a = 1 and gr = 105 with ε = 0.4. it is observed that for horizontal cavity (φ = 0°), where the buoyancy force is acting only in the y-direction, two recirculation cell is formed and the solution is symmetric about the vertical midline due to the symmetry of the problem geometry and boundary conditions. for the inclined enclosure this symmetry is completely destroyed due to the buoyancy force components acting in both x and y directions. the effect of cavity inclination is clearly visible on both the flow patterns and isotherms. this is evident at φ = 30°, when the left recirculating vortex becomes dominating in the enclosure while the right vortex is squeezed thinner and ultimately is divided into two minor corner vortices. this circulation inside the cavity is greater near the center and least at the wall due to no slip boundary conditions. when gr increases, the convection roll located at the left half of the square enclosure tends to merge in order to form a single large recirculation cell compared to two minor corner vortices. the isotherms are also adjusted according to the changes in the flow field and pushed towards the lower part of the right sidewall indicating the presence of a large temperature gradient there. g. saha, s. saha, m.q. islam, m.a.r. akhanda / journal of naval architecture and marine engineering 4(2007) 1-13 natural convection in enclosure with discrete isothermal heating from below 11 2 4 6 8 10 12 14 16 1000 10000 100000 1000000 gr n u a = 0.5 a = 1.0 φ = 10°, ε = 0.4 2 5 8 11 14 17 1000 10000 100000 1000000gr n u ε = 0.2 ε = 0.4 ε = 0.6 ε = 0.8 φ = 20°, a = 1.0 (a) (b) 2 5 8 11 14 17 20 1000 10000 100000 1000000gr n u 0º 10º 20º 30º a = 1.0, ε = 0.2 (c) fig. 6: variation of the nu at the heated surface with gr for various (a) aspect ratios, (b) heat source sizes, (c) inclination angles. 5.5 heat transfer: next attention is focused upon the influence of the aspect ratio, inclination angle and discrete heat source size on the heat transfer rate across the discretely heated enclosure. the variation of the average nusselt number, nu, at the heated surface with grashof number, gr, for the entire set of the heated surface lengths (ε), aspect ratios (a), and cavity inclination angles (φ) investigated are shown in fig. 6(a), (b) and (c), respectively, from which some interesting trends are observed. in general, the average nusselt number remains invariant up to a certain value of grashof number and then increases briskly with increasing grashof number. for low grashof number, the curves maintain a flat trend that means low temperature gradients but nu increases rapidly with gr especially for gr > 104. for a particular grashof number, the average nusselt number increases with increasing aspect ratio. these variations of the average nusselt number differ greatly at higher grashof number and vice versa. from these observations, it can be concluded that the overall heat transfer process improves as the aspect ratio increases until the cavity becomes square. the variation of the average nusselt number against gr is shown in fig. 6(b) for various values of ε. concentrating on each plot separately for a particular value of ε, a trend of nu increasing with gr, is observed. when gr ≥ 104, the buoyancy aids more and more in the heat transfer process which results in more rapid increase of nu. an important information obtained from this analysis is the effect of heat g. saha, s. saha, m.q. islam, m.a.r. akhanda / journal of naval architecture and marine engineering 4(2007) 1-13 natural convection in enclosure with discrete isothermal heating from below 12 source length on the heat transfer rate. due to the symmetrical boundary conditions, two symmetric convection cells are generated and their interface behaves like an insulator. the centre of the heat source surface becomes the stagnation point of the heat transfer area, and attains the maximum temperature and minimum heat transfer rate. an increase in ε increases the rate of formation of convection cells which in turns decreases the average nusselt number from the central area of the cavity. maximum nu is obtained at small heat source size for higher value of gr while φ = 0° for a = 0.5 and φ = 30° for a = 1.0. 6. conclusion natural convection in two-dimensional rectangular enclosure where the top wall is considered adiabatic, two vertical walls are maintained at constant low temperature, and the bottom wall is maintained at high temperature has been analyzed numerically using the finite element method. the resulting processes are investigated to yield quantitative results regarding the cooling effects. the main parameters of interest are grashof number, gr, the dimensionless heat source length, ε, the inclination angle with horizontal axis, φ, and the aspect ratio of the cavity, a. the resulting flow consists of two counter-rotating vortices. as far as the temperature field is concerned, at low values of grashof number, the temperature is found to be more evenly distributed within the enclosure, and a relatively large region of the enclosure is affected by the heat source. as grashof number increases and natural convection dominates, the temperature variation is restricted over a gradually diminishing region around the heat source. it is also noticed that the heat-affected region becomes larger with the increasing heat source length. the average or overall nusselt number increases mildly with cavity inclination for gr = 104 while it increases much more rapidly at φ = 30º for higher grashof number. the effect of enclosure aspect ratio on the average nusselt number of the discrete heaters tends to improve with the increase of the grashof number. references aydin, o. and yang, j. 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(2000): the finite element method, 5th edition. oxford: butterworthheinemann. microsoft word p7_14.doc 1813 8535 © aname publication. all right reserved journal of naval architecture and marine engineering june, 2006 http://jname.8m.net thermal radiation interaction with unsteady mhd flow past a vertical porous plate immersed in a porous medium md abdus samad and mohammad mansur rahman department of mathematics, university of dhaka, dhaka-1000, bangladesh, email: mansurdu@yahoo.com, fax: 0088 02 8615583 abstract a study of unsteady mhd free convection flow through a porous vertical flat plate immersed in a porous medium in presence of magnetic field with radiation has been analyzed. introducing a time dependent suction to the plate, a similarity procedure has been adopted by taking a time dependent similarity parameter. in this analysis we consider a darcy-forchhemier model and the corresponding momentum and energy equations have been solved numerically, for cooling and heating of the plate by employing nachtsheim-swigert iteration technique along with the sixth order runge-kutta integration scheme. non-dimensional velocity and temperature profiles are then presented graphically for different values of the parameter entering into the problem. during the process of numerical computations the skin-friction coefficient (viscous drag) and the rate of heat transfer (nusselt number), which are of physical interest, are sorted out and presented in the form of tables. keywords: thermal radiation, mhd, unsteady, suction, porous medium nomenclature fc skin friction coefficient pc specific heat at constant pressure da local darcy number ec eckert number fs local forchhemier number 1fs modified forchhemier number gr local grashof number g acceleration due to gravity m local magnetic field parameter n radiation parameter nu nusselt number n nonnegative integer pr prandtl number rq radiative heat flux re local reynolds number t temperature within the boundary layer )(tt temperature at the plate ∞t temperature of the ambient fluid t time u velocity along x -axis v velocity along y -axis 0v suction parameter )(0 tv time dependent suction velocity x coordinate along the plate y coordinate normal to the plate greek α thermal diffusivity β coefficient of volume expansion δ characteristic length scale ρ density of the fluid µ coefficient of dynamic viscosity υ coefficient of kinematic viscosity 1σ stefan-boltzmann constant κ thermal conductivity η similarity parameter θ dimensionless temperature η∆ step size 1. introduction the effect of free convection on the accelerated flow of a viscous incompressible fluid past an infinite vertical porous plate with suction has many important technological applications in the astrophysical, geophysical and engineering problems. the heating of rooms and buildings by the use of radiators is a familiar example of heat transfer by free convection. heat losses from hot pipes, ovens etc surrounded by cooler air, are at least in part, due to free convection. the problem of heat transfer in a vertical channel has been studied in recent years as a model for the re-entry problem. this is due to the significant role of thermal radiation in surface heat transfer when convection heat transfer is similar, particularly in free convection problems involving absorbing emitting fluids. soundalgekar and takhar (1981) studied radiation effects on free convection flow of a gas past a semimd abdus samad and mohammad mansur rahman / journal of naval architecture and marine engineering 3(2006) 7-14 8 infinite flat plate. hossain and takhar (1996) studied the effect of radiation using the rosseland diffusion approximation on mixed convection along a vertical plate with uniform free stream velocity and surface temperature. ali et al. (1984) studied radiation effect on natural convection flow over a vertical surface in a gray gas. following ali et al. mansour (1990) studied the interaction of mixed convection with thermal radiation in laminar boundary layer flow over a horizontal, continuous moving sheet with suction/injection. whereas albraba et al. (1992) studied the same problem considering magnetic effect taking into account the binary chemical reaction and soret doufour effects. seigel (1958) first studied transient free convection flow past a semi-infinite vertical plate by an integral method. since then many researchers have been published papers on free convection flow past a semi-infinite vertical plate. soundalgekar et al. (1981) studied free convection flow past a vertical porous plate. yamamoto et al. (1976) investigated the acceleration of convection in a porous permeable medium along an arbitrary but smooth surface. raptis (1983) studied free convection in a porous medium bounded by an infinite plate. raptis and perdikis (1985) studied numerically free convection flow through a porous medium bounded by a semi-infinite vertical porous plate. sattar (1992) studied the same problem and obtained analytical solution by the perturbation technique adopted by singh and dikshit (1988). sattar et al. (2000) studied unsteady free convection flow along a vertical porous plate embedded in a porous medium. very recently, alam and rahman (2005) studied mhd free convection flow and mass transfer along a vertical porous plate in a porous medium considering soret-dofour effects. sattar and kalim (1996) studied the effects of unsteady free convection interaction with thermal radiation in a boundary layer flow. el-arabawy (2003) studied the effect of suction/injection on a micropolar fluid past a continuously moving plate in the presence of radiation. recently, ferdows et al. (2004) investigate numerically the thermal radiation interaction with convection in a boundary layer flow at a vertical plate with variable suction. in the present paper we investigate the thermal radiation interaction on an absorbing emitting fluid permitted by a transversely applied magnetic field past a moving vertical porous plate embedded in a porous medium with time dependent suction and temperature. the similarity solutions are then obtained numerically for various parameters entering into the problem and discussed them from the physical point of view. 2. mathematical formulation let us consider the problem of an unsteady mhd free convection flow of a viscous, incompressible and electrically conducting fluid along a vertical porous flat plate under the influence of a uniform magnetic field. the flow is assumed to be in the x -direction, which is taken along the plate in the upward direction and y -axis normal to the plate. initially it is assumed that the plate and the fluid are at a constant temperature ∞t at all points. at time 0>t the plate is assumed to be moving in the upward direction with the velocity )(tu and there is a suction velocity )(0 tv taken to be a function of time, the temperature of the plate raised to )(tt where ∞> ttt )( . the plate is considered to be of infinite length, all derivatives with respect to x vanish and so the physical variables are functions of y and t only. the flow configuration and coordinate system are shown in fig. 1. x u v0(t) t ∞t v ∞u u(t) y fig. 1: flow configuration and coordinate system. the fluid is considered to be gray; absorbing-emitting radiation but non-scattering medium and the roseland approximation is used to describe the radioactive heat flux in the x-direction is considered negligible in comparison to the y-direction. assuming that the boussinesq and boundary-layer approximations hold and using the darcy-forchhemier model, the governing equations for the problem are as follows: md abdus samad and mohammad mansur rahman / journal of naval architecture and marine engineering 3(2006) 7-14 9 continuity equation 0= ∂ ∂ y v (1) momentum equation 2 2 0 02 2 )( u k b u k u b ttg y u y u v t u −−−−+ ∂ ∂ = ∂ ∂ + ∂ ∂ ∞ υ ρ σ βυ (2) energy equation y q cy u cy t y t v t t r pp ∂ ∂ −⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ ∂ ∂ + ∂ ∂ = ∂ ∂ + ∂ ∂ ρ υ α 1 2 2 2 (3) where vu,( ) are the components of velocity along the x -and y -directions respectively, t is the time, υ is the kinematic viscosity, ρ is the density of the fluid, 0g is the acceleration due to gravity, β is the coefficient of volume expansion, 0b is the magnetic induction, t and ∞t are the temperature of the fluid within the boundary layer and in the free stream respectively, σ is the electric conductivity, α is the thermal diffusivity and pc is the specific heat at constant pressure, k is the permeability of the porous medium. the corresponding boundary conditions for the above problem are given by 0 .,0 ,0)(),(),( 0 > ⎭ ⎬ ⎫ ∞→== ==== ∞ tfor yasttu yatttttvvtuu (4) by using rosseland approximation rq takes the form y t k q r ∂ ∂ −= 4 1 1 3 4σ (5) where 1σ , the stefan-boltzamann constant and 1k , the mean absorption coefficient. it is assumed that the temperature differences within the flow are sufficiently small such that 4t may be expressed as a linear function of temperature. this is accomplished by expanding 4t in a taylor series about ∞t and neglecting higher-order terms, thus 434 34 ∞∞ −≈ tttt . (6) using (5) and (6) in equation (3) we have 2 2 1 3 1 2 2 2 4 y t kc t y u cy t y t v t t pp ∂ ∂ +⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ ∂ ∂ + ∂ ∂ = ∂ ∂ + ∂ ∂ ∞ ρ συ α (7) in order to obtain a similarity solution in time of the problem, we introduce a similarity parameter δ as )(tδδ = , (8) such that δ is a length scale. with this similarity parameter, a similarity variable is then introduced as δ η y = . (9) in terms of this length scale, a convenient solution of the equation (1) can be taken as 0)( vtvv δ υ −== , where 0v is the mass transfer parameter, which is +ve for suction and –ve for injection. following sattar and hossain (1992) )(tu and )(tt are now consider to have the following form: ⎪⎭ ⎪ ⎬ ⎫ −+= = ∞∞ + ,)()( )( 2 *0 22 *0 n n ttttt utu δ δ (10) md abdus samad and mohammad mansur rahman / journal of naval architecture and marine engineering 3(2006) 7-14 10 where n is a non-negative integer and 0u , 0t are respectively the free stream velocity and mean temperature. here 0 * δ δ δ = , where 0δ is the value of δ at 0tt = . now to make the equations (2) and (7) dimensionless, we introduce the following transformations: ⎪⎭ ⎪ ⎬ ⎫ −+= == ∞∞ + ).()( ),()()( 2 *0 22 *0 ηθδ ηδ n n tttt futftuu (11) using equations (8), (9), and (11) the equations (2) and (7) are become [using the analysis of hashimoto (1957), sattar et al. (2000) and sattar and maleque (2000)] 0) 1 44(')2('' 210 =−++++−++ fda fs grf da mnfvf θη (12) 0' 43 pr3 43 pr12 ' 43 pr3 )2('' 20 =⎟ ⎠ ⎞ ⎜ ⎝ ⎛ + +⎟ ⎠ ⎞ ⎜ ⎝ ⎛ + −⎟ ⎠ ⎞ ⎜ ⎝ ⎛ + ++ ecf n n n nn n n v θθηθ (13) where 0 2 000 )( u ttg gr υ δβ ∞−= is the local grashof number, υρ δσ 220bm = is the local magnetic parameter, α υ =pr is the prandtl number, 2δ k da = is the local darcy number, υ δ0re v = is the local reynolds number, δ b fs = is the forchhemier number and re 22 0 1 + ⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ = n b fs δ δ δ is the modified forchhemier number, 3 1 1 4 ∞ = t n σ κκ is the radiation number. the corresponding boundary conditions for 0>t are given by ⎭ ⎬ ⎫ ∞→== === .0,0 ,01,1 ηθ ηθ asf atf (14) 4. numerical computation the numerical solutions of the nonlinear differential equations (12)-(13) under the boundary conditions (14) have been performed by applying nachtsheim-swigert (1965) iteration technique along with the sixth order runge-kutta integration scheme. we have chosen a step size of 01.0=∆η to satisfy the convergence criterion of 610− in all cases. the value of ∞η was found to each iteration loop by ηηη ∆+= ∞∞ . the maximum value of ∞η to each group of parameters 0v , m , n , pr , gr , da , and 1fs determined when the value of the unknown boundary conditions at 0=η not change to successful loop with error less than 610− . in order to verify the effects of the step size ( )η∆ we ran the code for our model with three different step sizes as 01.0=∆η , 005.0=∆η , 001.0=∆η and in each case we found excellent agreement among them. fig. 2 shows the velocity profiles for different step sizes. 5. results and discussion for the purpose of discussing the results, the numerical calculations are presented in the form of non-dimensional velocity and temperature profiles. numerical computations have been carried out for different values of the parameters entering into the problem. the values of grashof number ( )gr are taken to be large from the physical point of view. the large grashof number values correspond to free convection problem. the effects of suction parameter 0v on the velocity and temperature profiles are shown in fig. 3 and fig. 4 respectively. from fig. 3 we found that the velocity decreases with the increase of suction for cooling of the plate and increases for the heating of the plate. it is also clear that suction stabilizes the boundary layer growth. fig. 4 reveals that temperature decreases with the increase of the suction parameter. md abdus samad and mohammad mansur rahman / journal of naval architecture and marine engineering 3(2006) 7-14 11 0 1 2 3 4 5 0 0.2 0.4 0.6 0.8 1 η f curves ∆η = 0.01 ∆η = 0.005 ∆η = 0.001 fig. 2: velocity profiles for different step sizes 0 1 2 3 4 -0.5 0 0.5 1 η f v0 = 0.0, 0.5, 1.0, 2.0 v0 = 0.0, 0.5, 1.0, 2.0 gr = -10 gr = +10 fig. 3: velocity profiles for different values of suction parameter ( 0v ) fig. 5 and fig. 6 show the effects of prandtl number ( pr ) on the velocity as well as temperature profiles. from fig. 5 we see that for cooling plate velocity profiles decrease with the increase of pr whereas these profiles increase with the increase of pr for a heating plate. for cooling plate pr has decreasing effect on the temperature profiles. the effects of radiation parameter ( n ) on the velocity profile is shown in the figure 6 for both cooling and heating plates. this figure shows that velocity decreases with the increase of the radiation parameter. this parameter has reverse effects on the heating plate. fig. 8 shows the effect of n on the temperature profiles. for large n , it is clear that temperature decreases more rapidly with the increase of n . therefore using radiation we can control the flow characteristic and temperature distribution. the effect of magnetic field parameter on the velocity profiles are shown in fig. 9. it is observed from this figure that the magnetic field has decreasing effect on the velocity field for cooling plate and increasing effect for heating plate. magnetic field lines act as a string to retard the motion of the fluid as –consequence the rather heat transfer increases. fig. 10 and fig. 11 show the effect of non-negative integer n on the velocity and temperature profiles. from fig. 10 we see that velocity profiles decrease for the cooling plate while it increases for the heating plate with the increase of n . here n 0= case represents the velocity as well as temperature is time independent. the nonzero values of n represents the case of time dependent velocity and temperature. analyzing figs. 10 and 11 we can say flow characteristics strongly depend on the values of n . fig. 12 shows the effects of darcy number ( da ) on the velocity profiles for cooling as well as heating of the plate. for a cooling plate fluid velocity increases, whereas for a heating plate it decreases with increase of da . darcy number is the measurement of the porosity of the medium. as the porosity of the medium increases, the value of da increases. for large porosity of the medium fluid gets more space to flow as a consequence its velocity increases. the effect of the modified forchhemier number on the velocity field is shown in fig. 13. it is observed from this figure that modified forchhemier number has decreasing effect on the velocity field for the cooling plate while increasing effect for the heating plate. finally, the effects of the above mentioned parameters on the skin-friction coefficient and the nusselt number are shown in tables i-ii. these effects as observed from the tables i-ii are found to agree with the effects on the velocity and temperature profiles hence any further discussions about them seem to be redundant. 6. conclusions in this paper we have studied the thermal radiation interaction with unsteady mhd boundary layer flow past a continuously moving vertical porous plate immersed in a porous medium. from the present study we can make the following conclusions: (i) the suction stabilizes the boundary layer growth. (ii) the velocity profiles increase whereas temperature profiles decrease with an increase of the free convection currents. (iii) using magnetic filed we can control the flow characteristics and heat transfer. (iv) radiation has significant effects on the velocity as well as temperature distributions. md abdus samad and mohammad mansur rahman / journal of naval architecture and marine engineering 3(2006) 7-14 12 (v) flow characteristics strongly depend on the nonnegative integer n. (vi) large darcy number leads to the increase of the velocity profiles. 0 1 2 3 4 0 0.2 0.4 0.6 0.8 1 η θ v0 = 0.0, 0.5, 1.0, 2.0 gr = +10 fig. 4: temperature profiles for different values of suction parameter ( 0v ) 0 1 2 3 -1 -0.5 0 0.5 1 pr = 0.1, 0.71, 1.0, 7.0, 10.0 pr = 0.1, 0.71, 1.0, 7.0, 10.0 gr = +10 gr = -10 η f fig. 5: velocity profiles for different values of prandtl number ( pr ) 0 1 2 3 4 0 0.2 0.4 0.6 0.8 1 pr = 0.1, 0.71, 1.0, 7.0, 10.0 gr = +10 η θ fig. 6: temperature profiles for different values of prandtl number ( pr ) 0 1 2 3 4 -0.8 -0.4 0 0.4 0.8 η f n = 0.01, 0.10, 0.50, 1.0, 5.0 n = 0.01, 0.10, 0.50, 1.0, 5.0 gr = 10 gr = + 10 fig. 7: velocity profiles for different values of radiation (n) 0 1 2 3 4 0 0.2 0.4 0.6 0.8 1 η θ n = 0.01, 0.10, 0.50, 1.0, 5.0 gr = + 10 fig. 8: temperature profiles for different values of radiation parameter (n) 0 1 2 3 4 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 η f m = 0.0, 1.5, 3.0, 5.0 m = 0.0, 1.5, 3.0, 5.0 gr = +10 gr = 10 fig. 9: velocity profiles for values of magnetic parameter (m) md abdus samad and mohammad mansur rahman / journal of naval architecture and marine engineering 3(2006) 7-14 13 0 1 2 3 4 -0.4 0 0.4 0.8 η f n = 0.0, 0.5, 1.0, 2.0 n = 0.0, 0.5, 1.0, 2.0 gr = +10 gr = -10 fig. 10: velocity profiles for different values nonnegative integer (n) 0 1 2 3 4 0 0.2 0.4 0.6 0.8 1 η n = 0.0, 0.5, 1.0, 2.0 gr = +10 θ fig. 11: temperature profiles for different values of non-negative integer (n) 0 1 2 3 4 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 η gr = +10 f da = 0.1, 0.25, 0.5, 1.0 da = 0.1, 0.25, 0.5, 1.0 gr = -10 fig. 12: velocity profiles for different values of darcy number (da) 0 1 2 3 4 -0.4 0 0.4 0.8 η gr = +10 f fs = 0.0, 0.5, 1.0, 2.0 fs = 0.0, 0.5, 1.0, 2.0 gr = -10 fig. 13: velocity profiles for different values of forchhemier number ( 1fs ). table i: skin-friction coefficients and rate of heat transfer for different values of ec =0.2, n =0.5, m =0.5, 1fs =1.0, n=1.0 and da =0.25. 0v gr pr fc un 0.0 10 0.71 -2.0239 0.96755 0.5 10 0.71 -2.20139 1.01875 1.0 10 0.71 -2.39638 1.07183 2.0 10 0.71 -2.84188 1.18337 0.0 -10 0.71 -6.16298 0.84126 0.5 -10 0.71 -6.51201 0.88650 1.0 -10 0.71 -6.87269 0.93390 2.0 -10 0.71 -7.62243 1.03513 0.5 10 0.1 -1.84548 0.37878 0.5 10 0.71 -2.20139 1.01875 0.5 10 1.0 -2.29305 1.21177 0.5 10 7.0 -2.94205 3.25812 0.5 10 10.0 -3.06843 3.91936 0.5 -10 0.1 -7.04351 0.35009 0.5 -10 0.71 -6.51201 0.88650 0.5 -10 1.0 -6.39626 1.04436 0.5 -10 7.0 -5.66090 2.75038 0.5 -10 10.0 -5.52584 3.32275 table ii: skin-friction coefficients and rate of heat transfer for different values of 0v =0.5, gr =10, pr =0.71, ec =0.2, 1fs =1.0. m n n da fc un 0.0 0.5 1 0.25 -2.11351 1.01995 1.5 0.5 1 0.25 -2.37175 1.01636 3.0 0.5 1 0.25 -2.61501 1.01279 5.0 0.5 1 0.25 -2.91936 1.011811 0.5 0.01 1 0.25 -1.71445 0.17387 0.5 0.10 1 0.25 -1.92589 0.51118 0.5 0.50 1 0.25 -2.20139 1.01875 0.5 1.0 1 0.25 -2.32422 1.28088 0.5 5.0 1 0.25 -2.51338 1.74547 0.5 0.5 0.0 0.25 -1.16364 0.54941 0.5 0.5 0.5 0.25 -1.73516 0.81536 0.5 0.5 1.0 0.25 -2.20139 1.01875 0.5 0.5 2.0 0.25 -2.95768 1.33587 0.5 0.5 1 0.1 -3.68945 0.99792 0.5 0.5 1 0.25 -2.20139 1.01875 0.5 0.5 1 0.5 -1.53694 1.02612 0.5 0.5 1 1.0 -1.14466 1.02969 md abdus samad and mohammad mansur rahman / journal of naval architecture and marine engineering 3(2006) 7-14 14 references alabraba, m. a., bestman, a. r. and ogulu, a. (1992): laminar convection in binary mixer of hydromagnetic flow with radiative heat transfer. astophys. sapce sci., vol.195, pp. 431-445. alam, m. s. and rahman, m. m. (2005): dufour and soret effects on mhd free convective heat and mass transfer flow past a vertical porous flat plate embedded in a porous medium. j. nav. arc. mar. eng., vol.2(1), pp.55-65. ali, m. m., chen, t. s. and armaly, b. f. ( 1984): natural convection radiation interaction in boundary layer flow over horizontal surfaces. aiaa journal., vol.22(12), pp. 797-1803. el-arabawy, h. a. m. (2003): effect of suction/injection on a micropolar fluid past a continuously moving plate in the presence of radiation. intl. j. heat mass trans., vol.46, pp.1471-1477. ferdows, m., sattar, m. a. and siddiki, m. n. a. 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(1976): flow with convection acceleration through a porous medium. j. eng. math., vol.10(1), pp.41-54. microsoft word jname415.doc 1813-8535 © 2007 aname publication. all rights reserved. journal of naval architecture and marine engineering june, 2007 http://jname.8m.net ultimate strength of square plate with rectangular opening under axial compression m. suneel kumar1*, p. alagusundaramoorthy2 and r. sundaravadivelu3 1former doctoral student, department of ocean engineering, iitmadras, chennai – 600 036, india, e-mail: suneel78@rediffmail.com (* corresponding author) 2associate professor, composites technology centre, iitmadras, chennai – 600 036, india, e-mail: aspara0@iitm.ac.in 3professor and head, department of ocean engineering, iitmadras, chennai – 600 036, india, e-mail: rsun@iitm.ac.in abstract unstiffened plates are integral part of ship structures, offshore oil platforms, lock gates and floating docks. openings are provided in these plates for access and maintenance. provision of opening influences the ultimate strength of plate elements. in this paper the effect of increase in the size of rectangular opening along the loading direction on the ultimate strength is determined using nonlinear finite element analysis. a general purpose finite element software ansys is used for carrying out the study. the software is validated for the ultimate strength of unstiffened plate under axial compression. a parametric study is done for different plate slenderness ratios and by varying the area ratio of opening to plate to determine the effect of ultimate strength on the size of rectangular opening. it is found that increase in area ratio along the loading direction decreases the ultimate strength. the variation in ultimate strength varies linearly for plate slenderness ratio less than 2.23 and varies nonlinearly for plate slenderness ratio beyond 2.23 for area ratio ranging between 0.02 – 0.18. based on nonlinear regression analysis, a design equation is proposed for square plate with rectangular opening under axial compression. keywords: unstiffened plate, ultimate strength, rectangular opening, axial compression, design equation nomenclature a depth of opening b width of opening a length of plate ac area of opening ap area of plate ar area ratio (area of opening to area of plate) b width of plate e young’s modulus of elasticity t thickness of plate w lateral deflection at the centre of plate β plate slenderness ratio υ poisson’s ratio σu ultimate stress of plate σy yield stress of plate 1. introduction thin plates in between stiffeners of a stiffened plate are integral part of ship structures, offshore oil platforms, lock gates and floating docks. these stiffened plates are designed to withstand the axial compression due to sagging and hogging moments. a typical ship deck in between the bulkheads and deep longitudinals is shown in figure 1. the analysis of typical stiffened plate structure in a ship can be performed at grillage level, stiffened panel level between two adjacent transverses, and bare plate element level between longitudinal and transverse stiffeners. local buckling and collapse of plating between is a basic failure mode and is important to evaluate the exact strength for safe design. the m. suneel kumar et al., / journal of naval architecture and marine engineering 4(2007) 15-26 ultimate strength of square plate with rectangular opening under axial compression 16 bending rigidities of the boundary edges of plates in between transverse frames and between longitudinal stiffeners are quite high compared to that of the plate itself. the rotational restraints along the plate edges can be considered to be small for plates subjected to axial compression. hence, the plate elements in the present study are considered as simply supported along all the edges. rectangular openings are necessitated for piping, ducts and other accessories for maintenance purposes. it is evident that opening reduces the ultimate strength of plates. but, the effect of rectangular opening is to be included in the ultimate strength formulations for efficient and safe design purposes. figure 1: typical ship deck 2. literature review a brief review of literature on the ultimate strength of plate elements with different openings and subjected to various loads is presented here. bradfield (1980) presented the details of tests conducted on plates under axial compression with control on initial out-of-flatness and residual stresses. a large deflection elastic plastic analysis is carried out for tested specimens that showed good comparison. narayanan and chow (1984) developed design charts based on ultimate capacity of uniaxially compressed perforated plates with square and circular openings. roberts and azizian (1984) generated interaction curves for ultimate strength of square plates with central square and circular holes subjected to uniaxial compression, biaxial compression and pure shear. narayanan and chan (1985) presented design charts based on ultimate strength of plates containing circular holes under linearly varying edge displacements. yettram and brown (1985) studied the stability behaviour of flat square plates with central square perforations. guedes soares (1988) derived two design equations for merchant ships and warships which are weighted by the probability density function. guedes soares (1992) presented review of simple design methods for plate under uniaxial loads by incorporating partial safety factor which are applied on ro-ro ships, container ships and bulk carriers. jwalamalini et al., (1992) developed the design charts for the stability of simply supported square plate with opening under in-plane loading as uniform compression and trapezoidal loading. madasamy and kalyanaraman (1994) presented the analysis of plated structures with rectangular cutouts and internal supports using the spline finite strip method. motok (1997) carried out stress concentration studies on the contour of a plate opening of an arbitrary corner radius of curvature. shanmugam (1997) reviewed the effects of openings in plate elements subjected to uniaxial compression, biaxial compression and pure shear in stiffened plates, shear webs and cold formed steel sections. shanmugam et al., (1999) presented the design formula for axially compressed perforated plates with circular openings under axial compression for simply supported and clamped boundary conditions. paik et al., (2001) presented ultimate strength formulations for ship plating under combined biaxial compression/tension, edge shear, and lateral pressure loads. toulios and caridis (2002) carried out a numerical study on the effect of aspect ratio on the buckling and collapse behaviour of flat bar stiffened plates loaded in uniaxial compression. khaled el-sawy et al., (2004) employed finite element method to determine the elastoplastic buckling stress of uniaxially loaded simply supported square and rectangular plates with circular rectangular opening m. suneel kumar et al., / journal of naval architecture and marine engineering 4(2007) 15-26 ultimate strength of square plate with rectangular opening under axial compression 17 openings. the study recommended avoiding cutouts near the plate edge since this decreases considerably the critical buckling stress, especially when the failure occurs in the elasto-plastic buckling mode. based on the literature review, it is observed that there is lack of studies on the ultimate strength of square plate with central rectangular opening along the loading direction under axial compression. also, there is need for a design equation for the centrally located rectangular opening and that motivated the present study. 3. numerical study ultimate strength of unstiffened plate without opening is found to be maximum for an aspect ratio of a/b = 1.0 (toulios and caridis, 2002). so, an unstiffened plate of size 500 mm x 500 mm (a x b) is considered for the study. the thickness of plate is varied as 5 mm, 6 mm, 8 mm, 10 mm, 12 mm and 15 mm to obtain plate slenderness ratio ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ = e σ t b β y in the practical range of 1.0 4.5 used in ship construction. rectangular opening is provided in the centre of the plate as shown in the figure 2. the depth of opening (a) is kept constant as 100 mm throughout the study. the width of opening (b) is varied as 50 mm, 100 mm, 150 mm, 200 mm, 250 mm, 300 mm, 350 mm, 400 mm and 450 mm. area ratio (ar) of rectangular opening is defined as the ratio of area of opening (ac) to area of plate (ap). in this study, the area ratio (ar) is varied as 0.02, 0.04, 0.06, 0.08, 0.10, 0.12, 0.14, 0.16 and 0.18. the details of the parametric study are given in table 1. the yield strength of plate ( )yσ is assumed as 250 n/mm2 with young’s modulus of elasticity (e) as 2 x 105 n/mm2 and poisson’s ratio ( )υ of 0.3. all the edges of the plate are assumed to be simply supported. the unloaded edges are allowed to deform inplane but remains straight. this is achieved by coupling the deformation of nodes in that direction. this condition is to generate the actual situation of unstiffened plate between longitudinal and transverse stiffeners. the reaction edge is constrained to obtain an equal force caused due to loading edge. figure 2: unstiffened plate with centrally positioned rectangular opening b b a a m. suneel kumar et al., / journal of naval architecture and marine engineering 4(2007) 15-26 ultimate strength of square plate with rectangular opening under axial compression 18 table 1: details of parametric study sl. no. specimen width of opening (b) mm thickness of plate, (t) mm plate slenderness ratio, (β) area of opening to plate (ar = ac/ap) ultimate load (pu) kn σu/σy 5 3.54 356.64 0.57 6 2.93 459.56 0.61 8 2.27 719.32 0.72 10 1.77 1117.48 0.89 12 1.48 1373.70 0.92 1 p1 50 15 1.17 0.02 1717.21 0.92 5 3.54 342.32 0.55 6 2.93 441.60 0.59 8 2.27 684.60 0.68 10 1.77 1028.67 0.82 12 1.48 1238.94 0.83 2 p2 100 15 1.17 0.04 1548.74 0.83 5 3.54 325.61 0.52 6 2.93 419.33 0.56 8 2.27 652.74 0.65 10 1.77 935.47 0.75 12 1.48 1104.08 0.74 3 p3 150 15 1.17 0.06 1380.09 0.74 5 3.54 303.45 0.49 6 2.93 391.37 0.52 8 2.27 622.29 0.62 10 1.77 833.25 0.67 12 1.48 964.21 0.64 4 p4 200 15 1.17 0.08 1205.31 0.64 5 3.54 277.05 0.44 6 2.93 360.09 0.48 8 2.27 549.87 0.55 10 1.77 674.34 0.54 12 1.48 809.24 0.54 5 p5 250 15 1.17 0.10 1011.68 0.54 5 3.54 247.25 0.40 6 2.93 317.86 0.42 8 2.27 422.28 0.42 10 1.77 533.30 0.43 12 1.48 639.99 0.43 6 p6 300 15 1.17 0.12 800.10 0.43 5 3.54 206.86 0.31 6 2.93 272.97 0.31 8 2.27 310.21 0.31 10 1.77 387.78 0.31 12 1.48 465.36 0.31 7 p7 350 15 1.17 0.14 581.80 0.31 5 3.54 153.17 0.20 6 2.93 183.81 0.20 8 2.27 198.96 0.20 10 1.77 248.71 0.20 12 1.48 298.46 0.20 8 p8 400 15 1.17 0.16 373.12 0.20 5 3.54 75.78 0.10 6 2.93 90.93 0.10 8 2.27 98.58 0.10 10 1.77 123.22 0.10 12 1.48 147.87 0.10 9 p9 450 15 1.17 0.18 184.8499 0.10 m. suneel kumar et al., / journal of naval architecture and marine engineering 4(2007) 15-26 ultimate strength of square plate with rectangular opening under axial compression 19 4. nonlinear finite element analysis a general purpose finite element software ansys is used for modeling, analysis and post processing of unstiffened plate with rectangular opening under axial compression. modeling of unstiffened plate involves generation of a square of size 500 mm x 500 mm. to create the rectangular opening, a rectangle of size (a x b) is generated using key points and connecting it by means of area command available in preprocessor. using the ‘subtract areas’ option available in the ‘booleans’ operation under the ‘modeling’ part, the rectangular area is deleted. thus the geometry of an unstiffened plate with rectangular opening at the centre of the plate is developed. the lines are meshed set using the ‘size controls’ available with the ‘mesh tool’ in ‘meshing’ part. four noded finite linear strain element (shell181) available in the ansys element library is used for discretisation of unstiffened plate. the element has six degrees of freedom per each node; three translations (ux, uy and uz) and three rotations (rx, ry and rz). this element is well suitable for analysing the linear, large rotation, and/large strain nonlinear applications. the finite element model of the square plate with rectangular opening is shown in figure 3. simply supported boundary conditions along all the edges of the plate are used in the analysis as shown in (figure 3). all the nodes along the four edges of the plate are constrained for deflection and rotation along the thickness direction (uz, rz = 0). apart from it, the reactive edge is constrained against axial deformation (uy = 0). all the nodes along the unloaded edges are coupled for inplane displacement (ux) such that the displacements along the length of the plate are uniform. both geometric and material nonlinearities are considered in the analysis. large displacement static analysis with stress stiffening option is activated in geometric nonlinear analysis. bilinear isotropic rate independent hardening with von mises yield criteria is used in material nonlinear analysis. figure 3: finite element model of specimen p5 equal increments of axial deformation (uy) of magnitude 0.1 mm are applied along the loading direction. nonlinear equilibrium equations are solved using newton raphson iteration process. loading edge: uz, rz = 0 x y z reactive edge: uy, uz, rz = 0 unloaded edge: uz, rz = 0 ux along this edge is coupled unloaded edge: uz, rz = 0 ux along this edge is coupled m. suneel kumar et al., / journal of naval architecture and marine engineering 4(2007) 15-26 ultimate strength of square plate with rectangular opening under axial compression 20 summation of axial force at all the nodes along the loading edge for every displacement increment gives the axial load acting on the specimen. validation of the developed fe model is done with the published results of paik et al., 2001. for this purpose, an unstiffened plate under axial compression is considered for the study. the size of the plate (a x b) is 500 mm x 500 mm. the thickness of the plate (t) is 3.2 mm. the yield strength of plate ( )yσ is considered as 264.6 n/mm2 with young’s modulus of elasticity (e) as 2.058 x 105 n/mm2 and poisson’s ratio ( )υ of 0.3. convergence study is performed to obtain an optimal element size to be used for the study. the four meshes used are coarse (5 x 5), medium (10 x 10), fine (20 x 20) and very fine (40 x 40) with the number of elements being 25, 100, 400 and 1600 respectively. the ultimate load and the cpu time for the varied meshes are given in table 2. the results indicate a satisfactory convergence for fine mesh based on the ultimate load. also much variation in ultimate is not observed for fine mesh with higher computing time. thus an element of size of 25 mm x 25 mm is used for the mesh discretisation of the unstiffened plate with varied size of openings for the entire study. table 2: convergence study figure 4: load/deflection curves for unstiffened plate under axial compression sl. no. mesh no. of elements ultimate load (kn) cpu time (sec) 1 coarse 25 400.11 5.86 2 medium 100 393.66 8.19 3 fine 400 392.93 68.22 4 very fine 1600 392.26 1478.98 0 0.1 0.2 0.3 0.4 0.5 0 1 2 3 4 5 w/t σ av g / σ y analytical (paik et al., 2001) ansys (present study) m. suneel kumar et al., / journal of naval architecture and marine engineering 4(2007) 15-26 ultimate strength of square plate with rectangular opening under axial compression 21 a comparison of load/deflection plot for the developed model with the published result is shown in figure 4 and is found to be in good comparison. a typical plot for the specimen with thickness of 5 mm ( 3.54β = ) and varied area ratio is shown in figure 5. ultimate load of the specimens are determined from the peak of axial load/axial deformation plots (figure 5). similarly, ultimate load for the remaining specimens are obtained from axial load/axial deformation plots. the values of ultimate load for all the specimens with varied plate slenderness ratio, and area ratio are given in table 2. the axial deformation contour, axial stress contour and von mises equivalent stress contour for specimen p5 ( 3.54β = ) at ultimate load is shown in figures 6, 7 and 8 respectively. figure 5: axial load/axial deformation plot for specimen p5 ( 3.54β = ) with varied area ratio figure 6: axial deformation contour for specimen p5 ( 3.54β = ) at ultimate load 0 50 100 150 200 250 300 350 400 0 0.5 1 1.5 2 2.5 axial deformation (mm) a xi al l oa d (k n ) ar = 0.02 ar = 0.04 ar = 0.06 ar = 0.08 ar = 0.10 ar = 0.12 ar = 0.14 ar = 0.16 ar = 0.18 m. suneel kumar et al., / journal of naval architecture and marine engineering 4(2007) 15-26 ultimate strength of square plate with rectangular opening under axial compression 22 figure 7: axial stress contour for specimen p5 ( 3.54β = ) at ultimate load figure 8: von mises stress contour for specimen p5 ( 3.54β = ) at ultimate load 5. results and discussion design charts are prepared from the values of ultimate load obtained for all the specimens subjected to axial compression. the various charts developed are (i) effect of plate slenderness ratio on ultimate stress for varied a/b, (figure 9) (ii) effect of area of opening to plate on ultimate stress for varied plate slenderness ratio, (figure 10) (iii) effect of plate slenderness ratio on ultimate stress for varied b/b and (figure 11) (iv) effect of ratio of opening on ultimate stress for varied plate slenderness ratio (figure 12). it is observed that the specimens with rectangular opening of ratio a/b less than 0.33 for all plate slenderness ratios fail by yielding (figure 9). also, it is observed that for all plate slenderness ratios less than 1.77 fail by yielding irrespective of ratio of rectangular opening. the plot for σu/σy vs ac/ap (figure 10) indicates that for plate slenderness ratio less than 2.23 the variation of ultimate strength varies linearly for area ratio ranging between 0.02 0.18, while the variation varies nonlinearly for m. suneel kumar et al., / journal of naval architecture and marine engineering 4(2007) 15-26 ultimate strength of square plate with rectangular opening under axial compression 23 figure 9: effect of plate slenderness ratio on ultimate stress for varied a/b figure 10: effect of area of opening to plate on ultimate stress for varied plate slenderness ratio plate slenderness ratio more than 2.23. it is also observed that for plate slenderness ratio ranging 1.17 3.54. from figure 11, it is observed that the specimens with rectangular opening of ratio b/b more than 0.60 for all plate slenderness ratios fail by yielding. also, it is found that for all plate slenderness ratios less than 1.77 fail by yielding irrespective of ratio of rectangular opening. the plot for σu/σy vs a/b (figure 12) indicates that for plate slenderness ratio the variation of ultimate strength varies 0 0.2 0.4 0.6 0.8 1 0 1 2 3 4 5 plate slenderness ratio (β ) σ u / σ y a/b=0.22 a/b=0.25 a/b=0.29 a/b=0.33 a/b=0.40 a/b=0.50 a/b=0.67 a/b=1.00 a/b=2.00 0 0.2 0.4 0.6 0.8 1 0 0.05 0.1 0.15 0.2 area of cutout/area of plate (ac /ap) σ u / σ y β = 3.54 β = 2.93 β = 2.23 β = 1.77 β = 1.48 β = 1.17 m. suneel kumar et al., / journal of naval architecture and marine engineering 4(2007) 15-26 ultimate strength of square plate with rectangular opening under axial compression 24 nonlinearly. it is also observed that for ratio of opening (a/b) less than 0.33 the variation of ultimate stress is linear for plate slenderness ratio ranging 1.17 3.54 beyond which it is nonlinear up to a/b = 1. significant increase in ultimate strength for all plate slenderness ratios is not observed beyond a/b = 1.0. figure 11: effect of plate slenderness ratio on ultimate stress for varied b/b figure 12: effect of ratio of opening on ultimate stress for varied plate slenderness ratio 0 0.2 0.4 0.6 0.8 1 0 1 2 3 4 5 plate slenderness ratio (β ) σ u / σ y b/b=0.10 b/b=0.20 b/b=0.30 b/b=0.40 b/b=0.50 b/b=0.60 b/b=0.70 b/b=0.80 b/b=0.90 0 0.2 0.4 0.6 0.8 1 0 0.5 1 1.5 2 2.5 ratio of cutout (a/b) σ u / σ y β = 3.54 β = 2.93 β = 2.23 β = 1.77 β = 1.48 β = 1.17 m. suneel kumar et al., / journal of naval architecture and marine engineering 4(2007) 15-26 ultimate strength of square plate with rectangular opening under axial compression 25 shanmugam et al., (1999) used best fit regression analysis in developing the design formula to predict the ultimate load of square plates with centrally placed square or circular shapes and subjected to uniaxial and biaxial compression. similar approach is employed to develop the ultimate strength design formula for unstiffened plates with rectangular openings subjected to axial load. from the parametric study, it is ascertained that interaction between the parameters like plate slenderness ratio β)( , ratio of opening (a/b) and area ratio (ar) influence the strength of unstiffened plates. the effect of yield stress (σy) and young’s modulus of elasticity (e) of the plate are taken care in the calculation of plate slenderness ratio. the interaction of these parameters is important and there is a need for simple formulae for the design. interaction equations are derived for the ultimate strength design of unstiffened plates subjected to axial load from the parametric studies using statistical analysis software spss. from the present study it is observed that relationship between plate slenderness ratio β)( , ratio of opening (a/b) and area ratio (ar) on the ultimate strength of unstiffened plate under axial load varies nonlinearly. hence nonlinear regression analysis is adopted to predict the relationship these variables. a nonlinear regression can estimate models with arbitrary relationship between the dependent variable (σu/σy) and a set of independent variables (β, a/b, ar). this is accomplished using iterative estimation algorithms. for each iteration, parameter estimates and residual sum of squares is obtained. for the assumed model, sum of squares for regression, residual, uncorrected total and corrected total, parameter estimates, asymptotic standard errors, and asymptotic correlation matrix of parameter estimates are evaluated. the best fit for any assumed relationship between dependent and independent parameters can be ensured only if r-squared [1-(residual sum of squares/corrected sum of squares)] value is more than 0.95. the following design equations are developed using nonlinear regression analysis based on the present study: (1) (2) (3) it is found that for the above mentioned proposed design equations (1), (2) and (3), the r-squared value is found to be 0.98006, 0.98983 and 0.98983 respectively and hence best fits the data obtained using nonlinear finite element analysis. the developed formula is simple and reliable, and can be used for the purpose of design by practical engineers. 6. summary and conclusions the effect of rectangular central opening of a square plate on the ultimate strength under axial compression is found. the effect of plate slenderness ratio, area ratio of opening to plate and ratio of opening on ultimate strength is determined using nonlinear finite element analysis. the variation of ultimate strength is found to be linear for plate slenderness ratio less than 2.23 and nonlinear for plate slenderness ratio beyond 2.23 for area ratio ranging between 0.02 – 0.18. a design equation is proposed based on nonlinear regression analysis for rectangular opening under axial compression. the study has to be further extended to determine the effect of initial imperfections and residual stresses. 1.168 b a 0.147β b a 0.0290.137β σ σ 1.2970.459 y u +⎟ ⎠ ⎞ ⎜ ⎝ ⎛ −⎟ ⎠ ⎞ ⎜ ⎝ ⎛ −−= − − ( ) ( ) 1.234ar0.874βar231.70.321β σ σ 384.0849.0 y u ++−−= 1.234 b b 0.471β b b 446.10.321β σ σ 384.0849.0 y u +⎟ ⎠ ⎞ ⎜ ⎝ ⎛ +⎟ ⎠ ⎞ ⎜ ⎝ ⎛ −−= m. suneel kumar et al., / journal of naval architecture and marine engineering 4(2007) 15-26 ultimate strength of square plate with rectangular opening under axial compression 26 references bradfield, c. d. (1980): tests on plates loaded in in-plane compression, journal of constructional steel research, vol. 1, no. 1, pp. 27-37. khaled m. el-sawy, aly s. nazmy and mohammad ikbal martini. (2004): elasto-plastic buckling of perforated plates under uniaxial compression, thin-walled structures, vol. 42, no. 8, pp. 1083-1101. guedes soares, c. (1988): a code requirement for the compressive strength of plate elements, marine structures, vol. 1, no. 1, pp. 71-80. guedes soares, c. (1992): design equation for ship plate elements under uniaxial compression, journal of constructional steel research, vol. 22, no. 2, pp. 99-114. jwalamalini, r., sundaravadivelu, r., vendhan, c.p. and ganapathy, c. (1992): stability of initially stressed square plates with square openings, marine structures, vol. 5, no. 1, pp. 71-84. madasamy, c.m. and kalyanaraman, v. (1994): analysis of plated structures with rectangular cutouts and internal supports using the spline finite strip method, computers and structures, vol. 52, no. 2, pp. 277-286. motok, m. d. (1997): stress concentration on the contour of a plate opening of an arbitrary corner radius of curvature, marine structures, vol. 10, no. 1, pp. 1-12. narayanan, r. and chow, f. y. (1984): ultimate capacity of uniaxially compressed perforated plates, thin-walled structures, vol. 2, no. 3, pp. 241-264. narayanan, r. and chan, s.l. (1985): ultimate capacity of plates containing holes under linearly varying edge displacements, computers and structures, vol. 21, no. 4, pp. 841-849. paik, j.k., thayamballi, a.k. and kim, b.j. (2001): advanced ultimate strength formulations for ship plating under combined biaxial compression/tension, edge shear, and lateral pressure loads, marine technology, vol. 38, no. 1, pp. 9-25. roberts, t. m. and azizian, z. g. (1984): strength of perforated plates subjected to in-plane loading, thin-walled structures, vol. 2, no. 2, pp. 153-164. shanmugam, n.e. (1997): openings in thin-walled steel structures, thin-walled structures, vol. 28, no. 3-4, pp. 355-372. shanmugam, n.e., thevendran, v. and tan, y.h. (1999): design formula for axially compressed perforated plates, thin-walled structures, vol. 34, no. 1, pp. 1-20. toulios, m. and caridis, p.a. (2002): the effect of aspect ratio on the elastoplastic response of stiffened plates loaded in uniaxial edge compression, computers and structures, vol. 80, no. 14-15, pp. 1317-1328. yettram, a. l. and brown, c. j. (1985): the elastic stability of square perforated plates, computers and structures, vol. 21, no. 6, pp. 1267-1272. microsoft word p49_58.doc journal of naval architecture and marine engineering december, 2006 http://jname.8m.net 1813-8535 © aname publication. all rights reserved “opti-marine-ware”(optimization of vessel's parameters through spreadsheet model) abhijit de1 and ashish kumar2 1research scholar , department of marine engineering, jadavpur university, kolkata:700032, india, tel.:91-3323730312, e-mail: abhijitde549128@gmail.com 2research scholar, department of marine engineering, jadavpur university, kolkata :700032, india, tel.:91-9830582762, e-mail: ashishjha5615@gmail.com abstract the objective of this paper is to describe and evaluate a scheme of engineering-economic analysis for determining optimum ship’s main dimensions and power requirement at basic design stage. we have divided the optimization problem into five main parts, namely, input, equation, constraint, output and objective function. the constraints, which are the considerations to be fulfilled, become the director of this process and a minimum and a maximum value are set on each constraint so as to give the working area of the optimization. the outputs (decision variables) are optimized in favor of minimizing the objective function. microsoft excel-premium solver platform (a spreadsheet modeling tool is utilized to model the optimization problem). this paper is commenced by the description of the general optimization problems, and is followed by the model construction of the optimization. a case study on the determination of ship’s main dimensions and its power requirement is performed with the main objective to minimize the economic cost of transport (ect). after simulating the model and verifying the results, it is observed that the spreadsheet model yields considerably comparable results with the main dimensions and power requirement data of the real operated ships (tanker). it is also experienced that this kind of optimization process needs no exhaustive efforts in producing programming codes, if the problem and the optimization model have been well defined. keywords: optimization, design , ship power requirement nomenclature psp premium solver platform ect economic cost of transport dwt dead weight tonnage nlp non-linear programming grg generalized reduced gradient gui graphical user interface lp linear programming b/t breadth by draft ratio bhp brake horse power lpp ship’s length between perpendiculars dhp delivered horse power t draft hfo heavy fuel oil do diesel oil lo lub oil me main engine ge generator sfoc specific fuel oil consumption rfr freight rate atc annual tons of cargo lwl length of water line l/b length by breadth ratio ap blade area sim simulation loc lub oil consumption 1. introduction the problems in designing ship and marine machinery appear due to numerous considerations that must be taken into account. these conditions increase the capital cost and the complexity of the design option. therefore, ship’s design and its selected machinery must guarantee that the ship and its machinery will operate with low level of failure, safely and efficiently, with high level of availability and will deliver an optimum rate of return on the capital being employed. “opti-marine-ware”(optimization of vessel’s parameters through spreadsheet model) 50 thorp and armstrong (thorp et al, 1982) utilized a comprehensive method to select the machinery arrangement for a panamax-size bulk carrier of 70,000 dwt. their economic assessment was only focused on two alternatives of slow speed diesel installation and medium speed diesel installation. some parameters that were included in their study are also taken in our study. one of the major differences with their study is that our study tackles the problem at the basic design process allowing the optimization process to determine the ship’s main dimension and its machinery characteristics within the given constraints. this paper proposes an alternative method for optimizing marine designs, particularly in determining ship’s main dimension and its power requirement at basic design stage. spreadsheet modeling is utilized and non-linear programming (nlp) can express our problem. the generalized-reduced gradient (grg) method can work in conjunction with the nlp problems. basic diagrammatic concepts of the optimization process and a case study are also given comprehensively 2. premium solver platform and the basic optimization model the determination of ship’s main dimensions and its machinery power requirement encounters many constraints and considerations in its synthesized process (sen, 1998). a number of methods are available to solve the multi constraints and multi variables optimization problem such as those are summarized in (rao, 1991).furthermore, the optimization of ship’s design can be defined as an attempt to resolve the conflicts of a design situation, in such a way that the variables under the control of the decision-maker take their best possible value. generally, a classic multiple constrained optimization problems can be represented as follows. x1 find x = which minimize/maximize f(x) xn subject to constraints g(lb)i < gi(x) < g(ub) i for i = 1,2,3,…,m and x(lb)j < xj < x(ub)j for j = 1,2,3,…,p where x is a vector of n variables and the function g1,….,gm all depend on x. lb and ub stand for low bound and upper bound respectively. this paper employs the microsoft excel-psp software (psp) to deal with the above general expression of optimization problem. psp combines the function of a graphical user interface (giu), an algebraic modeling language and optimizers for linear, non-linear, and integer program. each of these functions is integrated into the host spreadsheet program, which allows us to specify an objective function, constraints and other supporting features interactively. the psp then makes the complete optimization model and produces the matrix form required by the optimizers. the optimizers itself employ the simplex (for lp model), the grg (for nlp), and branch and bound methods to find an optimal solution and sensitivity information. for the lp problem, the focus of this model representation is the lp coefficient matrix. this is the jacobian matrix of partial derivatives of the objective function and constraints with respect to the decision variables. in lp problems, the matrix entries are constant and need to be evaluated only once at the start of the optimization. on the other hand, in nlp problems, the jacobian matrix entries are variable and must be recomputed at each new trial point. assuming linear model for a certain problem, the psp uses a straightforward implementation of simplex method with bounded variables to find the optimal solution. for a nlp, the psp uses the grg method, as implemented in the grg2 code (ladson et al, 1978) & (ladson et al, 1992). grg requires function values and the jacobian matrix, which is not constant for nlp models. the psp approximates the jacobian matrix using finite difference method. the basic format of the offered optimization process is given in figure 1. there are five folders within the optimization, namely the input folder, equation folder, constraint folder, output folder and the objective function. the input folder consists of all the parameters that are used in the entire optimization process. for a complex problem, such parameters can be classified into several directories, which will make fault identification easier. “opti-marine-ware”(optimization of vessel’s parameters through spreadsheet model) 51 all basic calculations of the optimization are located in the equation folder. the result of each equation is continuously updated, since the process in the constraint folder and the output folder always affect the variables employed in the equation folder. the constraint folder contains all considerations that must be satisfied and becomes the director of the optimization process. a minimum and a maximum value are set on each constraint to give the working area of the optimization. the optimum values are located in the center of the form. the determination of the minimum and the maximum values depend on the characteristics of the constraints. figure 1: basic format of the optimization process inputs example: input 1 = c1 input 2 = c2 input 3 = c3 ………. input n = cn equations example: eq. 1 = c1 x c2 eq. 2 = sqrt (c3) eq. 3 = eq. 1 xeq. 2 …… eq. n = ln(eq. 3) min value example: constr. 1 min value constr. 2 min value constr. 3 min value …….. constr.n min value constraints example: constr. 1 = (eq. 1-eq. 2) x x1 constr. 2 =eq. 2 x (eq 3 ^x2) constr. 3 =eq. n–eq. 2– x3 …… constr. n = sqrt (eq 1 xxn) max value example: constr. 1 max value constr. 2 max value constr. 3 max value …….. constr. n max value min value example: dec. var 1 min value dec var2 min value dec. var3 min value …….. dec. var n min value outputs (decisionvar) example: decision variable 1 (x1) decision variable 2 (x2) decision variable 3 (x3) ……….. decision variable n (xn) max value example: dec. var 1 max value dec. var 2 max value dec. var 3 max value …….. dec. var n max value objective function example: minimize x1 + x2 + x3+…..+xn “opti-marine-ware”(optimization of vessel’s parameters through spreadsheet model) 52 3. case study: basic design optimization process for tanker with specified throughput 3.1 problem statement at the basic design stage, it is required to design a numbers of series ships (tanker) delivering contract of a certain throughput, which have optimum main dimension and optimum specified power. ect is utilized as the objective of the optimization problem. port characteristics require such constraints, as the ship must not exceed 200m in length and 11m in draught. the conceptual problem is shown in figure 2. some economic data are employed during the optimization process, as shown in table 1(refer appendix) 3.2. model structure to simplify the optimization problem, the input folder and the equation folder are grouped into several directories. in this particular optimization, the input folder covers: the ship data, machinery data, and reliability data. each directory represents collection of parameters that are used in the calculation process. the equation folder consists of several directories such as the ship coefficient, machinery, reliability, loading and unloading, fuel, operating cost and the economic considerations. the constraint folder comprises of the estimated annual throughput economic life machinery and ship owner equity steel, fuel, lub. oil, tax, interest rate, port service charge rate, and other basic costs depreciation period etc. expected repl. cost reliability function average cargo weight per ship total pumping cap. pump cap. %rated bhp req. req. freight rate midship coefficient max allowable ship length at port etc. number of ships draught b/t ratio l/b ratio block coefficient service speed propeller rpm port time per trip number of unloading pump/host etc. input constraints output what is the optimum basic design output, which minimizes the economic cost of transport (ect) during the economic life cycle of the ship and machinery? objective function? port a port b figure 2: problem statement “opti-marine-ware”(optimization of vessel’s parameters through spreadsheet model) 53 expected replacement cost, reliability index, unloading pump capacity, specific fuel oil consumption (sfoc) for maine engine (me) and and the maximum allowable ship length in port. the output folder yields the optimum preventive maintenance interval, block coefficient, optimum design draught, optimum, b/t ratio, and the number of ships. these values are sought with the main objective to minimize the ect of the ship. ect, the objective for this particular optimization problem is composed by several variables, namely the required freight rate (rfr), the inventory cost of cargo and the annual tons of cargo carried (atc) (hunt et al ,1995). the optimum value of rfr itself depends on the annual capital recovery of the vessel cost, the annual operating cost, and the annual throughput (gransberg et al, 1998). the sequence of this design process indicates strict relationship among each design consideration. figure 3: interdependency between variables for instance, it might be not a simple work to relate the optimum number of shore connection, which must be fitted on a tanker with the resulted rfr or outcomes of the loan repayment scheme. however, it is believed that those variables somehow interconnect and affect each other. hence, the basic nature of ships and its machinery design optimization process would lie on the ability of the engineers to accommodate all of the design considerations and to provide adequate flexibility in altering the decision variables, while fulfilling the main objective of the optimization process. no. of voyage interest rate cargo cost unit ect throughput rfr constant reliability vessel cost ann. operat. cost total cost ann. dry dock cost ann. port cost ann. m/r cost ann. expt. repl. cost ann. insur. cost ann. adm. cost ann. over head cost ann. crew cost no. of opr. ship no. of opr. ship no. of crew unit insur. cost no. of opr. ship voyage per year annual hfo cost annual do cost no. of opr. ship constant reliability unit port cost voyage per year etc. unit crew cost grt annual lo cost no. of opr. ship constant “opti-marine-ware”(optimization of vessel’s parameters through spreadsheet model) 54 figure: 4 optimization model structure figure 4 shows the general structure of this optimization problem. the optimization process is commenced by setting the initial value of the decision variables. using relevant basic parameters located in the input folder, all basic calculations are executed in the equation folder. the results are then exported to the constraint folder to calculate all constraints accordingly.the optimization problem can be mapped as shown in table 2 (refer appendix). the objective is to minimize f (x), which is the ect while determining the optimum value of x1 to x12 subject to constraint g1 (x) to g16 (x) , (refer appendix table 2). the basic ship design and ship resistance formulae are mainly taken form clarke (1975), oosterveld et al (1975, harvald, (1983) & sname ( 1967) and the economic parameters and major assumptions related to cost calculation from hunt et al , (1995) and kiss (1992). 3.3. further description of the directories set starting value of decision variables optimum: no. of req. ship, b/t ratio draught, cb, vs, propeller rpm, prop. diameter, pitch ratio, min values max values output has the min. ect achieved? objective function reduce gradient, set another decision variable values replacement cost reliability cargo weight pumping capacity sfoc loc cavitation number bhp req. rfr max allowable lpp l/b ratio min value max value constraint ship coefficient fuel consmpt. calculation markov evaluation voyage calculation lubrication oil powering calculation vessel cost estimation r.f.r calculation time value of money operating cost loan repayment resistance calculation equation machinery data adjustment factor cargo load data economic data voyage data ship data port data input “opti-marine-ware”(optimization of vessel’s parameters through spreadsheet model) 55 lpp-dwt & lpp-t verification 0 10000 20000 30000 40000 50000 60000 0 50 100 150 200 250 300 lpp (meters) d w t (to n) 0 2 4 6 8 10 12 14 16 t( m et er s) lpp-dwt real lpp-dwt sim lpp-t real lpp-t sim the input folder consists of given parameters and grouped into several directories. the ship data directory takes the cargo density of 915 kg/m 3 . appendages factor, which influences the resistance calculation, is assumed to have value of 0.03. this directory also allocates the need to use a reduction gear for engine speed reduction. the machinery data directory allows the alternative of using either single main engine or multiple main engines. the model also provides flexibility in employing number of generator set. their reliability model is assumed to be represented by weibull distribution, and its related parameters (γ,β,η) must be defined accordingly. the weibull analysis is then used to find the best period/interval to carry out the maintenance program. the unit cost of failure replacement and unit cost of preventive replacement is also assumed before the optimization process can be executed (jardine, 1973) & (rasmussen,1990). the voyage data directory is one of the vital directories in the optimization model. optional trip distance and number of intermediate port make the model flexible. the assumed outbound and inbound load factors allow the model to be more realistic. the economic data directory, as shown in table 1 is gathered from many different sources and plays a very important role within the optimization model. the annual adjustment factor provides more realistic calculation of the operating cost. the equation folder is also divided into several directories. the coefficient and ship directory collects all equations for determining the main dimensions of the ship. since such equations usually stand as empirical formula, then the interpolation process takes part when some values lie beyond the original range (kiss, 1992). the determination of ship resistance and power prediction is carried out using harvald power prediction method (harvald sv. aa, 1983). the propeller design and its cavitation prediction are based on the wageningen b-series propellers (clarke, 1975, oosterveld et al,1975, harvald, 1983). the vessel cost director y allows us to perform a basic hull cost, outfit cost, machinery cost and estimated overhead cost (hunt et al, 1995). the sfoc-speedpower directory estimates the optimum percentage of rated bhp to be used during the service condition. the reliability directory determines failure rate, reliability and unreliability of the main engine based on the given weibull parameters. this directory also estimates the expected length of operating hours before failure cycle. the number of voyage per year, which strongly influences the ect, is optimized in the trip per year directory. the fuel and lubricating oil directory estimates the annual fuel and lubricating oil requirement. since the model does not refer to any particular engine, the calculation is then made empirically. the operational cost directory determines the annual operational cost for all ships. because the investment scheme also affects the value of the optimized ect, the loan repayment directory and the time value of money directory are then allocated to give flexibility for determining the preferred investment scenario. figure 8: lpp-dwt & lpp-t verification “opti-marine-ware”(optimization of vessel’s parameters through spreadsheet model) 56 d wt -bhp & d wt -lpp verification 0 50 100 150 200 250 300 0 20000 40000 60000 80000 100000 120000 dwt (ton) lp p (m et er ) 0 5000 10000 15000 20000 25000 30000 b h p (h p) dwt-lpp real dwt-lpt real dwt-lpp real dwt-lpp sim 3.5. results verification to verify the performance of this optimization program, comparison on bhp, dwt, and t (draught) has been made on several tanker data (obtained from different shipping companies). the comparisons are shown in figure 8 and 9. generally, it is observed that the results of the simulation very closely conform to the real data. at some points the optimization result drastically shifts to a new point. this is caused by any adjustment made to the optimization program, which is different from that of the previous one. for instance, if the throughput is less than 300,000 ton, then we could set the maximum cargo carrying capacity of the constraint at the value of 25,000 ton. once we increase the throughput, the optimization cannot produce optimum results, until we increase the maximum value of the cargo carrying capacity. figure 9: dwt-bhp & dwt-lpp verification 4. conclusion for basic design stage or feasibility study purposes, this method could be employed before commencing any further design stage. the case study presented here shows how this optimization program can effectively and precisely become consistent with the real ship’s design. moreover the most challenging part of the optimization problem is to express the problem in mathematical expressions which can be executed by the psp. the ship main dimensions and its power requirement that are obtained through this method can be further traced down into a more detail analysis to design the machinery system on board. additional task can easily be added within the optimization program by inserting a new directory within the input and the equation folder. associated constraints and expected output can be attached with the objective either to minimize or to maximize the objective function. this kind of optimization process can also be utilized to select marine machinery from a certain number of available alternatives or to determine maintenance management scheme, as utilized by authors in reference (artana kb et al, 2000, 2001) acknowledgement the authors gratefully acknowledge the cooperation of retired prof. j.p.kundu, department of naval architecture and ocean engineering, iit kharagpur. “opti-marine-ware”(optimization of vessel’s parameters through spreadsheet model) 57 references artana, k. b. and ishida, k. (2001): determination of ship machinery performance and its maintenance management scheme using markov process analysis, marine technology iv, wit press: 379-389 based marine machinery selection: a study case on main engine cooling system. proceedings: sixth international symposium on marine engineering (isme 2000), tokyo. 2: 791-796 clarke, a.c.f. (1975): regression analysis of ship data, international shipbuilding progress.22: 227249 gransberg, d. and basilotto, j.p. (1998): cost engineering optimum seaport capacity, journal of cost engineering, 4 harvald sv. a.a. (1983): resistance and propulsion of ships, john wiley & sons. hunt, c.e. and butman, b.s. (1995): marine engineering economics and cost analysis, cornell maritime jardine a.k.s. (1973): maintenance, replacement and reliability, pitman publishing kiss rk (1992): ship design and construction, the society of naval architects and marine engineers, new york lasdon, l.s., waren, a.d. jain, a. and ratner, m (1978): design and testing of a generalized reduced gradient code for nonlinear programming, acm transactions on mathematical software, 4: 34-49 lasdon, l.s. and smith, s. (1992): solving large sparse nonlinear programs using grg, orsa journal on computing, 4: 2-15 oosterveld, m.w.c. and oossanen, p.v. (1975): further computer-analyzed data of the wageningen bscrew series, international shipbuilding progress, 22: 251-261 thorp, i. and armstrong, g. (1982): the economic selection of main and auxiliary machinery, transaction of imare. 951: 2-7 rao, s.s. (1991): optimization theory and application, 2nd edition, willey eastern limited, new delhi appendix table 1 : economic data input* economic life of machinery years 20.00 loan repayment period years 20.00 interest rate % 0.10 rate of return on equity % 0.12 economic life of ship years 20.00 ship depreciation period years 15.00 machinery depreciation period years 15.00 tax rate % 0.30 annual inflation rate % 0.01 average fuel price (hfo/do) us$/lb. 0.08 average crew cost per month us$/month 1,250.00 * source: mainly obtained from reference (hunt et al ,1995) table 2: optimization statement find x1 min value < time (t) independent variable < max. value x2 min value < number of ships < max. value x3 min value < draught < max. value x4 min value < b/t ratio < max. value x5 min value < block coefficient < max. value x6 min value < service speed < max. value x7 min value < propeller rpm < max. value x8 min value < diameter propeller < max. value x9 min value < pitch ratio < max. value x10 min value < time required for preventive replacement < max. value “opti-marine-ware”(optimization of vessel’s parameters through spreadsheet model) 58 x11 min value < port time per trip (loading) < max. value x12 min value < number of unloading pump/host < max. value which minimizes: economic cost of transport (ect) (f(x)) rfr total cost annual port cost f (unit cost, grt, voyage per year, no. of operated ship) annual insurance cost f (voyage per year, weight of cargo, unit insurance, no. of ship) annual overhead cost f (constant, no. of ship) annual crew cost f (unit of crew cost, no. of crew, no. of ship) annual expected replacement cost f (reliability, no. of ship) annual m/r cost f (reliability, no. of ship) annual dry docking expenses f (constant, no. of ship) annual administration cost f (constant, no. of ship) annual operating cost f (lo cost, do cost, hfo cost, etc) owner equity constant throughput given cargo cost unit constant number of voyage operating day f(docking days, unscheduled maintenance days, time at port) turn round time interest rate constant subject to g1(x) min value < exptd. replacement cost, f(reliability index,cost of fail. rep,cost of prev. rep)<max. value g2(x) min value < reliability function, f(failure distribution parameters) <max. value g3(x) min value < ave. cargo wt./ship, f(throughput,no. of ship, voy./ year,load factor) <max. value g4(x) min value < total pumping capacity, f(pump capacity, no. of req. pump) <max. value g5(x) min value < pump capacity f(cargo weight, port time, cargo density) <max. value g6(x) min value < sfoc for full load me f(dhp, engine rpm) <max. value g7(x) min value < sfoc for full load ge f(dhp, diesel generator rpm) <max. value g8(x) min value < cavitation no f(thp,projected blade area(ap),dyn. press. at tip radius) < max. value g9(x) min value < local cavitation no. f(press. at the screw centerline, dyn.press. at tip radius) <max. value g10(x) min value < %rated bhp requirement f(min resulted sfoc at feasible region) <max. value g11(x) min value < required freight rate f(ann.vessel cost,total opr.cost, throughput) <max. value g12(x) min value < midship coefficient f(displacement, breadth, draught, lpp) <max. value g13(x) min value < l/b ratio f(lpp/breadth) <max. value g14(x) min value < max allowable ship length at port f(vol.displ,breadth,draught,block coef.) <max. value g15(x) min value < length of water line (lwl) f (loa) <max. value g16(x) min value < length between perpendicular (lpp) f (lwl) <max. value microsoft word jname414.doc 1813-8535 © 2007 aname publication. all rights reserved. journal of naval architecture and marine engineering june, 2007 http://jname.8m.net numerical investigation of unsteady flow past a circular cylinder using 2-d finite volume method md. mahbubar rahman1, md. mashud karim2 and md. abdul alim3 1department of natural science, stamford university bangladesh, dhaka-1209, 2dept. of naval architecture and marine engineering, buet, dhaka-1000,3department of mathematics, buet, dhaka-1000, bangladesh. e-mail: 1mahbub_stamford@yahoo.com, 2mmkarim@name.buet.ac.bd, 3maalim@math.buet.ac.bd abstract the dynamic characteristics of the pressure and velocity fields of unsteady incompressible laminar and turbulent wakes behind a circular cylinder are investigated numerically and analyzed physically. the governing equations, written in the velocity pressure formulation are solved using 2-d finite volume method. the initial mechanism for vortex shedding is demonstrated and unsteady body forces are evaluated. the turbulent flow for re = 1000 & 3900 are simulated using k-ε standard, k-ε realizable and k-ω sst turbulence models. the capabilities of these turbulence models to compute lift and drag coefficients are also verified. the frequencies of the drag and lift oscillations obtained theoretically agree well with the experimental results. the pressure and drag coefficients for different reynolds numbers were also computed and compared with experimental and other numerical results. due to faster convergence, 2-d finite volume method is found very much prospective for turbulent flow as well as laminar flow. keywords: viscous unsteady flow, laminar & turbulent flow, finite volume method, circular cylinder. nomenclature: a radius of the circular cylinder cp pressure coefficient d diameter of the circular cylinder ωw wall vorticity ρ density of the fluid cd drag coefficient µ coefficient of viscosity cdp drag coefficient due to pressure t time cdv drag coefficient due to viscosity u,v velocity components pw dimensionless wall pressure → u velocity vector re reynolds number ∞u free stream velocity t the period of vortex shedding. l turbulence length f frequency of the vortex shedding k turbulence kinetic energy i turbulence intensity ε turbulence dissipation cµ empirical constant st strouhal number ω specified dissipation rate 1. introduction flow around a circular cylinder is a fundamental fluid mechanics problem of practical importance. it has potential relevance to a large number of practical applications such as submarines, off shore structures, bridge piers, pipelines etc. the laminar and turbulent unsteady viscous flow behind a circular cylinder has been the subject of numerous experimental and numerical studies, especially from the hydrodynamics point of view. according to the observation of sumer (1997); the flow field over the circular cylinder is symmetric at low values of reynolds number. as the reynolds number increases, flow begins to separate m. m. rahman, m. m. karim and m. a. alim / journal of naval architecture and marine engineering 4(2007) 27-42 numerical investigation of unsteady flow past a circular cylinder using 2-d finite volume method 28 behind the cylinder causing vortex shedding which is an unsteady phenomenon. for the 40 < re < 200, there is a laminar vortex shedding in the wake of the cylinder. the laminar wake transient to turbulence in the region of re = 200 to 300. in the subcritical region 300 < re < 3×105 the wake behind the circular cylinder becomes completely turbulent and a laminar boundary layer separation occurs. the unsteady flow was first studied by payne (1958) for reynolds number equal to 40 & 100. the numerical study and physical analysis of the pressure and velocity fields in the near wake of a circular cylinder has been investigated by braza et al (1986). tuann & olson (1978), martinez & minh (1978), loc (1980), coutanceau & defaye (1991) and recently lakshmipathy (2004) and reichl et al. (2005) have investigated this problem for different reynolds numbers. braza et al (1990) predicted the large-scale transition features in the wake of a circular cylinder for 2000 < re < 10000. the turbulent flow over cylinder was also investigated by rai & moin (1993) and mittal (1995) for high reynolds number. the common points of interest of this work are the development of the primary unsteady wake behind the circular cylinder and the evaluation of the drag coefficient and the separation angle with time. most of the experimental studies investigated the steady and unsteady behaviors of the alternating vortices in the wake. the work of tritton (1971), lourenco & shih (1993),and anderson (1995) should be mentioned. besides these theoretical and numerical investigations, some experimental visualizations have been described by honji & taneda (1969), beaudan & moin (1994) and coutanceau & bouard (1977). all of the above numerical studies have some common characteristics: they solve the unsteady navierstokes equation in two dimensional helmholdtz (vorticity & stream function) formulation; they described the relevant flow by the global parameters such as strouhal number as a main feature of the unsteady wake, drag and lift coefficients in the wall region; nevertheless, poor analysis is provided for the near wake characteristics. the main goal of the present study is consequently to (i) visualize the laminar and turbulent unsteady flow field for different reynolds numbers; (ii) check the capability of different turbulent models for the simulation the unsteady flow over circular cylinder. (ii) compute drag coefficients along the surfaces of the cylinder for different reynolds numbers; (iii) plot contours of stream function, velocity vectors and static pressure to visualize vortex shedding at different time for re=100, 1000 & 3900. numerical results presented here are compared with experimental measurements and other numerical results and the agreements are found satisfactory. 2. physical model the flow field around the cylinder is modeled in two dimensions with the axes of the cylinder perpendicular to the direction of flow. the cylinder is modeled as a circle and a square flow domain is created surrounding the cylinder. the flow from left to right with the cylinder of diameter, d submersed in an incompressible fluid is considered. the computational domain consists of an upstream 23 times the radius to downstream 40 times the radius and the width of the domain is 50 times the radius of the cylinder. the problem setup together with the important dimensions is shown in fig.1. a typical rectangular mesh used for simulation of this study. this particular mesh has approximately 15659 nodes, 30924 faces and 15380 quadrilateral cells with considerable mesh concentration both around the cylinder and in the wake. to facilitate meshing, a square with side length of three times diameter of the cylinder is created around the cylinder. 3. mathematical model 3.1 governing equations the governing equations of the unsteady flow of an incompressible viscous fluid past a circular cylinder are the classical continuity and navierstokes equations. they are written in the following form: 0)( = → udiv (1) m. m. rahman, m. m. karim and m. a. alim / journal of naval architecture and marine engineering 4(2007) 27-42 numerical investigation of unsteady flow past a circular cylinder using 2-d finite volume method 29 ( ) ( )udivuudiv t u ∇γ=⎟ ⎠ ⎞ ⎜ ⎝ ⎛+ ∂ ∂ → ρ 1 (2) equation (2) has the rate of change term and the convective term in the left hand side and the diffusion term (γ= diffusion coefficient) on the right hand side. the equation is used as the starting point for computational procedure in finite volume method. the key step of the finite volume method (versteeg & malalsekera,1995) is the integration of equation (2) over a two dimensional control volume cv yielding ( )∫ ∫∫ ∇γ=⎟⎠ ⎞ ⎜ ⎝ ⎛+ ∂ ∂ → cv cvcv dvudivdvuudivdv t u ρ 1)( (3) the volume integrals in the 2nd term on the left hand side, the convective term and in the right hand side the diffusion term can be rewritten as integrals over the entire boundary surface of the control volume by using gauss divergence theorem. then equation (3) becomes ( ) ( )∫ ∫∫ ∇γ=⎟⎠ ⎞ ⎜ ⎝ ⎛+⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ ∂ ∂ → a acv daundauundvu t . 