Microsoft Word - MATH090622-f -edited_corrected.doc SQU Journal For Science, 15(2010) 55-79 © 2010 Sultan Qaboos University 55 Convective Hydromagentic Slip Flow with Variable Properties Due to a Porous Rotating Disk Mohammad M. Rahman Department of Mathematics and Statistics, College of Science, Sultan Qaboos University, P.O. Box 36, Postal code 123, Muscat, Sultanate of Oman, Email:mansurdu@yahoo.com. انسياب الحمل اإلنزالقي لمائع متغير الخصائص ، ناتج عن دوران قرص مسامي رحمان. محمد م في هذا البحث ندرس خصائص االنتقال الحراري بالحمل في االنسياب أالنزالقي المستتب لمائع حول قرص :خالصة ل الحراري علي درجة الحرارة، و في وجود مسامي دوار، آخذين في االعتبار اعتماد كل من الكثافة و اللزوجة والتوصي لتحويل " فون كارمان"نستخدم تحويالت التشابه ل ".جول"و تبديد طاقة عبر اللزوجة، و كذلك تسخين " هول"تيار .المعادالت الحاكمة لمسألتي الحركة و االنتقال الحراري إلي نظام مترابط من المعادالت التفاضلية العادية عالي الالخطية و قد أظهرت ". سفيجرت"و " ناختسهايم"م حل المعادالت الالبعدية المتحصل عليها عدديا باستخدام الطريقة التكرارية ل يت النتائج إن نموذج الطبقة الحدية الحرارية في المائع ذي الخصائص المعتمدة علي الحرارة ال يؤدي إلي نتائج مقبولة كما تبين النتائج إن معامل االنزالق يتحكم . و لذلك يجب اعتباره متغيرا داخلهاثابتا في الطبقة،" براندل"عندما يؤخذ عدد .بشكل جوهري في خصائص االنزالق و النقل الحراري ABSTRACT: In this paper we investigate convective heat transfer characteristics of steady hydromagnetic slip flow over a porous rotating disk taken into account the temperature dependent density, viscosity and thermal conductivity in the presence of Hall current, viscous dissipation and Joule heating. Using von-Karman similarity transformations we reduce the governing equations for flow and heat transfer into a system of ordinary differential equations which are highly nonlinear and coupled. The resulting nondimensional equations are solved numerically by applying Nachtsheim-Swigert iteration technique. The results show that when modeling a thermal boundary layer, with temperature dependent fluid properties, consideration of Prandtl number as constant within the boundary layer, produces unrealistic results. Therefore it must be treated as variable throughout the boundary layer. Results also show that the slip factor significantly controls the flow and heat transfer characteristics. KEYWORDS: Rotating disk; Heat transfer; Convection; Slip flow; Variable properties. MOHAMMAD M. RAHMAN 56 1. Introduction n recent years, the flow dynamics due to a rotating disk, originating from the early formulation of von Karman (1921), has been a popular area of research. Since then many researchers (Cochran, 1934; Roger and Lance, 1960; Benton, 1965; Kuiken, 1971; Owen and Rogers, 1989; Herrero et al 1994; Kelson and Desseaux, 2000; Andrsson and Korte, 2002; Takhar et al 2002) have studied and reported results on disk-shaped bodies with or without heat transfer. Flow due to a rotating disk is encountered in many industrial, geothermal, geophysical, technological and engineering applications. A few of them are rotating heat exchangers, rotating disk reactors for bio-fuels production, computer disk drives, and gas or marine turbines. Nomenclature a constant b constant B magnetic field vector 0B applied magnetic field Cf skin friction coefficient pC specific heat at constant pressure d constant E electric field Ec Eckert number e− charge of electron F dimensionless radial velocity G dimensionless tangential velocity H dimensionless axial velocity Ha Hartman number J electric current density Kn Knudsen number m Hall current parameter Nu Nusselt number en electron concentration per unit volume p pressure within the boundary layer p∞ pressure of the ambient fluid ep electronic pressure Pr variable Prandtl number Pr∞ ambient Prandtl number q velocity vector wq surface heat flux Re rotational Reynolds number r cylindrical radial coordinate T temperature within boundary layer tU target velocity u velocity along radial direction v velocity along tangential direction w velocity along axial direction sw non-dimensional suction velocity ww suction velocity x , y , z Cartesian coordinates Greek Symbols β Hall factor γ relative temperature difference parameter ρ density of the fluid ∞ρ density of the ambient fluid µ coefficient of dynamic viscosity µ∞ dynamic viscosity of the ambient fluid υ∞ kinematic viscosity of the ambient fluid σ electric conductivity κ thermal conductivity κ∞ thermal conductivity of the ambient fluid η similarity parameter ξ target momentum accommodation coefficient λ mean free path θ dimensionless temperature φ tangential coordinate Ω angular velocity ε slip factor I CONVECTIVE HYDROMAGNETIC SLIP FLOW 57 wT temperature at the surface of the disk ∞T temperature of the ambient fluid Φ viscous dissipation function τ shear stress The effects of an applied magnetic field on the steady flow due to the rotation of a disk of infinite or finite extent were studied by El-Mistikawy et al. (1991) and El-Mistikawy and Atia (1990). Atia and Aboul-Hassan (1997) studied steady hydromagnetic flow due to an infinite disk rotating with uniform angular velocity in the presence of an axial magnetic field. In their analysis they neglected the induced magnetic field but considered Hall current. Attia (1998) studied the effects of suction as well as injection in the presence of a magnetic field on the unsteady flow past a rotating porous disk. It was found that the combined effect of a magnetic field with strong injection may stabilize the growth of the boundary layer. Sparrow et al. (1971) studied the flow of Newtonian fluid due to the rotation of a porous-surfaced disk with a set of linear slip-flow conditions. A substantial reduction in torque then occurred as a result of surface slip. Miklavcic and Wang (2004) further revisited the problem of Sparrow et al. and pointed out that the slip- flow boundary conditions could also be used for slightly rarefied gases or for flow over grooved surfaces. Arikoglu and Ozkol (2006) studied MHD slip flow over a rotating disk with heat transfer. It is observed that both the slip factor and the magnetic flux decrease the velocity in all directions and thicken the thermal boundary layer. Recently, Osalusi et al. (2008) studied thermal-diffusion and diffusion-thermo effects on MHD slip flow due to a rotating disk. In classical treatment of thermal boundary layers, fluid properties (such as density, viscosity, thermal conductivity) are assumed to be constant; however, experiments indicate that this assumption only makes sense if temperature does not change rapidly for the application of interest. To predict the flow behavior accurately, it may be necessary to take into account these variable properties. Zakerullah and Ackroyd (1979) investigated free convection flow above a horizontal circular disk considering variable fluid properties. In the case of fully developed laminar flow in concentric annuli, the effect of the variable property has been investigated by Herwig and Klemp (1988). Atia (2006) studied unsteady hydromagnetic flow due to an infinite rotating disk, considering temperature dependent viscosity in a porous medium with Hall and ion-slip currents. Maleque and Sattar (2005a) studied the effect of variable properties on the steady laminar convective flow due to a rotating disk while Maleque and Sattar (2005b) further investigated the same problem in the presence of Hall current. Osalusi and Sibanda (2006) revisited the problem of Maleque and Sattar (2005a), considering magnetic effect. When fluid properties such as viscosity and thermal conductivity vary with temperature, Prandtl number (see section 2) varies too. All of these afore-mentioned works considered Prandtl number as constant within the boundary layer, although viscosity and thermal conductivity depends on temperature. Hence one of the motivations behind this study is also to investigate how variable Prandtl number affects the flow and heat transfer characteristics. In the present study we extend the work of Maleque and Sattar (2005b) and analyze the flow and heat transfer characteristics in the presence of viscous dissipation and Joule heating, considering slip flow boundary condition at the surface of a uniformly heated rotating disk. The resulting governing equations are solved numerically applying Nachtsheim-Swigert (1965) iteration technique. Graphical results for non-dimensional velocity and temperature profiles including skin-friction coefficient and the Nusselt number in tabular form are presented for a range of values of the parameters characterizing the flow. The accompanying discussion provides physical interpretations of the results. 