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 x- and y- axes, 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 Runge- Kutta 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. 1097- 1120.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, McGraw- Hill 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/s10483-010- 1355-6 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