1 . ρ (4) in the steady state case, the rate of change term is equal to zero. this leads to the integrate form of the steady transport equation ( ) ( )∫ ∫ ∇γ= a a daundauun . 1 . ρ (5) in time dependent (unsteady) case, it is also necessary to integrate with respect to time t over a small time interval ∆t from (say) t to t+ ∆t . this yields the most general integral form of the transport equation. ( ) ( )∫ ∫∫ ∫∫∫ ∆ ∆ ∆ → ∇γ=⎟ ⎠ ⎞ ⎜ ⎝ ⎛+⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ ∂ ∂ t t a t acv dadtundadtuundtdvu t . 1 . ρ (6) r fig.1: schematic diagram of the flow field around circular cylinder 23 r 40 r 50 r periodic pr es su re o ut le t periodic v el oc ity in le t m. m. rahman, m. m. karim and m. a. alim / journal of naval architecture and marine engineering 4(2007) 27-42 numerical investigation of unsteady flow past a circular cylinder using 2-d finite volume method 30 in finite volume method, flow domain is divided into a number of control volumes or cells. discritized form of eq. (6) must be set up at a nodal point placed within each control volume in order to solve the problem. for control volumes adjacent to the domain boundaries, the general equation is modified to incorporate boundary conditions. the resulting system of linear algebraic equations is then solved to obtain the velocity and pressure distribution at each nodal point. finally, drag and lift coefficients are computed as follows: du d c d ∞ = 25.0 ρ ∫= π2 0 cos dxpc wdp ∫= π ω 2 0 sin re 2 xdxc wdv (7) du l c l ∞ = 25.0 ρ ∫= π2 0 sin xdxpc wlp ∫= π ω 2 0 cos re 2 xxdxc wlv (8) where, d & l represent drag and lift force. also, the pressure coefficient is defined as: ( ) duppc p 22 1 / ρ∞−= . (9) where, the subscripts p and v respectively represent the contributions of the pressure and viscous force. pw is the dimensionless wall pressure and ωw is the dimensionless wall vorticity defined as ∞= urw /ωω , (here r = 0.5). the dimensionless reynolds number is given by µρ /re ∞= du (10) also, the dimensionless strouhal number is expressed as ∞= ufdst / (11) where, f is the frequency of the vortex shedding (=1/t), d is the diameter of the cylinder and ∞u the free stream velocity. in case of turbulent flow, the viscous runs are done using standard k-ε, realizable k-ε and shear-stress transport (sst) k-ω model (lakshmipathy, 2004). the standard k-ε model is a semi-empirical model based on model transport equations for the turbulence kinetic energy (k) and its dissipation rate (ε). the model transport equation for k is derived from the exact equation, while the model transport equation for ε was obtained using physical reasoning and bears little resemblance to its mathematically exact counterpart. in the derivation of the kε model, it was assumed that the flow is fully turbulent, and the effects of molecular viscosity are negligible. the standard k-ε model is therefore valid only for fully turbulent flows. on the other hand the term “realizable" means that the model satisfies certain mathematical constraints on the normal stresses, consistent with the physics of turbulent flows. the most straightforward way to ensure the realizability is to make cµ variable by sensitizing it to the mean flow (mean deformation) and the turbulence (k, ε). the turbulence kinetic energy k is given by ( )2 2 3 iuk avg= (12) where, uavg is the mean flow velocity and i is the turbulence intensity defined by ( ) 8 1 re16.0 −=i (13) turbulence length can be written as dl 07.0= and turbulence dissipation rate ε as m. m. rahman, m. m. karim and m. a. alim / journal of naval architecture and marine engineering 4(2007) 27-42 numerical investigation of unsteady flow past a circular cylinder using 2-d finite volume method 31 l k c 2 3 4 3 µε = (14) where, cµ is an empirical constant specified in the turbulent model (approximately 0.09). in this study, another turbulent model is used to simulate the turbulent flow named shear-stress transport (sst) k-ω model. it is so named because the definition of the turbulent viscosity is modified to account for the transport of the principal turbulent shear stress. it is such a feature that gives the sst kω model an advantage in terms of performance over both the standard k-ε and realizable k-ε model. other modification includes the addition of a cross-diffusion term in the ω-equation and a blending function to ensure that the model equations behave appropriately in both the near-wall and far-field zones. in sst k-ω model the specific dissipation rate ω can be found by lc k 4 1 2 1 µ ω = (15) the control volume technique consists of integrating the governing equations about each control volume, yielding discrete equations that conserve each quantity on a control volume basis. the coupling of pressure and velocity is done using piso algorithm (versteeg & malalsekera ,1995). 3.2 boundary condition the wall boundary conditions used in this steady are those of impermeability and non-slip condition, i.e., u = 0, v = 0 in the physical domain the flow is not confined. nevertheless, a fictitious external rectangle boundary is needed at a large distance from the cylinder (fig.1) in order to solve the governing equations numerically. uniform free stream condition with velocity ∞u =1.0m/s are applied at the inlet boundary. the periodic condition is considered at the lateral boundaries. also the flow exit is treated as a pressure outlet. 4. results and discussion 4.1. the vortex shedding: 4.1.1 the laminar flow (re = 100) experimental result shows that, for reynolds number less than 40 the flow over the circular cylinder reaches the steady state within 15 second and two symmetric vortices are formed behind the cylinder ( braza et al, 1986). as reynolds number becomes higher than 40 the flow reports a loss of symmetry in the wake and alternating eddies are formed and convected in the wake. this generates the alternating separation of vortices, which are convected and diffused away from the cylinder, forming the well-known karman vortex streets. such destabilizing effects always occur during any physical experiment on the flow around a circular cylinder. the strouhal number (st) is found to be 0.164 for re=100. this result agrees well with the experimental value (0.164~0.165) reported by tritton (1959). the results of unsteady laminar flow for re = 100 are presented in figs. 2-4. the instantaneous stream lines for re = 100 are shown in fig.2 at the four phases during one cycle of vortex shedding. in fig. 3 & fig.4 the directions of the velocity vectors and corresponding contours of pressure are plotted respectively at those phases for the same reynolds number. the alternating formation, convection and diffusion of the vortices are clearly shown. m. m. rahman, m. m. karim and m. a. alim / journal of naval architecture and marine engineering 4(2007) 27-42 numerical investigation of unsteady flow past a circular cylinder using 2-d finite volume method 32 (a) (b) (c) (d) fig. 2: contours of stream function at (a) t = 48.2s, (b) t = 51.2s, (c) t = 54.2s & (d) t = 57.2s for re=100 m. m. rahman, m. m. karim and m. a. alim / journal of naval architecture and marine engineering 4(2007) 27-42 numerical investigation of unsteady flow past a circular cylinder using 2-d finite volume method 33 (a) (b) (c) (d) fig. 3: contours of velocity vectors at (a) t = 48.2s, (b) t = 51.2s, (c) t = 54.2s & (d) t = 57.2s for re=100 m. m. rahman, m. m. karim and m. a. alim / journal of naval architecture and marine engineering 4(2007) 27-42 numerical investigation of unsteady flow past a circular cylinder using 2-d finite volume method 34 (a) (b) (c) (d) fig. 4: contours of static pressure at (a) t = 48.2s, (b) t = 51.2s, (c) t = 54.2s & (d) t = 57.2s for re=100 m. m. rahman, m. m. karim and m. a. alim / journal of naval architecture and marine engineering 4(2007) 27-42 numerical investigation of unsteady flow past a circular cylinder using 2-d finite volume method 35 4.1.2 the turbulent flow (re = 1000 and 3900) the experimental work of roshko (1954) locates the beginning of the laminar-to-turbulent transition at reynolds numbers 200 – 300. beyond this reynolds number but less than 3×105 the wake of the cylinder is completely turbulent and the boundary layer separation is laminar (sumer, 1997). it is known that the flow around the cylinder is two dimensional only when re < 200. for larger reynolds number, the vortex shedding occurs in cells and therefore the flow is generally simulated in 3-d. but the 3-d simulation is very much complicated and treated as a numerically costly procedure. therefore, the turbulent flow over a circular cylinder for reynolds numbers 1000 & 3900 is investigated here using 2-d finite volume method. in this research, the turbulent flow for both reynolds numbers is simulated using three turbulence models namely, standard k-ε, realizable k-ε and shear-stress turbulence (sst) k-ω turbulence model. the vortex shedding is observed and the drag & lift forces is also calculated using these methods. for unsteady case, time interval have been chosen small enough to capture the vortex shedding. the easiest way to set up the right time step size is one that ensures the maximum courant number very near to unity. to avoid running the case for very long time, a small perturbation has been introduced in the domain. this perturbation is generally in the form of very small velocity in perpendicular direction to the flow direction. this assists the solver to start vortex shedding. since there is nothing to sustain this perturbation in the domain for very long time, it dies out and does not affect the calculation of cl and cd. the dimensionless frequency of vortex shedding is the strouhal numbers found by different methods are shown in fig.5. the vortex shedding is visualized by means of the contours of stream function, velocity vectors and static pressure. fig. 6 show the stream function during one cycle of lift coefficient for re = 1000. at the same time the velocity vector and pressure fields are also shown in fig. 7. the alternating vortex shedding clearly observed in these figures. fig. 8 shows the stream function and fig. 9 shows the velocity vector & pressure field during one cycle of lift coefficient for re = 3900. the alternating formation, convection and diffusion of the vortices are clearly shown again. fig.-5: strouhal number vs reynolds number 0 1000 2000 3000 4000 0.05 0.10 0.15 0.20 0.25 0.30 0.35 (laminar) (standard k-epsilon) (realizable k-epsilon) (sst k-omega) (experimental) s tr ou ha l n um be r reynolds number m. m. rahman, m. m. karim and m. a. alim / journal of naval architecture and marine engineering 4(2007) 27-42 numerical investigation of unsteady flow past a circular cylinder using 2-d finite volume method 36 the results in this section show that the vortex shedding is an intrinsic phenomenon of the flow, well predicted by the solution of the navierstokes equations. the vortex shedding is generated by a loss of symmetry of the two dimensional symmetric structures in the wake of the circular cylinder. (a) (b) (c) (d) fig.-6: stream function (a) t=120.12s,(b) t=121.80 (c)t=123.48s, (d) t=125.16s for re=1000 m. m. rahman, m. m. karim and m. a. alim / journal of naval architecture and marine engineering 4(2007) 27-42 numerical investigation of unsteady flow past a circular cylinder using 2-d finite volume method 37 (a) (b) fig.-7: (a) velocity vector (b) static pressure at t=120.12s, t=121.80 t=123.48s, t=125.16s for re=1000 m. m. rahman, m. m. karim and m. a. alim / journal of naval architecture and marine engineering 4(2007) 27-42 numerical investigation of unsteady flow past a circular cylinder using 2-d finite volume method 38 (a) (b) (c) (d) fig.-8: stream function (a)t=75s, (b) t=76.23s, (c)t=77.46s and (d)t=78.79s for re=3900 4.2. the forces: the drag force is a result of the convective motion of the cylinder through the fluid. because of this motion and of the non-slip condition of the wall, a tangential velocity gradient is created in the direction normal to the wall. the mean value of the drag coefficient calculated by the present method for different reynolds numbers is very close to the experimental results of tritton (1959), tritton (1971) and anderson (2005). the total drag is also due to two effects: the pressure field (cdp) and the viscous forces (cdv) calculated by the eqs. (7-8). for re = 100 the mean value of the pressure drag, cdp is 0.917, which corresponds to 75% of the total drag. this value is closer to the experimental result (0.995) of m. m. rahman, m. m. karim and m. a. alim / journal of naval architecture and marine engineering 4(2007) 27-42 numerical investigation of unsteady flow past a circular cylinder using 2-d finite volume method 39 roshko (1954) than that (1.02) of braza et al.(1986). numerical values of the components of total drag coefficient are shown in table 1. (a) (b) fig.-9: (a) velocity vector (b) static pressure at t=75s, t=76.23s, t=77.46s and t=78.79s for re=3900 m. m. rahman, m. m. karim and m. a. alim / journal of naval architecture and marine engineering 4(2007) 27-42 numerical investigation of unsteady flow past a circular cylinder using 2-d finite volume method 40 the pressure and drag coefficients of the turbulent flow at re = 1000 & 3900 have also been calculated using three turbulent models. the predicted cd values by standard k-epsilon model are quite closely to the experimental values shown in table 2. but for the visualization of vortex shedding the realizable k-epsilon turbulence model is more effective, as this model captures the separating flow better than standard kepsilon model. on the other hand, the sst k-omega model is much more recommendable for high reynolds numbers. the global periodic character of the flow is found to be essentially the same as the reynolds number increases. for the reynolds numbers of 1000 and 3900 it should be recalled that the unsteady lift and drag coefficient oscillates periodically. fig. 10 shows the lift and drag coefficient for re=1000. the drag coefficients as a function of different reynolds numbers are compared with experimental and other numerical results in fig. 11 and in table 2. in most of the cases, the computed results show better agreement with experimental data compared to other predicted values. fig.-10: time history of (a) lift coefficient and (b) drag coefficient at re = 1000 table 1: components of total drag coefficient. re pressure coefficient viscous coefficient total drag coefficient 100 0.917 0.329 1.245 1000 0.876 0.119 0.995 3900 0.877 0.12 0.997 5. conclusion the complex problem of the unsteady laminar and turbulent flow around a circular cylinder at reynolds numbers of 100, 1000 and 3900 is studied using 2-d finite volume method. a second-order-accurate numerical scheme is used, based upon a velocity-pressure formulation and conservative schemes. three different turbulent models are used to tackle the complex behavior of turbulent flow. from this study it is found that the two-dimensional finite volume method computes hydrodynamic forces and captures vortex shedding very well. even at high reynolds number, the method is very much applicable without loss of m. m. rahman, m. m. karim and m. a. alim / journal of naval architecture and marine engineering 4(2007) 27-42 numerical investigation of unsteady flow past a circular cylinder using 2-d finite volume method 41 accuracy. it is also observed that standard k-epsilon model computes drag coefficients accurately, where the realizable k-epsilon turbulence model is more effective for visualization of vortex shedding. the sst komega model is much more recommendable for high reynolds numbers. most of the results presented in this research are compared with experimental data and are in better agreement with respect to other numerical results. in this research, 2-d method is found very prospective one for turbulent flow due to its faster convergence. however, the method is restricted here only at two reynolds numbers (1000 & 3900) to analyze turbulent flow around circular cylinder; more studies are required to make it applicable at higher reynolds numbers. fig.-11: drag coefficient vs reynolds number table2: drag coefficient for different reynolds numbers with experimental and some numerical results. re 100 1000 3900 experimental 1.24-1.26 (tritton,1959) 0.9 (anderson, 2005) 0.98+/-0.05 lourenco & shih (1993) tuann & olson (1978) 1.25 -- martinez et al. (1978) 1.1 -- loc (1980) 1.15 -- braza et al. (1986) 1.17 1.15 - jordon & fromm(1972) -1.2 - beaudan & moin (1994) --1.74 lakshmipathy --0.87 breuer (1998) --1.08 present 1.245 0.995 0.997 0 800 1600 2400 3200 4000 0.80 0.85 0.90 0.95 1.00 1.05 1.10 1.15 1.20 1.25 1.30 1.35 1.40 (experimental) (martinez et al) (loc) (braza et al) (jordan & fromm) (mittal) (lakshmipathy) (breuer) (present) d ra g c oe ffi ci en t c d reynolds number m. m. rahman, m. m. karim and m. a. alim / journal of naval architecture and marine engineering 4(2007) 27-42 numerical investigation of unsteady flow past a circular cylinder using 2-d finite volume method 42 references anderson, j.d. (2005): fundamentals of aerodynamics, 4th ed., mc grow-hill, columbus, usa. braza, m., chassaing, p. and minh, h. h.(1986): the numerical study and physical analysis of the pressure and velocity fields in the near wake of a circular cylinder, j. fluid mech., 165, 79. braza, m., chassaing, p. and minh,h. h.(1990): prediction of large-scale transition features in the wake of a circular cylinder, phys. fluids. a2 (8), 1461-1471. beaudan, p. & moin, p. (1994): numerical experiments on the flow past circular cylinders at sub-critical reynolds numbers. report no. tf-62, thermosciences div., dept. of mech. engr., stanford university. coutanceau, m. and bouard (1977): experimental determination of the main features of the viscous flow in the wake of the circular cylinder in uniform translation, j. fluid mech. 79, 231 & 257. coutanceau, m. and defaye, j.r. (1991): circular cylinder wake configurationsa flow visualization survey, appl. mech. rev., 44(6). honji, h. and taneda, s. (1969): unsteady flow past a circular cylinder, j. phys. soc. japan, 27, 16681677. lakshmipathy, s. (2004): pans method for turbulence simulation of high and low reynolds number flow past a circular cylinder, m.sc. thesis, dept. of aerospace eng. texas a. & m. university, usa. loc, t. p. (1980): numerical analysis of unsteady secondary vortices generated by an impulsively started circular cylinder. j. fluid mech, 100, 111. lourenco, l. m. & shih, c. (1993): characteristics of the plane turbulent near wake of a circular cylinder. a particle image velocimetry study, private communication. martinez, g. & ha minh, h.(1978): in proc. intl. conf. on numerical methods in laminar and turbulent flow, swansea, pineridge. mittal, r., (1995): large-eddy simulation of floe past a circular cylinder, annual research briefs, center fo turbulence research, stanford university. payne, r.b. (1958): calculations of unsteady viscous flow past a circular cylinder, j. fluid mech. 4, 81. rai, m. m. & moin, p. (1993): direct numerical simulation of transition and turbulence in a spatially evolving boundary layer, j. comp. phys. 109, no. 2, 169-192. reichl, p.,hourigan, k. and thompson, m.c. (2005): flow past a circular cylinder close to a free surface, j. fluid mech. 533, 269-296. roshko, a (1954): vortex shedding from circular cylinder at low reynolds number, naca rep. no.1191. sumer, b. m. (1997): hydrodynamics around cylindrical structures, world scientific, singapore. tritton, d.j. (1959): experiments on the flow around a circular cylinder at low reynolds number, j. fluid mech. 6, 547. tritton, d. j.(1971): a note on vortex streets behind circular cylinders, j. fluid mech. 45,203. tuann, s. y.& olson, m. d. (1978): numerical studies of the flow around a circular cylinder by finite element method, computers and fluids, 6, 219 versteeg,h.k & malalasekera, w. (1995): an introduction to computational fluid dynamics, longman scientific & technical, england. separation points of magnetohydrodynamic boundary layer flow along a vertical plate with exponentially decreasing free stream velocity¬ journal of naval architecture and marine engineering june, 2023 https://doi.org/10.3329/jname.v20i1.64301 http://www.banglajol.info 1813-8535 (print), 2070-8998 (online) © 2023 jname publication. all rights reserved. received on: feb., 2023 design and testing of a compact inexpensive prototype remotely operated underwater vehicle for shallow water operation a. tanveer1,2 and s. m. ahmad3,4 1department of mechanical & aerospace engineering, institute of avionics & aeronautics, air university, islamabad 44000, pakistan, ahsan.tanveer@mail.au.edu.pk 2faculty of mechanical engineering, ghulam ishaq khan institute of engineering sciences and technology, topi 23640, pakistan, ahsantanveer3883@gmail.com 3control & instrumentation engineering department, king fahd university of petroleum and minerals, dhahran 31261, saudi arabia, sarvat.ahmad@kfupm.edu.sa 4interdisciplinary research center for intelligent manufacturing and robotics, king fahd university of petroleum and minerals, dhahran 31261, saudi arabia abstract: this article focuses on a small and cost-effective remotely operated unmanned underwater vehicle (rov) that has the ability to move through shallow waters. the proposed rov measures 35 cm × 26 cm × 23 cm and weighs 1.59 kg. the primary objective is to remotely communicate in real time using serial protocol to record the vehicle's orientation data. the three dc motors powering the rov are controlled by a multi-parameter root-locus proportional-integral (mrpl-pi) controller. utilizing the processing capabilities of the atmega328 microcontroller, heading, depth control, and water condition monitoring are all accomplished simultaneously. the arduino development environment is used to establish the rov's remote control. the problem with human divers who can only descend to a depth of 30 meters is resolved by the proposed rov's ability to dive up to 100 meters underwater. also, the suggested vehicle can be beneficial in marine applications such as measurement, maintenance, operations, and surveillance. furthermore, the preliminary testing carried out in the pool indicates that the vehicle is fully operational and responds to the pilot's commands efficiently. keywords: remotely operated underwater vehicle, marine robotics, proportional—integral controller, affordable rovs, underwater exploration nomenclature 𝐾𝑝 proportional gain 𝑃 pressure/depth 𝛼 weighing factor for complementary filter 𝜌 density of water 1. introduction human divers and manned underwater vehicle have increasingly been supplanted by rovs due to the hazards to human life that come with underwater work. the fundamental reason for the increasing popularity of rovs over the past several years is that, in addition to safety, they have provided a more efficient and affordable way for undersea research. numerous rov studies have recently been published with applications in ocean research (tanveer & ahmad, 2022), (li et al., 2023), (dong & duan, 2023). rovs have been employed for underwater maintenance operations, naval defense, and marine research (capocci et al., 2017), (khojasteh & kamali, 2017). the majority of research efforts, however, have been directed at underwater surveillance (ferri et al., 2018), (shahani et al., 2018). other research studies have centered on gathering tidal data (brito & griffiths, 2018). additionally, rovs have been used to take images underwater (sánchez-ferreira et al., 2019). considering the vast variety of specializations that are involved in obtaining optimal performance, research continues to be actively pursued despite substantial advancements that have been made in the development of rovs. a prevalent topic in the marine robotic community right now is the development of affordable vehicles without compromising functionality (teague et al., 2018). some of the domains where current research is impacting include the autonomy of the rov, energy efficiency, and communication reliability (cely et al., 2019), https://doi.org/10.3329/jname.v20i1.64301 http://www.banglajol.info/ mailto:ahsan.tanveer@mail.au.edu.pk mailto:ahsantanveer3883@gmail.com mailto:sarvat.ahmad@kfupm.edu.sa a. tanveer, s.m. ahmad / journal of naval architecture and marine engineering, 20(2023) 1-10 design and testing of a compact inexpensive prototype remotely operated underwater vehicle for shallow water operation 2 (martorell-torres et al., 2018), (xu et al., 2018). however, in the last few years, there has been an increasing trend towards creating affordable and easily accessible marine robotic platforms. in recent times, there have been various attempts to create affordable underwater robotic platforms, such as the one in (laidani et al., 2022), and (zhang et al., 2022). various pakistani researchers have also made efforts to develop a low-cost rov, with success achieved by (rafi et al., 2018), and (shah et al., 2021). while these systems offer costeffective solutions, it remains difficult to design a system that is suitable for academic purposes and uses widely accessible components. in this article, we propose a low-cost and easy-to-replicate rov that is capable of functioning as a fully operational underwater vehicle in calm waters. we suggest utilizing an arduino nano as an onboard computer since it is affordable, open-source, and supports c++. it is also recommended to add a complementary filter to the mprl-pi controller. this contributes to the improved performance of the controller utilized for trajectory tracking. the remainder of this article is structured as follows: section 2 details hardware organization, mprlpi controller development, and rov mechanical design. the experimental findings are presented in section 3, along with a comparison to commercial rovs and details on rov’s performance in actual underwater environment. the conclusions of the article are finally summarized in section 4. 2. rov design and construction 2.1 electronic architecture design the hardware block diagram, which comprises of two components—the rov and the ground station—is shown in fig. 1. an ethernet cable is used to transmit data between these two components. an arduino nano microcontroller, which forms the central part of the rov hardware, is in charge of performing operations like measuring and recording various variables associated with the sensors and controlling motors. additionally, the system includes two power sources: one is an ethernet wire that provides 5 v to the arduino nano microcontroller and digital sensors, and the other is a dedicated line that provides 12 v to power two dc motors. the rov has virtually limitless endurance with these power sources. motor drivers, which input pulse width modulation (pwm) signals produced by arduino nano microcontrollers, control dc motors. typically, a bi-directional motor driver is needed for surge and yaw motion. however, it is often large in size. unidirectional motor drivers, on the other hand, are fairly small in size but can only run motors in one direction. onboard the rov employed in this study are two distinct types of motor drivers. first a bi-directional l298dn which can power two motors simultaneously. second a uni-directional irf 520, capable of handling a single motor at a time. l298dn is used to operate the two side-mounted thrusters, while irf 520 drives the top-mounted thruster. fig. 1: rov electronic architecture. a. tanveer, s.m. ahmad / journal of naval architecture and marine engineering, 20(2023) 1-10 design and testing of a compact inexpensive prototype remotely operated underwater vehicle for shallow water operation 3 for effective control and maneuvering, the precise orientation and position of the rov must be recorded. this is done by using an mpu9250 sensor, which has nine degrees-of-freedom (dofs). the degrees of freedom indicate that the sensor contains three gyroscopes, three magnetometers, and three accelerometers. the data from mpu9250 is processed on the arduino nano using a complimentary filter (brzozowski et al., 2018), (yang & sun, 2018) to decrease noise and filter undesirable effects. the cutoff-frequency of the low-pass filter (lpf) is set by taking into account the data from the accelerometer, whilst the design of the high-pass filter is based on the signal obtained from the gyroscope measurement. fig. 2 illustrates a schematic for the complimentary filter. by mixing data from the accelerometer and gyroscope, the expression for the complementary filter is obtained: 𝜃𝑛 = (1 − 𝛼)(𝜃𝑛−1 + �̇�𝑔𝑑𝑡) + (𝛼𝜃𝑎) (1) 𝛼 = 𝑡𝑐 /(𝑡𝑐 + 1/𝑓𝑠 ) (2) where, 𝜃𝑛 is the current output, 𝜃𝑛−1 is the previous output, �̇�𝑔 is the gyroscope data, 𝜃𝑎 is the accelerometer data, 𝑡𝑐 is time constant and 𝑓𝑠 is the sampling frequency. 2.1.1 control design since it is widely known that rovs are vulnerable to parametric uncertainties and external disturbances, the control subsystem employed in the proposed rov is a multi-parameter root-locus pi (mprl-pi) controller (tanveer & ahmad, 2021), which has the capability of disturbance rejection. to enhance the performance of the pi controller and the stability of the rov, it is suggested in this study that a complementary filter be added to the mpu9250 sensor output. the pi algorithm has the following expression in the time domain (nise, 2020): 𝑢(𝑡) = 𝐾𝑝𝑒(𝑡) + 𝐾𝑖 ∫ 𝑒(𝑡) 𝑑𝑡 (3) where 𝑢(𝑡) indicates the control signal, 𝑒(𝑡) = 𝑦(𝑡) − 𝑟(𝑡) defines the position tracking error, 𝑟(𝑡) is the intended trajectory, 𝑦(𝑡) denotes the real trajectory, and 𝐾𝑝 and 𝐾𝑖 signify the proportional, and integral gains, respectively. the block diagram of the pi controller is shown in fig. 3. it is obvious that using the complementary filter shown in fig. 2 decreases noise levels at the mpu 9250 sensor's output. fig. 2: an illustration of a complementary filter in use. a. tanveer, s.m. ahmad / journal of naval architecture and marine engineering, 20(2023) 1-10 design and testing of a compact inexpensive prototype remotely operated underwater vehicle for shallow water operation 4 the response of a mprl-pi controller to a unit step input in heading is simulated in fig. 4 with a sampling rate of 45 ms. it can be shown that the proposed rov stabilizes itself when an abrupt change is induced in the input. 2.2 mechanical design the maximum depth capability of the rov is determined by the mechanical properties of the polyvinyl chloride (pvc) utilized as a structural element in this study. equation (4) is used to compute depth; however, it requires knowledge of the material's fracture point, equates to a pressure of 1.73 mpa. the maximum depth for a pvc pipe under these conditions is h = 125.65 m. 𝑃 = 𝜌𝑔ℎ (4) where 𝑃 denotes pressure, 𝜌 = 1000 𝑘𝑔/𝑚3 signifies water density, 𝑔 represents gravity, and ℎ connotes depth in meters. fig. 5 depicts the 3d structural architecture of the proposed rov. the design considers buoyancy and hydrostatic factors. the structure shown incorporates a weight balancing configuration to enable implementation in the motor topology, and it was created in the solidworks 3d figure simulator. it measures 35 cm × 26 cm × 23 cm, weighs 1.66 kg, and an estimated volume of 𝑉 = 1.3 𝑥 10−3 𝑚3. this information fig. 3: pi controller block diagram with complementary filter and actuation subsystem. fig. 4: simulated response of a pid controller to a unit step input in heading. a. tanveer, s.m. ahmad / journal of naval architecture and marine engineering, 20(2023) 1-10 design and testing of a compact inexpensive prototype remotely operated underwater vehicle for shallow water operation 5 indicates that the buoyancy is 13.19 n, which can be calculated using equation (5), where 𝑉 is the volume, and 𝐸 is the total thrust. therefore, a maximum thrust force of 7.5 n is considered in the projected topology for the three motors. 𝐸 = −𝜌𝑔𝑉 (5) fig. 6 depicts the placements of the three dc motors: two are situated on the front for translational and rotational motions, and one is located on top for ascending and descent movements. the materials utilized to build the mechanical framework of the rov are listed in table 1. the placement of the three dc motors is made possible by the mechanical structure's design. table 2 lists the parts and their estimated weights, which results in a total weight of 1.59 kg. fig. 5: 3d rendering of the envisioned rov. fig. 6: rov movements: (a) surge/translational; (b) heave/ascent-descent; (c) yaw/rotation. a. tanveer, s.m. ahmad / journal of naval architecture and marine engineering, 20(2023) 1-10 design and testing of a compact inexpensive prototype remotely operated underwater vehicle for shallow water operation 6 table 1: materials used to build the rov structure. quantity (no. of pieces) material description size (in cm) 2 pvc pipe ∅1.27 x 13.9 2 pvc pipe ∅1.27 x 11.4 2 pvc pipe ∅1.27 x 10.1 1 pvc pipe ∅1.27 x 8.8 2 pvc pipe ∅1.27 x 7.6 2 pvc pipe ∅1.27 x 5.7 4 pvc pipe ∅1.27 x 4.4 6 pvc pipe ∅1.27 x 3.8 9 pvc elbow ∅1.27 9 pvc tee ∅1.27 1 polypropylene netting 30 x 18 3 plastic pipe strap ∅1.27 table 2: the material weights for the rov. item quantity weight (kg) total weight (kg) pvc structure 1 0.770 0.770 motors 3 0.250 0.750 motor drivers 3 0.025 0.075 arduino nano 1 0.001 0.001 sensors 2 0.001 0.002 total weight 1.59 3. experimental results this section provides an overview of the experimental findings utilizing the arduino nano as the rov's onboard computer. the data that was recorded in a controlled aquatic setting is also presented. additionally, the key characteristics are contrasted with those of other commercial rovs, and the results of the implementation of the mprl-pi controller are provided. 3.1 rov performance 3.1.1 motor sealing and thruster test rig design for the rov under consideration, a custom-built low-powered thrusters are developed using an off-the-shelf 12v dc motor. the choice of a dc motor over a bldc motor was motivated by its low cost, ease of operation, low-cost motor drivers, and ease of sealing. fig. 7: the motor housed within a pvc casing. a. tanveer, s.m. ahmad / journal of naval architecture and marine engineering, 20(2023) 1-10 design and testing of a compact inexpensive prototype remotely operated underwater vehicle for shallow water operation 7 the motors are initially wrapped in insulating tape to cover the openings at the top and sides of the motor, leaving just the shaft and connection terminals exposed. the motor housing is custom manufactured from pvc pipes, and the endcaps for the housing are fabricated from acrylic, as seen in fig. 7. from the interior, the motor housing is covered with a thick coating of grease. molten wax is poured into the housing once the motor is installed to fill any gaps and offer an extra layer of water protection. a test rig is built to test the motors, and the motor step response and thrust force table are recorded. as shown in fig. 8, a simple rectangular iron frame is built with a nut-and-bolt fastening in the centre to connect the load cell. an s-shaped 5-kg load cell is used to measure the amount of force that the motor generates underwater. 3.1.2 stability performance the rov's orientation is tracked by the mpu9250 sensor, which is also extremely helpful in stabilizing the rov. a smoothed signal is produced from using the complimentary filter. the data from the gyroscope is utilized in the short term since it is accurate and resistant to external influences. the accelerometer's data is used in the long-term since it did not drift. in the pool tests, the full operating ranges for gyroscope and accelerometer were +/-2000 deg/s and +/-16g, respectively. fig. 9 depicts motion in 𝑥 − 𝑦 plane. it is clear that no erratic accelerometer and gyroscope data was found, proving that they did not drift off. the complementing filter also assisted in enhancing the rov's stability and the quality of the data that was collected. the response of the mprl-pi controller for yaw and heave motion can be seen in fig. 10. it is evident that the actual trajectory followed the anticipated one. it is clear that the error was reasonably small in trajectory tracking. as a result, the rov's stability is enhanced, which improves the quality of the gathered data. fig. 8: motor test rig. fig. 9: analyzing the output of the complementary filter while doing rov stability tests. a. tanveer, s.m. ahmad / journal of naval architecture and marine engineering, 20(2023) 1-10 design and testing of a compact inexpensive prototype remotely operated underwater vehicle for shallow water operation 8 3.1.3 real-time tests in controlled pool environment fig. 11 depicts the submerged rov in a controlled aquatic setting. in the first phase of experimentation, openloop heave and yaw experiments were performed. rov is given known input commands from a ground station pc. the response of the rov against these commands is observed in real-time and stored on ground-pc. these tests also confirmed that water does not seep into the hull and that the vehicle is buoyant in water. furthermore, the rov has a payload capacity equal to 5% of its entire weight. 3.1.4 in comparison to other rovs table 3 compares the capabilities of the proposed rov to commercial rovs. some characteristics can be seen to be similar. however, the proposed rov is far less expensive, allowing it to be utilized as an educational test vehicle and a low-cost inspection platform. the use of a complementary filter on imu signals improves the performance of the mprl-pi controller used for tracking control. the rov's remote control is written in c++, fig. 10: mprl-pi controller response. fig. 11: a view of the submerged rov. a. tanveer, s.m. ahmad / journal of naval architecture and marine engineering, 20(2023) 1-10 design and testing of a compact inexpensive prototype remotely operated underwater vehicle for shallow water operation 9 which is compatible with a variety of operating systems. finally, because of the characteristics of the hardware design, the user of the proposed rov can maintain its technological independence through the repair and replacement of mechanical and hardware components. table 3: specifications of the envisaged rov in comparison to commercial rovs. features proposed rov standard blue rov uabc rov (aguirre-castro et al., 2019) open rov architecture open open open open maximum depth 100 100 100 100 endurance (in hrs) unlimited 2-3 2-3 2-3 processing type series unknown parallel unknown communication ethernet (cat-5e) ethernet ethernet ethernet control algorithm mprl-pi pid smart pid pid payload capacity (% of total weight) 05 20 20 35 total weight (kg) 1.59 11 15.64 2.9 total cost ($) 80 2780 600 1790 4. conclusions the article introduces the design and real-time testing of an inspection class rov, that weighs 1.59 kg. the envisioned rov can move in three axes of translation, ascent/descent, and rotation. the rov was designed to reach depths of up to 100 meters, addressing the problem of human divers who can only reach depths of 30 meters. moreover, the rov is propelled by three dc motors controlled by a mprl-pi controller. improved rov stability is achieved by using a complementary filter to reduce noise from the mpu 9250 sensor. the rov is programmed using open-source software tools, allowing for the incorporation of additional sensors and functionality as per the operational requirements. additionally, the adaptability of mechanical design and inexpensive hardware expands the range of possible applications, including monitoring and the research of maritime flora and fauna, without sacrificing the accuracy of the data collected. acknowledgements the authors gratefully acknowledge the research facilities and financial support provided by faculty of mechanical engineering, ghulam ishaq khan institute of engineering 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architecture and marine engineering june, 2023 https://doi.org/10.3329/jname.v20i1.61155 http://www.banglajol.info 1813-8535 (print), 2070-8998 (online) © 2023 aname publication. all rights reserved. received on: aug. 2022 mhd casson and carreau fluid flow through a porous medium with variable thermal conductivity in the presence of suction/injection babitha1, c.v. ramana murthy2 and g.venkata ramana reddy3 1department of mathematics, presidency university, bangalore, india-560073, email: babithaatiwale16@gmail.com 2department of engineering mathematics, college of engineering, koneru lakshmaiah education foundation, andhra pradesh, india-522302, email: gvrr1976@kluniversity.in. 3department of engineering mathematics, college of engineering, koneru lakshmaiah education foundation, andhra pradesh, india-522302., abstract: the primary interest of this study is to investigate the results of magnetohydrodynamics casson and carreau fluid heat via a porous medium at the aspect of a stretching sheet. variable thermal physical phenomena and suction/injection parameter results are also taken into account. similarity transformations are applied to convert the governing equations of the magnetohydrodynamics casson and carreau fluid glide model into dimensionless non-linear normal differential equations. the results on velocity, concentration, temperature profiles, surface drag, nusselt number, and sherwood number are discussed for various flow parameters and presented the results through plots and tables. the fluid flow is analyzed for every suction and injection case. from the analysis it is noted that the velocity profile reduces by increasing the magnetic parameter and the reverse trend is observed for increasing values of weissenberg number in the case of suction and injection. the temperature space and heat transfer rate enhances by increasing the variable thermal conductivity and more in the case of suction. keywords: casson and carreau fluid, suction.injection, variable thermal conductivity, magnetohydrodynamics, thermal radiation. nomenclature greek symbols 𝐵0 magnetic field (𝑇) 𝛤 characteristic time constant 𝐶𝑝 specific heat (𝐽 𝑘𝑔 −1𝐾) 𝜈 kinematic viscosity (𝑚2𝑠−1) 𝑞𝑟 radiative heat flux 𝜙 dimensionless concentration 𝑊𝑒 weissenberg number 𝜃 dimensionless temperature r radiation parameter 𝑘∗ rosseland mean absorption coefficient dct soret kind diffusivity(𝑚2𝑠−1) 𝜇 dynamic viscosity (𝑘𝑔 𝑚−1𝑠 −1) 𝑀 magnetic parameter 𝜎 electric conductivity (ω−1𝑚−1) 𝑘 thermal conductivity of the fluid (𝑊 𝑚−1𝐾 −1) 𝜌 density (𝑘𝑔 𝑚−3) 𝐵0 magnetic field (𝑇) 𝜎 ∗ stefan-boltzmann constant (𝑊 𝑚−2𝐾 4) 𝐶𝑝 specific heat (𝐽 𝑘𝑔 −1𝐾) 𝛾 casson fluid parameter 1. introduction non-newtonian fluids have drawn significant attention in recent decades on account of their tremendous applications in the fields of engineering, environment, and geophysics. also, the study of non-newtonian fluid motion and heat transfer past a stretching sheet has renewed great attention from several researchers due to its significance in many industrial and engineering applications, like liquid crystal solidification, exotic lubricants, wire drawing, fibers spinning, petroleum production, continuous cooling, and glass-fiber production, etc, in recent decades, several researchers (2016,2017,2018) reported their work on flow past a stretching sheet under various circumstances. https://doi.org/10.3329/jname.v20i1.61155 http://www.banglajol.info/ mailto:babithaatiwale16@gmail.com mailto:gvrr1976@kluniversity.in babitha, c.v.r. murthy and g.v.r. reddy3/ journal of naval architecture and marine engineering,20(2023) 25-35 mhd casson and carreau fluid flow through a porous medium with variable thermal conductivity… 26 the concept of magnetohydrodynamics (mhd) is adopted in the design of pumps, heat exchangers, space vehicles, thermal production, etc. many researchers were interested in studying mhd non-newtonian fluids flow through a permeable medium because of its potential uses in alloy and metal solidification, nuclear fuel debris treatment, and other fields ([2020,2021,2019,2020,2023,2022,2014]). in view of these applications and the importance of soret together with dufour on fluids that have a light molecular weight together with fluids with medium molecular weight, investigators have published many works whom alam et al. (2005) examined the effects of soret together with dufour on free convection mhd as well as mass transport flow using numerical approach. hasanuzzaman et al (2021) considered soret together with dufour on mhd unsteady mixed convection flow past a radiative porous vertical plate. cheng (2011) has elucidated soret together with dufour’s impacts on mixed convection heat together with mass transport through a downward-pointing vertical wall. sharma et al. (2014) examine the impact of soret together with dufour on heat as well as mass transport for mhd mixed convection flow with ohmic heating. the problem of mixed convective flow together with incompressible flow under the action of buoyancy as well as transverse magnetic field simultaneously in the presence of soret and dufour effects has been examined by makinde (2011). in the non-newtonian fluids regime, a substantial amount of research has been done, and many more are needed in a range of non-newtonian models. since the power-law model is quite simple, few researchers are used to investigating non-newtonian effects. however, due to the limits of the power-law model, this model has limitations, particularly for high and low shear rates. to investigate non-newtonian effects, we can use various viscosity models. the carreau fluid is a generalized newtonian fluid in which fluid viscosity is dependent on shear rate. the carreau fluid model is useful for describing fluid flow behavior at high shear rates. this model was first introduced by pierre carreau in 1972. salman et al. (2018) have discussed the effect of variable thermal conductivity, and suction/injection on carreau fluid flow past a stretching sheet. under various conditions, the flow and heat transport characteristics of mhd carreau fluid are examined by shah et al.