2. Mathematical Model Let us consider a steady hydromagnetic laminar flow of an electrically conducting fluid due to a porous rotating disk of infinite extent in the presence of an external uniform magnetic field directed perpendicular to the disk. The fluid properties are taken as strong functions of temperature. A uniform suction or injection through the disk is considered for the whole range of suction or injection velocities. MOHAMMAD M. RAHMAN 58 2.1 Basic Equations The equations governing the steady hydromagnetic laminar convective flow are: Equation of continuity: .( ) 0ρ∇ =q , (1) Navier–Stokes equation: [ ]( . ) .( ) ( )pρ µ∇ = −∇ + ∇ ∇ + ×q q q J B , (2) Ohm’s law for a moving conductor with Hall currents: [ ]( ) epσ β β= + × − × + ∇J E q B J B , (3) Maxwell electromagnetic equations: . 0, , . 0∇ = ∇× = ∇ =J E 0 B , (4) Energy equation: 2 ( . ) .( )p J C T Tρ κ µ σ ∇ = ∇ ∇ + + Φq , (5) 2 2 22 2 2 u v w u v v w x y z y x z y ⎡ ⎤⎛ ⎞ ⎛ ⎞ ⎛ ⎞∂ ∂ ∂ ∂ ∂ ∂ ∂⎛ ⎞ ⎛ ⎞ Φ = + + + + + +⎢ ⎥⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟∂ ∂ ∂ ∂ ∂ ∂ ∂⎝ ⎠ ⎝ ⎠⎢ ⎥⎝ ⎠ ⎝ ⎠ ⎝ ⎠⎣ ⎦ ( ) 2 22 . 3 w u x z ∂ ∂⎛ ⎞ + + − ∇⎜ ⎟∂ ∂⎝ ⎠ q . (6) Here q is the velocity vector, B is the magnetic field vector, E denotes the electrical field vector which results from charge separation and is in the z -direction, J is the current density vector, p is the pressure, ρ is the density of the fluid, µ is the viscosity of the fluid, σ is the electrical conductivity of the fluid, κ is the thermal conductivity of the fluid, pC is the specific heat of the fluid, T is the temperature of the fluid, and Φ is the viscous dissipation function. In equation (3) the term ( )σβ ×J B denotes the Hall effects where 1 een β = designates the Hall factor, e− is the charge of electron, en is the electron concentration per unit volume and ep is the electronic pressure. In equation (5) the term 2J σ represents Joule heating whereas µΦ is the viscous dissipation or frictional heating effects. 2.2 Governing equations In non-rotating cylindrical polar coordinates ( , , )r zφ , let us consider a disk which rotates with constant angular velocity Ω about the z -axis. The disk is placed at 0z = , and the fluid occupies the region 0z > , CONVECTIVE HYDROMAGNETIC SLIP FLOW 59 where z is the vertical axis in the cylindrical coordinates system with r and φ as the radial and tangential axes respectively. The components of the flow velocity q are ( , , )u v w in the directions of increasing ( , , )r zφ respectively. The surface of the rotating disk is maintained at a uniform temperature wT and far away from the wall, the free stream is kept at a constant temperature T∞ and at a constant pressure p∞ . The fluid is assumed to be Newtonian, viscous and electrically conducting. An external uniform magnetic field is applied in the z - direction. The electron–atom collision frequency is assumed to be relatively high so that the Hall effect cannot be neglected. Ion-slip effects are however ignored in the present analysis. From equation (4), using the relation 0∇⋅ =B for the magnetic field ( , , )x y zB B B=B , we obtain that 0 (constant)zB B= everywhere in the fluid. This assumption is valid only when the magnetic Reynolds number is very small so that magnetic induction effects can be ignored. For the current density ( , , )x y zJ J J=J we obtain from the relation 0∇⋅ =J that constantzJ = . Hence we consider that the disk is non-conducting and therefore 0zJ = at the disk and hence zero everywhere. Finally we consider the case of a short circuit problem in which the applied electric field =E 0 and also assume that the induced magnetic field is negligible in comparison with the applied magnetic field. In the absence of electric field E and electron pressure ep equation (3) becomes 0 0 2 2 ( ) ( ) , , 0 1 1 B v mu B mv u m m σ σ+ −⎡ ⎤ = ⎢ ⎥+ +⎣ ⎦ J , (7) where 0m Bσβ= is called Hall current parameter. It can be further shown that 2 2 0 0 2 2 ( ) ( ) , , 0 1 1 B mv u B v mu m m σ σ⎡ ⎤− + × = −⎢ ⎥+ +⎣ ⎦ J B . (8) We also assume that the fluid