(2017). the flow and heat transport characteristics of mhd carreau fluid past a permeable stretching sheet are studied by reddy and sandeep (2018) and they noted that dufour and soret parameters regulate the rate of heat and mass transfers. madhura et al. (2022) have examined the effect of nonlinear thermal radiation on the mhd flow of micropolar carreau fluid. casson fluid is a plastic fluid that produces shear stress in constitutive equations. soup, jelly, honey, paints, tomato sauce, coal in water, concentrated fruit juice, blood, and other fluid models are examples of casson fluid models. casson (1959) developed this model to estimate the flow behavior of printing ink pigment oil suspensions. priam and nasrin (2022) have carried out a numerical study to examine the time-dependent peristaltic duct flow of four different casson fluids and they noticed that 14.05% magnification of thermal distribution and 64.91% diminishment of total entropy due to rising internal thermal radiation from 0.5 to 2. sohail et al. (2020) have analyzed the effect of variable thermal conductivity on mhd casson fluid flow past a nonlinear stretching surface with entropy generation. the radiative heat transfer effect on a mixed convective flow of casson fluid past a permeable sheet is discussed by waqas et al (2017). the impact of couple stresses on mixed convective flow of casson fluid through microchannel is studied by felicita et al. (2016). extensive researchers examined the nature of casson fluid under various situations to analyze the flow and heat transfer characteristics (2020,2014, 2016). raju and sandeep (30) have examined the effect of nonlinear thermal radiation on the threedimensional flow of casson carreau fluid in the presence of homogeneous–heterogeneous reactions. the significance of non-newtonian fluids flowing through a stretching sheet is explained by the literature review described above. in this study, the magnetohydrodynamics casson and carreau fluid (ccf) model was taken into account when analyzing the flow behavior. the primary goal of this work is to investigate the heat transfer properties of mhd ccf flow past a stretching sheet via a porous media. the study includes the effects of thermal radiation, fluctuating thermal conductivity, and suction/injection. the nonlinear dimensionless governing equations are numerically solved using the shooting method to examine how embedded parameters affect processes for mass, heat, and fluid transport. the findings are displayed using plots and tables. 2. flow analysis and mathematical formulation consider two-dimensional steady, mhd boundary layer flow of viscous, incompressible casson and carreau nanofluid over the heated permeable porous elongated sheet in the presence of chemical reaction were babitha, c.v.r. murthy and g.v.r. reddy3/ journal of naval architecture and marine engineering,20(2023) 25-35 mhd casson and carreau fluid flow through a porous medium with variable thermal conductivity… 27 examined. the sheet is taken into consideration within the x-axis with stretching velocity ,wu ax wherein 0a  is a constant. it was assumed that the flow of magnetic reynolds variety is very small in liquid metals and partially ionized and the results of polarised prices are not taken into account. so, a prompted magnetic field is negligible in assessment with an applied magnetic area. the cartesian coordinate gadget has been utilized in a manner that the x-axis is along the stretching sheet, y-axis is ordinary to the sheet, the beginning is positioned at the space, and the flow inside the place is considered. the geometry of the drift phenomena is depicted in fig-1. fig-1. schematic diagram of the flow the governing equations for the fluid flow of the model are formulated as [2022]: 0 x y u v  , (1)     2 22 0 * 3 11 1 2 x y x yy n b uu vu u u u k                           , (2)    1 1 x y yy ry tc yy p p ut vt k t t q d c c c      , (3)  *x y sm yy ct yyuc vc d c d t kr c c     , (4) with the following boundary conditions that physically go along , , , at 0, 0, , as . w w w w u u ax v v t t c c y u t t c c y              (5) in this study we incorporated the variable thermal conductivity, and it is expressed as   1 w t t k t k t t              . in this analysis, the radiative heat flux is assumed to be >>ry rxq q since the heat flux diverges only towards y direction. hence, the heat flux ryq dominates the fluid flow. considering that the difference in temperature throughout the flow is small in a way that 4 t is evaluated as a linear function of the ambient temperature t . simplifying 4 t in taylor’s approach in t and forgone terms of higher order to obtain: 4 3 4 4 3t t t t     . (6) utilizing rosseland approximation [26], the heat flux in terms y gives 4 0 4 = 3 r t q ke y     , (7) here 0 signifies stefan-boltzmann constant and ke signifies mean absorption coefficient. since the rosseland approximation is utilized in this analysis, the tangent hyperbolic liquid is assumed to be optically thick liquids. linearizing the (7) above and utilizing the outcome on the energy equation to obtain babitha, c.v.r. murthy and g.v.r. reddy3/ journal of naval architecture and marine engineering,20(2023) 25-35 mhd casson and carreau fluid flow through a porous medium with variable thermal conductivity… 28    30 161 3 x y yy yy tc yy p p ut vt k t t t t d c c c ke         . (8) the governing equations (2),(4),(8) and (5) are transformed into ordinary differential equations by introducing the dimensionless variables are given by :               , , , . w w t t c c a av xf y t t c c v                    (9) the stream function velocity  can be defined as , y x u v    so that equation (5) satisfies the continuity equation.  f  denote the injection and suction,  is the dimensionless space variable,    and    are the dimensionless of temperature and concentration of the fluid respectively. in view of the above-mentioned transformations equations (2), (4) and (8) are reduced to the following odes:     22 3 11 1 1 0 2 n f ff f we f f m f k                       , (10)  2 4 1 pr pr 0 3 f nd r                      , (11) 0 nd le krle ld          . 12) the transformed boundary restrictions as: , 1, 1, 1 at 0, 0, 0, 0 as . f s f f                  (13) where , and f   are the dimensionless velocity, temperature and concentration respectively, the prime denotes differentiation with respect to . where the physical parameters are:     2 * 0 d magnetic parameter, -permeabelity parameter, n brownian motion, n thermoporetic parameter leiws number, pr pramdtl number, reynolds number; 4 r x w b w t w b e b b x k a m k u x d c c d t t t ax le r d                              3 radiation parmater. t k k      the skin friction f c , local nusselt number x nu and sherwood number x sh are the important physical quantities they can be defined as follows    2 , , ,w w m f x x w sm ww xq xq c nu sh k t t d c cu          (14) here 3 21 1 1 , , . 2 w w m sm u n u t c q k q d y y y y                               (15) using (9) and (12), the above quantities can be transformed form are:          1 1 1 3 2 2 2 1 1 4 re 1 0 0 , re 1 0 , re 0 . 2 3 f x x x x x n c f we f nu sh r                              where 2 re x ax   is the local reynolds number. 3. numerical computation: the non-dimensional governing boundary layer equations (10)-(12) subject to the boundary conditions of (15) are solved numerically by using the runge-kutta fourth-order method in conjunction with the shooting babitha, c.v.r. murthy and g.v.r. reddy3/ journal of naval architecture and marine engineering,20(2023) 25-35 mhd casson and carreau fluid flow through a porous medium with variable thermal conductivity… 29 technique. first, the higher-order non-linear differential equations have been converted into simultaneous linear differential equations of first order and are further transformed into the initial value problem and it is solved numerically by applying the runge-kutta fourth order along with the shooting technique. the skin-friction, the nusselt and the sherwood numbers have been discussed in detail and various physical parameters have been illustrated graphically. 4. validation: skin friction coefficient values for different hartmann numbers are compared in table 1 with the work presented by salman et al. [5] for analyzing the accuracy of our results which are showing close agreement with each other. 5. results and discussion the main goal of this research is to investigate and discuss a complete analysis of the steady, mhd boundary layer ccf flow past an infinite stretching sheet embedded with porous media. similarity and shooting schemes were employed to obtain the solutions for velocity, temperature, and concentration profiles. thermal radiation, suction/injection, and variable thermal conductivity effects are considered in the study. the variation of nondimensional fluid motion, concentration, and temperature is discussed in detail for different physical parameter values, including the magnetic parameter, weissenberg number, porous parameter, carreau fluid parameter, casson fluid parameter, radiation parameter, prandtl number, chemical reaction parameter, modified dufour parameter, and dufour solutal lewis number is discussed in detail. the variation in fluid flow, temperature, and concentration are observed for the two cases 𝑆 > 0 and 𝑆 < 0. in plots, 𝑆 > 0 indicates the suction with the solid line, and 𝑆 < 0 is for injection with dashed lines. physically 𝑆 is utilized to control the fluid flow in the channel. figure 2 shows the impact of magnetic parameters on dimensionless velocity profiles for both cases of suction and injection. increasing 𝑀 values promote diminution in fluid flow for both cases. physically 𝑀 corresponds to lorentz force because of which greater values of 𝑀 enhance lorentz force and this force is one type of resistive force acting against the motion of the fluid therefore fluid velocity decreases. in plots, the value of power law index n value is fixed as n=1.5 and n=1.6, because the considered ccf fluid is nonnewtonian. generally, the power law index which categorizes the fluids into pseudo-plastic or shear thinning non-newtonian for 𝑛 < 1, newtonian fluid for 𝑛 = 1, and dilatant or shear thickening non-newtonian fluid for 𝑛 > 1. the range of physical parameters considered in this study is as follows: 0 ≤ 𝑀 ≤ 4, 0.71 ≤ pr ≤ 7, 0 ≤ 𝑅 ≤ 3, 0.1 ≤ 𝑁𝑑 ≤ 0.5, 0.1 ≤ 𝐿𝑑 ≤ 0.5, 0.1 ≤ 𝛽 ≤ 3, 1 ≤ 𝑊𝑒 ≤ 4, 0.1 ≤ 𝛾 ≤ 0.5, 0.1 ≤ 𝐾𝑟 ≤ 2, −0.5 ≤ 𝑆 ≤ 0.5, 0.5 ≤ 𝐾 ≤ 4. fig. 2. velocity profile versus magnetic parameter. fig. 3. velocity profile versus weissenberg number the influence of the weissenberg number 𝑊𝑒 on the velocity profile is shown in figure 3. the fluid motion improves as the weissenberg number is increased. in terms of physics, the weissenberg number is the ratio of relaxation time to fluid viscosity. the viscosity of the fluid decreases as the weissenberg number rises, and the fluid motion rises as a result. the velocity profile is reversed when the porous parameter 𝐾 is increased and is depicted in figure 4. the increase in the porosity parameter of the fluid is owing to an increase in the viscosity babitha, c.v.r. murthy and g.v.r. reddy3/ journal of naval architecture and marine engineering,20(2023) 25-35 mhd casson and carreau fluid flow through a porous medium with variable thermal conductivity… 30 of the fluid, or a drop in the permeability at the edge, or a decrease in the stretching rate of the accelerating surface, resulting in a progressive decrease in the fluid's flow velocity. in figure 5, the velocity profile is plotted against the casson fluid parameter 𝛾. increased 𝛾 slowed the fluid flow in the direction of the stretching surface. this is because of the fall in yield stress at greater values of the 𝛾 which leads the fluid to behave more like a newtonian fluid, resulting in a decrease in fluid velocity. figure 6 shows the impact of the nonlinear thermal conductivity parameter 𝛽 on the thermal profile. the temperature rises as a function of 𝛽. this is due to the fact that the higher the thermal conductivity, the higher the kinetic energy of the fluid particles, which raises the fluid temperature. fig. 4. velocity profile versus porous parameter. fig. 5. velocity profile versus casson fluid parameter. fig. 6. temperature profile versus thermal conductivity parameter. fig. 7. temperature profile versus radiation parameter. figure 7 displays temperature curves for various radiation parameter values. the radiation parameter 𝑅 specifies how much conduction heat transfer contributes to thermal radiation transfer. it is evident that increasing the radiation parameter causes the temperature within the boundary layer to rise. figure 8 depicts the impact of prandtl's number 𝑃𝑟 on the temperature field. the numerical results reveal that when the prandtl number increases, the temperature decreases. a rise in the prandtl number is associated with a decrease in the thickness of the thermal boundary layer and a lower average temperature within the boundary layer. because lower 𝑃𝑟 values imply higher thermal conductivities, heat can diffuse away from the heated surface more quickly than with higher 𝑃𝑟 values. as a result, the boundary layer is thicker, and the rate of heat transmission is lowered in the case of smaller prandtl numbers. figure 9 shows the impact of the dufour number 𝑁𝑑 on the fluid's temperature. variation in the dufour number has only minor effects on the fluid temperature, according to this diagram. the dufour effect is magnified, which results in a modest increase in the fluid's temperature because the viscosity of fluid decreases and particles take momentum and the average temperature of fluid enhances and also noticed that the decrement in heat transfer rate. since dufour number and heat transfer rate are inversely related, i.e., 𝑁𝑑 ⋉ 1 𝑇𝑤−𝑇∞ . so by rising 𝑁𝑑 temperature gradient and thermal potential decays due to which heat flux reduction. figure 10 depicts the effect of the dufour solutal lewis number 𝐿𝑑 on the solute profile. with an increase in babitha, c.v.r. murthy and g.v.r. reddy3/ journal of naval architecture and marine engineering,20(2023) 25-35 mhd casson and carreau fluid flow through a porous medium with variable thermal conductivity… 31 the number of dufour solutal lewis, the solute distribution increased. within the solute boundary layer, increasing the 𝐿𝑑 increases the fluid concentration. fig. 8. temperature profile versus prandtl number. fig. 9. temperature profile versus modified dufour parameter. fig. 10. concentration profile versus lewis number. fig. 11. concentration profile versus chemical reaction parameter fig. 12. concentration profile versus modified dufour parameter. the influence of a chemical reaction parameter 𝐾𝑟 on dimensionless concentration profiles is depicted in figure 11. the concentration of the fluid is suppressed as the 𝐾𝑟 is increased. higher 𝐾𝑟 values result in a decrease in chemical molecule diffusivity. as a result, they are obtained by species transfer. the species concentration decreases as 𝐾𝑟 rises. with an enhancement in the 𝐾𝑟, the concentration distribution decreases at all places of the flow field. the influence of the dufour solutal lewis number 𝐿𝑑 on concentration is shown in figure 12. this parameter enhances the thickening of the solutal boundary layer while it reduces that fluid concentration. surface drag, heat, and mass transfer rates of the model are discussed in detail with the help of tables 2-3. in point of physics, these quantities have great significance due to their vast applications in engineering and industrial fields. from table 2, it is clear that surface drag is an increasing function of the magnetic parameter, weissenberg number, porous parameter, and casson fluid parameter. heat and mass transfer rates are rises with the rise in thermal conductivity parameter, radiation parameter, and chemical reaction parameter, and heat transfer rate is decreasing function of modified dufour number. babitha, c.v.ramana murthy and g.v. r. reddy/ journal of naval architecture and marine engineering,20(2023) 25-35 mhd casson and carreau fluid flow through a porous medium with variable thermal conductivity… 32 table 1. comparison of the skin friction for various values of m, and pr = nd = ld = le = we = n ==0 m salman et al.(2012) present study 0 1.0000 1.00000 0.5 -1.1181 -1.118012 1 -1.4142 -1.414189 table 2. variation of friction drag for different values of 𝑀, 𝑊𝑒, 𝐾, and 𝛾 m we k 𝜸 s skin friction 1 0.3 2 0.5 -0.5 0.966957 2 1.355020 3 1.822203 4 2.309703 1 0.5 1.134600 2 1.516732 3 1.977366 4 2.457719 1 0.5 0.698427 2 0.764641 3 0.847183 4 0.939871 1 -0.5 0.626586 2 0.675865 3 0.733013 4 0.790057 1 -0.5 0.609604 2 0.815128 3 0.981782 4 1.124990 1 0.5 0.785761 2 0.986217 3 1.149899 4 1.290979 0.1 0.5 0.448695 0.2 0.604424 0.3 0.716865 0.4 0.803702 0.1 -0.5 0.401367 0.2 0.517181 0.3 0.596134 0.4 0.654575 babitha, c.v.ramana murthy and g.v. r. reddy/ journal of naval architecture and marine engineering,20(2023) 25-35 mhd casson and carreau fluid flow through a porous medium with variable thermal conductivity… 33 table 3. impact of different physical parameters on heat and mass transfer rates:  r kr ld nd s nusselt number sherwood number 0 0.5 0.855719 0.686429 1 0.954112 0.735279 2 1.10133 0.761286 0 -0.5 0.371158 0.699791 1 0.372821 0.69994 2 0.381621 0.700731 0.1 0 0.5 0.3 0.1 0.5 0.266182 0.705174 1 0.287431 0.706006 2 0.325364 0.70641 0 -0.5 0.495053 0.746041 1 0.610984 0.774109 2 0.847285 0.812811 0.5 0.5 0.29759 1.232912 1 0.319851 1.259533 1.5 0.345339 1.290779 0.5 -0.5 0.704927 0.693342 1 0.992564 0.996611 1.5 1.215909 1.232086 0.1 0.5 0.37583 0.700602 0.2 0.376263 0.70267 0.3 0.376662 0.704927 0.1 -0.5 1.354313 0.511013 0.2 1.342295 0.60333 0.3 1.330601 0.693342 6. concluding remarks mhd boundary layer flow, heat, and mass transport characteristic of ccf are studied numerically. thermal radiation, suction/injection, and variable thermal conductivity effects are taken into account in the study. the significant outcomes of the study are drawn as follows:  the fluid motion decreases for an enhancing value of the magnetic parameter, porous parameter, and time constant parameter, while intensifies for the weissenberg number.  the increase in thermal radiation parameter, variable thermal conductivity, and dufour number leads to an increase in temperature in both suction and injection cases, on the other hand, fluid temperature decreases with an increase in prandtl number.  the fluid concentration decreases with an increase in chemical reaction parameter and increases with an increase in dufour number and dufour solutal lewis number.  the surface drag is intensifying with enhancing values of 𝑀, 𝑊𝑒, 𝐾, and 𝛽.  heat transfer is an increasing function of variable thermal conductivity and thermal radiation and decreasing function of dufour number. babitha, c.v.ramana murthy and g.v. r. reddy/ journal of naval architecture and marine engineering,20(2023) 25-35 mhd casson and carreau fluid flow through a porous medium with variable thermal conductivity… 34 further, the impact of various physical phenomena on the ccf model can be discussed and also one can obtain 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architecture and marine engineering december, 2006 http://jname.8m.net 1813-8535 © aname publication. all rights reserved. effect of pressure stress work and viscous dissipation in natural convection flow along a vertical flat plate with heat conduction md. m. alam1, m. a. alim2 and md. m. k. chowdhury2 1department of mathematics, dhaka university of engineering and technology, gazipur-1700, bangladesh. 2department of mathematics, bangladesh university of engineering and technology, dhaka-1000, bangladesh. email: maalim@math.buet.ac.bd abstract in this paper, the effect of viscous dissipation and pressure stress work on free convection flow along a vertical flat plate has been investigated. heat conduction due to wall thickness b is considered in this investigation. with a goal to attain similarity solutions of the problem posed, the developed equations are made dimensionless by using suitable transformations. the non-dimensional equations are then transformed into non-linear equations by introducing a non-similarity transformation. the resulting non-linear similar equations together with their corresponding boundary conditions based on conduction and convection are solved numerically by using the finite difference method along with newton’s linearization approximation. numerical results for the details of the velocity profiles, temperature profiles, skin friction coefficients and the surface temperature distributions are shown both on graphs and tabular form for different values of the parameters entering into the problem. keywords: free convection, viscous dissipation, pressure work and conduction. nomenclature b plate thickness ∞t fluid asymptotic temperature cp specific heat vu , velocity components along yx, directions respectively. d ( ) ∞∞− tttb / vu, dimensionless velocity components f dimensionless stream function yx, cartesian coordinates g acceleration due to gravity yx, dimensionless cartesian coordinates l reference length, 3/13/2 / gν greek symbol l length of the plate β co-efficient of thermal expansion n viscous dissipation parameter ψ stream function p coupling parameter, ( )( ) 4/1// dlbkkp sf= κf , κs fluid and solid thermal conductivities pr prandtl number η dimensionless similarity variable t temperature ρ density of the fluid tb temperature at outer surface of the plate θ dimensionless temperature ts solid temperature m. m. alam, m. a. alim, m. k. chowdhury/journal of naval architecture and marine engineering 3(2006) 69-76 effect of pressure stress work and viscous dissipation in natural convection flow along a vertical flat plate… 70 1. introduction free convection flow is often encountered in cooling of nuclear reactors or in the study of the structure of stars and planets. the study of temperature and heat transfer is of great importance to the engineers because of its almost universal occurrence in many branches of science and engineering. although heat transfer analysis is most important for the proper sizing of fuel elements in the nuclear reactors cores to prevent burnout. the viscous dissipation effect plays an important role in natural convection in various devices which are subjected to large deceleration or which operate at high rotational speeds and also in strong gravitational field processes on large scales (on large planets) and in geological processes. the discussion and analysis of natural convection flows, pressure and viscous stress work effects are generally ignored but here we have considered the effects of viscous dissipation and pressure work on a natural convection flow along a vertical flat plate with heat conduction. the influence and importance of viscous stress work effects in laminar flows have been examined by gebhart (1962) and gebhart and mollendorf (1969). in both of the investigations special flows over semi-infinite flat surfaces parallel to the direction of body force were considered. gebhart (1962) considered flows generated by the plate surface temperatures, which vary as powers of ξ (the distance along the plate surface from the leading edge), and mollendorf (1969) considered flows generated by plate surface temperatures, which vary exponentially in ξ. zakerullah (1972) has been investigated the viscous dissipation and pressure work effects in axisymmetric natural convection flows. ackroyd (1974) studied the stress work effects in laminar flat plate natural convection flow. takhar and soundalgekar (1980) have studied the effects of viscous and joule heating on the problem posed by sparrow and cess (1961), using the series expansion method of gebhart (1962). joshi and gebhart (1981) have shown that the effect of pressure stress work and viscous dissipation in some natural convection flows. pozzi and lupo (1988) studied the coupling of conduction with laminar natural convection along a flat plate. miyamoto et al. (1980) has been investigated the effect of axial heat conduction in a vertical flat plate on free convection heat transfer. in the present work, we have investigated the viscous dissipation and pressure work effect on the skin friction and the surface temperature distribution in the entire region from leading edge to down stream of a viscous incompressible flow along a semi-infinite vertical flat plate. the entire thermo-fluid dynamic field resulting from the coupling of natural convection along and conduction inside the heated plate has been considered. the transformed non similar boundary layer equations governing the flow together with the boundary conditions based on conduction and convection were solved numerically using the keller box (implicit finite difference) method, cebeci and bradshaw (1984) along with newton's linearization approximation method in the entire region. we have studied the effect of the prandtl number pr, the viscous dissipation parameter n and pressure work parameter ∈ on the velocity and temperature profiles as well as on the skin friction and surface temperature. 2. governing equations of the flow steady two dimensional laminar free convection boundary layer flow of a viscous incompressible fluid along one side of a semi-infinite vertical flat plate of thickness ‘b’ insulated on the edges with temperature tb. the flow configuration and the coordinates system are shown in fig. 1. the mathematical statement of the basic conservation laws of mass, momentum and energy for the steady viscous incompressible flow are: 0. =∇ q r (1) ( ) 2.q q p q fρ µ∇ = −∇ + ∇ + rr r r (2) ( ) ( ) ( ). . .pc q t q p tρ κ µ φ∇ − ∇ = ∇ ∇ + r r (3) where ( , )q u v= r , u and v are the velocity components along the x and y axes respectively, f r is the body force per unit volume which is defined as -ρg, t is the temperature of the fluid in the boundary layer, g is the acceleration due to gravity, κ is the thermal conductivity and cp is the specific heat at constant pressure and µ is the viscosity of the fluid. in the energy equation the viscous dissipation and pressure work terms are included. after introducing boussinesq approximation, ( )[ ]1 t tρ ρ β∞ ∞= − − the basic equations (1) to (3) become: 0 u v x y ∂ ∂ + = ∂ ∂ (4) m. m. alam, m. a. alim, m. k. chowdhury/journal of naval architecture and marine engineering 3(2006) 69-76 effect of pressure stress work and viscous dissipation in natural convection flow along a vertical flat plate… 71 ( )∞−+∂ ∂ = ∂ ∂ + ∂ ∂ ttg y u y u v x u u βν 2 2 (5) x p c ut y u cy t cy t v x t u ppp ∂ ∂ + ∂ ∂ + ∂ ∂ = ∂ ∂ + ∂ ∂ ρ βν ρ κ 2 2 2 )( (6) the appropriate boundary conditions to be satisfied by the above equations are 0== vu at 0y = 0 ,u t t∞→ → as y → ∞ (7) the temperature and the heat flux are considered continuous at the interface for the coupled conditions and at the interface we must have 0( ) s so y f k t t k y y = ∂ ∂ = ∂ ∂ (8) where ks and kf are the thermal conductivity of the solid and the fluid respectively. the temperature tso in the solid as given by pozzi and lupo (1988) is { }( , 0) ( , 0)so b y t t x t t x b = − − (9) where t( x ,0) is the unknown temperature at the interface to be determined from the solutions of the equations. we observe that the equations (4) (6) together with the boundary conditions (8) (9) are non-linear partial differential equations, which have been solved numerically and are described in the following sections. 3. transformation of the governing equations equations (4)–(6) may now be non-dimensionalized by using the following dimensionless dependent and independent variables: x x l = , 1/ 4 y y d l = , 1/ 4u d u l ν = , 1/ 4v d v l ν = , b t t t t θ ∞ ∞ − = − , 2 / 3 1/ 3l g ν = , ( )bd t tβ ∞= − (10) as this natural convection problem is of parabolic nature the characteristic length, l has been defined in terms of ν and g, which are the intrinsic properties of the system. the reference length along the ‘y’ direction has been modified by a factor d-1/4 in order to eliminate this quantity from the dimensionless equations and the boundary conditions. using the above relations (10) the non-dimensional form of the governing equations are: fig.1: physical configuration and coordinates system v l b ks g t∞ lower surface interface t=tb upper surface kf o u v x y ( , 0 )t x m. m. alam, m. a. alim, m. k. chowdhury/journal of naval architecture and marine engineering 3(2006) 69-76 effect of pressure stress work and viscous dissipation in natural convection flow along a vertical flat plate… 72 0 u v x y ∂ ∂ + = ∂ ∂ (11) 2 2 u u u u v x y y θ ∂ ∂ ∂ + = + ∂ ∂ ∂ (12) ( ){ } ( ) 2 2 2 1 ( ) pr b p b g t t tu u v n x y y y c t t β θθ θ θ ∞ ∞ ∞ + −∂ ∂ ∂ ∂ + = + − ∂ ∂ ∂ ∂ − (13) where pr = fpc κµ is the prandtl number and n = )( 22 ∞− ttcld bpν , the dimensionless viscous dissipation parameter. for exterior conditions, we know hydrostatic pressure, gxp eρ=∂∂ and eρρ = , and the pressure work parameter pcxgβ∈= which is less than one as suggested by gebhart (1962). the corresponding boundary conditions (7) (8) take the following form: u = v = 0, 1 p y θ θ ∂ − = ∂ at y =0 (14) u → 0 v →0 as y → ∞ (15) where p is the pressure and p is the conjugate conduction parameter given by p = (κf / κs) (b/l) d 1/4. here the coupling parameter 'p' governs the described problem. the order of magnitude of ‘p’ depends actually on b/l and κf / κs, d 1/4 being the order of unity. the term b/l attains values much greater than one because of l being small. in case of air, κf / κs becomes very small when the vertical plate is highly conductive i.e. κs > > 1and for materials, o (κf / κs) = 0.1 such as glass. therefore in different cases ‘p’ is different but not always a small number. in the present investigation we have considered p = 1 which is accepted for b/l of o (κf / κs). to solve the equations (12) – (13) subject to the boundary conditions (14) to (15), the following transformations were introduced for the flow region starting from up stream to down stream. ( )4 / 5 1/ 20 1/ 5 1/ 20 1/ 5 1/ 5(1 ) ( , ), (1 ) , (1 ) ,x x f x yx x x x h xψ η η θ η− − − −= + = + = + (16) here η is the dimensionless similarity variable and ψ is the stream function which satisfies the equation of continuity and x v y u ∂ ∂ −= ∂ ∂ = ψψ , and h(η,x) is the dimensionless temperature. then the equations (12) and (13) transformed to the following non dimensional forms: 216 15 6 5 ( ) 20(1 ) 10(1 ) x x f f f ff f h x f f x x x x ′+ + ∂ ∂ ′′′ ′′ ′ ′ ′′+ − + = − + + ∂ ∂ (17) ( ) ( ) 2 1/ 5 1 16 15 1 pr 20 1 5 1 1 b x h fh f h nxf x x tx h f f hf x f h x t t x x ∞ ∞ + ′′ ′ ′ ′′+ − + = + ⎧ ⎫⎛ ⎞+ ∂ ∂⎪ ⎪⎛ ⎞ ⎛ ⎞′ ′ ′ ′− ∈ + = −⎨ ⎬⎜ ⎟⎜ ⎟ ⎜ ⎟− ∂ ∂⎝ ⎠ ⎝ ⎠⎪ ⎪⎝ ⎠⎩ ⎭ (18) in the above equations the primes denote differentiation with respect to η. the boundary conditions (14)-(15), take the following form 0),(,0),( )0,()1()1()0,(,0)0,()0,( 20/15/14/1 =∞′=∞′ +++−=′=′= xhxf xhxxxxhxfxf (19) 4. method of solution the numerical methods used is finite difference method together with keller box scheme which is described in details by keller and cebeci (1971), cebecci and bradshow (1984) and widely used by hossain and alim (1997) and hossain et al. (1998). m. m. alam, m. a. alim, m. k. chowdhury/journal of naval architecture and marine engineering 3(2006) 69-76 effect of pressure stress work and viscous dissipation in natural convection flow along a vertical flat plate… 73 5. results and discussion the viscous dissipation and pressure work effect on the skin friction and the surface temperature distribution of a viscous incompressible flow along a semi-infinite vertical flat plate has been investigated and the coupling of natural convection along and conduction inside the heated plate has been considered. solutions are obtained for the values of prandtl number pr = 0.05, 0.72, 1.0, 1.74 and for a range of values of the viscous dissipation parameter n = 0.1, 0.3, 0.5, 0.9 and the pressure work parameter ∈ = 0.1, 0.4, 0.7, 0.9. if we know the values of the functions f (η, x), h (η, x) and their derivatives for different values of the pertinent parameters then it is possible to calculate the numerical values of the surface temperature θ (0, x) and the velocity gradient (skin frictions) f '' (0, x) at the surface that are important from the physical point of view. fig.2 (a) and fig.2 (b) show the effects of the viscous dissipation parameter n (=0.1, 0.3, 0.5, 0.9) on the velocity and the temperature profiles for pr = 0.72 and ∈ = 0.5. from fig. 2(a), it is revealed that the velocity profile f '(η ,x) increases with the increase of the viscous dissipation parameter n which indicates that viscous dissipation increases the fluid motion slightly. in fig. 2(b) the similar behavior has also been observed for the temperature profiles h (η, x) with the similar values of controlling parameters n, pr and ∈. we observe from fig. 3(a), that an increase in the pressure work parameter, ∈, is increases the velocity profiles but near the surface of the plate the velocity increases and become maximum and then decreases and finally approaches to zero. the maximum values of the velocities are 0.5129, 0.5228, 0.5352, 0.5542 for ∈ = 0.1, 0.4, 0.7, 0.9 respectively and each of which occurs at η = 1.3025 for the first maximum value and η = 1.3693 for the 2nd, 3rd, fourth maximum values. here we found that the velocity increases by 8.0523% as ∈ increases from 0.1 to 0.9. however fig. 3(b) shows the distribution of the temperature profiles h (η, x) against η for some values of the pressure work parameter ∈ (= 0.1, 0.4, 0.7, 0.9). clearly it is seen that the temperature distribution increases owing to increasing values of the pressure work parameter ∈ and becomes maximum at the wall. the local maximum values of the temperature profiles are 0.8962, 0.9283, 0.9641, 0.9937 for ∈ (= 0.1, 0.4, 0.7, 0.9) respectively and each of which attains at the surface. thus the temperature profiles increase by 10.8793% as ∈ increases from 0.1 to 0.9. figs. 4 (a) and (b) illustrates the velocity and temperature profiles for different values of prandtl number in presence of viscous dissipation and pressure work terms. from fig. 4 (a), we may conclude that the velocity profile decreases when the value of the prandtl number pr increases. but it is seen that near the surface of the flat plate the velocity increases considerably and become maximum and then decreases slowly and finally approaches to zero. the maximum values of the velocities are 0.3679, 0.4476, 0.4987, 0.9749 for pr = 1.74, 1.0, 0.77, 0.05 respectively which occur at η = 1.3025 for the first maximum value, η = 1.3693 for the second and third maximum values and at η = 1.6593 for the last maximum value. here it is found that the velocity profiles decrease by 62.263% while pr increases from 0.05 to 1.74. on the other hand, in the case of temperature field, from fig. 4(b), we observed that the temperature distribution over the whole boundary layer decreases significantly when the values of the prandtl fig.2: (a) velocity and (b) temperature profiles are shown against η for different values of viscous dissipation parameter n for controlling parameter pr = 0.72, ∈ = 0 .5. 0 2 4 6 η 0 0.1 0.2 0.3 0.4 0.5 0.6 v el oc ity pr of ile s n=0.9 n=0.5 n=0.3 n=0.1 pr = 0.72 0 2 4 6 η 0 0.2 0.4 0.6 0.8 1 t em pe ra tu re pr of ile s n = 0.9 n = 0.5 n = 0.3 n = 0.1 pr = 0.72 ∈ = 0.5∈ = 0.5 (a) (b) m. m. alam, m. a. alim, m. k. chowdhury/journal of naval architecture and marine engineering 3(2006) 69-76 effect of pressure stress work and viscous dissipation in natural convection flow along a vertical flat plate… 74 number pr increases. here we observed that the maximum values of the temperature profiles are 1.0555, 0.8522, 0.8271, 0.7844 for pr = 0.05, 0.7, 1.0, 1.74 respectively and each of which attains at the surface of the plate. thus in this case temperature profiles decrease by 28.684% numerical values of the velocity gradient f '' (0, x) and the surface temperature θ (0, x) are depicted graphically in fig.5 (a) and 5(b) respectively against the axial distance x for different values of the viscous dissipation parameter n (=0.1, 0.3, 0.5, 0.9) for the fluid having prandtl number pr = 0.72 and pressure work parameter ∈ = 0.5. from fig. 5 (a), it can be seen that an increase in the dissipative heat is associated with the enhancing of the skin friction coefficient. same result is seen in fig. 5 (b) for surface temperature distribution. in fig.6 (a), the shear stress coefficient f''(0, x) and fig.6 (b), the surface temperature distribution θ (0, x) are shown graphically for different values of the prandtl number pr (= 0.05, 0.72, 1.0, 1.74) with fixed parameter n = 0.05 and ∈ = 0.5. four values 0.05 (mercury), 0.72 (air), 1.0 (salt water), 1.74 (theoretically) are taken for pr, the prandtl number. from the first figure, 6(a) it can be seen that for the increased values of prandtl number, pr decrease the shear stress coefficient and the same result is observed from fig. 6(b) for surface temperature distribution. fig.3: (a) velocity and (b) temperature profiles are shown against η for different values of pressure work parameter ∈ for controlling parameter pr = 0.72, n = 0.05. 0 2 4 6 η 0 0.1 0.2 0.3 0.4 0.5 0.6 v el oc ity pr of ile s = 0.9 = 0.7 = 0.4 = 0.1 pr = 0.72 n = 0.05 ∈ (a) 0 2 4 6 η 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 t em pe ra tu re pr of ile s = 0.9 = 0.7 = 0.4 = 0.1 pr = 0.72 n = 0.05 ∈ (b) fig. 4: (a) velocity and (b) temperature profiles are shown against η for different values of prandtl number pr for controlling parameter n = 0.05, ∈ =0 .5. 0 4 8 12 16 20 η 0 0.2 0.4 0.6 0.8 1 v el oc ity pr of ile s pr = 1.74 pr = 1.00 pr = 0.72 pr = 0.05 n = 0.05 ∈ = 0.5 (a) 0 4 8 12 16 20 η 0 0.2 0.4 0.6 0.8 1 t em pe ra tu re pr of ile s pr = 1.74 pr = 1,00 pr = 0.72 pr = 0.05 n = 0.05 ∈ = 0.5 (b) m. m. alam, m. a. alim, m. k. chowdhury/journal of naval architecture and marine engineering 3(2006) 69-76 effect of pressure stress work and viscous dissipation in natural convection flow along a vertical flat plate… 75 in fig.6 (a), the shear stress coefficient f''(0, x) and fig.6 (b), the surface temperature distribution θ (0, x) are shown graphically for different values of the prandtl number pr (= 0.05, 0.72, 1.0, 1.74) with fixed parameter n = 0.05 and ∈ = 0.5. four values 0.05 (mercury), 0.72 (air), 1.0 (salt water), 1.74 (theoretically) are taken for pr, the prandtl number. from the first figure, 6(a) it can be seen that for the increased values of prandtl number, pr decrease the shear stress coefficient and the same result is observed from fig. 6(b) for surface temperature distribution. 6. conclusions the effect of viscous dissipation and the pressure work on natural convection boundary layer flow from a vertical flat plate with heat conduction inside the heated plate has been studied. the transformed non-similar boundary layer equations governing the flow together with the boundary conditions based on conduction and convection were solved numerically using the implicit finite difference method together with keller box scheme. the coupled effect of natural convection and conduction required that the temperature and the heat flux be continuous at the interface. the numerical values of skin friction, the surface temperature distribution, velocity and temperature profiles have been presented graphically and in tabular form. from the present investigation, the following conclusions may be drawn: fig.5: (a) shear stress and (b) surface temperature are depicted against x for different values of viscous dissipation n with others fixed controlling parameters pr and ∈. ∈ = 0.5 0 5 10 15 20 25 x 0 1 2 3 4 5 6 7 s ki n fr ic tio n n= 0,1 n = 0.3 n = 0.5 n = 0.9 pr = 0.72 (a) 0 2 4 6 x 0 0.5 1 1.5 2 2.5 s ur fa ce te m pe ra tu re n = 0.9 n = 0.5 n = 0.3 n = 0.1 pr = 0.72 ∈ = 0.5 (b) fig. 6: (a) shear stress and (b) surface temperature are depicted against x for different values of prandtl number pr with others controoling parameters n and ∈. 0 2 4 6 8 10 12 x 0 0.5 1 1.5 2 s ur fa ce te m pe ra tu re pr = 1.74 pr = 1,00 pr = 0.72 pr = 0.05 n = 0.05 ∈ = 0.5 (b) 0 2 4 6 8 10 12 x 0 1 2 3 4 s ki n fr ic tio n pr = 1.74 pr = 1.00 pr = 0.72 pr = 0.05 n = 0.05 ∈ = 0.5 (a) m. m. alam, m. a. alim, m. k. chowdhury/journal of naval architecture and marine engineering 3(2006) 69-76 effect of pressure stress work and viscous dissipation in natural convection flow along a vertical flat plate… 76 • the skin friction, the velocity and the temperature distributions increase for increasing value of the viscous dissipation parameter n. • the surface temperature, the velocity and the temperature distributions increase for increasing values of the pressure work parameter ∈. • it has been observed the skin friction, the temperature distribution over the whole boundary layer and the velocity distribution decrease with the increase of the prandtl number pr. table-1: skin friction coefficient and surface temperature against x for different values of pressure work parameter ∈ with other controlling parameters pr = 0.72, n=0.05. ∈ = 0.90 ∈ = 0.70 ∈ = 0.40 ∈ = 0.10 x f //(0,x) θ (0,x) f //(0,x) θ (0,x) f //(0,x) θ (0,x) f //(0,x) θ (0,x) 0.0000 0.3150 0.7090 1.0409 2.0369 3.1340 4.0635 4.9876 6.1118 7.1132 10.1191 0.0155 0.5399 0.7143 0.8230 1.0859 1.3443 1.5642 1.7942 2.0986 2.3999 3.5492 0.2052 0.7063 0.7791 0.8206 0.9350 1.0609 1.1795 1.3155 1.5114 1.7221 2.6499 0.0155 0.5322 0.6929 0.7876 0.9974 1.1776 1.3135 1.4416 1.5934 1.7280 2.1465 0.2052 0.6943 0.7517 0.7793 0.8453 0.9034 0.9495 0.9960 1.0545 1.1098 1.2966 0.0155 0.5266 0.6776 0.7629 0.9394 1.0756 1.1687 1.2493 1.3365 1.4070 1.5928 0.2052 0.6855 0.7326 0.7514 0.7902 0.8162 0.8325 0.8471 0.8628 0.8761 0.9099 0.0155 0.5257 0.6751 0.7589 0.9303 1.0602 1.1475 1.2219 1.3011 1.3641 1.5253 0.2052 0.6841 0.7295 0.7470 0.7819 0.8036 0.8164 0.8274 0.8388 0.8482 0.8698 references ackroyd, j. a. d. (1974): stress work effects in laminar flat-plate natural convection, j. fluid mech., vol. 62, pp. 677-695. cebeci, t. and bradshaw, p. (1984): physical and computational aspects of convective heat transfer, springer, new york. gebhart, b. (1962): effects of viscous dissipation in natural convection, j. fluid mech., vol. 14, pp. 225-232. gebhart, b., and mollendorf, j. (1969): viscous dissipation in external natural convection flows. j. fluid mech, vol.38, pp. 97-107. hossain, m. a. and alim, m. a. (1997): natural convection-radiation interaction on boundary layer flow along a thin cylinder, j. heat and mass transfer vol. 32, pp.515-520. hossain, m. a., alim, m. a., and rees, d.a.s. (1998): effects of thermal radiation on natural convection over cylinders of elliptic cross section. acta mechanica, vol. 129, pp. 177-186. joshi,y. and gebhart, b. (1981): effect of pressure stress work and viscous dissipation in some natural convection flows, int. j. heat mass transfer, vol. 24, no. 10, pp. 1377-1388. keller, h. b., & cebeci, t. (1971): accurate numerical methods for boundary layer flows, part-i. two– dimensional laminar flows”, proceedings of the second international conference on numerical methods in fluid dynamics, p. 92. springer, new york. miyamoto, m., sumikawa, j., akiyoshi, t. and takamura, t. (1980): the effect of axial heat conduction in a vertical flat plate on free convection heat transfer. int. j. heat mass transfer, v. 23, pp. 1545-1553. pozzi, a. and lupo, m. (1988): the coupling of conduction with laminar natural convection along a flat plate, int. j. heat mass transfer. vol. 31, no. 9, pp. 1807-1814. sparrow, e.m. & cess, r. d. (1961): the effect of magnetic field on free convention heat transfer, int. j. heat and mass transfer, vol. 3, pp. 267. takhar, h. s. and soundalgekar, v. m. (1980): dissipation effects on mhd free convection flow past a semiinfinite vertical plate, applied scientific research. vol. 36 pp.163-171. zakerullah, m. (1972): viscous dissipation and pressure work effects in axisymmetric natural convection flows, ganit(j. bangladesh math. soc.) vol. 2. no.1, pp. 43. microsoft word p1_6.doc 1813 8535 © aname publication. all right reserved journal of naval architecture and marine engineering june, 2006 http://jname.8m.net effects of process parameters on tensile strength of jute fiber reinforced thermoplastic composites h.m.m.a. rashed, m. a. islam and f. b. rizvi department of materials and metallurgical engineering, bangladesh university of engineering and technology, dhaka 1000, bangladesh abstract: for environmental concern on synthetic fibers (such as glass, carbon, ceramic fibers, etc.) natural fibers such as flax, hemp, jute, kenaf, etc. are widely used. in this research work, jute fiber reinforced polypropylene matrix composites have been developed by hot compression molding technique with varying process parameters, such as fiber condition (untreated and alkali treated), fiber sizes (1, 2 and 4 mm) and percentages (5%, 10% and 15% by weight). the developed jute fiber reinforced composites were then characterized by tensile test, optical and scanning electron microscopy. the results show that tensile strength increases with increase in the fiber size and fiber percentage; however, after a certain size and percentage, the tensile strength decreases again. compared to untreated fiber, no significant change in tensile strength has been observed for treated jute fiber reinforcement. fractographic observation suggests the fracture behavior to be brittle in nature. keywords: natural fiber, jute fiber, polypropylene, composite, tensile strength. 1. introduction now-a-days, newer polymer matrix composites reinforced with fibers such as glass, carbon, aramid, etc. are getting a steady expansion in uses because of their favorable mechanical properties. however, they are quite expensive materials. for this, natural fibers such as jute, flux, hemp, etc. can be alternately used to reduce the cost of the composites (mohanty et al., 2002). moreover, production of environmentally friendly materials is another important issue. natural fiber composites focus well into this ecological image. the use of natural fibers, derived from annually renewable resources, as reinforcing fibers in both thermoplastic and thermoset matrix composites provide positive environmental benefits with respect to ultimate disposability and raw material utilization. the prominent advantages of natural fibers include acceptable specific strength properties, low cost, low density, high toughness, good thermal properties, and so on. low specific weight, which results in a higher specific strength and stiffness than glass is a benefit especially in parts designed for bending stiffness. in the fields of automotive industries, reduction of energy consumption in production of motor vehicles and improvement of their day to day fuel economy are growing upwards due to accelerating use of natural fiber composites. in the case of thermoplastic composites, adhesion between the hydrophilic fiber (such as jute fiber) and hydrophobic matrix (such as polypropylene) is poor (karmaker and youngquist, 1996). therefore, the bond between them needs to be improved. this may be improved by alkali treatment. it is believed that the alkali treatments results in an improvement in the interfacial bonding by giving rise to additional sites for mechanical interlocking, hence promoting more matrix/fiber interpenetration at the interface (gassan and bledzki, 2000). in this project, jute fiber reinforced polypropylene composites were prepared under various processing parameters using hot compression molding technique. the goal of this work is to understand the changes of tensile strength under various process parameters. optical microscopy was done to show the conditions of fiber with increased fiber loading. fracture surfaces of tensile test specimens were examined under scanning electron microscope to get an idea about the fracture behavior. 2. experimental 2.1 materials the composites were produced using treated (jute fibers were treated by bangladesh jute research institute, dhaka, bangladesh, with 20% sodium hydroxide) and untreated jute fiber and polypropylene pellets. the treated h.m.m.a. rashed, m. a. islam and f. b. rizvi / journal of naval architecture and marine engineering 3(2006) 1-6 2 and untreated jute fibers were chopped into various lengths of 1, 2 and 4 mm. for all lengths of fibers, composites were developed with 5, 10 and 15% (by weight) of jute. 2.2 methods 2.2.1 composite fabrication the chopped fibers were sieved with 1, 2 and 4 mm sieves for obtaining the desired variation in jute fiber length. the fibers were conditioned at 110 0c for 24 hours to remove moisture and polypropylene was also conditioned at the same temperature. proper proportion of fibers (5, 10 and 15% by weight for each of 1, 2 and 4 mm length) and polypropylene were then properly blended in the blender to get a homogeneous mixture for each length type. the mixture was placed in a mold and composites were made with 50 kn load at 180 0c. 2.2.2 tensile test tensile testing of the specimens was performed according to astm d 638-98 on a universal test machine operated at a crosshead speed 3 mm/min. three test specimens from every composition (combination of predefined fiber length and wt percentage with polypropylene) were tested at the same time and the averages of results were used. 2.2.3 metallography and fractography conventional metallography of selected specimens was done under metallurgical microscope. fracture surfaces of the tensile test specimens were observed under philips xl 30 scanning electron microscope operated at 10 kv. samples were mounted with carbon tape on aluminum stubs and then sputter coated with gold for 30 seconds to make them conductive. (a) (b) (cont.) (c) fig. 1: optical microscopy of jute fiber reinforced polymer composites reinforced with various fiber wt. percentages. (a) 5% fiber (b) 10% fiber (c) 15% fiber. entanglement of fibers is shown by the arrow. it is clear that 15% fiber composite has highest fiber entanglement compared to 5 & 10% fiber composites 3. results and discussion in this research work, at first selected specimens were observed under metallurgical microscope. then tensile specimens were prepared according to astm specification and were tested. using a scanning electron microscope, fracture surfaces of the tested specimens were observed. h.m.m.a. rashed, m. a. islam and f. b. rizvi / journal of naval architecture and marine engineering 3(2006) 1-6 3 3.1 metallographic observation the examination under metallurgical microscope shows the variation of fiber wt. percentages in different jute fiber composites (fig. 1). as evident from the figure, with the increase of fiber percentage in the composite, probability of entanglement of fiber increases. this is due to strong inter-hydrogen bonding between fibers (alam et al., 2004). 3.2 tensile strength the typical load-stroke curve obtained from the tensile test is shown in fig. 2. from this curve, it can be predicted that the failure behavior of the jute fiber reinforced thermoplastic composite is brittle type. fig. 2: an observed load-stroke curve obtained from tensile test the tensile test results have been plotted in figs. 3 & 4 as a function of fiber length. from these figures, it is clear that as the fiber length increases, the value of tensile strength increases and then decreases. this observation is true for almost all cases (with exception in the case of 5% treated specimens). figs. 5 & 6 represent the relationship between fiber percentage (wt %) and tensile strength value. as per these plots, in general, as the fiber percentage increases, the tensile strength also increases and then decreases. as observed from the curves, tensile strength was increased to a maximum at 2 mm fiber length and then dropped. also, tensile strength was found to increase to a maximum at 10% fiber (by weight) and then decreased. 2 mm & 10% treated fiber composites gave better results than untreated fiber composites but not so distinguishable. fiber length has profound impact on the properties of composites. besides holding the fibers together, the matrix has the important function of transferring applied load to the fibers. the efficiency of a fiber reinforced composite depends on the fiber-matrix interface and the ability to transfer stress from the matrix to the fiber (karnani et al., 1997). in small fiber size (here, 1 mm), tensile strength is low due to the fact that length may be not sufficient enough for proper distribution of load. as proper length is not available for stress distribution, failure occurs easily. on the other hand, for the composites of longer fiber size (here, 4 mm), tensile strengths were decreased compared to 2 mm fiber reinforced composites. the probable reason is that a long fiber may not become compatible with the matrix properly. thus improper bonding occurs between the fibers and the matrix. moreover, fibers may be folded and there is no bonding between the folded and unfolded portion of fiber which resulted in a lower strength. fiber entanglement may also contribute to reduce the strength (joseph et al., 2002). for 5% treated fiber composites, the exceptional behavior is probably that 4 mm size of fiber is still not enough to create fiber entanglement or folding inside the matrix. in phenol formaldehyde/banana fiber composites, with the increase of fiber length tensile strength was found to be increased (joseph et al., 2002). the trend of increase followed by decrease of tensile strength observed in current project was found in sisal/polypropylene composites (jayaraman, 2003). according to figs. 5 & 6, after 10 wt. percent fiber as reinforcement in the composites, tensile strength was decreased with higher percentages of fiber. the incorporation of fibers into thermoplastics leads to poor dispersion of fibers due to strong inter fiber hydrogen bonding which holds the fibers together. improper adhesion hinders the considerable increment of tensile strength (beckermann et al., 2004). thus, as fiber percentage increases, gathering of fibers takes place instead of dispersion and melted polypropylene can not wet h.m.m.a. rashed, m. a. islam and f. b. rizvi / journal of naval architecture and marine engineering 3(2006) 1-6 4 them properly due to non entrance of melt through the adjacent two fibers. since no adhesion is present between the fibers and fibers are also not bonded with matrix, failure occurs before attaining the theoretical strength of composite. thus high fiber content was limited by the incompatibility issue unless coupling agent is used (wollerdorfer and bader, 1998). fig. 3: tensile strength of untreated jute fiber composites at different lengths* fig. 4: tensile strength of treated jute fiber composites at different lengths* fig. 5: tensile strength of untreated jute composites at different fiber wt. percentages* fig. 6: tensile strength of treated jute fiber composites at different fiber wt. percentages* * (uf: untreated fiber, tf: treated fiber) it has been reported that initially strength may decrease after a slight increase in strength and then at very high fiber content it may again increased (wambua et al., 2003, jayaraman, 2003). in polypropylene/wood composites, tensile strength was found to decrease after a certain percentage of fiber (beg and pickering, 2004). the tensile strengths of the uncoupled composites have values in close range for all fiber percentage levels (karmaker and schneider, 1996, rowell and stout, 1998). without coupling agent, fiber content and fiber length do not have significant effects on composite tensile properties (sameni et al., 2003). there exist incompatibilities between the different surface properties of the polar fibers and non-polar polypropylene. due to presence of hydroxyl and other polar groups in various constituents of natural fiber, the moisture uptake is high for dry fibers. all these lead to poor wettability with matrix and weak interfacial bonding between the fiber and relatively more hydrophobic matrices. to improve affinity and adhesion between fiber and thermoplastic, chemical coupling agents can be used so that tensile strength increases (khan et al., 2001, saheb and jog, 1999). as a coupling agent, mapp may be used to enhance interfacial adhesion that may react or interact favorably with the hydroxyl group on the fiber surface (mohanty et al., 2004). use of coupling agent reduces the number of fiber pull-out (gassan and bledzki, 1997). as evident from figs. 3-6, tensile strength was not significantly improved by alkali treatment. but, alkali treatment generally increases the strength of natural fiber composites (dieu et al., 2004, gañán and mondragon, 2004, razera and frollini, 2004). a strong sodium hydroxide treatment may remove lignin, hemicellulose and other alkali soluble compounds from the surface of the fibers to increase the numbers of reactive hydroxyl groups on the fiber surface available for chemical bonding. so, strength should be higher than untreated fiber h.m.m.a. rashed, m. a. islam and f. b. rizvi / journal of naval architecture and marine engineering 3(2006) 1-6 5 composites. the probable cause of this unlike phenomenon may be, alkali react on the cementing materials of the fiber specially hemicellulose which leads to the splitting of the fibers into finer filaments. as a result, wetting of fiber as well as bonding of fiber with matrix may improve which consequently make the fiber more brittle. under stress, these fibers break easily. therefore, they can not take part in stress transfer mechanism (ray et al., 2001). so, high concentration of sodium hydroxide may increase the rate of hemicellulose dissolution which will finally lead to strength deterioration. moreover, unnecessary extra time in treatment may also cause increment of hemicellulose dissolution. 3.3 fractography the main reason for poor mechanical properties in jute fiber composites is weak bonding between the fiber and matrix. this is evident in the micrographs obtained from scanning electron microscopy (fig. 7). the important feature of these micrographs is that a clear picture of fracture is shown. there is a clear evidence of brittle failure in fig. 7 a. also crack initiation site and propagation through the matrix were observed in fig. 7a. according to the sem fractographs (fig. 7b-d), fiber pull-out and debonding predominate in fracture surfaces with fairly clean and recognizable fiber surface without matrix adherence. in fig. 7b, clear fracture surface shows poor fiber/matrix interfacial bonding. fiber pull-out was also clearly seen in fig. 7c. fiber debonding was observed in fig. 7d. (a) a probable crack initiation site b crack propagation site (b) (c) a fiber pull-out (d) a fiber debonding fig. 7: scanning electron microscopy of fracture surfaces of jute fiber composites after tensile test. evidence of brittle fracture (a). indication of poor interfacial adhesion: clean fracture surface (b), fiber pull-out (c) and fiber debonding (d) 4. conclusions a. optical microscopy clearly shows that with the increase of fiber wt. percentage, entanglement of fibers occurs. b. in the case of fiber length, 2 mm jute fiber composites give better tensile strength over 1 & 4 mm jute fiber composites. a a b a a h.m.m.a. rashed, m. a. islam and f. b. rizvi / journal of naval architecture and marine engineering 3(2006) 1-6 6 c. in the case of fiber amount, 10 percent fiber (by weight) composites has better tensile strength compared to 5 & 15 wt. percent fiber composites. d. scanning electron microscopy of fracture surfaces indicates the fracture behavior to be brittle type. acknowledgement the authors are very much grateful to bangladesh university of engineering and technology (buet) to provide necessary fund for this research and also to the authority of bcsir, dhaka-1205, bangladesh, for allowing them to use the laboratory facilities there. references alam, s. n., pickering, k. l. and fernyhough, a. (2004): the characterization of natural fibers & their interfacial & composite properties, proceedings of sppm, 25-27 february 2004, dhaka, pp. 248-256 beckermann, g. w., pickering, k. l. and foreman, n. j. (2004): the processing, production and improvement of hemp-fiber reinforced polypropylene composite materials, proceedings of sppm, 25-27 february 2004, dhaka, pp. 257-265 beg, m. d. h. and pickering, k. l. (2004): effect of fiber pretreatment on the mechanical properties of wood/polypropylene composites, proceedings of sppm, 25-27 february 2004, dhaka, pp. 240-247 dieu, t. v., phai, l. t., ngoc, p. m., tung, n. h., thao, l. p. and quang, l. h. (2004): study on preparation of polymer composites based on polypropylene reinforced by jute fibers, jsme international journal, series a: solid mechanics and material engineering, vol. 47, no. 4, pp. 547-550. gañán, p. and mondragon, i. (2004): fique fiber-reinforced polyester composites: effects of fiber surface treatments on mechanical behavior, journal of materials science, vol. 39, no. 9, pp. 1573-4803. gassan, j. and bledzki, a. k. (1997): influence of fiber-surface treatment on the mechanical properties of jutepolypropylene composites, composites part a: applied science and manufacturing, vol. 28, no. 12, pp. 1001-1005. gassan, j. and bledzki, a. k. (2000): possibilities to improve the properties of natural fiber reinforced plastics by fiber modification jute polypropylene composites, applied composite materials, vol. 7, no. 5-6, pp. 373-385. jayaraman, k. (2003): manufacturing sisal-polypropylene composites with minimum fibre degradation, composites science and technology, vol. 63, no. 3-4, pp. 367-374. joseph, s., thomas, s., sreekala, m. s., oommen, z. and koshy, p. (2002): a comparison of the mechanical properties of phenol formaldehyde composites reinforced with banana fibres and glass fibres, composites science and technology, vol. 62, no. 14, pp. 1857-1868. karmaker, a. c. and schneider, j. p. (1996): mechanical performance of short jute fibre reinforced polypropylene, journal of materials science letters, vol. 15, no. 3, pp. 201-202. karmaker, a. c. and youngquist, j. a. (1996): injection molding of polypropylene reinforced with short jute fibers, journal of applied polymer science, vol. 62, no. 8, pp. 1147-1151. karnani, r., krishnan, m. and narayan, r. (1997): biofiber-reinforced polypropylene composites, polymer engineering and science, vol. 37, no. 2, pp. 476-483. khan, m. a., hinrichsen, g. and drzal, l. t. (2001): influence of novel coupling agents on mechanical properties of jute reinforced polypropylene composite, journal of materials science letters, vol. 20, no. 18, pp. 1711-1713. mohanty, a. k., misra, m. and drzal, l. t. (2002): sustainable bio-composites from renewable resources: opportunities and challenges in the green materials world, journal of polymers and the environment, vol. 10, no. 12, pp. 19-26. mohanty, s., nayak, s. k., verma, s. k. and tripathy, s. s. (2004): effect of mapp as a coupling agent on the performance of jute-pp composites, journal of reinforced plastics and composites, vol. 23, no. 6, pp. 625-637. ray, d., sarkar, b. k., rana, a. k. and bose, n. r. (2001): effect of alkali treated jute fibres on composite properties, bulletin of materials science, vol. 24, no. 2, pp. 129-135. razera, i. a. t. and frollini, e. (2004): composites based on jute fibers and phenolic matrices: properties of fibers and composites, journal of applied polymer science, vol. 91, no. 2, pp. 1077-1085. rowell, r. m. and stout, h. p. (1998): jute and kenaf, in lewin, m. and pearce, e. m. (eds.) handbook of fiber chemistry. 2nd ed. new york, marcel dekker. saheb, d. n. and jog, j. p. (1999): natural fiber polymer composites: a review, advances in polymer technology, vol. 18, no. 4, pp. 351-363. sameni, j. k., ahmad, s. h. and zakaria, s. (2003): mechanical peoperties of kenaf-thermoplastic natural rubber composites, polymer plastics technology and engineering, vol. 42, no. 3, pp. 345-355. wambua, p., ivens, j. and verpoest, i. (2003): natural fibres: can they replace glass in fibre reinforced plastics?, composites science and technology, vol. 63, no. 9, pp. 1259-1264. wollerdorfer, m. and bader, h. (1998): influence of natural fibres on the mechanical properties of biodegradable polymers, industrial crops and products, vol. 8, no. pp. 105-112. microsoft word 6945-34248-3-le journal of naval architecture and marine engineering december, 2011 doi: 10.3329/jname.v8i2.6945 http://www.banglajol.info 1813-8235 (print), 2070-8998 (online) © 2011 aname publication. all rights reserved. received on: january 2011 a flexible system for initial ship design parameters estimation using a system of neural networks hamada senousy1 and mahmoud abou-el-makarem2 marine engineering department, faculty of engineering, alexandria university, egypt 1email: snosy_ship_2010@yahoo.com 2email: apoelmakarem@yahoo.org abstract: to initialize ship design process, it is very important to be able to develop an initial estimate of ship parameters to satisfy designer required specifications. for new emerging designs, this estimate has to be made based on a limited available set of examples. moreover, a practical estimate prediction strategy should be flexible enough having no distinction between input (specified constraints) and outputs (parameters required to be estimated), since these vary from one design case to another. conventional regression-based techniques, which are usually employed to provide the required estimates, suffer from low accuracy in case of a small number of available examples. in addition to that, they fail to capture the interrelation between different design parameters. to overcome these limitations and others, the present paper proposes a new approach based on a system of artificial neural-networks (anns). the new approach not only overcomes regression limitations but is also capable of providing a reliable estimate of initial design offset table based on different ann outputs. the paper uses a case study for demonstrating the merits of the proposed approach. keywords: ship design, regression, ship series, artificial neural networks (anns), multilayer perceptrons (mlps), normalized gaussian modified lagrangian (ngml) 1. introduction prediction based design remains to be an important initial step in the design process of complex systems. this is especially exemplified in the complex process of ship design. prediction-based design uses a set of user specifications for a certain class of systems (ships) to predict the rest of the design parameters. this prediction depends on available data of existing designs of the same class. this prediction problem is usually performed using conventional regression techniques. despite the advances in ship design software, prediction based design remains to be indispensable. all design softwares proceed by evaluating a design input by the user (calculates resistance, assess fatigue, estimate needed power, weight, stability). thus, it remains the role of prediction based design to provide a good near-optimal initial design point, which can be verified and further optimized using available software tools. ideally the procedure used to predict a suitable initial design point should exhibit the following desirable features: 1. the procedure should be able to respond to queries with varying inputs and of varying length. this is because the inputs to the prediction procedure vary from one case study to another dependent on the nature of the area of application. to clarify the importance of this point in particular, consider the following arguments. any designer when designing a new ship must have input data from the owner, which are considered as design constraints. he/she must satisfy these constraints, while keeping the design as optimal as possible (large dead weight with small dimension and low power). for example, an owner may specify a ship with a certain dead weight and certain draft (design constraints). these parameters are very important parameters for a ship that will pass through suez canal. the suez canal has a certain draft that ships must not exceed to avoid additional resistance and grounding. in another example, however, an owner may specify a ship with certain dead weight and certain beam. these are the most important parameters in case the designed ship will pass through panama canal. panama canal has restricted breadth. thus, ships must have a certain beam that does not exceed the panama breadth. it is clear from these examples that having a design strategy with fixed predetermined inputs and outputs is highly undesirable and unpractical, since it will not be possible to employ it in different design situations. 2. the prediction procedure should take into account the inter-relation between the different design parameters. series-based design clearly lacks this advantage, since the ship lines are extracted based on a hierarchical design procedure that take into account the ratios between the different design parameters hamada senousy and mahmoud abou-el-makarem / journal of naval architecture and marine engineering 2(2011) 1-12 a flexible system for initial ship design parameters estimation using a system of neural networks 72 rather than their actual values (for example, the length to breadth ratio, the breadth to draft ratio). this is natural to prediction procedures that rely on single-variable regression. 3. the procedure should be able to make use of examples with partial available information and should be able to provide accurate estimates based on information from a limited number of design examples. this is common with new emerging designs. with such designs there are usually very limited design examples with detailed information made public. 4. the prediction procedure decisions should be transparent to the user i.e. the reasoning performed on available data (example) to produce the required estimates should be clear to the user. moreover, the estimates should be accompanied by a degree of confidence that gives the user an idea of how confident the procedure is in a particular estimate. 5. whenever, more examples become available, there should be an easy way of incorporating information from them within the prediction procedure with out having to re-build the prediction system from scratch. unfortunately, existing methods require thousands of entries to build the database. in addition to that traditional regression variants used to produce estimates for design variables lack most of the advantages stated above. although advanced methods such as artificial neural networks (anns) and bayesian networks successfully capture some of these advantages, they fail to satisfy them all. thus, the authors present an efficient accurate alternative strategy for providing initial ship design estimates. the strategy uses a multi-ann-based approach. the approach can be considered an adapted extension of the computational strategies proposed in literature (nelwamondo et al. 2007), (polikar et al. 2001) so as to handle the requirements of ship design discipline. the proposed strategy makes it possible to construct reliable initial ship design parameters estimates from only a few data points (ship data). this is due to the generalization property of anns. fuzzy logic interpretation of neural networks decisions may be used in future research to guarantee the transparency of the system decisions to designers (hamid et al. 2008). using a multi-ann approach helps us to cope with three problems. the first is handling incomplete entries which may meet designers during series building or when employing them in` design. the second is that they allow for incremental learning which improves the quality of prediction by adding new data entries (ship examples) if they become available without having to re-train the series anns as a whole. furthermore, multiple anns can be used to eliminate the input output distinction and lets decision making rely on available evidence. it is important to note that the use of neural networks in preliminary ship design itself is not new. several authors have pointed out the importance of using anns in place of traditional regression techniques (gougoulidis, 2008), (bertram, 2004). anns have been used in various aspects of ship design and stability. some developments are summarized below. for preliminary ship design, clausen et al. (2001) have developed multilayer perceptrons (mlp) and bayesian networks for the determination of the main particulars of ships at the initial design stage. a single hidden layer mlp network has been developed with three neurons in the hidden layer. the loading capacity of the vessel is the input to the network which estimates six parameters, namely, length, breadth, speed, draft, depth and displacement. alkan et al. (2004) propose two anns for determining initial stability particulars of fishing vessels. the architecture is mlp with two hidden layers. seven neurons in the first hidden layer and six in the second hidden layer have been used. inputs to the first layer are the block coefficient, beam, depth and length to displacement ratio. the output is the vertical centre of gravity. in their second network, the inputs are the length overall, moulded beam, design draught, moulded depth, block coefficient, prismatic coefficient, water line area coefficient and displacement at the design waterline. since it is important to estimate the metallic hull weight in primary design of the ship in order to control the weight and cost of the ships built,. wu et al. (1999) have developed an mlp network for this purpose. they have used 10 neurons in the hidden layer. inputs to the network are: length between perpendiculars (l), depth (d), draught (d), breadth (b), block coefficient (cb), l/d, b/d, and d/d. output is the metallic hull weight. islam et al. (2001) have used anns for automatic hull form generation. they have used a three layer mlp network. the network has four inputs: length, breadth, draft and type of ship. three hidden parameters are the water plane area, sectional area and midship area and the four outputs are the displacement, breadth, draft and speed. for the purpose of hull optimization, several authors (schmitz,2004),(abramowski,2010) used anns as response surfaces i.e. an ann is trained to take the design particulars as input and predict its performance (value of the objective function of the optimization procedure) as output. this speeds up the optimization procedure. however, all these efforts merely concentrated on using anns to predict a certain parameter based on some design constraints. they were not flexible enough to simultaneously handle the variation of design constraints hamada senousy and mahmoud abou-el-makarem / journal of naval architecture and marine engineering 2(2011) 1-12 a flexible system for initial ship design parameters estimation using a system of neural networks 73 from application to another. moreover, they rely mainly on using anns trained with backpropagation algorithm. the backpropagation algorithm has several disadvantages. training is slow and there is a large probability of getting trapped in a local minimum. moreover, although the work of hansen (2000) offers some flexibility in parameters values specification due to the use of bayesian networks in conjunction with anns, it still requires a huge training database. in this work, however, an alternative ann structure based on the work of (abdelsalam 2009) called normalized gaussian modified lagrangian (ngml) and use a system of anns and not just one to add flexibility to our system and make it suitable for different application scenarios and also allow users to easily make use of all available data in training the system (even partial information can be used to train some of the anns). also using ngml anns makes it easy to include additional examples whenever they are available. unlike multilayer perceptrons (mlps) trained with the conventional iterative backpropagation algorithm, ngml anns (as explained in section ii) are one-shot trained anns. thus, the addition of new examples does not involve lengthy retraining procedure. in addition to that ngml anns do not require many examples to give satisfactory results. 2. mathematical formulation 2.1 a brief overview of normalized gaussian modified lagrangian (ngml) artificial neural networks (anns) conventional anns often suffer from local minima trapping and require long training and trial and error parameter-tuning (haykin,2008). to avoid all this, abdelsalam et.al. (abdelsalam,2009) proposed a new oneshot trained ann called gaussian modified lagrangian (gml) ann. the architecture of the proposed ann is shown in fig.1. fig.1: architecture of the feedforward ngml ann used in this research. the shown architecture is for a general multi-input multi-output curve fitting problem. there are h hidden neurons. with gmls the number of hidden neurons is equal to the number of training examples. the hidden neurons activation function is given by: .......................................................................... (1) where jj = 1,2,………h, j = 1,2,….,h (2) while the activation function of the output neurons is given by: hamada senousy and mahmoud abou-el-makarem / journal of naval architecture and marine engineering 2(2011) 1-12 a flexible system for initial ship design parameters estimation using a system of neural networks 74 (3) where k=1,2,………no no is the number of output neurons the normalizing denominator in ks is optional. the performance of ngml anns is further improved by replacing conventional euclidean norm by a weighted euclidean norm: (4) in case of using the normalizing denominator, we shall denote the ann as normalized gaussian modified lagrangian ann (ngml) the gml/ngml anns can be trained using a one shot training procedure as follows. first a representative set of (input, output) centroid patterns are chosen to be memorized by the ann. the chosen patterns may be determined based on expert's knowledge or using an unsupervised clustering algorithm. the chosen patterns are encoded in the ann parameters as follows. each input pattern is stored as a centroid for one of the hidden neurons. the corresponding target is stored as the weight connecting that node to the output nodes. for example, to store ]y,[x jj , set one of the centroids of the hidden neurons to jx and set the weight connecting it to each output node k to jky . (note that by centroid we mean the pattern at which the neuron outputs 1). just as with conventional lagrangians, this is done by removing the function (x)g j centered at jx from the product in the numerator and denominator of jl , with the denominator being the result of substituting with jx=x in the numerator. moreover, note that conventional have been replaced by the euclidean norm so that the proposed ann can handle multi-input/multi-output case.) it is noteworthy that the basis functions in (1) are not the only possible choice. researchers (adel et. al, 2011) have proposed alternative choices and reported good results using those. 2.2 the proposed procedure for initial ship design generation the proposed system for initial ship design parameters estimation makes use of complete information of examples of ships belonging to a particular category of interest to the designer. ngml anns can be trained to find the relation between different design parameters. thus, after training, they can be used in design by simply presenting them with design constraints. the anns generalization capability guarantees that they will produce as output reasonable estimates of the unspecified design parameters. in this section, the different stages for preparing and testing the multi-anns based design parameters prediction system (madpps) will be discussed. a. construction phase the different steps of constructing and testing the proposed madpps are illustrated in the flow chart in fig.2. first, enough examples that represent a particular ships category are gathered. these examples will be used to form the database that will used to train, test and validate the anns. the information that is stored in the database for each example is the hull displacement (d), length overall (loa), draft (t), the maximum beam width (b) as well as the offset table. throughout the remaining of the paper, {d,l,t,b,p} will be referred to as the "design parameters string". in addition the water plan areas (wpa) and sectional areas (sa) are stored in the database. as has been illustrated in the introduction, different design cases, dictate different design constraints. thus, it is desirable that regardless of which parameters values are specified (these are considered constraints), the madpps is able to produce reliable estimates of the rest of the unspecified parameters, such that they are consistent with the main theme of the category of ships of interest. to achieve this goal, a system of anns is trained using the information in the database. each ann is trained to learn the relation between different    i ijii xxw 2 hamada senousy and mahmoud abou-el-makarem / journal of naval architecture and marine engineering 2(2011) 1-12 a flexible system for initial ship design parameters estimation using a system of neural networks 75 combinations of inputs and outputs. for example, as shown in fig.3, one ann is trained to learn the relation between d, l,t (inputs) and b, p (outputs), while another ann will be trained to learn the relation between d,t,b (inputs) and l,p (outputs). thus, the number of anns trained should be such that they cover all reasonable combinations of inputs and outputs. fig.2: a flow chart illustrating the steps of constructing and validating the proposed initial design parameters estimation system. only some of the examples are used for training (for example, 30%). a different set of examples (another 30%) is used to test the ann. the error based on the anns response to the test examples (that were unseen during training) is a measure of the ann generalization capability and reliability of the estimates that will be produced when these anns are employed. the rest of the database examples are used to validate the madpps as a whole. in addition to the anns trained to learn the relation between the different parameters, another ann is trained to learn the relation between d,l,t,b,p and the corresponding offset table. thus, this ann serves as offset table predictor; it takes as input the design parameters string and gives as output the corresponding offset table. a similar successful use of ngmls to recover offset tables based on some of the hulls parameters have been demonstrated in (el-bastawesy et. al., 2011). hamada senousy and mahmoud abou-el-makarem / journal of naval architecture and marine engineering 2(2011) 1-12 a flexible system for initial ship design parameters estimation using a system of neural networks 76 b. validation and deployment phase a typical scenario that illustrates the flow of the validation stage is shown in fig.3. in the validation stage, the designer makes up different scenarios that depict how the madpps will be used in design. this is done by setting at random some of the design parameters of the examples used for validation to zero. the system understands that this means that the system is required to produce estimates for these parameters. an "annselector" passes the design parameters "string" to the anns that are capable of producing estimates of the remaining parameters. the capability of the ann to give an estimate of a certain parameter depends on which combination of inputs/outputs has been used to train it. for example, in fig.3 in one of the validation examples, the values of l,t,p are specified. thus, the system is required to give estimates for d,b that are consistent with those specified parameter. the "ann-selector" decides to pass the specified parameters values to ann2 (which is trained to predict b,p based on specified values for l,t), ann3 (which is trained to predict b based on specified values for l,t,p) and ann5 (which is trained to predict d,t,b based on specified values for l, p). ann1 is not used because it requires d to be known, while ann4 is not considered because it predicts l, p, which are already specified in this example. it can be noted though that some of the anns used to produce the desired estimates (ann2, ann5) give estimates for parameters that are already specified (p in case of ann2, t in case of ann5). these estimates are simply ignored by the madpps (however, the difference between these estimates and their specified values can be outputed to the user as a measure of the consistency and reliability of the madpss for the example under consideration: a low difference indicates a high degree of consistency and reliability). moreover, it is clear that anns system will produce three different estimates for b (from ann2, ann3, ann5). this is resolved using the "merger" block. this block produces a weighted average of multiple estimates of the same parameter. the weights are taken to be inversely proportional to the anns errors on the test data. for example, if the normalized error on test data of ann2 is 0.1, while that of ann3 is 0.4 and that of ann5 is 0.2 then the weighted average estimate of b is computed as follows: fig.3: an illustration of how the proposed parameters estimation system is used to generate an initial offset table for a certain case study-a dashed arrow indicates an ann that is inactive for the current input parameters. this may be because it is only capable of producing estimates for parameters that are already known or because it requires as input a parameter that is unspecified for the current case study. a subset of the design parameters is specified as constraints, (as an example consider only l, t,p specified as constraints) ann selector: this part of the system passes the inputs to suitable anns that are capable of producing estimates for unspecified parameters based on available parameter values (in the present example these are d,b) ann1 d,l,t--->b,p ann2 l,t---> b,p ann3 l,t,p--->b ann4 d,t,b--->l,p ann5 l,p--->d,t,b merger: this block merges different estimates for the same parameter produce by different anns based on their respective degree of confidence offset table estimator ann completed set of parameters hamada senousy and mahmoud abou-el-makarem / journal of naval architecture and marine engineering 2(2011) 1-12 a flexible system for initial ship design parameters estimation using a system of neural networks 77 )(+)(+)( )b(+)b(+)b( =b 0.210.410.11 0.210.410.11 532   (5) where 53 b,b,b2 are the estimates for b produced by ann2, ann3,ann5, respectively. once the system produces an estimate for d,b the whole design parameters string is completed. the complete design parameters string is passed to the ann offset table predictor. note that this same sequence of operations used in the validation will be followed when the madpps is used for design. however, during design, the designer will have to trust the system, since the unspecified parameters are truly unknown. however, in the validation stage, the true values of the unspecified parameters are available in the database (they are simply hidden from the system to test the quality of its performance). this gives the designer the chance to assess the quality of estimates produced by the system. thus, the validation stage is very important to judge the reliability of the system and assess whether or not it can be used in design. in case the validation errors are unsatisfactory, more training examples should be sought and added to the database, the anns should be retrained and the system re-validate. in what follows, three different measures for assessing the performance of the madpps as a whole based on its response to the validation examples are discussed. c. assessment of obtained estimates from the validation stage. to be able to follow the different assessment measures that will be discussed, three different of parameter values are defined. first, there are the true parameters values available in the database (tv). second, there are the values of the madpps estimates of the parameters (mv). third, there are the actual values of the parameters (av). the actual values of the parameters are found by taking the values of d,l,t, and the offset table to the computational fluid dynamics (cfd) software. michlet software is used in this work (tuck,2008). the software gives the actual values of b and p consistent with these inputs. power (p) is computed as the product of the total resistance at a certain speed and this speed value. michlet predicts the total resistance to steady motion of a ship as the sum of a skin friction estimated by the standard ittc 1957 line and a wave resistance computed by michell’s integral yields quite good results compared to model experiments (tuck,2008). i. direct errors direct errors are computed by comparing the tvs and mvs of the unspecified parameters and the offset table. this is summarized in the following formula:   nup upup =direct nup =i mv i tv i error   1 2 (6) where mv i tv i up,upnup, are, respectively, the number of unspecified parameters, the tv of the i th unspecified parameter and the mv of the unspecified parameter. in this study the direct error of the parameters in the design parameters string and that of the offset table are calculated separately. ii. effective error the effective error is a measure of the similarity of certain aspects of the actual performance (computed using cfd software) of the predicted offset table with those of the true values stored in the database. the compared aspects are the values of tv and av of b, wpa, sa and p. for most applications, a sufficiently low effective error is enough to certify that the madpps can be reliably used in design. the average effective error is computed using the following formulas: (7)   nsa sasa =sae nsa =i av i tv i  1 2   nwp wpawpa =wpae nwp =i av i tv i  1 2 hamada senousy and mahmoud abou-el-makarem / journal of naval architecture and marine engineering 2(2011) 1-12 a flexible system for initial ship design parameters estimation using a system of neural networks 78 (wpae avi tv i wpa,wpanwp, are the effective error in the wpa, the number of waterlines, the tv and the av of the wpa of the ith waterplane, sae , avi tv i sa,sansa, are the effective error in the sa , number of stations, the tv and the av of the sa of the ith station, respectively) the total average effective error is given by:         4 4 1 22   =i avtvavtv error sae+wpae+pp+bb =effective (8) iii. inconsistency error the proposed madpps produces estimates of the unspecified design parameters and the offset table based on the specified parameter values. thus, the madpps ouput can be interpreted as the following statement: "the output offset table has mvmv p=p,b=b ". for these estimates to be consistent, their values should be close enough to avav p,b that are computed using the cfd software when it simulates the output offset table produced by the madpps. thus, the inconsistency error may be defined using the following formula:     2 2 1 22   =i avmvavmv error pp+bb =yconsistenc (9) 3. results and discussions to verify the effectiveness of the proposed madpps, the following case study has been adopted. 