properties, viscosity ( µ ), thermal conductivity (κ ) and density ( ρ ) are functions of temperature alone and obey the following laws (see Jayaraj, 1995; later used by Malek and Sattar, 1995b; Osalusi and Sibanda, 2006) [ ]/ aT Tµ µ∞ ∞= , [ ]/ b T Tκ κ∞ ∞= , [ ]/ d T Tρ ρ∞ ∞= , (9) where a , b and d are arbitrary exponents while µ∞ , κ∞ and ρ∞ are the viscosity, thermal conductivity and density of the ambient fluid respectively. The flow configurations and geometrical coordinates are shown in Figure 1. Due to steady axially symmetric, compressible hydromagnetic laminar flow of a homogeneous fluid the governing equations take the following form (see Malek and Sattar, 1995b): ( ) ( ) 0ru rw r z ρ ρ ∂ ∂ + = ∂ ∂ , (10) MOHAMMAD M. RAHMAN 60 22 0 ( ),2(1 ) Bu v u p u u u u w mv u r r z r r r r r z z m σ ρ µ µ µ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ − + = − + + + + − ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ + ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠⎝ ⎠ (11) 2 0 ( ),2(1 ) Bv uv v v v v u w v mu r r z r r r r z z m σ ρ µ µ µ ∂ ∂ ∂ ∂ ∂ ∂ ∂ + = + + + ∂ ∂ ∂ ∂ ∂ ∂ ∂ + ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ + −⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠ (12) Figure 1. Flow configurations and coordinate system ( )1 ,w w p w wu w w r z z r r r r z z ρ µ µ µ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂⎛ ⎞ ⎛ ⎞ ⎛ ⎞ + = − + + +⎜ ⎟ ⎜ ⎟ ⎜ ⎟∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂⎝ ⎠ ⎝ ⎠ ⎝ ⎠ (13) 2 2 p T T T T T u v C u w r z r r r z z z zr ρ κ κ κ µ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ = + + ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ⎡ ⎤⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ + + + +⎢ ⎥⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠⎢ ⎥⎣ ⎦ 2 2 20 2 ( )(1 ) B u v m σ + + (14) If the mean free path of the fluid particles is comparable to the characteristic dimensions of the flow field domain, the assumption of continuum media is no longer valid, and as a consequence Navier–Stokes equation breaks down. In the range 0.1 10Kn< < of Knudsen number, the high order continuum equations (Burnett equations) should be used. For the range of 0.001 0.1Kn< < , no-slip boundary conditions cannot be used and should be replaced with the following expression (Gad-el-Hak, 1999): 2 t u U z ξ λ ξ − ∂ = ∂ , (15) φ rΩ z wT T∞ 0B w uv P∞ x y CONVECTIVE HYDROMAGNETIC SLIP FLOW 61 where tU is the target velocity, ξ is the target momentum accommodation coefficient and λ is the mean free path. For 0.001Kn < , the no-slip boundary condition is valid; therefore, the velocity at the surface is equal to zero. In this study the slip and the no-slip regimes of the Knudsen number that lies in the range 0 0.1Kn< < are considered. 2.3 Boundary conditions By using equation (15), the appropriate boundary conditions for our model are (i) On the surface of the disk ( 0z = ): tu U= , tv r U= Ω + , ww w= (slip flow and permeable surface conditions), (16a) wT T= (uniform surface temperature). (16b) (ii) Matching with the quiescent free stream ( z → ∞ ): 0u = , 0v = , T T∞= , p p∞= . (16c) 3. Transformation of the model To obtain the solutions of the governing equations (10)-(14) together with the boundary conditions (16) we introduce a dimensionless normal distance from the disk, ( )1/ 2zη υ∞= Ω along with the von-Karman transformation ( )1/ 2( ), ( ), ( ), 2 ( ), ( ), u rF v rG w H p p P T T T η η υ η ρ υ η θ η ∞ ∞ ∞ ∞ ∞ ⎫= Ω = Ω = Ω ⎪ ⎬ − = Ω − = ∆ ⎪⎭ (17) where υ∞ is the kinematic viscosity of the ambient fluid and wT T T∞∆ = − . Now substituting (17) into (10)-(14) we obtain the following nonlinear ordinary differential equations 12 (1 ) 0H F dHγ θ γθ −′ ′+ + + = , (18) 2 1 2 2 2(1 ) [ ](1 ) ( )(1 ) 01 d a aHaF a F F G HF mG F m γ γθ θ γθ γθ− − −′′ ′ ′ ′+ + − − + + + − + = + , (19) 2 1 2(1 ) [2 ](1 ) ( )(1 ) 01 d a aHaG a G FG HG G mF m γ γθ θ γθ γθ− − −′′ ′ ′ ′+ + − + + − + + = + , (20) 2 1 2 2 2 2(1 ) Pr (1 ) Pr (1 ) ( )1 d b bHab H Ec F G m θ γ γθ θ θ γθ γθ− − −∞ ∞′′ ′ ′+ + − + + + + ++ 2 2Pr (1 ) ( ) 0a bEc F Gγθ −∞ ′ ′+ + = , (21) where ( )1/ 20Ha B σ ρ∞= Ω is the Hartmann number, Pr pCµ κ∞ ∞ ∞= is the ambient Prandtl number, 2( ) pEc r C T= Ω ∆ is the Eckert number and T Tγ ∞= ∆ is the relative temperature difference parameter, which is positive for a heated surface, negative for a cooled surface and zero for uniform properties. Thus by using (17) boundary conditions (16) become , 1 , , 1 at 0,sF F G G H wε ε θ η′ ′= = + = = = (22a) MOHAMMAD M. RAHMAN 62 0, 0, 0, 0 as ,F G P θ η= = = = → ∞ (22b) where 1/ 2 2 ( ) ξ ε λ υ ξ ∞ − = Ω is the slip factor and 1/ 2( )sw w υ − ∞= Ω represents a uniform suction when 0sw < and uniform injection when 0sw > at the surface of the disk. 3.1 Particular cases A number of special cases can be derived from the full transformed momentum and energy equations (18)-(21) with the boundary conditions (22) which are as follows: i 0swγ = = without heat transfer no-slip condition Hassan and Attia (1997) ii 2 0Ha m Ecγ = = = = heat transfer without Joule heating no-slip condition Kelson and Desseaux (2000) iii 2 0Ha mγ = = = without heat transfer slip condition Miklavcic and Wang (2004) iv 0sw m Ecγ = = = = heat transfer without Joule heating slip condition Arikoglu and Ozkol (2006) v 2 0Ha m Ec= = = heat transfer without Joule heating no-slip condition Maleque and Sattar (2005a) vi 0m Ec= = heat transfer without Joule heating no-slip condition Osalusi and Sibanda (2006) vii 0Ec = heat transfer without Joule heating no-slip condition Maleque and Sattar (2005b) 4. Variable Prandtl Number The Prandtl number is a function of viscosity and as viscosity varies across the boundary layer, the Prandtl number varies, too. The assumption of constant Prandtl number inside the boundary layer may produce unrealistic results. Therefore, Prandtl number related to the variable viscosity is defined by (1 ) Pr (1 ) Pr (1 ) (1 ) a p p p a b a b b C C Cµ µ γθ µ γθ γθ κ κ γθ κ ∞ ∞ − − ∞ ∞ ∞ + = = = + = + + (23) At the surface ( 0η = ) of the disk, this can be written as Pr Pr (1 )a bw γ − ∞= + . (24) From equation (23) it can be seen that for 0γ → , the variable Prandtl number Pr equals the ambient Prandtl number Pr∞ . For η → ∞ that is outside the boundary layer, ( )θ η becomes zero. Therefore Pr equals Pr∞ regardless of the values of γ . Table 1 shows the variation of the Prandtl number at the surface of the disk for several values of γ for a fixed value of the ambient Prandtl number Pr 0.64∞ = and the exponents 0.7a = , 0.83b = . From this table we see that for a positive value of γ , Prandtl number at the surface of the disk Prw decreases as γ increases. The opposite effect is observed when γ is negative. It must be noted that for 1γ ≤ − no physically viable solutions exist. CONVECTIVE HYDROMAGNETIC SLIP FLOW 63 Table 1. Values of Pr versus γ for Pr 0.64∞ = , 0.7a = , 0.83b = at 0η = . γ -0.8 -0.5 -0.2 0.0 0.2 0.5 1.0 3.0 5.0 Pr 0.789 0.700 0.659 0.640 0.625 0.607 0.585 0.534 0.507 In light of the above discussions, using (23) the non-dimensional temperature equation (21) can be rewritten as 1 2(1 ) Pr(1 )d ab Hθ γ γθ θ γθ θ− −′′ ′ ′+ + − + + 2 2 2 2 2 2(1 ) Pr (1 ) ( ) Pr ( ) 0aHa m Ec F G Ec F Gγθ − ′ ′+ + + + + = . (25) Equation (25) is the corrected non-dimensional form of the energy equation in which Prandtl number is treated as variable. It is mentionable that this correction does not appear in the literature. 5. Parameters of engineering interest The parameters of engineering interest for the present problem are the skin-friction coefficient (Cf ) and the Nusselt number ( Nu ) which indicate physically wall shear stress and rate of heat transfer respectively. The action of the variable properties in the fluid adjacent to the disk sets up a tangential shear stress, which opposes the rotation of the disk. As a consequence, it is necessary to provide a torque at the shaft to maintain a steady rotation. The radial shear stress rτ and tangential shear stress tτ are defined by: 1/ 2 0 (1 ) Re (0)ar z u w F z r τ µ µ γ∞ = ⎡ ⎤∂ ∂⎛ ⎞ ′= + = + Ω⎜ ⎟⎢ ⎥∂ ∂⎝ ⎠⎣ ⎦ , (26) 1/ 2 0 1 (1 ) Re (0)at z v w G z r τ µ µ γ φ ∞ = ⎡ ⎤⎛ ⎞∂ ∂ ′= + = + Ω⎢ ⎥⎜ ⎟∂ ∂⎝ ⎠⎣ ⎦ . (27) Hence the skin-frictions ( 2 2Cf rτ ρ∞= Ω ) along radial and tangential directions are obtained as 1/ 2(1 ) Re (0)arCf Fγ − ′= + , (28) 1/ 2(1 ) Re (0)atCf Gγ − ′= + . (29) The rate of heat transfer from the disk surface to the fluid is computed by the application of