100 ship examples (offset tables) have been generated using a mathematical series (refer to http://www.cyberiad.net/michlet.htm). a random number generator has been used to produce the eight parameters for each of the 100 examples as well as the associated values for d, l,t . each example was presented to michlet cfd software to estimate the values of the corresponding p and b values. the range of d is [61 300] tons. the range of l is [25 91] meters. the range of t is [0.78 5.5] meters. 5 anns were trained using different input output combinations as shown in fig.3. the inputs to the anns were normalized by dividing them by the maximum value of each parameter through the 100 examples. the normalizing factors for the design string {d,l,t,b,p} were {300, 91.,5.5, 8.533699, 4822.5612}(the normalizing factors for b and p are based on the results of simulating all of the 100 hulls using michlet software). the power p for each example was calculated by averaging the power at three speeds (14.6, 15,15.4 m/sec). 33 examples were used for training and a different 33 example were used for test. the average errors for the 5 anns on the test data were: 0.0232234, 0.0227662, 0.0424491, 0.0185591, 0.0219834. the average error of the offset table predictor ann is 0.0019351. the remaining 34 examples were used for validation. the overall average error in the offset tables generated in validation is 0.0018. (please note that offset table matrices were also normalized by dividing them by the database half maximum beam width). due to normalization multiplying these errors by 100 gives an indication of a semi-percentage error. table.1 lists the details of the results of the validation stage (otde, dpde, effe, coe are the offset table direct error, the design parameters direct error, the effective error and the consistency error, respectively). as has been explained in earlier sections, in the validation stage some of the design parameters are set to zero at random. this are called the unspecified parameters. these unspecified parameters are indicated by placing "0" in their places in the input specifications string. the proposed madpps completes the design parameters string and passes the complete design parameters string to the offset table predictor ann. the true information in the database along with the output of the madpps and the simulator (michlet) output in response to the offset table and the values of l, t, d of the completed string are used to compute the direct, effective and consistency errors, respectively. the highlighted cells in table.1 indicate an unspecified parameter whose value has been estimated by the proposed madpps. the errors are considered to be satisfactory since the primary interest of the present work resides in generating a suitable initial design point, whose performance can be fine tuned using available ship design software packages. the reliability of the offset table estimated by the ann is demonstrated through the effective error computed in table.1. it is clear that the produced offset tables by our multi-ann systems hamada senousy and mahmoud abou-el-makarem / journal of naval architecture and marine engineering 2(2011) 1-12 a flexible system for initial ship design parameters estimation using a system of neural networks 79 results in hulls whose performance is close to that dictated by the designer's specifications (low effective error). lower errors can be attained if the ranges of the design parameters strings are narrowed or if more examples are included. it is also clear from the table that the more the unspecified parameters, the more the deviation from the true values. this is to be expected due to the effect of accumulation of errors in the different predicted parameters on the madpps output offset table. table.1 details of the results of the validation stage hamada senousy and mahmoud abou-el-makarem / journal of naval architecture and marine engineering 2(2011) 1-12 a flexible system for initial ship design parameters estimation using a system of neural networks 80 this point is further clarified through fig.4. figure 4 shows the results of the validation stage 2nd ,3rd and 19th examples (top to bottom). to the left are the true parameters of the hulls generated using the mathematical series, whereas to the right are the parameters of the hulls produced by the madpps. by comparing these results and taking into account the number of unspecified parameters in each example (refer to the highlighted cells in table.1), the impact of this number on the accuracy of the produced lines and hull parameters is easily noticed. moreover, during training and testing it was clear that power prediction (p) was the major contributor to the error. this can be improved in future work by training the ann to predict the power over a wider speed range and taking the average. to quantitatively assess the merits of our proposed approach over existing approaches, we compared the estimates produced by our proposed strategy with those that would be produced by a conventional design strategy. conventionally, given a database of existing ship designs and a string of specifications for a new design, a designer finds the closest design in the database (based on euclidean distance between the specifications of the new design and those in the database) and takes it as the initial design. ship design software fig.4: top to bottom are the results of examples 2, 3, 19 in the validation stage. to the left are the true parameters of the hulls generated using the mathematical series, whereas to the right are the parameters of the hulls produced by the madpps hamada senousy and mahmoud abou-el-makarem / journal of naval architecture and marine engineering 2(2011) 1-12 a flexible system for initial ship design parameters estimation using a system of neural networks 81 is then used to fine tune it. in table.1, the columns effec, dpdec,otdec contain the effective error, direct error in design parameters estimates, direct error in offset table using this conventional design strategy, respectively. it is clear by comparing effec (effective error using the conventional strategy) with effe (effective error using our proposed strategy) and dpdec,otdec (direct errors using the conventional strategy) with dpde,otde (direct errors using our proposed strategy) that our proposed method is capable of producing initial designs whose specifications are closer to those required by a designer. (note with the conventional design strategy, there is no room for inconsistency since the computed estimates are the actual parameters of the new design's nearest neighbor in the database). 4. conclusion the primary aim of the present paper was to propose a suitable strategy for producing an initial ship design based on available specified parameters. the main contributions are summarized as follows. first, we adapted the multi-classifier ann approach that is typically used in pattern recognition to suite the needs of naval architects. second, the choice of the ann architecture allowed for the extremely fast construction of a prediction strategy of satisfactory performance based on a very limited number of examples compared to those required by traditional methods employed in literature. third, the proposed estimates assessment measures (direct, effective and consistency errors) are expected to prove to be useful for researchers of similar and even different design interests. future research directions include repeating the design strategy for different series, trying more sophisticated parameters normalization schemes, investigating the effect of the choice and number of considered input/output combinations as well as offering a fuzzy-logic based interpretation of the system decision to the designer so that he/she can assess the experience learned by the system and thus its expected reliability. acknowledgement thanks to allah almighty whose love and support guided and motivated this work. thanks to all his prophets and messengers who taught us to prove this love by putting much effort in all we do. the authors also would like to thank engineers ahmed el-bastawesy and mohey el-deen ahmed for their supports. references abdel-salam a., b. salah, a. el-bastaweesy, islam adel, a. el-sayed, m. el-laffy, b. magdy, m. tarek, m. mahmoud, i. abdel-majeed (2009): designing high speed monohull small crafts (hsmsc) using neural networks guided cfd based optimization, ieee oceans conference bremen, germany. abramowski t. 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(1999): a neural network approach for estimating the metallic hull weight of transport ships, international shipbuilding progress, vol. 46, no. 446, pp. 141-150. wave overtopping characteristics of non-perforated and seaside perforated emerged quarter-circle breakwater journal of naval architecture and marine engineering june, 2023 https://doi.org/10.3329/jname.v20i1.55046 http://www.banglajol.info 1813-8535 (print), 2070-8998 (online) © 2023 aname publication. all rights reserved. received on: aug., 2021 wave overtopping characteristics of nonperforated and seaside perforated emerged quarter-circle breakwater vishwanatha mane1* , subba rao1, lokesha2, and arkal vittal hegde3 1,3department of water resources and ocean engineering, national institute of technology karnataka, surathkal–575025. india, e-mail: vishwanathmane@gmail.com, surakrec@gmail.com, arkalvittal@gmail.com 2director, research and consultancy wing, visual and transparent infra pvt ltd., mysore–570016. india, e-mail: lokesha.iitm@gmail.com, loki@vintrans.co.in abstract: a breakwater is a structure used to dissipate the wave energy in order to protect the shore and maintain tranquility inside the harbor basin. the quarter-circle breakwater (qbw) constitutes a quarter circular front wall facing incident waves, a vertical rear wall, and a horizontal base slab placed on a rubble mound foundation. in this study, a comprehensive experimental investigation is carried out in order to examine the wave overtopping characteristics of an emerged non-perforated and seaside perforated emerged quarter-circle breakwater subjected to regular waves. a model scale of 1:30 is selected based on the limitations of testing facilities. for the current investigation, an emerged qbw models of the radius 0.50 m is utilized. the model is tested for six different perforations ranging between 0% and 20%, with a constant perforation radius of 0.016 m. the paper highlights the influence of wave steepness (hi/gt2), relative crest freeboard (rc/hi), relative water depth (d/gt2) on the wave overtopping performance of the seaside perforated and non-perforated qbw models. an increase in wave steepness is found to increase the dimensionless mean wave overtopping discharge. also, an exponential decrease in dimensionless mean wave overtopping discharge is observed with an increasing relative freeboard. the relative freeboard is found to be one of the predominant parameters influencing the wave overtopping discharge rate. keywords: emerged quarter-circle breakwater; perforations; mean wave overtopping discharge; relative freeboard; and wave steepness nomenclature d water depth q overtopping volume d diameter of perforations r radius of structure g acceleration due to gravity t wave period hs height of structure q/ghit overtopping discharge hi incident wave height d/gt 2 depth parameter l wave length hi/gt 2 incident wave steepness p percentage of perforations rc/hi relative freeboard 1. introduction a breakwater is a structure used to attain calm conditions on its lee side. throughout the world, different types of breakwaters are used to protect the coastal region and harbours. research accomplishments are evolving to analyze the performance of hydrodynamic characteristics of new innovative breakwaters, which can be recommended for the prevailing economic and environmental conditions (aburatani et al., 1996; mane et al., 2013; rajendra et al., 2017). the quarter circular breakwater was proposed by xie et al. (2006) based on the concept of the semi-circular breakwater (sbw), and the construction of qbw is almost similar to sbw, which is generally provided with a base of rubble mound foundation. a conceptual 3d view of non-perforated qbw and seaside perforated qbw is as shown in figure 1. https://doi.org/10.3329/jname.v20i1.55046 http://www.banglajol.info/ mailto:vishwanathmane@gmail.com mailto:surakrec@gmail.com mailto:arkalvittal@gmail.com v. mane, s. rao, lokesha, a. v. hegde / journal of naval architecture and marine engineering, 20(2023) 11-23 wave overtopping characteristics of non-perforated and seaside perforated emerged quarter-circle breakwater 12 as the bottom width of qbw is half of the base width of sbw, the volume of its rubble mound foundation would be reduced to nearly half of the sbw. however, qbw still has advantages such as reducing wave force on the seaside surface against the incoming wave, easy installation as it is prefabricated on land, and an excellent aesthetic view similar to sbw. the most common difference between qbw and sbw is their rubble foundation width, which causes different stress distributions on the foundation. qbw is suitable for the places where a stronger subsoil is available, as the magnitude of stress on the foundation soil will be more. the wave overtopping phenomenon is generally the flow of seawater over a crest of the coastal structure due to wind action, wave run-up, and wave breaking (van der meer et al., 2016). fig. 1: isometric view of non-perforated and perforated qbw many researchers studied the hydrodynamic performance characteristics of qbw, wherein they focused mainly on the investigation of dynamic wave pressures, transmission, and reflection characteristics (x. l. jiang et al., 2017; liu et al., 2006). also, the characteristics of different types of breakwater models (rubble mound and vertical breakwater) were analyzed, focusing on the wave overtopping performance (gil et al., 2015; tuan, 2013; van bergeijk et al., 2019). as expected, shi et al. (2011) observed that the loss of wave energy for emerged breakwater is more than that for the submerged breakwater. they have concluded that the hydrodynamic performances of sbw and qbw are almost similar, resulting in the identical wave profiles of both breakwaters. hegde and ravikiran (2013) examined the impact of wave structure interaction for submerged qbw of the different radii, wave height, and submergence ratios. they concluded that the wave reflection increased with an increase in wave steepness. further, qie et al. (2013) conducted a study on the development of a wave force formula to design quartercircular caisson breakwater. they suggested a simplified method to calculate the wave forces based on the goda formula. pedersen (1996) carried out experimental work to examine the crown wall's performance against the wave forces and wave overtopping. the authors developed a newly designed empirical formula to predict the mean overtopping discharge over a crown wall structure. franco et al. (1994) measured the wave overtopping response on various caisson breakwaters and studied the probability distribution of individual overtopping waves. the authors concluded that the overtopping discharges on deepwater vertical walls are considerably greater than that of those projected by tanimoto and goda (2015) and moderately lesser than those for a corresponding sloping arrangement. the arrangement of a perforated wall with a recurved crest (nose) on the front wall creates a significant overtopping drop, while rock shelter in front of the caisson up to the sea level can increase overtopping. an experimental study was carried by reis et al. (2008) on a two-dimensional breakwater model to investigate the impact of the test duration on mean wave overtopping. the effect of the higher waves on the overtopping discharge is strong. even a low variance in the elevation of the involved waves in a wave train may powerfully influence the mean overtopping discharge, especially for lesser overtopping rates with a small test duration. another researcher, bruce et al. (2007) compared the overtopping performance for different armour units of rubble mound breakwater through experimental investigation. the inquiry concluded that the wave period has a greater influence on wave overtopping, and a larger wave period contributes more to wave overtopping. the investigators binumol et al. (2017) and dhinakaran et al. (2002) explored the hydrodynamic characteristics of qbw and sbw. the authors found that the dimensionless wave run-up increases with an increase in wave v. mane, s. rao, lokesha, a. v. hegde / journal of naval architecture and marine engineering, 20(2023) 11-23 wave overtopping characteristics of non-perforated and seaside perforated emerged quarter-circle breakwater 13 steepness for various values of height of the structure to the depth of water (hs/d) and water depth parameter (d/gt2). also, the non-dimensional stability parameter is always decreasing with an increase in wave steepness. another observation was that the wave run-up (ru/hi) decreases with an increase in the water depth parameter (d/gt2). it is expected that curvature influence is more pronounced due to higher water depths, which results in a lower run-up. jiang et al. (2018) examined the flow separation and vortex dynamics phenomenon during wave overtopping on submerged qbw. they concluded that the instant and mean value of time vorticity fields expose a couple of vortices of conflicting marks at the breakwater structure that is likely to affect transportation, suspension, and sediment entrainment. thus, resulting in scour on the lee side of the breakwater. further, salauddin and pearson (2020) studied the comprehensive two-dimensional experimental study on the sloping walls overtopping performance undertaken on both impermeable and permeable foreshore slopes. they proposed a revised forecast tool to predict the overtopping performance at sloping structures on porous rock foreshores. kerpen et al. (2020, 2019) developed a reduction coefficient for a ventured revetment roughness for a broad utility scope considering its progression proportion. the obtained reduction coefficient for ventured revetments did not base on the prototype model scale. the correction factor for the prototype model scale effect for ventured revetments has not been considered, subsequently, which is likely to be influenced by scale effects. many researchers studied the hydrodynamic characteristics of qbw, focusing mainly on wave transmission and reflection. the available literature confirms that there are limited studies on wave overtopping characteristics of emerged qbw. wave overtopping is an essential factor as it plays a significant role in the design of emerged qbw structure. the objective of the current study is to investigate the wave overtopping performance of nonperforated and perforated emerged qbw using physical models. the studies are conducted on emerged qbw models for varying percentages of perforation at different water levels against different wave conditions. 2. experimental setup and methodology 2.1 testing facility a detailed investigation is carried out in a 2-dimensional wave flume equipped with a generation system for regular waves only of length 50 m, width 0.74 m, and depth 1.1 m at marine structures laboratory, department of water resources and ocean engineering, national institute of technology, surathkal, karnataka, india. the proposed quarter-circle breakwater models are tested with the existing amenities of the wave flume and suit the mangaluru coast characteristics (dattatri, 1993). in the wave flume, regular waves of heights varying from 0.02 m to 0.24 m and wave periods ranging from 0.8 s to 4 s can be generated. detailed sectional views of the wave flume along with the location of the qbw model and wave probes, are shown in figure 2. fig. 2: wave flume arrangement (not to scale) v. mane, s. rao, lokesha, a. v. hegde / journal of naval architecture and marine engineering, 20(2023) 11-23 wave overtopping characteristics of non-perforated and seaside perforated emerged quarter-circle breakwater 14 2.2 model casting a typical cross-section of seaside perforated and non-perforated qbw is shown in figure 3. the breakwater model comprises two sections, the bottom slab and the top quarter-circle formed by a metal sheet. the model is built in two steps, with the first phase involving casting the base slab and the second involving casting the qbw to the necessary dimensions. the base slab is provided to increase the total weight of the qbw in order to form a stable base for the superstructure and the dimension of the slab is chosen accordingly to serve the purpose. a thick galvanized iron (g.i) sheet of thickness 0.002 m is used to fabricate the quarter-circular breakwater of radius 0.5 m and coated with cement slurry. then, the g.i sheet is fixed to the base slab with the help of stiffeners. the dimensions for qbw are chosen to serve as an emerging type for all water depths and facilitate the overtopping of incident waves. the breakwater model is then positioned over the foundation with a rubble mound of thickness 0.05 m (minimum thickness as per cem, 2001) and stones weighing from 50 g to 100 g. fig. 3a: typical cross-section of non-perforated qbw fig. 3b: typical cross-section of a seaside perforated qbw the studies carried out by binumol et al., (2017) on perforated qbw of varying sizes (0.016 m and 0.02 mm) indicated that the influence of perforation size is incognizable on the performance of the structure. hence, under the constant size of perforations (d = 0.016 m), the percentage of perforations (p) is varied from 1.25 to 20%. the size of the perforation considered in the present study is in line with the other investigations (binumol et al. 2017; dhinakaran et al. 2002; hegde and ravikiran, 2013) on similar types of structures. 2.3 mechanism of overtopped water collecting tank the overtopped water volume per wave is collected in a tray attached to the breakwater models on the lee side using stiffeners. the water collecting tray length, breadth, and depth are 0.87 m, 0.73 m, and 0.13 m, respectively. initially, trial cases are run for the maximum water depth (d = 0.50 m) and the dimensions of the water collecting tray are arrived at based on the maximum water collected by including free board. the v. mane, s. rao, lokesha, a. v. hegde / journal of naval architecture and marine engineering, 20(2023) 11-23 wave overtopping characteristics of non-perforated and seaside perforated emerged quarter-circle breakwater 15 maximum water collected in the collection tank is 0.0825 cubic meters. the waves are generated in a short burst and the total volume of overtopped water is measured after each burst. hence, the overtopping discharge per wave is calculated as the ratio of the total volume of overtopped water collected to the number of waves overtopped. the overtopping discharge is expressed in m3/s per m width of the breakwater. the collected water is disposed of at the end of each wave burst and the next trial is conducted. the correctness and consistency of the outcomes are confirmed by repeating all the cases by three times, according to the studies by zhao and ning (2018). to prevent spillage losses, a thin sheet of rubber is placed between the structure corners and the flumes side wall. figure 4 shows the mechanism of collecting overtopped water over the quarter circle breakwater structure. fig. 4a: a typical front and rear view of an empty tray fig. 4b: a typical front and rear view of water collected in a tray 2.4 wave characteristics the regular waves with different wave periods and wave heights are considered for the present study is shown in table 1. a burst of five waves is produced in order to prevent wave distortion due to wave reflection and a small amount of re-reflection from the breakwater assembly and the wave paddle. for each consecutive test run, a considerable amount of time-lapse is given in order to attain calmness with respect to still water level. the model is positioned in the wave flume at a distance of 30 m from the wave generator flap. in order to measure the incident wave heights, capacitance type wave probes are employed, and the calibration is carried out for each model setup. the spacing of the probes is in accordance with the methodology proposed by isaacson (1991), is a function of wavelength (l) and is kept at the distance of l/3 for a particular water depth. the spacing of the first probe on the seaside is measured 1 m from the breakwater model. the wave probes arranged at suitable intervals will measure and record the incident wave heights. the wave parameters of the mangalore coast of the arabian sea are used for the present study. a geometric model scale of 1:30 is considered based on the limitations of the testing facility. the wave periods ranging between 1.4 s and 2.2 s are considered in the current investigation. a maximum of six wave heights ranging from 0.08 m to 0.16 m was considered for each wave period. the models were tested for three different water depths, such as 0.45 m, 0.475 m, and 0.50 m. v. mane, s. rao, lokesha, a. v. hegde / journal of naval architecture and marine engineering, 20(2023) 11-23 wave overtopping characteristics of non-perforated and seaside perforated emerged quarter-circle breakwater 16 table 1: summary of the wave and structural design parameters parameters unit range of investigation incident wave height (hi) m 0.08, 0.1, 0.12, 0.14, 0.16 wave period, (t) s 1.4, 1.6, 1.8, 2.0, 2.2 water depth, d (m) m 0.45, 0.475, and 0.50 structure radius (r) m 0.5 diameter of perforation (d) m 0.016 percentage of perforation (p) % 0, 1.25, 5, 10, 15, and 20 3. results and discussion 3.1 general in the present study, the experimental tests are conducted for emerged non-perforated qbw and the seaside perforated qbw in order to compare their overtopping performance. the dimensional analysis is carried out for various wave and structural design components using buckingham's π theorem in arriving at non-dimensional parameters. the parameters such as radius of the structure (r), the height of the structure (hs), the diameter of perforations (d), mean wave overtopping discharge (q), water depth (d), wavelength (l), incident wave height (hi), wave period (t), the mass density of water (ρ) and acceleration due to gravity (g) are considered for dimensional analysis. the non-dimensional π terms used in the discussion are mean wave overtopping discharge (q/ghit), wave steepness parameter (hi/gt 2), relative water depth (d/gt2), relative freeboard (rc/hi), and percentages of perforation (p). the experiments were conducted for non-perforated (0% perforation) and sea-side perforated (1.25%, 5%, 10%, 15%, and 20% perforation) qbw model of radius 0.50 m. the overtopping breakwater model is tested for different heights; wave periods with varying water depths, and freeboards. the effect of various sea state parameters and structural parameters on the wave overtopping of the emerged qbw models are discussed in detail. the results obtained are plotted for wave overtopping discharge against wave steepness, relative freeboard, and relative water depth. the graphs were plotted to aid in understanding the effect of influencing parameters on the mean overtopping for seaside perforated and non-perforated models. 3.2 effect of wave steepness on wave overtopping characteristics the mean wave overtopping discharge is plotted against wave steepness for non-perforated qbw (0%), and perforated qbw models (1.25%, 5%, 10%, 15%, and 20%) is shown in figure 5. the results are plotted for each percentage of perforations and rc/hi = 0.625 to 1.875 with three water depths. the range of d/gt 2 values are 0.0095 to 0.0234, 0.0090 to 0.0221, and 0.0084 to 0.0208 corresponding to the water depths of 0.45 m, 0.475 m, and 0.50 m respectively. figure 5 shows that q/ghit increases with an increase in hi/gt 2 for all the water depths, i.e., 0.45 m, 0.475 m, and 0.50 m. this is maybe due to an increase in wave height and a decrease in relative freeboard for a particular water depth that allows the waves to pass over the model resulting in an increasing mean overtopping rate. the overtopping discharge is found to be more pronounced at higher wave steepness, in other ways, for lower wave periods and higher wave heights. the increase in wave period (wavelength) for higher wave heights with a particular relative freeboard and water depth admits the incoming waves on the lee side of the models. also, when the wave height increases, there is an increase in wave energy, and hence overtopping. it is also observed that q/ghit increases with an increase in the water depth and decrease of the relative freeboard. this may be because, at higher water depths, the curvature effect is more pronounced, resulting in higher overtopping rates. figure 5 shows that the wave overtopping discharge increases with hi/gt 2 for all the considered depth parameters and perforations cases. in case of non-perforated model, the maximum and minimum values of q/ghit observed are 3.56×10 -3 at hi/gt 2 = 8.32×10-3 for d/gt2 = 0.026 and 3.21×10-4 at hi/gt 2 = 1.68×10-3 for d/gt2 = 0.009 respectively. the maximum and minimum values of overtopping discharge rates for qbw with various percentages of perforations considered are shown in table 2. v. mane, s. rao, lokesha, a. v. hegde / journal of naval architecture and marine engineering, 20(2023) 11-23 wave overtopping characteristics of non-perforated and seaside perforated emerged quarter-circle breakwater 17 fig. 5: variation of q/ghit with hi/gt 2for different perforations table 2: summarised results percentage of perforation q/ghit hi/gt2 rc/hi d/gt2 1.25 min. 1.91e-04 1.68e-03 1.875 0.009 max. 2.49e-03 8.32e-03 0.625 0.026 5 min. 2.74e-04 3.15e-03 1.500 0.014 max. 1.92e-03 8.32e-03 0.625 0.026 10 min. 1.82e-04 3.15e-03 1.500 0.014 max. 1.66e-03 8.32e-03 0.625 0.026 15 min. 9.12e-05 3.15e-03 1.500 0.014 max. 1.40e-03 8.32e-03 0.625 0.026 20 min. 3.70e-04 3.06e-03 1.250 0.011 max. 1.13e-03 8.32e-03 0.625 0.026 3.3 effect of relative freeboard on wave overtopping characteristics figure 6 shows the plot of q/ghit against the relative freeboard parameter (rc/hi) for non-perforated qbw (0%) and perforated qbw models (1.25%, 5%, 10%, 15%, and 20%) with a fixed radius of 0.50 m. figure 6 shows that q/ghit decreases with an increase in rc/hi for all the water depths considered. the variations observed from the data points shown have the same trend as described by troch et al. (2014) and bradbury et al. (1988). as water depth increases, the relative freeboard decreases; the decrease in water depth increases the relative freeboard makes a lesser contribution to the overtopping discharge rate. also, it becomes heavier for the waves to cling over the structure and overtop. the variation in q/ghit is found to be primarily dependent on the freeboard. in general, the relative freeboard (rc/hi) is found to be indirectly proportional to the wave overtopping discharge (q/ghit). it is expected that because of the higher water depth, the influence of curvature is more pronounced, which results in higher overtopping. for non-perforated qbw, the maximum value for v. mane, s. rao, lokesha, a. v. hegde / journal of naval architecture and marine engineering, 20(2023) 11-23 wave overtopping characteristics of non-perforated and seaside perforated emerged quarter-circle breakwater 18 q/ghit observed is 3.56×10 -3 at rc/hi = 0.625 for d/gt 2 = 0.026. similarly, the minimum q/ghit observed is 3.21×10-4 at rc/hi = 2.5 for d/gt 2 = 0.009. the maximum and minimum rates of q/ghit at both parameters for varying d/gt2 values are summarised in table 2. fig. 6: variation of q/ghit with rc/hi for different perforations 3.4 effect of perforations and relative water depth on wave overtopping characteristics for a particular water depth, the mean wave overtopping discharge rate is plotted against the relative water depth for all the percentages of perforation considered in order to examine the effect of perforation. the variation of q/ghit plotted as the function of d/gt 2 for all the percentages of perforation (p) is shown in figure7. it is observed that an increase in d/gt2 increases q/ghit for all the water depths considered. the increase in the percentages of perforation from 0 to 10% has a larger influence on decreasing q/ghit. a further increase in the percentages of perforation has a lesser impact on q/ghit. the decrease in overtopping discharge (q/ghit) with an increase in the perforations (p) maybe because of the dissipation of wave energy due to the turbulence inside the chamber. as the water enters the qbw through the perforations, it flows back out of the perforations, which encounters the next incoming wave resulting in partial energy dissipation, accomplishes even before that wave reaches the breakwater. the other reason would be the waves of smaller wave periods ride over the arched surface upon which most of the incident wave energy gets reflected. thus, lesser overtopped discharge rates are available. fig. 7: variation of q/ghit with d/gt 2for different perforations v. mane, s. rao, lokesha, a. v. hegde / journal of naval architecture and marine engineering, 20(2023) 11-23 wave overtopping characteristics of non-perforated and seaside perforated emerged quarter-circle breakwater 19 also, it is observed that an increase in water depth decreases rc/hi resulting in a larger mean overtopping rate. the values of q/ghit are higher for non-perforated qbw compared with perforated qbw due to less dissipation of wave energy in the non-perforated breakwater. in the case of non-perforated qbw, the range of variation of q/ghit is found to be varying from 1.18×10 -4 to 3.56×10-3 with a range of hi/gt 2= 1.69×10-3 to 8.32×10-3 and rc/hi = 1.875 to 0.625. for hi/gt 2 = 1.69×10-3 to 8.32×10-3 and rc/hi = 1.875 to 0.625, the maximum and minimum discharge rates varies from 1.91×10-4 to 2.49×10-3 for 1.25% perforation, 2.74×10-4 to 1.92×10-3 for 5% perforation, 1.82×10-4 to 1.66×10-3 for 10% perforation, 9.12×10-5 to 1.40×10-3 for 15% perforation and 3.70×10-4 to 1.13×10-3 for 20% perforation. 3.5 regression analysis the experimental results on mean overtopping discharge rates under regular waves are subjected to multiple regression analysis based on the least square method. the empirical equations have arrived for q/ghit as a function of independent variables hi, t, rc, d, and g for the non-perforated model. for the perforated models, in addition to the above said independent variables, a new independent variable p is considered in the arriving empirical equation. these developed empirical equations can be elementary used to forecast overtopping rates. the empirical equation of q/ghit for non-perforated and perforated models is given in equations (3.1) and (3.2). these equations are based on regular wave test conditions only. od={0.146 (hi/gt 2)}+{0.0164(rc/hi)}-{0.01(d/gt 2)} + 0.0019………………...(3.1) od={0.083 (hi/gt 2)}-{0.0006(rc/hi)}-{0.001 (d/gt 2)} 0.0003p + 0.0018…….(3.2) where od is a dimensionless wave overtopping discharge (q/ghit), the non-dimensional test ranges for the above equations are mentioned and shown in table 3. table 3: test ranges parameters range wave steepness (hi/gt 2) 1.69×10-3 to 8.32×10-3 relative freeboard (rc/hi) 1.875 to 0.625 relative depth (d/gt2) 0.009 to 0.026 percentage of perforation (p) 0% to 20% fig. 8: comparison of measured and predicted q/ghit for non-perforated and perforated qbw the comparison of measured and predicted q/ghit from the above-derived equations is shown in figure 8a for the non-perforated and figure 8b for the perforated models. the line of equality is superposed in the plot. it is observed that the measured and predicted q/ghit for both the models are reasonably good in agreement. the correlation coefficient for the predicted values of q/ghit is found to be 0.89 for the non-perforated model and 0.91 for the perforated model. v. mane, s. rao, lokesha, a. v. hegde / journal of naval architecture and marine engineering, 20(2023) 11-23 wave overtopping characteristics of non-perforated and seaside perforated emerged quarter-circle breakwater 20 3.6 uncertainty analysis uncertainty is an evaluation of experimental error. generally, whenever experimentation is involved, there is a possibility of some errors creeping in a while making measurements. with the help of uncertainty analysis, it is possible to conduct experiments scientifically and predict the accuracy of the result (s.c. misra, 2001). the width of the confidence interval is a measure of the overall quality of the regression line. the 95% confidence interval limits must always be estimated, and this concept of confidence level is fundamental to uncertainty analysis. the methodology used is the method of confidence bands. confidence interval may be constructed from the mean response at a specified value x, say xo. this is a confidence interval about         o x y e = 0x y   and is often called a confidence interval about the regression line. a 100(1-α) percent confidence interval about the mean response at the value of x = xo, say 0x y   , is given by, ……………………………. (3.3) where, 0x y   = βo + β1xo computed from the fitted regression model, α = significance level used to compute the confidence level, 2  = variance, x = sample size, x = sample mean. ………………………………… (3.4) the 95% confidence and prediction band for variation of wave overtopping discharge (q/ghit) with incident wave steepness (hi/gt 2)for perforated and non-perforated quarter-circle breakwater models is shown in figure. 