Fourier’s law as given below 1/ 2 0 (1 ) (0).bw z T q T z κ κ γ θ υ∞= ∞ ⎛ ⎞∂ Ω⎛ ⎞ ′= − = − ∆ + ⎜ ⎟⎜ ⎟∂⎝ ⎠ ⎝ ⎠ (30) Hence the Nusselt number ( w rq Nu Tκ∞ = ∆ ) is obtained as 1/ 2(1 ) Re (0),bNu γ θ′= − + (31) where 2Re r υ∞= Ω is the rotational Reynolds number. Thus from equations (28), (29) and (31) we see that skin-friction coefficient and Nusselt number are proportional to the numerical values of (0)F′ , (0)G′ and (0)θ′− which are calculated in the process of integration when solving the corresponding differential equations. MOHAMMAD M. RAHMAN 64 6. Method of solutions The set of equations (18)-(20) and (25) are highly nonlinear and coupled and therefore the system cannot be solved analytically. The system of transformed governing equations (18)-(20) and (25) with boundary conditions (12) is solved numerically using shooting method similar to that described by Nachtsheim-Swigert (1965). In equation (22) there are three asymptotic boundary conditions and hence three unknown surface conditions ( )0F′ , ( )0G′ and ( )0θ′ . Nachtsheim-Swigert developed an iteration technique to overcome the difficulties of determining the guess values of the unknown surface boundary conditions required for the shooting method. Within the context of the initial value method and the Nachtsheim-Swigert shooting iteration technique the outer boundary conditions may be functionally represented by ( )max( ) (0), (0), (0) , 1, 2 6,j j jF G jη θ δ′ ′ ′Ψ = Ψ = = (26) where 1 FΨ = , 2 GΨ = , 3 θΨ = , 4 F′Ψ = , 5 G′Ψ = , 6 θ′Ψ = . The last three of these represents asymptotic convergence criteria. Choosing ( ) 10F g′ = , ( ) 20G g′ = and ( ) 30 gθ′ = and expanding in a first-order Taylor’s series after using equations (26) yields 3 max , max 1 ( ) ( ) ,jj j C i j i i g g η η δ = ∂Ψ Ψ = Ψ + ∆ = ∂∑ 62,1=j (27) where subscript ‘C’ indicates the value of the function at maxη determined from the trial integration. Solution of these equations in a least-square sense requires determining the minimum value of 6 2 1 j j δ = ∏ = ∑ (28) with respect to ig ( 3,2,1=i ). Now differentiating ∏ with respect to ig we obtain ∑ = = ∂ ∂6 1 0 j i j j g δ δ . (29) Substituting equation (27) into (29) after some algebra we obtain 3 1 , 1, 2, 3,ik k i k a g b i = ∆ = =∑ (30) where 6 6 , 1 1 . , ; , 1, 2, 3. j j jik i j C j ji k i a b i k g g g= = ∂Ψ ∂Ψ ∂Ψ = = − Ψ = ∂ ∂ ∂∑ ∑ (31) Now solving the system of linear equations (30) we obtain the missing (unspecified) values of ig as i i ig g g≅ + ∆ . (32) Thus adopting this numerical technique aforementioned, a computer program was set up for the solutions of the governing non-linear ordinary differential equations (18)-(20) and (25) of our problem where the integration technique was adopted as a sixth-order Runge-Kutta method of integration. The velocity and temperature are determined as a function of the coordinate η and displayed graphically. CONVECTIVE HYDROMAGNETIC SLIP FLOW 65 0 2 4 6 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 η F ws = 0.5, 0, -0.5, -1.0, -2.0 (a) 0 2 4 6 0 0.2 0.4 0.6 0.8 η G ws = 0.5, 0, -0.5, -1.0, -2.0 (b) 0 2 4 6 -0.5 0 0.5 1 1.5 2 η -H w s = 0 w s = -0.5 w s = -1.0 w s = -2.0 w s = 0.5 (c) 0 2 4 6 8 10 12 14 0 0.2 0.4 0.6 0.8 1 η θ w s = 0.5, 0, -0.5, -1.0, -2.0 (d) 0 3 6 9 12 15 0.625 0.63 0.635 0.64 η Pr ws = 0.5, 0, -0.5, -1.0, -2.0 (e) Figure 2. Variation of (a) radial velocity, (b) tangential velocity, (c) axial velocity, (d) temperature profile, and (e) variable Prandtl number for several values of sw . MOHAMMAD M. RAHMAN 66 6.1 Numerical experiment In this paper, the effects of Hall current, viscous dissipation and Joule heating on a steady hydromagnetic convective slip flow of a viscous, Newtonian, electrically conducting fluid with variable properties over a rotating porous disk have been investigated numerically by using Nachtsheim-Swigert shooting iteration technique. It can be seen that the solutions are affected by the seven parameters, namely suction (or injection) parameter sw , magnetic field parameter (or Hartmann number) Ha , Hall current parameter m , relative temperature difference parameter γ , Prandtl number Pr , Eckert number Ec and slip parameter ε . Since experimental data of the physical parameters are not available, in the numerical simulations the choice of the values of the parameters was dictated by the values chosen by the previous investigators. For the present investigation we considered our working fluid as flue gas. For flue gases (ambient Prandtl number, Pr 0.64∞ = ) the values of the exponents a , b and d are taken as 0.7a = , 0.83b = and 1.0d = − (see Jayaraj, 1995). The default values of the other parameters which we considered are 1.0sw = − , 2 0.5Ha = , 0.5m = , 0.2γ = , Pr 0.625= , 0.2Ec = , and 0.2ε = unless otherwise specified. 6.2 Code verification To assess the accuracy of the present code, we reproduced the values of (0)F′ , (0)G′ , ( )H ∞ and (0)θ′ for constant property models of Kelson and Desseaux (2000) (herein and after referred as KD2000) (see case-ii in section 3.1) and Arikoglu and Ozkol (2006) (herein and after referred as AO2006) (see case-iv in section 3.1). Tables 2-4 show the comparisons of the data produced by the present code and those of KD2000 and AO2006. In fact the results show a close agreement, and hence justify the use of the present code for the current model. Table 2. Numerical values of (0)F′ , (0)G′− and (0)θ′− for various values of sw with 2 0Ha m Ec γ= = = = and Pr 0.71= . (0)F′ (0)G′− (0)θ′− sw Present KD2000 Present KD2000 Present KD2000 4 0.24304404 0.243044 0.02892121 0.0289211 0.00001075 0.0000107 3 0.30914768 0.309147 0.06028945 0.0602893 0.00057793 0.000576 2 0.39893387 0.398934 0.13595275 0.135952 0.01103604 0.011013 1 0.48948057 0.489481 0.30217432 0.302173 0.08504687 0.084884 0 0.51022378 0.510233 0.61592380 0.615922 0.32637889 0.325856 -1 0.38954065 0.389569 1.17526180 1.175222 0.79393633 0.793048 -2 0.24241310 0.242421 2.03859590 2.038527 1.43876482 1.437782 -3 0.16558828 0.165582 3.0122231 3.012142 2.13677058 2.135585 -4 0.12475268 0.124742 4.00526266 4.005180 2.84369011 2.842381 CONVECTIVE HYDROMAGNETIC SLIP FLOW 67 Table 3. Numerical values of (0)F′ and (0)G′− for various values of ε with 2 0sw Ha m Ec γ= = = = = and Pr 0.71= . (0)F′ (0)G′− ε Present AO2006 Present AO2006 0.0 0.51022378 0.51023261 0.61592380 0.61592201 0.1 0.42144560 0.42145363 0.60583699 0.60583524 0.2 0.35257377 0.35258100 0.58367858 0.58367676 0.5 0.22384294 0.22384820 0.50281179 0.50280970 1.0 0.12792035 0.12792364 0.39492982 0.39492759 2.0 0.06100834 0.06101009 0.27337241 0.27337013 5.0 0.01858796 0.01858852 0.14339025 0.14338820 10 0.00681240 0.00681255 0.08103175 0.08103008 20 0.00236161 0.00236159 0.04378973 0.04378846 Table 4. Numerical values of ( )H− ∞ and (0)θ′− for various values of ε with 2 0sw Ha m Ec γ= = = = = and Pr 0.71= . ( )H− ∞ (0)θ′− ε Present AO2006 Present AO2006 0.0 0.88344324 0.8844741 0.32637889 0.32586063 0.1 0.88055012 0.8813642 0.33402796 0.33349695 0.2 0.87334256 0.8739572 0.33732324 0.33678090 0.5 0.84230103 0.8423926 0.33521597 0.33465287 1.0 0.79003973 0.7894772 0.32099888 0.32043299 2.0 0.71185974 0.7103133 0.29357940 0.29299798 5.0 0.58730981 0.5837646 0.24466400 0.24440461 10 0.49317208 0.4875846 0.20570012 0.20504924 20 0.40816322 0.3999758 0.16953552 0.16882963 6.3 Effect of step size To see the effects of the integration step size η∆ , we ran the code for our model with three different step sizes namely ,01.0=∆η ,005.0=∆η and 001.0=∆η . In each case, we found excellent agreement among the results. It was also found that 001.0=∆η provided sufficiently accurate (error less than 610− ) results and further refinement of the grid size was therefore not warranted. 7. Results and discussion For the purpose of discussing the results, the numerical calculations are presented in the form of non- dimensional velocity (radial, tangential and axial) and temperature profiles. In the calculations the values of the parameters namely suction (or injection) parameter sw , magnetic field parameter (or Hartmann number) Ha , Hall current parameter m , relative temperature difference parameter γ , Prandtl number Pr , Eckert number Ec and slip parameter ε are varied. MOHAMMAD M. RAHMAN 68 0 2 4 6 0 0.02 0.04 0.06 0.08 0.1 η F Ha 2 =0, 0.5, 0.8, 1.0 (a) 0 2 4 6 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 η G Ha 2 =0, 0.5, 0.8, 1.0 (b) 0 2 4 6 1 1.04 1.08 1.12 1.16 1.2 η -H Ha 2 =0, 0.5, 0.8, 