9. the wave parameters tested with a range of t = 1.2 s to 2.2 s, h = 0.06 m to 0.18 m, and d = 0.45 m and 0.475 m are considered in the current investigation. fig. 9: a plot of 95% confidence and prediction bands for the variation of q/ghit for both non-perforated and perforated (1.25%) qbw            xx n x y s xx n t 2 0 2 2,2/ )(1 0        n i n i i xx n x xs 1 1 2 2 0 )( v. mane, s. rao, lokesha, a. v. hegde / journal of naval architecture and marine engineering, 20(2023) 11-23 wave overtopping characteristics of non-perforated and seaside perforated emerged quarter-circle breakwater 21 the figures show that the trend line showing q/ghit variation with hi/gt 2 lies within the 95% confidence bands, and data points lie within the 95% prediction bands drawn. also, from the figures, it is observed that more than 80% of experimental data lie within the 95% confidence bands. the regression coefficient, r2, is found to be 0.87. therefore, the results obtained may be analyzed with 95% confidence, i.e., the conclusions drawn from these graphs are 80% reliable. further, from the above-said figure 9, it may be visualized that all the experimental data points are found to be close to the 95% confidence level limits. 3.7 comparative analysis this section deals with the comparison of the results of current experimental work with results obtained by other researchers on similar work collected from the literature. shankar and jayaratne (2002) describe the wave steepness demonstrates a suitable parameter for defining the combined influence of wave height and wave period on wave overtopping discharge along with the relative crest height parameter. their objectives are to explore the effect of wave and structural parameters on wave overtopping discharge for the sloped permeable and impermeable breakwater. within the laboratory facilities limitations, the author chosen wave parameters ranges are wave height (h) = 0.05 to 0.12 m, wave period (t) = 0.8 to 1.2 s, depth parameter (d/gt2) = 0.019 to 0.073, and wave parameter (hi/gt 2) = 0.006 to 0.011. figure 10shows the comparative analysis of overtopping discharges on an impermeable breakwater model (1:2) with the current experimental results. from figure 10, it can be noticed that the present experimental data points are shown in good agreement with the smooth, impermeable experimental data points. fig.10: comparative analysis of present work with sloped and impermeable breakwater 4. conclusions the study explored the investigation of mean wave overtopping discharges on the emerged seaside perforated and non-perforated quarter-circle breakwater subjected to regular waves of different wave heights and wave periods. based on the analysis of the results of the current study, the following conclusions are drawn. the present study observed that an increase in the percentage of perforations results in a decrease in the mean wave overtopping discharge rate. the mean wave overtopping discharge increases with an increase in wave steepness and increases with the relative water depth parameter. also, the mean overtopping discharge decreases with an increase in relative freeboard for all the water depths. the values of q/ghit are higher for non-perforated qbw when compared with perforated qbw due to the lesser dissipation of wave energy in the non-perforated breakwater. the percentage of reduction in q/ghit for 1.25% perforated qbw is varied from 7% to 38% compared to the non-perforated model. similarly, the decrease in q/ghit for 5% perforated qbw is varied from 13% to 63%. whereas, for 10%, 15%, and 20% perforated qbw, the decrease in q/ghit is varied from 30% to 72%, 40% to 73%, and 43% to 74% respectively. the developed empirical equation has reproduced the experimental results with desirable accuracy. the proposed empirical equation can be extended for predicting the overtopping discharge of qbw within the test limit with an appropriate engineering judgment based on the site conditions. v. mane, s. rao, lokesha, a. v. hegde / journal of naval architecture and marine engineering, 20(2023) 11-23 wave overtopping characteristics of non-perforated and seaside perforated emerged quarter-circle breakwater 22 acknowledgments the authors are thankful to the director, national institute of technology karnataka (nitk), surathkal, and the head, department of water resources and ocean engineering, nitk, surathkal, for the encouragement laboratory and testing facilities provided to carry out the investigations. references aburatani, s., 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(2018): experimental investigation of breakwater-type wec composed of both stationary and floating pontoons, energy, vol. 155, pp. 226-233. https://doi.org/10.1016/j.energy.2018.04.189 https://doi.org/10.1007/s11804-013-1176-7 https://doi.org/10.1142/s057856340800182x https://doi.org/10.1016/j.oceaneng.2019.106866 https://doi.org/10.1016/s0029-8018(02)00016-1 https://doi.org/10.1007/s13344-011-0038-1 https://doi.org/10.1680/csab.16729.0013 https://doi.org/10.9753/icce.v34.structures.2 https://doi.org/10.1142/s0578563413500137 https://doi.org/10.1016/j.coastaleng.2019.03.001 https://doi.org/10.1016/j.energy.2018.04.189 microsoft word jname413r.doc 1813-8535 © 2007 aname publication. all rights reserved. journal of naval architecture and marine engineering june, 2007 http://jname.8m.net dufour and soret effects on steady mhd free convection and mass transfer fluid flow through a porous medium in a rotating system nazmul islam and md. mahmud alam mathematics discipline, khulna university, khulna-9208, bangladesh, e-mail: alam_mahmud2000@yahoo.com abstract the numerical studies are performed to examine the steady mhd free convection and mass transfer fluid flow through a continuously moving porous medium with thermal diffusion and diffusion thermo past a semi-infinite vertical porous plate in a rotating system. impulsively started plate moving in its own plane is considered. with appropriate transformations the boundary layer equations are transformed into nonlinear ordinary differential equations. the local similarity solutions of the transformed dimensionless equations for the flow, heat and mass transfer characteristics are evaluated using shooting iteration technique. numerical results are presented in the form of velocity, temperature and concentration within the boundary layer for different parameters entering into the analysis. also the effects of the pertinent parameters on the local skin-friction coefficients and rate of heat transfer as well as rate of mass transfer in terms of the local nusselt number and sherwood number respectively are also discussed. keywords: mhd free convection, porous medium, rotation effect, dufour effect, soret effect. nomenclature zyx ,, cartesian coordinates rg grashof number wf transpiration parameter mg modified grashof number wvu ,, velocity components k nondimensional permeability parameter 0v ( x ) suction velocity rp prandtl number b magnetic field intensity cs schimidt number )(0 xb constant magnetic field intensity r s soret number ),,( zyx jjj=j current density fd dufour number ω angular velocity m magnetic parameter k thermal conductivity of the medium r rotation parameter 0g gravitational acceleration k ′ permeability of the porous medium f ′ non-dimensional primary velocity m d coefficient of mass diffusivity g non-dimensional secondary velocity p c specific heat at constant pressure 0u uniform velocity mt mean fluid temperature n. islam and m. m. alam / journal of naval architecture and marine engineering 4(2007) 43-55 dufour and soret effects on steady mhd free convection and mass transfer fluid flow through a porous medium... 44 t temperature of the flow field 0c mean concentration wt temperature at the plate tk thermal diffusion ratio ∞t temperature of the fluid outside the boundary layer s c concentration susceptibility c species concentration un nusselt number wc concentration at the plate hs sherwood number ∞c species concentration outside the boundary layer ),( zx ττ skin-friction coefficients greek µ coefficient of viscosity ρ fluid density η similarity variable σ′ electrical conductivity υ coefficient of kinematics viscosity β coefficient of thermal expansion θ dimensionless fluid temperature *β coefficient of concentration expansion ϕ dimensionless fluid concentration 1. introduction the science of magnetohydrodynamics (mhd) was concerned with geophysical and astrophysical problems for a number of years. in recent years the possible use of mhd is to affect a flow stream of an electrically conducting fluid for the purpose of thermal protection, braking, propulsion and control. from the point of applications, model studies on the effect of magnetic field on free convection flows have been made by several investigators. some of them are georgantopoulos (1979), nanousis et al. (1980) and raptis and singh (1983). along with the effects of magnetic field, the effect of transpiration parameter, being an effective method of controlling the boundary layer has been considered by kafousias (1979) and singh (1982). on the other hand, along with the free convection currents, caused by the temperature difference, the flow is also effected by the difference in concentrations on material constitution. gebhart and pera (1971) made extensive studies of such a combined heat and mass transfer flow to highlight the insight of the flow. in the above mentioned works, the level of concentration of foreign mass is assumed very low so that the soret and dufour effects can be neglected. however, exceptions are observed therein. the soret effect, for instance, has been utilized for isotope separation, and in mixture between gases with very light molecular weight ),( 2 ehh and of medium molecular weight )air,( 2n . the dufour effect was found to be of order of considerable magnitude such that it cannot be ignored (eckert and drake, 1972). in view of the importance of above mentioned effects, kafoussias and williams (1995) studied the soret and dufour effects on mixed free-forced convective and mass transfer boundary layer flow with temperature dependent viscosity. anghel et al. (2000) vestigated the dufour and soret effects on free convection boundary layer flow over a vertical surface embedded in a porous medium. quite recently, alam and rahman (2006) investigated the dufour and soret effects on mixed convection flow past a vertical porous flat plate with variable suction. in consequence of the above studies, several investigators disclosed that the coriolis force is very significant as compared to viscous and inertia forces occurring in the basic fluid equations. it is generally admitted that the coriolis force due to earth’s rotation has a strong effect on the hydromagnetic flow in the earth’s liquid core. the study of such fluid flow problem is important due to its applications in various branches of geophysics astrophysics and fluid engineering. in light of, singh and singh (1989), singh (1983,1984) and raptis and singh (1985) initiated a few studies by taking various aspects of the flow phenomena. from the point of application in solar physics and cosmic fluid dynamics, it is important to consider the effects of the electromagnetic n. islam and m. m. alam / journal of naval architecture and marine engineering 4(2007) 43-55 dufour and soret effects on steady mhd free convection and mass transfer fluid flow through a porous medium... 45 and rotation forces on the flow. but no works of the simultaneous effects of the electromagnetic and rotation forces on the hydromagnetic free convection without dufour and soret effects have been reported in the literature. hence, our objective is to investigate the dufour and soret effects on steady mhd free convection and mass transfer flow through a porous medium past a semi-infinite vertical porous plate in a rotating system. 2. formulation of the problem and similarity analysis consider a steady mhd free convection and mass transfer flow of an electrically conducting viscous fluid flow through a porous medium past a continuously moving semi-infinite vertical porous plate 0=y in a rotating system. the flow is also assumed to be in the x -direction which is taken along the plate in the upward direction and y -axis is normal to it. initially the fluid as well as the plate is at rest, after that the whole system is allowed to rotate with a constant angular velocity ω about the y -axis. the temperature and the species concentration at the plate are constantly raised from wt and wc to ∞t and ∞c respectively, which are thereafter maintained constant. ∞t and ∞c are the temperature and species concentration of the uniform flow respectively. a uniform magnetic field b is taken to be acting along the axisy − which is assumed to be electrically non-conducting. we assumed that the magnetic reynolds number of the flow is taken to be small enough so that the induced magnetic field is negligible in comparison with applied one (pai, 1962), so that )0,,0( 0b=b and the magnetic lines of force are fixed relative to the fluid. the equation of conservation of charge 0. =∇ j gives =yj constant, where the current density )( ,, zyx jjj=j . since the plate is electrically non-conducting, this constant is zero and hence =yj 0 at the plate and hence zero everywhere. the physical configuration considered here is shown in the following fig. 1. fig. 1. physical configuration and coordinate system. thus accordance with the above assumptions relevant to the problem and boussinesq’s approximation, the basic boundary layer equations are given by the continuity equation: 0= ∂ ∂ + ∂ ∂ y v x u (1) y )(0 xb )x(v0 x ω u v wt wc ∞t ∞c w n. islam and m. m. alam / journal of naval architecture and marine engineering 4(2007) 43-55 dufour and soret effects on steady mhd free convection and mass transfer fluid flow through a porous medium... 46 the momentum equations: ( ) ( ) ρ συ ββυ ub u k wccgttg y u y u v x u u 2 0* 002 2 2 ′ − ′ −ω+−+−+ ∂ ∂ = ∂ ∂ + ∂ ∂ ∞∞ (2) ρ συ υ wb w k u y w y w v x w u 2 0 2 2 2 ′ − ′ −ω− ∂ ∂ = ∂ ∂ + ∂ ∂ (3) the energy equation: 2 2 2 2 y c cc kd y t c k y t v x t u ps tm p ∂ ∂ + ∂ ∂ = ∂ ∂ + ∂ ∂ ρ (4) the concentration equation: 2 2 2 2 y t t kd y c d y c v x c u m tm m ∂ ∂ + ∂ ∂ = ∂ ∂ + ∂ ∂ (5) the boundary conditions for the present problem are given by: ⎭ ⎬ ⎫ ∞→→→=== ====== ∞∞ yatccttwvu yatccttwxvvuu ww ,,0,0,0 0,,0),(, 00 (6) where all physical quantities are defined in the nomenclature. we now introduce the following dimensionless variables ( ) ( ) ( ) ( ) 0 0 0 0 2 w u y x u f ( ) u w g u t t t t c c x c c η υ η η θ η φ η ∞ ∞ ∞ ∞ ⎧ ⎪ = ⎪ ⎪ ′ =⎪ ⎪ ⎪⎪ =⎨ ⎪ ⎪ − =⎪ −⎪ ⎪ − =⎪ −⎪⎩ (7) now for reasons of similarity, the plate concentration is assumed to be ( )∞∞ −+= ccxcxcw 0)( , (8) where 0c is considered to be mean concentration and υ 0xux = . introducing the relations (7)-(8) into the equations (2)-(5), we obtain the following local similarity equations 02 =−′−′−++′′+′′′ rgfmfkggfff mr φθ (9) 02 =′+−−′+′′ frmgkggfg (10) 0=′′+′+′′ φθθ frr dpfp (11) 02 =′′+′+′−′′ θφφφ rccc ssfsfs (12) n. islam and m. m. alam / journal of naval architecture and marine engineering 4(2007) 43-55 dufour and soret effects on steady mhd free convection and mass transfer fluid flow through a porous medium... 47 where ( ) x u ttg g wr 22 0 0 ∞−= β , ( ) 2 0 0 * 0 2x u ccg gm υ β ∞−= , 0 2 uk x k ′ = υ , 0 2 0 2 u xb m ρ σ ′ = , 0 2 u x r ω = , k c p pr ρυ = , ( ) ( )∞ ∞ − − = tt ccxu cc kd d wps tm f 00 υυ , m c d s υ = and ( ) ( )∞ ∞ − − = cc tt xut kd s w m tm r 00 υ υ . the corresponding boundary conditions are ⎭ ⎬ ⎫ ∞→====′ =====′= ηφθ ηφθ asgf atgfff w 0,0,0,0 01,1,0,1, (13) where 0 0 2 )( u x xvf w υ −= is taken to be transpiration parameter. 3. skin-friction coefficients, nusselt number and sherwood number the quantities of physical interest are the skin friction coefficients, the nusselt number and the sherwood number. the equations defining the wall skin frictions are 0= ⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ ∂ ∂ = y x y u µτ , 0= ⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ ∂ ∂ = y z y w µτ which are proportional to 0 2 2 = ⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ ∂ ∂ η η f and 0 g =η ⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ η∂ ∂ . the nusselt number denoted by un is proportional to 0= ⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ ∂ ∂ − y y t , hence we have )0(θ′−∞un . the sherwood number hs is proportional to 0= ⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ ∂ ∂ − y y c , hence we have )0(ϕ′−∞hs . the numerical values of the local skin-friction coefficients, the local nusselt number and the local sherwood number are sorted in tables 1-3. 4. numerical solution the set of non-linear ordinary differential equations (9)-(12) with boundary conditions (13) have been solved by using sixth order runge-kutta method along with the nacthsheim-swigert (1965) shooting iteration technique. 5. results and discussion in this paper, the effect of different parameters entering into steady two-dimensional mhd free convection and mass transfer fluid flow through a porous medium past a continuously moving semiinfinite vertical porous plate 0y = in a rotating system has been investigated using nacthsheimswigert shooting iteration technique. for the purpose of discussing the effects of various parameters on the flow behavior, some numerical calculations have been carried out for non-dimensional primary velocity )(ηf ′ , secondary velocity )(ηg , temperature )(ηθ and concentration )(ηϕ . n. islam and m. m. alam / journal of naval architecture and marine engineering 4(2007) 43-55 dufour and soret effects on steady mhd free convection and mass transfer fluid flow through a porous medium... 48 0 1 2 3 0.1 0.4 0.7 1 0 1 2 3 -0.04 -0.03 -0.02 -0.01 0 fig. 2: primary velocity profiles for different values of wf with 0.10=rg , 0.4=mg , 5.0=m , 2.0=r , 71.0=rp , 0.1=rs , 6.0=cs , 2.0=fd , 5.0=k . fig. 3: secondary velocity profiles for different values of wf with 0.10=rg , 0.4=mg , 5.0=m , 2.0=r , 71.0=rp , 0.1=rs , 6.0=cs , 2.0=fd , 5.0=k . 0 1 2 3 0.1 0.4 0.7 1 0 1 2 3 -0.15 -0.125 -0.1 -0.075 -0.05 -0.025 0 fig. 4: primary velocity profiles for different values of r with 0.3=wf , 0.10=rg , 0.4=mg , 5.0=m , 71.0=rp , 0.1=rs , 6.0=cs , 2.0=fd , 5.0=k . fig. 5: secondary velocity profiles for different values of r with 0.3=wf , 0.10=rg , 0.4=mg , 5.0=m , 71.0=rp , 0.1=rs , 6.0=cs , 2.0=fd , 5.0=k . the velocity profiles for x and z components of velocity, commonly known as non-dimensional primary ( f ′ ) and secondary ( g ) velocities, are shown in figs. 2 17 for different values of suction parameter ( 0v ), the magnetic parameter ( m ), the rotation parameter ( r ), the prandtl number ( rp ), the soret number ( rs ), the schmidt number ( cs ), the dufour number ( fd ) and the permeability parameter ( k ) and for fixed values of grashof number ( rg ) and modified grashof number ( mg ). η f ′ η g 0.5,0.4,0.3=wf η f ′ 6.0,4.0,2.0=r η 6.0,4.0,2.0=r g 0.5,0.4,0.3=wf n. islam and m. m. alam / journal of naval architecture and marine engineering 4(2007) 43-55 dufour and soret effects on steady mhd free convection and mass transfer fluid flow through a porous medium... 49 0 1 2 3 0.1 0.4 0.7 1 1.3 0 1 2 3 -0.06 -0.05 -0.04 -0.03 -0.02 -0.01 0 fig. 6: primary velocity profiles for different values of rs with 0.3=wf , 0.10=rg , 0.4=mg , 5.0=m , 2.0=r , 71.0=rp , 6.0=cs , 2.0=fd , 5.0=k . fig. 7: secondary velocity profiles for different values of rs with 0.3=wf , 0.10=rg , 0.4=mg , 5.0=m , 2.0=r , 71.0=rp , 6.0=cs , 2.0=fd , 5.0=k . 0 1 2 3 0.1 0.4 0.7 1 0 1 2 3 -0.06 -0.04 -0.02 0 fig. 8: primary velocity profiles for different values of fd with 0.3=wf , 0.10=rg , 0.4=mg , 5.0=m , 2.0=r , 71.0=rp , 0.1=rs , 6.0=cs , 5.0=k . fig. 9: secondary velocity profiles for different values of fd with 0.3=wf , 0.10=rg , 0.4=mg , 5.0=m , 2.0=r , 71.0=rp , 0.1=rs , 6.0=cs , 5.0=k . for prandtl number ( rp ), three values 0.7and0.1,71.0 which represent air at 20 deg c, electrolytic solution such as salt water and water respectively. the values 0.75and60.0,22.0 of the schmidt number ( cs ) are also considered for they represent specific conditions of the flow. in particular, 22.0 corresponds to hydrogen while 60.0 corresponds to water vapor that represents a diffusivity chemical species of most common interest in air and the value 75.0 represent oxygen. in the calculations wf , m , r , rs , fd , k and mg are chosen arbitrarily. η f ′ 8.0,5.0,2.0=fd g η 8.0,5.0,2.0=fd η 0.3,0.2,0.1=rs g η f ′ 0.3,0.2,0.1=rs n. islam and m. m. alam / journal of naval architecture and marine engineering 4(2007) 43-55 dufour and soret effects on steady mhd free convection and mass transfer fluid flow through a porous medium... 50 0 1 2 3 0.1 0.4 0.7 1 0 1 2 3 -0.04 -0.03 -0.02 -0.01 0 fig. 10: primary velocity profiles for different values of m with 0.3=wf , 0.10=rg , 0.4=mg , 2.0=r , 71.0=rp , 0.1=rs , 6.0=cs , 2.0=fd , 5.0=k . fig. 11: secondary velocity profiles for different values of m with 0.3=wf , 0.10=rg , 0.4=mg , 2.0=r , 71.0=rp , 0.1=rs , 6.0=cs , 2.0=fd , 5.0=k . 0 1 2 3 0.1 0.4 0.7 1 0 1 2 3 -0.04 -0.03 -0.02 -0.01 0 fig. 12: primary velocity profiles for different values of rp with 0.3=wf , 0.10=rg , 0.4=mg , 5.0=m , 2.0=r , 0.1=rs , 6.0=cs , 2.0=fd , 5.0=k . fig. 13: secondary velocity profiles for different values of rp with 0.3=wf , 0.10=rg , 0.4=mg , 5.0=m , 2.0=r , 0.1=rs , 6.0=cs , 2.0=fd , 5.0=k . with the above mentioned flow parameters, it is observed from figs. 2 and 3 that an increase in the suction parameter wf leads to decrease in the primary velocity and to increase the secondary velocity. the variations of the primary and secondary velocities for different values of rotation parameter r are shown in figs. 4 & 5. from these figures it is observed that the rotation parameter r has a minor decreasing effect on the primary velocity while it has quite a larger decreasing effect on the secondary velocity. in figs. 6 & 7 and 8 & 9, the variations of the primary and secondary velocities for different η f ′ 0.7,0.1,71.0=rp η g 0.7,0.1,71.0=rp η f ′ 5.1,0.1,5.0=m η g 5.1,0.1,5.0=m n. islam and m. m. alam / journal of naval architecture and marine engineering 4(2007) 43-55 dufour and soret effects on steady mhd free convection and mass transfer fluid flow through a porous medium... 51 values of soret number rs and dufour number fd are shown respectively. from these figures it is observed that the primary velocity increases with the increase of soret number rs and dufour number fd . the effects of soret number rs and dufour number fd on the secondary velocity are opposite to that of the primary velocity. in figs. 10 & 11, 12 & 13, 14 & 15 and 16 & 17, the variations of the primary and secondary velocities for different values of magnetic parameter m , prandtl number rp , schmidt number cs and permeability parameter k are shown respectively. it is observed from these figures that the magnetic parameter m has a decreasing effect on the primary velocity and increasing effect on the secondary velocity. it is also seen from these figures that the primary velocity decreases while the secondary velocity increases with the increase of prandtl number rp . the same effects are observed in the case of schmidt number cs and permeability parameter k . the effects of various parameters on non-dimensional temperature are shown in figs. 18-20. it is observed from fig. 18 that the temperature decreases with the increase of suction parameter wf increase. in fig. 19, the temperature profiles for different values of dufour number fd are shown. it is observed from this figure that the dufour number fd has an increasing effect. in fig. 20, the temperature profiles for different values of prandtl number rp are shown. this figure revels that the prandtl number rp has a large decreasing effect on temperature.the effects of various parameters on the concentration field are shown in figs. 21 24. it is observed from fig. 21 that the concentration decreases as the suction parameter wf increase. the concentration profiles for different values of soret number rs are shown in fig. 22. the figure shows that the concentrations increases as the soret number rs increase. 0 1 2 3 4 5 0.1 0.4 0.7 1 1.3 0 1 2 3 4 5 -0.06 -0.04 -0.02 0 fig. 14: primary velocity profiles for different values of cs with 0.3=wf , 0.10=rg , 0.4=mg , 5.0=m , 2.0=r , 71.0=rp , 0.1=rs , 2.0=fd , 5.0=k . fig. 15: secondary velocity profiles for different values of cs with 0.3=wf , 0.10=rg , 0.4=mg , 5.0=m , 2.0=r , 71.0=rp , 0.1=rs , 2.0=fd , 5.0=k . in figs. 23 and 24, the concentration profiles for different values of prandtl number rp and schmidt number cs are shown respectively. it is observed from these figures that the concentration increases as the prandtl number rp increase while the concentration decreases as the schmidt number cs increase. finally, the effects of various parameters on the components of the skin friction coefficient xτ and zτ , the nusselt number un and the sherwood number hs are shown in tables 1-3. η f ′ 75.0,60.0,22.0=cs η g 75.0,60.0,22.0=cs n. islam and m. m. alam / journal of naval architecture and marine engineering 4(2007) 43-55 dufour and soret effects on steady mhd free convection and mass transfer fluid flow through a porous medium... 52 0 1 2 3 0.1 0.4 0.7 1 0 1 2 3 -0.04 -0.02 0 fig. 16: primary velocity profiles for different values of k with 0.3=wf , 0.10=rg , 0.4=mg , 5.0=m , 2.0=r , 71.0=rp , 0.1=rs , 6.0=cs , 2.0=fd . fig. 17: secondary velocity profiles for different values of k with 0.3=wf , 0.10=rg , 0.4=mg , 5.0=m , 2.0=r , 71.0=rp , 0.1=rs , 6.0=cs , 2.0=fd 0 0.5 1 1.5 2 0.1 0.4 0.7 1 0 1 2 3 0.1 0.4 0.7 1 fig. 18: temperature profiles for different values of wf with 0.10=rg , 0.4=mg , 5.0=m , 2.0=r , 71.0=rp , 0.1=rs , 6.0=cs , 2.0=fd , 5.0=k . fig. 19: temperature profiles for different values of fd with 0.3=wf , 0.10=rg , 0.4=mg , 5.0=m , 2.0=r , 71.0=rp , 0.1=rs , 6.0=cs , 5.0=k . from table 1, we observe that the skin-friction component xτ decreases with the increase of suction parameter wf , but the skin-friction component zτ , the nusselt number un and the sherwood number hs increase with the increase of suction parameter wf . it is also observed from this table that the skin-friction component xτ , the nusselt number un and sherwood number hs decrease with the increase of magnetic parameter m , while the skin-friction component zτ increases with the increase of magnetic parameter m . η f ′ 5.1,0.1,5.0=k η g 5.1,0.1,5.0=k η θ 0.5,0.4,0.3=wf η θ 8.0,5.0,2.0=fd n. islam and m. m. alam / journal of naval architecture and marine engineering 4(2007) 43-55 dufour and soret effects on steady mhd free convection and mass transfer fluid flow through a porous medium... 53 again, from table 2, we observe that the skin-friction components xτ and zτ , the nusselt number un and the sherwood number hs decrease with the increase of rotation parameter r . it is also observed from this table that the skin-friction component xτ and the nusselt number un increase while the skin-friction component zτ and the sherwood number hs decrease with the increase of soret number rs . 0 0.5 1 1.5 2 2.5 0.1 0.4 0.7 1 0 0.5 1 1.5 2 2.5 0.1 0.4 0.7 1 fig. 20: temperature profiles for different values of rp with 0.3=wf , 0.10=rg , 0.4=mg , 5.0=m , 2.0=r , 0.1=rs , 6.0=cs , 2.0=fd , 5.0=k . fig. 21: concentration profiles for different values of wf with 0.10=rg , 0.4=mg , 5.0=m , 2.0=r , 71.0=rp , 0.1=rs , 6.0=cs , 2.0=fd , 5.0=k . 0 1 2 3 0.1 0.4 0.7 1 0 0.5 1 1.5 2 2.5 3 0.1 0.4 0.7 1 fig. 22: concentration profiles for different values of rs with 0.3=wf , 0.10=rg , 0.4=mg , 5.0=m , 2.0=r , 71.0=rp , 6.0=cs , 2.0=fd , 5.0=k . fig. 23: concentration profiles for different values of rp with 0.3=wf , 0.10=rg , 0.4=mg , 5.0=m , 2.0=r , 0.1=rs , 6.0=cs , 2.0=fd , 5.0=k . η θ 0.7,0.1,71.0=rp η ϕ 0.5,0.4,0.3=wf η ϕ 0.7,0.1,71.0=rp η ϕ 0.3,0.2,0.1=rs n. islam and m. m. alam / journal of naval architecture and marine engineering 4(2007) 1-13 dufour and soret effects on steady mhd free convection and mass transfer fluid flow through a porous medium... 54 from table 3, we observe that the skin-friction component xτ and the sherwood number hs increase while the skin-friction component zτ and the nusselt number un decrease owing to the increase of dufour number fd . it is also seen from this table that the skin-friction component xτ , the nusselt number un and the sherwood number hs decrease while the skin-friction component zτ increases with the increase of permeability parameter k . table 1: numerical values of xτ , zτ , un and hs for 71.0=rp , 0.10=rg , 0.4=mg , 2.0=r , 0.1=rs , 6.0=cs , 2.0=fd and 5.0=k . wf m xτ zτ un hs 3.0 0.5 2.1183684 -0.3123093 2.2126436 1.3606514 4.0 0.5 0.6630225 -0.2322516 2.8415285 1.3736072 5.0 0.5 -0.8398383 -0.1720828 3.4941526 1.3843751 3.0 1.0 1.8034374 -0.2837792 2.2074190 1.3379786 3.0 1.5 1.5115707 -0.2596410 2.2026285 1.3168868 table 2: numerical values of xτ , zτ , un and hs for 5.0=wf , 0.10=rg , 0.4=mg , 5.0=m , 71.0=rp , 6.0=cs , 2.0=fd and 5.0=k . r rs xτ zτ un hs 0.2 1.0 2.1183684 -0.3123093 2.2126436 1.3606514 0.4 1.0 2.0825316 -0.6208549 2.2119861 1.3572153 0.6 1.0 2.0241833 -0.9220832 2.2107869 1.3517746 0.2 2.0 2.5626587 -0.3425054 2.4041973 0.1109341 0.2 3.0 2.9919283 -0.3699451 2.6305160 -1.3989729 table 3: numerical values of xτ , zτ , un and hs for 5.0=wf , 0.10=rg , 0.4=mg , 71.0=rp , 2.0=r , 5.0=m , 0.1=rs and 6.0=cs . fd k xτ zτ un hs 0.2 0.5 2.1183684 -0.3123093 2.2126436 1.3606514 0.5 0.5 2.5017350 -0.3402315 1.8900020 1.5829858 0.8 0.5 2.8887150 -0.3663611 1.4347118 1.8786190 0.2 1.0 1.8034374 -0.2837792 2.2074190 1.3379786 0.2 1.5 1.5115707 -0.2596410 2.2026285 1.3168868 0 1 2 3 4 5 0.1 0.4 0.7 1 fig. 24: concentration profiles for different values of cs with 0.3=wf , 0.10=rg , 0.4=mg , 5.0=m , 2.0=r , 71.0=rp , 0.1=rs , 2.0=fd , 5.0=k . η ϕ 75.0,60.0,22.0=cs n. islam and m. m. alam / journal of naval architecture and marine engineering 4(2007) 1-13 dufour and soret effects on steady mhd free convection and mass transfer fluid flow through a porous medium... 55 references alam, m. s. and rahman, m. m. (2006): dufour and soret effects on mixed convection flow past a vertical porous flat plate with variable suction, nonlinear analysis: modeling and control, 11, 1. anghel, m., takhur, h. s. and pop, i. (2000): dufour and soret effects on free convection boundary-layer over a vertical surface embedded in a porous medium, studia universitatis babes-bolyai. mathematica, xlv(4), 11. eckert, e. r. g. and drake, r. m. ( 1972): analysis of heat and mass transfer, mcgraw-hill book co., new york. georgantopoulos, g. a. (1979): effects of free convection on the hydromagnetic accelerated flow past a vertical porous limiting surface. astrophysics and space science, 65, 433. gebhart, b. and pera, l. (1971): the nature of vertical convection flows resulting from combined buoyancy effects of the thermal and mass diffusion nature, international journal of heat and mass transfer, 14, 2025. kafousias, n. g., nanousis, n. and georgantopoulos, g. a. 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(1983): mhd free convection flow past an accelerated vertical plate, international communications in heat and mass transfer, 10, 313. raptis, a. and singh, a. k. (1985): rotation effects on mhd free convection flow past an accelerated vertical plate, mechanical resulation communication, 12, 31. singh, a. k. (1982): mhd free convection flow in the stokes problem for a porous vertical plate, astrophysics and space science, 87, 455. singh, a. k. and singh, j. n. (1989): transient mhd free convection in a rotating system, astrophysics and space science, 162, 85. singh, a. k. (1983): mhd free convection flow in the stokes problem for a vertical porous plate in a rotating system, astrophysics and space science, 95, 283. singh, a. k. (1984a): hydromagnetic free convection flow past an impulsively started vertical plate in a rotating system, international communications in heat and mass transfer, 11, 349. singh, a. k. (1984b): stokes problem for a porous vertical plate with heat sinks by finite difference method, astrophysics and space science, 103, 55. journal of naval architecture and marine engineering june, 2017 http://dx.doi.org http://dx.doi.org/10.3329/jname.v14i1.27967 http://www.banglajol.info 1813-8535 (print), 2070-8998 (online) © 2017 aname publication. all rights reserved. received on: june, 2016 a preliminary study of fluid-dynamic loads on a plane surface beneath a circular cylinder in the subcritical flow regime jian-cheng cai 1* , jie pan 2 , shi-ju e 1 , wei-dong jiao 1 and dong-yun wang 1 1 department of mechanical engineering, college of engineering, zhejiang normal university, jinhua 321004, china, *email: cai_jiancheng@foxmail.com 2 school of mechanical and chemical engineering, the university of western australia, crawley wa 6009, australia, jie.pan@uwa.edu.au abstract: this paper studies the fluctuating forces on a plane surface beneath a circular cylinder in the subcritical flow regime using two-dimensional computational fluid dynamics (cfd) approach. the turbulent flow fields were calculated via numerical solutions of the navier– stokes (n–s) equations without a turbulence model (laminar flow computation), large eddy simulation (les), and reynolds-averaged n-s equations (rans) approach with the shearstress transport (sst) turbulence model. the primary goal is to evaluate the correctness, accuracy, and applicability of 2-d approximation of turbulence simulation with different approaches for subcritical flow regime (re=5000). some preliminary knowledge of the forces on the plane which is important in studying scours and flow-induced vibration in ocean engineering is obtained. results show that the laminar approach with high mesh resolution can adequately simulate turbulent flows at this moderate reynolds number. specially, the fluctuating forces on the plane surface due to the flow are significant within three times the cylinder diameter in the downstream, and within one cylinder diameter in the upstream of the cylinder. the pressure fluctuations are approximately two orders of magnitude larger than the shear stress fluctuations. in the frequency domain, the fluctuating forces are significant under twice the vortex-shedding frequency. within one cylinder diameter in the downstream and upstream regions of the cylinder, the pressure fluctuations on the plane surface are well correlated whereas the shear stress is not so well correlated. keywords: fluctuating forces, surface pressure, surface shear stress, unsteady flow, circular cylinder, wall proximity, vortex shedding, numerical simulation nomenclature greek symbols d diameter of the cylinder ij  kronecker delta / 2 i i k u u  turbulent kinetic energy per unit mass ε turbulent kinetic energy dissipation rate p pressure  dynamic viscosity t time t  turbulent or eddy viscosity 0 u free-stream flow velocity ν kinematic viscosity i u velocity components  fluid density i x space coordinate components 1. introduction flow around a circular cylinder has been a subject of much research. particularly important aspects are the periodic fluctuating cross-flow (lift) and in-line (drag) forces that result from the vortex shedding of the flow, which are the sources of flow-induced oscillations of a cylindrical structure. although as a classic topic in fluid mechanics, it is actually quite a complex problem. as the reynolds number re=du0/ν increases, the flow pattern around the cylinder changes (sumer and fredsoe, 2006). when re < 5, no flow separation occurs, and for 5 < re < 40, a fixed pair of vortices forms in the wake of the cylinder. when the reynolds number is further increased, the wake becomes unstable and vortex shedding occurs. the flow of the wake behind the cylinder j. c. cai, j. pan, s. j. e, w. d. jiao, d. y. wang/journal of naval architecture and marine engineering, 14(2017) 9-24 a preliminary study of fluid-dynamic loads on a plane surface beneath a circular cylinder in the subcritical flow regime 10 enters the turbulence regime when re > 300; however, the boundary layer around the cylinder surface is laminar. with a further increase in re up to 3×10 5 , the transition to turbulence occurs in the boundary layer itself. for 300 < re < 3×10 5 , the flow regime is known as the subcritical flow regime with the laminar boundary layer separation, where re = 3×10 5 marks critical flow, and re > 3.5×10 5 defines the supercritical flow regime with the turbulent boundary layer separation. a plane boundary beneath the cylinder will add further complexity to the flow structure around the cylinder, as the plane boundary influences the flow via the kinematic and kinetic constraints on its rigid surface and the growing boundary layer along the surface. it is well known that the re, the gap-to-diameter ratio (g/d) and the ratio of the incident boundary layer thickness to the diameter (δ/d) are crucial to the flow pattern around the cylinder. here, g is the minimum distance between the bottom of the cylinder and the plane wall, and δ is the thickness of boundary layer of the inlet flow upstream of the cylinder. it is generally accepted that the vortex motion behind the cylinder and close to the boundary wall is suppressed for g/d < 0.3, and that the flow interference between the cylinder and the plane boundary becomes very weak when g/d > 1.0 (sumer and fredsoe, 2006). these observations were also confirmed in flow field visualizations by oner et al. (2008). oner et al. (2008) also measured the distribution of mean pressure around the cylinder and along a plane boundary, and their results showed that along the plane boundary the pressure is positive at the upstream part of the cylinder and negative downstream, and that the mean pressure over the cylinder showed that the cylinder experienced a net positive lift force, i.e., the force pushes the cylinder away from the plane wall. price et al. (2002) showed that the onset of vortex shedding can only be observed for g/d > 0.5, while it is completely suppressed when g/d < 0.125, in the reynolds number range of 1200 to 4960. in the intermediate region, 0.125 < g/d < 0.5, the flow is similar to that for g/d < 0.125, but there is a pronounced pairing between the lower shear layer shed from the bottom side of the cylinder and the wall boundary layer. kazeminezhad et al. (2010) numerically investigated the effects of the boundary layer of the plane boundary on the vortex shedding frequency of the flow around the cylinder and the forces acting upon the cylinder when exposed to a steady current (at re = 7000 and 9500) by solving the 2-d reynolds-averaged navier–stokes (rans) equations with the k–ε turbulence model. their model slightly over-predicted the mean force coefficients and strouhal number. it was concluded that the mean force coefficients and the root-mean-square (rms) lift coefficient are strongly affected by the gap-to-diameter ratio while the strouhal number is slightly affected by the gap ratio. ong et al. (2010) performed a numerical simulation of flow around a circular cylinder close to a flat seabed at high reynolds numbers (re = 1.31×10 4 and 3.6×10 6 ) using a 2-d standard k–ε model. they found that overall the employed approach is suitable for design purposes at high reynolds numbers, but that there existed differences between the experimental and numerical results. tutar and holdø (2001) performed 2-d simulation of flow around a cylinder in the subcritical regime using a k– ε turbulence model and a large eddy simulation (les). they found that les methods with the sub-grid scale (sgs) model yielded much more realistic pictures of the flow’s vortex shedding in the transitional flow regime, but slightly overestimated critical flow parameters. liang et al. (2005) studied the performance of the standard k–ε, wilcox k-ω, and smagorinsky’s sgs turbulence models against the flow around a circular cylinder located 0.37 diameters above a rigid surface. they demonstrated that an les with smagorinsky’s sgs model predicted much stronger vortex shedding as well as the fluctuation velocities and the rms lift coefficient. previous work on flow around a cylinder near a plane surface had largely focused on the forces on the cylinder. however, there appears to be a need to understand the fluctuating forces on the plane surface caused by the unstable flow generated by the flow–cylinder interaction, as fluctuating pressure and shear stress on a surface are important sources of surface vibration and sound radiation in structural acoustics (liu et al., 2012). this paper focuses studying the fluctuating forces on the surface of a rigid plane below a cylinder. the study is undertaken via a 2-d simulation of unsteady flow fields around a cylinder above a plane surface at two different gap-to-diameter ratios (g/d = 0.3 and 1.0) using different turbulence approaches. it is generally accepted that g/d = 0.3 corresponds to the critical situation where vortex shedding begins, while g/d = 1.0 is a case where the influence of the plane surface on the flow field behind the cylinder can be ignored (price et al., 2002). this study is limited to subcritical reynolds number, re = 5000, because the corresponding flow patterns are relatively simple for explaining the features of the surface forces. j. c. cai, j. pan, s. j. e, w. d. jiao, d. y. wang/journal of naval architecture and marine engineering, 14(2017) 9-24 a preliminary study of fluid-dynamic loads on a plane surface beneath a circular cylinder in the subcritical flow regime 11 a numeric solution of the navier–stokes (n–s) equations without any turbulence model (i.e., the laminar approach in commercial cfd packages), the rans equations with the k–ω shear-stress transport (sst) model, and an les approach with smagorinsky’s sgs model were used for this study. the numerical results were compared to existing experimental results. it should be pointed out that the turbulent flow around the cylinder is intrinsically three dimensional, nevertheless the 2-d simulation in flow around cylinders still has its applications because of its low calculation effort, especially of the rans approach (rahman et al., 2007, stringer et al., 2014), where the k-ε realizable and k-ω sst turbulence models were used. the application of 2-d les seems to be a controversial issue. some researchers argue that the 2-d les is inaccurate (murakami and mochida, 1995, rodi, 1993), while some studies show quite good results by using a fine grid resolution (bouris and bergeles, 1999, tutar and holdø, 2001).this contradiction can be explained by different conditions, under which the mentioned results have been obtained. it gives us the possibility to suppose that in the case of subcritical flow mode especially of low reynolds number flows, when boundary layers developing over cylindrical surface are laminar and turbulence in the wake behind cylinder just begins to develop, its spanwise direction (along the cylinder axis) component is not so effective or slightly correlated with other dominant directions of flow spreading, thus the 2-d level of turbulence approximation for this special case can be acceptable. this suggestion motivates the primary goal of this research: to evaluate the correctness, accuracy, applicability and performance of such 2-d level of approximation of turbulence simulation with different approaches for subcritical flow mode, and to obtain some preliminary knowledge of the forces on the plane which is important in studying scours and flow-induced vibration in ocean engineering. 2. mathematical formulation 2.1 problem description a schematic sketch of the flow around a circular cylinder close to a plane surface is shown in fig. 1. the diameter of the cylinder is d = 0.01 m. the upper boundary of the mesh is located at a distance from 8.7d from the top of the cylinder. this ensures that the boundary of the mesh has a negligible effect on the flow around the cylinder and the plane surface. the gap between the bottom of the cylinder and the plane surface is denoted by g, which is 0.3d or 1.0d related to g/d = 0.3 or 1.0. the flow inlet is located 13d upstream from the center of the cylinder, and the flow outlet is located at 32d. these distances are sufficient to eliminate far-field effects of the flow upstream and downstream of the cylinder. the computational region in the downstream dimension is larger than the regime enclosed by the inlet and symmetric boundaries, as suggested by ong et al. (2010), in order to further reduce the influence of the outlet, because backflow usually occurs in the near downstream region owing to the vortex motion carried out by the main flow. fig. 1: schematic of the flow region. unlike most previous studies in which the inlet velocity distribution, including the boundary layer shape, was given, this study applies a uniform velocity distribution at the inlet with a magnitude of u0 = 0.5 m/s. the dynamic viscosity μ = 10 −3 pa∙s, and the density ρ = 1000 kg∙m −3 , thus the kinematic viscosity ν = 10 −6 m 2 /s and re=du0/ν=5000. after a distance of 1d along the flow direction, the incoming flow hits the plane surface and a boundary layer is thereafter generated along the leading point of the plane. the ratio of boundary layer thickness to the cylinder diameter at a distance of 5d in front of the cylinder center, obtained from the computational fluid dynamics (cfd) simulation, is δ/d ≈ 0.2. one can expect that the interactions between the boundary layer and flow around the cylinder will be appreciable for g/d = 0.3, and very weak for g/d = 1.0. three mesh schemes with quadrilateral cells of different sizes were constructed. the total cell numbers of these three mesh schemes are approximately 80,000, 300,000, and 1,100,000, and some representative mesh pictures of g/d = 0.3 are shown in fig. 2. much finer meshes are used in the near-wall regions in order to keep the nonj. c. cai, j. pan, s. j. e, w. d. jiao, d. y. wang/journal of naval architecture and marine engineering, 14(2017) 9-24 a preliminary study of fluid-dynamic loads on a plane surface beneath a circular cylinder in the subcritical flow regime 12 dimensional wall distance y-plus of ~1 at most near-wall vertexes. the y-plus is obtained from the reynolds number, using the wall distance and the friction velocity as the characteristic length and velocity. lee et al. (2014) carried out a 3-d les calculation of the flow past a cylinder at re = 5000 with the span length of 2d using six different mesh levels ). judging from the y-plus values, it can be predicted that the fine mesh scheme in the paper is equivalent to the finest mesh level therein. (a) (b) (c) fig. 2: (a) coarse, (b) fine, and (c) very fine mesh schemes. 2.2 governing equations due to the low mach number flow (ma < 0.3) considered in this study, the flow is regarded as the incompressible two-dimensional flows assuming a constant fluid density. thus, the continuity and momentum equations of the navier-stokes equations can be solved independent of the energy equation: 0i i u t x        (1) 2 i ji i j j j j u uu p u t x x x x               , (2) although the n–s equations can theoretically describe turbulence phenomena, the direct numerical simulation of turbulent flow using n–s equations is unrealistic especially in the high reynolds number regime to resolve the kolmogorov microscales related to the smallest eddies with the reynolds number reη~1 based on their characteristic velocity υ and characteristic length η. typically, the smallest scale of motion in a turbulent flow is in the order of 0.1 to 0.01 mm and frequencies around 10 khz. resolving eddies in large engineering flow problems with today’s computers is a formidable task. therefore, in practice, turbulence models are often employed to describe turbulence phenomena. in this study, the laminar flow computation, the rans and the les approaches with their respective turbulence modes are utilized. the laminar flow computation is realized by discretizing the n–s equations with secondorder difference schemes without a turbulence model. it is known that the dependence on turbulence modelling is reduced as the grid size decreases, since more turbulence is directly resolved. in the limit of infinitely fine grid density, the eddy viscosity approaches zero, which means all scales of eddies are directly resolved (lee et al., 2014). therefore numerical solution of the n–s equations with sufficiently high precision can approximate turbulent flows to some extent. this is usually called the laminar approach (lam) in commercial cfd packages. singh and mittal (2005) used this approach to study the drag crisis of the flow past a single cylinder. they found the results are comparable with the les results. the rans and les approaches are introduced briefly as follows. the rans approach is probably the most popular method to simulate turbulent flow, because for most engineering purposes it is unnecessary to resolve the details of the turbulent fluctuations. the rans equations are obtained by replacing the flow variables  by the sum of a mean  and fluctuating components  (     ) in the n–s equations and time-averaging to produce the resultant equations: 0i i u t x       and (3) j. c. cai, j. pan, s. j. e, w. d. jiao, d. y. wang/journal of naval architecture and marine engineering, 14(2017) 9-24 a preliminary study of fluid-dynamic loads on a plane surface beneath a circular cylinder in the subcritical flow regime 13 2 ( ) i j i ji i j j j j j u u u uu up t x x x x x                    . (4) turbulence modes are used to describe the reynolds stresses, i j u u   , which can be expressed as follows based on the boussinesq presumption that there exists an analogy between the action of viscous stresses and reynolds stresses on the mean flow: 2 3 ji i j t ij j i uu u u k x x                 (5) the k–ε model is the most widely used and validated to represent 2 / t c k     . the k–ω model is used to give a more accurate prediction of the shedding of vortices than the k–ε model (liang and cheng, 2005). the k– ω model takes the turbulent frequency ω = ε/k as the second variable instead of the rate of dissipation of turbulence kinetic energy ε, and / t c k     . it is believed to have a better performance in the near-wall region because it does not require wall-damping functions in low reynolds number applications. menter (versteeg and malalasekera, 2007) suggested a hybrid sst model, using a transformation of the k–ε model to the k–ω model in the near-wall region and the standard k–ε model in the fully turbulent region far from the wall. this model offers improved performance with respect to adverse pressure gradients. in rans the collective behavior of all eddies must be described by a single turbulence model, but the problem dependence of the largest eddies complicates the search for widely applicable models. a different approach to the computation of turbulent flows accepts that the larger eddies need to be computed for each problem with a time-dependent simulation. the universal behavior of the smaller eddies, on the other hand, should be easier to capture with a compact model. this is the essence of the large eddy simulation (les) approach to the numerical treatment of turbulence. instead of time-averaging the n–s equations, the les uses spatial filtering to obtain the les continuity and momentum equations: 0i i u t x       and (6) 2 ( ) i j i j i ji i j j j j j j u u u u u uu up t x x x x x x                      . (7) the over bar indicates spatial filtering. similar to the rans approach, the sgs stresses: 31 2 ( ) ij i j i j i j i j i j i j i j u u u u u u u u u u u u u u                 , (8) including on the right hand side the 1st (called leonard stresses), 2nd (called cross-stresses), and 3rd (called les reynolds stresses) terms, need to be modeled in order to make the equations closed. for this purpose, the smagorinksy–lilly sgs turbulence model is employed: 1 1 2 3 3 ji ij sgs ij ii ij sgs ii ij j i uu s x x                      . (9) the sgs viscosity sgs  is evaluated as follows: 2 ( ) 2 sgs sgs ij ij c s s   , (10) where δ is the filter cutoff width, and the constant csgs is set to 0.1 in the present simulation, as is suggested by versteeg and malalasekera (2007). j. c. cai, j. pan, s. j. e, w. d. jiao, d. y. wang/journal of naval architecture and marine engineering, 14(2017) 9-24 a preliminary study of fluid-dynamic loads on a plane surface beneath a circular cylinder in the subcritical flow regime 14 2.3 boundary conditions and numerical methods a boundary condition of uniform velocity is applied at the inlet. for the laminar flow computation (lam), no extra turbulence information is required. for the rans and les approaches, such information is essential. similar to what was done by doolan (2010), 1% turbulence intensity and a 5% turbulent viscosity ratio (μt/μ) at the inlet were employed in the rans approach, while no velocity perturbations were introduced at the inlet in the les approach. doolan (2010) indicated that the free stream turbulence intensity has no influence on the frequency of the instabilities in the shear layer, although it does influence shear layer transition. in the case of the low and medium free stream turbulence intensities, the flow around and behind the cylinder is dominated by the intrinsic flow pattern around the cylinder rather than by the inflow turbulence. a symmetric boundary condition is applied to the top boundary by setting the vertical velocity to zero and applying a zero normal gradient condition for all other quantities (brørs, 1999, kazeminezhad et al., 2010). the symmetric boundary condition is also applied at the short boundary in front of the leading point of the plane surface. zero normal gradients for all parameters except pressure are applied at the outlet boundary. a no-slip velocity boundary condition is applied on the cylinder wall. the finite volume method is employed to discretize the governing equations. for the incompressible flow, the continuity equation is simplified into a form in which the divergence of the velocity vanishes. unlike the momentum equations, there is no transport or other equation for the pressure, though evidently the pressure field should be coupled with the velocity field. in the present study, a revised algorithm called simplec (versteeg and malalasekera, 2007) is used to solve the pressure–velocity coupling problem. for the spatial discretization of the governing equations in the convection–diffusion form, the diffusive fluxes are discretized using central differencing. for the convective fluxes, a second-order upwind scheme was employed in the lam and sst models, and a bounded central differencing scheme was used in the les model, as was suggested by kim (2006). an implicit second-order backward-differencing scheme for time-discretization was employed to advance the solution in time. the initial flow field used to start the simulation was a uniform velocity distribution with the incoming velocity of the inlet velocity and with a gauge pressure of 0 pa. using the well-recognized strouhal number st = fd/u0 ≈ 0.2, one can estimate the vortex shedding frequency as f ≈ 10 hz, i.e., the period t ≈ 0.1 s. a time step of δt = 0.05 ms is chosen, with the dimensionless time step δt * = u0δt/d=0.0025, which is 1/2000th of the vortex shedding period. lee et al. (2014) suggested at least 50 time steps per period are recommended to resolve the shedding physics correctly. kazeminezhad et al. (2010) found that a dimensionless time step of ~0.02, i.e. around 250 time steps per period, is sufficient to obtain a reliable transient flow field. the system of discretized governing equations is solved using a point-implicit, gauss–seidel relaxation along with an algebraic method to accelerate solution convergence. the solution was run for a non-dimensional time of u0t/d =250, which is larger than the value of 76 used by doolan (2010) and 100 suggested by kim (2006), to establish a statistically steady state flow in the near-wake region. after that, the flow data of 5 s were recorded, which are enough to offer sufficient statistical information about the fluctuating forces. the frequency resolution is 0.2 hz with a sampling duration of 5 s. 3. numerical results 3.1 forces on the cylinder first, the drag and lift forces on the cylinder, including the mean and fluctuation characteristics are examined to establish an understanding of the reliability and accuracy of flow simulation approaches. the drag and lift coefficients are defined by scaling the cylinder drag force fd and lift force fl with the free-stream velocity 0u as follows: 2 2 0 0 , / 2 / 2 d l d l f f c c u d u d    . (11) the mean drag and lift coefficients and their respective standard deviations are shown in fig. 3. the mean values are represented by the bars. the standard deviation, in some literatures called root mean square (rms), was used to indicate the fluctuation while counteracting the mean value, as indicated by the symbols ‘i’ in fig. 3. three bars in each turbulence approach represent the results based on three mesh schemes, i.e., the coarse, fine, and very fine mesh as mentioned in the previous section. overall, the numerical results of different j. c. cai, j. pan, s. j. e, w. d. jiao, d. y. wang/journal of naval architecture and marine engineering, 14(2017) 9-24 a preliminary study of fluid-dynamic loads on a plane surface beneath a circular cylinder in the subcritical flow regime 15 turbulence models and different meshes are in the same order of magnitude, except for some cases of the sst model, which will be explained further at a later stage when the detailed flow fields are examined. fig. 3: the mean values and standard deviations of the drag and lift coefficients. the mean drag force coefficients of the cylinder predicted with the coarse-mesh scheme are different from those from the finer-mesh schemes. this is because the coarse mesh cannot resolve the flow details. one can also notice that the drag forces obtained by the lam and les approaches are quite close to each other and that they have appreciable differences from the results of the sst model. for the case of g/d = 0.3, the mean drag coefficient obtained by the lam and les approaches is around 1.3–1.5, and it rises to 1.7–2.0 at g/d = 1.0. the results of the sst model show that the drag coefficient is approximately 1.0 at g/d = 0.3, and 1.8 at g/d = 1.0. this tendency agrees with most previous experimental and numerical results, such as those of lei et al.(1999) and kazeminezhad et al. (2010). experimental results for cd are 1.1–1.5 in the case of a single cylinder at the subcritical regime with the reynolds numbers in the range of 1.0×10 4 –1.0×10 5 (sumer and fredsoe, 2006) the numerical cd obtained in the present study is a little larger than the experimental data as was in the 2-d calculation results of tutar and holdø (2001), where cd = 1.4–1.47 was predicted. this may be due the fact that in a 2-d simulation the spanwise effect is ignored. the 2-d simulation assumes that the flow is uniform in the axial direction of the cylinder, which will cause an increase in the drag force. the standard deviation of the drag coefficients shows that the drag fluctuations predicted by the lam and les approaches are about 10–30% of the mean drag forces. the drag fluctuations predicted by the coarse mesh model are lower than those by models with fine and very fine meshes, indicating that the coarse mesh cannot adequately resolve the transient flow pattern. it is interesting to find that the standard deviations vanish for g/d = 0.3, showing no fluctuations at all. it will be seen in the following subsection that no vortex shedding was predicted by the sst approach at g/d = 0.3. unlike the mean drag coefficient, which is rather stable in all predictions, the mean lift coefficient varies significantly with the different mesh schemes and turbulence models. the drag is ascribed to the appreciable pressure difference between the front region of the cylinder, where flow stagnation occurs, and the wake region, while lift is ascribed to the pressure difference between the up and down surfaces of the cylinder. due to vortex shedding, the pressure distribution pattern on the up and down surfaces of the cylinder alternates, resulting in the lift force fluctuations. for an isolated cylinder, the mean lift is theoretically zero. however, for a cylinder above a plane surface, the mean lift force does not vanish for small g/d and is positive (directed away from the plane 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 2.2 2.4 2.6 v e ry f in e f in e g/d=0.3 sst d ra g c o e ff ic ie n t (c d ) lam les c o a rs e 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 2.2 2.4 2.6 c o a rs e f in e v e ry f in e sstleslam g/d=1.0 d ra g c o e ff ic ie n t (c d ) -2.0 -1.5 -1.0 -0.5 0.0 0.5 1.0 1.5 2.0 c o a rs e f in e v e ry f in e sstleslam g/d=0.3 l if t c o e ff ic ie n t (c l) -2.0 -1.5 -1.0 -0.5 0.0 0.5 1.0 1.5 2.0 c o a rs e f in e v e ry f in e sstleslam g/d=1.0 l if t c o e ff ic ie n t (c l) j. c. cai, j. pan, s. j. e, w. d. jiao, d. y. wang/journal of naval architecture and marine engineering, 14(2017) 9-24 a preliminary study of fluid-dynamic loads on a plane surface beneath a circular cylinder in the subcritical flow regime 16 surface), as observed in the previous experiment (sumer and fredsoe, 2006). the lift coefficients are shown in fig. 3. as can be seen, the lift coefficient reduces dramatically from g/d = 0.3 to g/d = 1.0. this phenomena was also found in the numerical results of kazeminezhad et al. (2010). the measured mean lift coefficients of g/d = 0.3 and g/d = 1.0 by buresti and lanciotti (1992) are approximately 0.2 and 0.03, respectively, in the subcritical regime. among all the simulated results, the les and lam show better agreement with the experimental data. the large standard deviation in the predicted lift coefficient in fig. 3 is due to the fact that the mean lift force is very small while its fluctuations are very large. thus, a precise estimation of the mean lift coefficient is difficult. this observation is also supported by previous experimental results. for example, the mean lift force was found to be all positive in the results of buresti and lanciotti (1992), while some negative lift forces were also detected in some cases (lei et al., 1999). the lift fluctuation predicted by the 2-d simulations is around 1.0~1.1 for g/d = 0.3, and 1.4~1.6 for g/d = 1.0, which are much larger than the experimental results, which are between 0.1~0.2 and 0.2~0.3 respectively (sumer and fredsoe, 2006). this may be ascribed to the intrinsic deficiency of a 2-d simulation, where the flow correlation in the spanwise direction is not taken into account. a 2-d simulation assumes that vortexes along the cylinder are shedding simultaneously, resulting in the increase of the lift fluctuations. a large lift fluctuation was also obtained in lee et al. (2014) where a small span length 2d was adopted. they found that change in the cylinder span length does not affect drag and strouhal number considerably, whereas it affects lift quite significantly. the spanwise correlations drop significantly as the span length increases, so is the lift coefficient. they argued that a span length of 16d is needed to predict physically correct spanwise correlations. there are no lift fluctuations in the sst results in the case of g/d = 0.3 owing to the fact that no vortex shedding is predicted. at g/d = 1.0, the fluctuation levels obtained by sst are in the same order as those obtained by the lam and les approaches. the spectra of the drag and lift fluctuations are shown in fig. 4 to fig. 7. the results of the sst approach in the case of g/d = 0.3 are omitted because no vortex shedding is predicted, leading to no fluctuations of the drag and lift forces on the cylinder. the frequency is non-dimensionalized by multiplying d/u0. the fluctuations are mainly located in the non-dimensional frequency range of 0.2–0.3, where the strouhal number is supposed to fall into. at g/d = 0.3, the fluctuation at one-half of the strouhal number is also obvious. for the drag force, fluctuations at the strouhal number and its second harmonic are predominant. the 2 nd harmonic of the vortex shedding frequency is due to the fact that the vortexes are shedding on both the up and down sides of the cylinders, leading to drag fluctuations at twice the vortex shedding frequency. especially in the case of g/d = 1.0 in the sst model, the drag spectra are very clean, with components at the vortex shedding frequency and its low harmonics. this indicates that the rans approach predicts a well-organized and periodic flow pattern at this reynolds number. this is due to the intrinsic property of rans approach: when the turbulence model is used, the diffusion term is increased by the added turbulent viscosity and the numerical stability is guaranteed, however high frequency fluctuations as well as small eddies (high wave-number fluctuations) will be smoothed out. it can be seen that the coarse mesh scheme also produces a very regular vortex shedding frequency in the lam and les models at g/d = 1.0. this demonstrates that the mesh resolution is not fine enough to capture the turbulent flow and therefore only potential flow results are arrived at. there were no fluctuations in the drag and lift spectra in the sst model at g/d = 0.3, indicating no vortex shedding was predicted. this will be further explained in the next subsection, where the flow patterns will be shown. from the drag and lift fluctuation spectra, it can also be seen that the results of the lam and les approaches are quite similar, indicating that, with a sufficiently high mesh resolution, a numerical solution of the n–s equations without turbulence models can yield the unsteady flow reasonably as was found by singh and mittal (2005). the estimation of the strouhal number and judgment of the onset and suppression of vortex shedding can be determined by different methods. in this study, the estimation is based on an observation of the spectrum of the lift coefficient (lei et al., 1999), since each vortex shedding is accompanied by a lift fluctuation. many studies have shown that the vortex shedding frequency is insensitive to g/d, once vortex shedding occurs (choi and lee, 2000, lei et al., 1999, oner et al., 2008); the strouhal number remains constant at ~0.2 and is identical with that in a uniform flow without the plane wall. from the lift spectra, it can be seen that the peaks fall in a normalized frequency range of 0.2–0.3, and, with an increase of the mesh resolution, more spikes appear around this range, indicating that smaller size vortices are captured. for g/d = 1.0, the predicted strouhal number decreases and is closer to 0.2 as the mesh resolution increases. this is especially obvious when going from the coarse mesh scheme to the fine mesh scheme, but from the fine scheme to the extremely fine scheme little j. c. cai, j. pan, s. j. e, w. d. jiao, d. y. wang/journal of naval architecture and marine engineering, 14(2017) 9-24 a preliminary study of fluid-dynamic loads on a plane surface beneath a circular cylinder in the subcritical flow regime 17 change in the strouhal number was found, showing the second mesh scheme is sufficiently fine to resolve the flow structures. fig. 4: fluctuations of the lift coefficient at g/d = 0.3. fig. 5: fluctuations of the drag coefficient at g/d = 0.3. fig. 6: fluctuations of the lift coefficient at g/d = 1.0. 0.2 0.4 0.6 0.8 1.0 0.0 0.3 0.6 0.9 0.00 0.15 0.30 0.45 0.60 0.00 0.15 0.30 0.45 0.60 0.2 0.4 0.6 0.8 1.0 normalized frequency (f*d/u 0 ) lam-coarse mesh l if t c o e ff ic ie n t f lu c tu a ti o n lam-fine mesh lam-very fine mesh 0.2 0.4 0.6 0.8 1.0 0.0 0.3 0.6 0.9 0.00 0.15 0.30 0.45 0.60 0.00 0.15 0.30 0.45 0.60 0.2 0.4 0.6 0.8 1.0 normalized frequency (f*d/u 0 ) les-coarse mesh l if t c o e ff ic ie n t f lu c tu a ti o n les-fine mesh les-very fine mesh 0.2 0.4 0.6 0.8 1.0 0.000 0.075 0.150 0.225 0.000 0.075 0.150 0.225 0.000 0.075 0.150 0.225 0.2 0.4 0.6 0.8 1.0 normalized frequency (f*d/u 0 ) lam-coarse mesh d ra g c o e ff ic ie n t f lu c tu a ti o n lam-fine mesh lam-very fine mesh 0.2 0.4 0.6 0.8 1.0 0.000 0.075 0.150 0.225 0.000 0.075 0.150 0.225 0.000 0.075 0.150 0.225 0.2 0.4 0.6 0.8 1.0 normalized frequency (f*d/u 0 ) les-coarse mesh d ra g c o e ff ic ie n t f lu c tu a ti o n les-fine mesh les-very fine mesh 0.2 0.4 0.6 0.8 1.0 0.00 0.25 0.50 0.75 1.00 0.00 0.25 0.50 0.75 1.00 0.00 0.25 0.50 0.75 1.00 0.2 0.4 0.6 0.8 1.0 normalized frequency (f*d/u 0 ) lam-coarse mesh l if t c o e ff ic ie n t f lu c tu a ti o n lam-fine mesh lam-very fine mesh 0.2 0.4 0.6 0.8 1.0 0.00 0.25 0.50 0.75 1.00 0.00 0.25 0.50 0.75 1.00 0.00 0.25 0.50 0.75 1.00 0.2 0.4 0.6 0.8 1.0 normalized frequency (f*d/u 0 ) les-coarse mesh l if t c o e ff ic ie n t f lu c tu a ti o n les-fine mesh les-very fine mesh 0.2 0.4 0.6 0.8 1.0 0.00 0.25 0.50 0.75 1.00 0.00 0.25 0.50 0.75 1.00 0.00 0.25 0.50 0.75 1.00 0.2 0.4 0.6 0.8 1.0 normalized frequency (f*d/u 0 ) sst-coarse mesh l if t c o e ff ic ie n t f lu c tu a ti o n sst-fine mesh sst-very fine mesh j. c. cai, j. pan, s. j. e, w. d. jiao, d. y. wang/journal of naval architecture and marine engineering, 14(2017) 9-24 a preliminary study of fluid-dynamic loads on a plane surface beneath a circular cylinder in the subcritical flow regime 18 0.2 0.4 0.6 0.8 1.0 0.000 0.075 0.150 0.225 0.000 0.075 0.150 0.225 0.000 0.075 0.150 0.225 0.2 0.4 0.6 0.8 1.0 normalized frequency (f*d/u 0 ) lam-coarse mesh d ra g c o e ff ic ie n t f lu c tu a ti o n lam-fine mesh lam-very fine mesh 0.2 0.4 0.6 0.8 1.0 0.000 0.075 0.150 0.225 0.000 0.075 0.150 0.225 0.000 0.075 0.150 0.225 0.2 0.4 0.6 0.8 1.0 normalized frequency (f*d/u 0 ) les-coarse mesh d ra g c o e ff ic ie n t f lu c tu a ti o n les-fine mesh les-very fine mesh fig. 7: fluctuations of the drag coefficient at g/d = 1.0. 3.2 flow structures the previous subsection shows that the numerical results are more sensitive to the turbulence models than the mesh resolution, as long as the mesh is sufficiently fine. therefore only the flow fields of the fine mesh scheme are discussed in this section. the mean pressure distribution and the stream traces based on the mean velocities are shown in fig. 8. the pressure in front of the cylinder is higher than in the wake region behind the cylinder as a result of stagnation of the flow. on the upper and down sides of the cylinder, low pressure regions exist due to the flow acceleration. the pressure distribution around the cylinder is not symmetric due to the plane wall influence in the case of g/d = 0.3 with the stagnation point on the lower front part, while for g/d = 1.0 it is almost symmetric with the stagnation point very close to the cylinder front tip. for g/d = 0.3, the flow structures behind the cylinder found with the lam and les approaches are more complicated than those from the sst approach. in fact, the lam and les results show that, besides the major eddy pair behind the cylinder, there are some secondary eddies, as via experiments (oner et al., 2008), while in the sst results based on the rans equations, no secondary eddies exist. the upwash created by the wall behind the cylinder is obvious: the wake regions in the lam and les results are mainly on the upper part of the cylinder indicating that the plane wall forces the wake flow upward; in absence of the plane wall, i.e., just a single cylinder, the time-averaged wake flow is symmetric about the cylinder. when the flow past the cylinder, it expands, but due to the restriction of the plane wall the lower part can be substantially affected. vortices exist under the wake region near the plane wall in the lam and les approaches, acting as obstacles push the wake up. vortices are absent in the sst turbulence approach, but the upwash of the wake by the plane wall is still noticeable through the slightly skewed asymmetric streamlines in the wake. for g/d = 1.0 the average streamlines of the three approaches are more similar compared with the case of g/d = 0.3, and the flow structure are close to symmetric meaning that the influence of the plane wall in this case is very weak. snapshots of the vorticity distribution are shown in fig. 9. the boundary layer formed along the cylinder contains a significant amount of vorticity, as can be easily seen in the vorticity contours. intense vorticity exists on the up and down sides of the cylinder. the vorticity on the up side is in the clockwise direction, while on the down side it is in the counter-clockwise direction. when the reynolds number is high enough, the boundary layer over the cylinder surface will separate owing to the adverse pressure gradient imposed by the divergent geometry of the flow environment at the rear side of the cylinder. due to the separation of boundary layers, a shear layer is formed, as can be seen clearly in vorticity contours. the vorticity is fed into the shear layer formed downstream of the separation point and may cause the shear layer to roll up into a vortex with a sign identical to that of the incoming vorticity. the pair of up and down shear layers is unstable, and vortex shedding can occur when the two shear layers interact with each other. when the larger vortex at one side becomes strong enough to draw the opposing vortex across the wake, the supply of vorticity to the larger vortex from its boundary layer will be cut off by the approach of the opposing vortex, resulting in the shedding of the larger vortex. this phenomenon occurs in almost all cases shown, as can be seen in the velocity vectors and vorticity contours, where the two shear layers bend toward each other. the exception is the sst model at g/d = 0.3, in which the shear layers are almost straight. 0.2 0.4 0.6 0.8 1.0 0.00 0.25 0.50 0.75 1.00 0.00 0.25 0.50 0.75 1.00 0.00 0.25 0.50 0.75 1.00 0.2 0.4 0.6 0.8 1.0 normalized frequency (f*d/u 0 ) sst-coarse mesh l if t c o e ff ic ie n t f lu c tu a ti o n sst-fine mesh sst-very fine mesh j. c. cai, j. pan, s. j. e, w. d. jiao, d. y. wang/journal of naval architecture and marine engineering, 14(2017) 9-24 a preliminary study of fluid-dynamic loads on a plane surface beneath a circular cylinder in the subcritical flow regime 19 fig. 8: the mean pressure distribution and stream traces. j. c. cai, j. pan, s. j. e, w. d. jiao, d. y. wang/journal of naval architecture and marine engineering, 14(2017) 9-24 a preliminary study of fluid-dynamic loads on a plane surface beneath a circular cylinder in the subcritical flow regime 20 fig. 9: an instantaneous distribution of the vorticity: left: lam; middle: les; and right: sst. 3.3 wall pressure and shear stress on the plane surface since the results predicted by the les have a better agreement with the experimental results for unsteady flow fields around the cylinder, the les is used to study the pressure and shear stress on the plane surface. the mean distributions of the pressure and the shear stress in the x direction on the plane surface in the duration of 1 minute are shown in fig. 10. the pressure and shear stress were non-dimensionalized by 2 0 / 2u . in front of the cylinder, the pressure on the wall boundary increases as the flow approaches the cylinder, while the amplitude of the shear stress decreases slightly. in the downstream region, both the pressure and the shear stress wiggle violently along the wall boundary, especially for the case of g/d = 0.3. previous study using the rans approach with the k-ε turbulence model (ong et al., 2010) showed that the mean pressure distribution was quite smooth. the above discussions indicate that the flow structure is complex, and this will lead to the fluctuation of the forces on the cylinder and plane surfaces as well. fig. 10. distributions of (left) mean pressure and (right) shear stress on the plane surface. six points with x-coordinates of −3.0d, −1.0d, 0.0d, 1.0d, 3.0d, and 6.0d from the center of the cylinder (x = 0) were selected on the plane surface for examining the spatial and frequency properties of the surface forces. the frequency spectra of the pressure and shear stress were obtained via fourier transforms of the recorded data, and are shown in figs. 10 and 11. the dc components were omitted in the figures, because only the fluctuating components are important in flow-induced boundary vibration and sound radiation. the pressure and shear stress components are found to have large amplitudes in the low frequency range. when the non-dimensional frequency is greater than 0.4, which is approximately twice the strouhal number, the amplitude becomes negligibly small. the amplitude of the pressure is approximately two orders of magnitude larger than that of the shear stress. it is known from boundary layer theory that the static pressure is constant across the thin boundary layer and is equal to the static pressure at the outer edge of the boundary layer, which can be approximately predicted by an inviscid flow solution. therefore, the pressure fluctuation on the plane boundary is directly related to the main flow where vortex motion exists. however, the shear stress fluctuation is j. c. cai, j. pan, s. j. e, w. d. jiao, d. y. wang/journal of naval architecture and marine engineering, 14(2017) 9-24 a preliminary study of fluid-dynamic loads on a plane surface beneath a circular cylinder in the subcritical flow regime 21 determined by μ∂ux/∂y=0 on the wall and, due to the viscosity in the boundary layer, it can be substantially smaller than that outside the boundary layer, where its fluctuation is expected to be of the same order as the pressure fluctuations. in the spectra of pressure fluctuation, one can see that at the locations of x = 0.0d and 1.0d, the components have peak values at the vortex-shedding frequency. for the positions beyond 3.0d, the components at the vortex-shedding frequency gradually decrease. at 6.0d, the fluctuations at frequencies lower than the vortexshedding frequency even overweigh the component near the strouhal number. upstream of the cylinder (at x = −1.0d), the component is also discernible at the vortex-shedding frequency, although its amplitude is much smaller compared with those at x = 0.0d and 1.0d. at the location of x = −3.0d upstream, the fluctuations become very weak. this phenomenon indicates that the effects of vortex shedding on the force fluctuations o n the plane wall surface are mainly confined to 3.0d downstream and 1.0d upstream, and beyond this region the effects become very small. fig. 11: spectra of pressure (left) and shear stress (right) fluctuations on the plane wall at g/d = 0.3. fig. 12: spectra of pressure (left) and shear stress (right) fluctuations on the plane wall at g/d = 1.0. it is worth pointing out that, to zero pressure gradient turbulent boundary layer flows, it is generally accepted that the fluctuation magnitude of wall-shear stress ,w rms  is around 0.4 of the mean shear stress w (colella and keith, 2003, örlü and schlatter, 2011). for g/d = 1.0, , / w rms w   at −3.0d, −1.0d, 0.0d, 1.0d, 3.0d, and 6.0d are 0.33, 0.34, 0.15, 11.0, 3.02, 2.41, respectively. the upstream values are close to 0.4 indicating that the effect of the cylinder on the upstream wall stress is small. the large values of , / w rms w   at the downstream demonstrating the cylinder have certain influence on the downstream wall shear stress. 0.2 0.4 0.6 0.8 1.0 0.0 0.1 0.2 0.3 0.0 0.1 0.2 0.3 0.0 0.1 0.2 0.3 0.0 0.1 0.2 0.3 0.0 0.1 0.2 0.3 0.0 0.1 0.2 0.3 0.2 0.4 0.6 0.8 1.0 normalized frequency (f*d/u 0 ) x=-3.0d x=-1.0d x=0.0d n o rm a li z e d w a ll p re s s u re ( p /0 .5 ρ u 2 0 ) x=1.0d x=3.0d x=6.0d 0.2 0.4 0.6 0.8 1.0 0.0000 0.0012 0.0024 0.0036 0.0000 0.0012 0.0024 0.0036 0.0000 0.0012 0.0024 0.0036 0.0000 0.0012 0.0024 0.0036 0.0000 0.0012 0.0024 0.0036 0.0000 0.0012 0.0024 0.0036 0.2 0.4 0.6 0.8 1.0 n o rm a li z e d i n -p la n e w a ll s h e a r s tr e s s ( s /0 .5 ρ u 2 0 ) normalized frequency (f*d/u 0 ) x=-3.0d x=-1.0d x=0.0d x=1.0d x=3.0d x=6.0d 0.2 0.4 0.6 0.8 1.0 0.0 0.1 0.2 0.3 0.0 0.1 0.2 0.3 0.0 0.1 0.2 0.3 0.0 0.1 0.2 0.3 0.0 0.1 0.2 0.3 0.0 0.1 0.2 0.3 0.2 0.4 0.6 0.8 1.0 normalized frequency (f*d/u 0 ) x=-3.0d x=-1.0d x=0.0d n o rm a li z e d w a ll p re s s u re ( p /0 .5 ρ u 2 0 ) x=1.0d x=3.0d x=6.0d 0.2 0.4 0.6 0.8 1.0 0.0000 0.0012 0.0024 0.0036 0.0000 0.0012 0.0024 0.0036 0.0000 0.0012 0.0024 0.0036 0.0000 0.0012 0.0024 0.0036 0.0000 0.0012 0.0024 0.0036 0.0000 0.0012 0.0024 0.0036 0.2 0.4 0.6 0.8 1.0 n o rm a li z e d i n -p la n e w a ll s h e a r s tr e s s ( s /0 .5 ρ u 2 0 ) normalized frequency (f*d/u 0 ) x=-3.0d x=-1.0d x=0.0d x=1.0d x=3.0d x=6.0d j. c. cai, j. pan, s. j. e, w. d. jiao, d. y. wang/journal of naval architecture and marine engineering, 14(2017) 9-24 a preliminary study of fluid-dynamic loads on a plane surface beneath a circular cylinder in the subcritical flow regime 22 to investigate the spatial correlations of the surface pressure and shear stress, the coherence functions of the pressure and shear stress at each monitoring position with respect to those at x = 0d are presented in fig. 13 for g/d = 1.0. the spatial coherence value of pressure is higher than that of the shear stress, indicating that the pressure fluctuation is more correlated than the shear stress fluctuation. this is clearly shown by the mean coherence value in the normalized frequency band of 0 – 2. if a threshold value of 0.55 is set, then only the pressure fluctuations at the locations of x = ±1.0d are correlated to x = 0.0d, and the shear stress fluctuations are not well correlated. for the pressure fluctuation, due to the fact that the pressure in the boundary layer is (almost) equal to that outside the boundary layer at the same x-coordinate (∂p/∂y=0), the correlation of the pressure fluctuations is retained. the coherence values of the pressure fluctuations at monitoring points x = −1.0d and 1.0d with the reference point x = 0.0d are close to 1.0, which shows that they are well correlated. the pressure fluctuations at x = −3.0d and 3.0d with the reference point x = 0.0d are weakly correlated, as shown by their low coherence value, especially in the higher frequency range (normalized frequency > 1.0). this is reasonable because, as discussed above, the range of influence of vortex shedding is mainly restricted to x = −1.0d to 3.0d, and beyond this range the random nature of turbulent flow plays an important role. thus, the force fluctuations in this range are expected to be uncorrelated with the one at the reference point of x = 0.0d. for the shear stress, the complex flow structures affect the wall shear stress through the boundary layer. although the flow structure outside the boundary layer is quite complex owing to the vortex shedding and the flow structure is correlated in the vortex region, due to the viscosity effect of the boundary layer this correlation between the transmitted flows into the boundary layer is dramatically reduced. fig. 13: coherence functions of the pressure and shear stress fluctuations on the plane wall at g/d = 1.0. 4. conclusion in this paper, the fluctuating pressure and shear stress on a plane surface located beneath a circular cylinder have been numerically studied in the subcritical flow regime in 2-d sense using different turbulence approaches, namely the computation without any turbulence model (or the laminar approach, lam), the large eddy simulation (les), and the reynolds-averaged navier-stokes equations (rans) approach with shear-stress transport (sst) model. the overall results are reasonable; the mean drag and lift coefficients of the cylinder, and the strouhal number were found to be consistent with previous experimental results. the 2-d large eddy simulation (les), though there exist controversial opinions with its application, displayed good performance in this study for the investigated subcritical regime of flow around cylinder with re = 5000. the main observations are summarized as follows:  the les approach produced the most accurate lift force on the cylinder and a detailed flow pattern in the wake region. the flow structure obtained via the rans approach was quite regular. at g/d = 1.0, vortices were shed systematically, leading to regular drag and lift fluctuations with distinct spikes at the strouhal number and its harmonics. however, at g/d = 0.3, no vortex shedding was predicted even though it 0.0 0.5 1.0 1.5 2.0 0.0 0.3 0.6 0.9 1.2 0.0 0.3 0.6 0.9 1.2 0.0 0.3 0.6 0.9 1.2 0.0 0.3 0.6 0.9 1.2 0.0 0.5 1.0 1.5 2.0 normalized frequency (f*d/u 0 ) -3d&0d -1d&0d c o h e re n c e o f p re s s u re f lu c tu a ti o n s 1d&0d 3d&0d 0.0 0.5 1.0 1.5 2.0 0.0 0.3 0.6 0.9 1.2 0.0 0.3 0.6 0.9 1.2 0.0 0.3 0.6 0.9 1.2 0.0 0.3 0.6 0.9 1.2 0.0 0.5 1.0 1.5 2.0 -3d&0d c o h e re n c e o f w a ll s h e a r s tr e s s normalized frequency (f*d/u 0 ) -1d&0d 1d&0d 3d&0d j. c. cai, j. pan, s. j. e, w. d. jiao, d. y. wang/journal of naval architecture and marine engineering, 14(2017) 9-24 a preliminary study of fluid-dynamic loads on a plane surface beneath a circular cylinder in the subcritical flow regime 23 occurred in the laminar (lam) and les models. for g/d = 0.3 the asymmetric mean flow fields clearly show the upwash created by the plane wall, while for g/d = 1.0 the almost symmetric mean flow fields indicate that the effect of the wall is very weak.  the mesh resolution had a noticeable effect on the flow structure. when the mesh resolution was not fine enough, no turbulent flow could be observed. with a more refined mesh, the strouhal number decreased and became closer to the experimental value of 0.2. when the mesh resolution was sufficiently high, the lam approach (without any turbulence model) could predict the turbulent flow around the cylinder at a moderate reynolds number reasonably.  the force fluctuation on the plane surface was mostly significant below twice the vortex-shedding frequency, and the amplitude of the pressure fluctuations was approximately two orders of magnitude larger compared with the amplitude of the stress force fluctuations. the fluctuations were mainly confined to a distance of three cylinder diameters downstream and one diameter upstream of the cylinder, indicating that the effects of vortex shedding behind the cylinder on the plane wall force were confined to this region. the surface pressure fluctuations in the range of one diameter upstream and downstream of the center of the cylinder were spatially well correlated, and became less coherent as the distance to the cylinder center increased. the in-plane shear force fluctuations were spatially less correlated compared with that of the pressure fluctuations. this study also showed that in a 2-d simulation the force fluctuations on the cylinder were much higher than those from experimental measurements. this may be attributed to the fact that the spanwise flow correlation effect was ignored, i.e., it assumed that all the vortexes along the cylinder were shed simultaneously. therefore, a more accurate and universal fully 3-d turbulent flow simulation, especially an les, will be conducted to study the characteristics of 3-d statement of flow consideration in the near future. acknowledgement this work was partially completed while the first author was on sabbatical leave at the centre for acoustics, dynamics and vibration of the university of western australian. the first author kindly acknowledges the support of the national natural science foundation of china (project numbers: 51306163, 51575497). the authors are grateful for the financial support from the crc for infrastructure engineering asset management (cieam). references bouris, d. and bergeles, g. 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