1.0 (c) 0 2 4 6 8 10 0 0.2 0.4 0.6 0.8 1 η θ Ha 2 =0, 0.5, 0.8, 1.0 (d) 0 2 4 6 8 10 0.625 0.63 0.635 0.64 η Pr Ha 2 =0, 0.5, 0.8, 1.0 (e) Figure 3. Variation of (a) radial velocity, (b) tangential velocity, (c) axial velocity, (d) temperature profile, and (e) variable Prandtl number for several values of Ha . CONVECTIVE HYDROMAGNETIC SLIP FLOW 69 The effects of the suction (or injection) parameter ( sw ) on the radial, tangential and axial velocity profiles are shown in Figures 2 (a)-(c) respectively. From Figures 2(a)-(b) we see that radial and tangential velocity profiles decrease very rapidly as the suction velocity ( 0sw < ) intensifies. The maximum of the radial velocity profiles moves towards the surface of the disk. It is also apparent that the thickness of the boundary layer decreases as suction velocity increases. Therefore, suction stabilizes the boundary layer growth. From Figure 2(c) we found that for strong suction, inward axial velocity is nearly constant. Figure 2(d) depicts the variation of the temperature profiles for various values of the suction parameter. The effect of the suction parameter on the thermal boundary layer is found to be similar to those of the radial and tangential velocity boundary layers. Applying suction, one can control the flow and heat transfer characteristics. In Figure 2(e) we have plotted variable Prandtl number as a function of η to show the variation of the Prandtl number throughout the boundary for several values of the suction parameter. From this Figure we see that within the boundary layer for a fixed value of η variable Prandtl number increases as the suction parameter increases while far away from the surface of the disk Pr equals its ambient value Pr∞ . An opposite effect is found for the case of fluid injection ( 0sw > ). The influence of the magnetic field parameter (Hartmann number) Ha on F , G , and H− distributions is depicted in Figures 3(a)-(c). An increase in Ha induces a significant decrease in radial and tangential velocity profiles throughout the boundary layer; this is due to fact that imposition of a magnetic field to an electrically conducting fluid creates a drag force called the Lorentz force that has a tendency to slow down the flow around the disk at the expense of increasing its temperature. From Figure 3(c) it is also apparent that inward axial velocity decreases substantially with the increase of the Hartmann number. An increase in Hartmann number increases temperature profiles and hence increases the thermal boundary layer as can be seen from Figure 3(d). The variation of the Prandtl number within the boundary layer for different values of the Hartmann number is depicted in Figure 3(e). This Figure reveals that variable Prandtl number decreases with the increase of the Hartmann number. In Figures 4(a)-(d), the influence of Hall current parameter ( m ) on F , G , H− and θ distributions across the boundary layer are given. The parameter m has remarkable effect on the velocity profiles. It is observed that radial as well as inward axial velocity profiles increase as the Hall current parameter increases up to a certain value of 1m < . Beyond this value of m , profiles of F and H− decrease with the further increase of m . It can be explained as follows: From equation (19) we see that the radial velocity term with Hall current is 2 2 (1 )1 aHa F m γθ −− + + . An increase in m ( 1< ) will induce very minor alterations in the expression 2 1 1 m+ . However in equation (20), the term 2 2 ( )(1 )1 aHa mF G m γθ −− + + + gives an effective contribution to the radial velocity through 2 2 (1 )1 aHa mF m γθ −− + + indicating that an increase in m ( 1< ) causes a direct increase in the radial velocity. But for 1m > an opposite scenario is observed. Conversely we observe that the tangential velocity ( G ) increases with an increase in Hall current parameter. From equation (20) we see that the tangential velocity is affected via the term 2 2 (1 )1 aHa G m γθ −− + + , thus a change in m produces very little effect, due to the inverse relationship of m and the tangential velocity G . This effect will impede the tangential MOHAMMAD M. RAHMAN 70 0 2 4 6 0 0.02 0.04 0.06 0.08 0.1 0.12 η F m = 0.5 m = 0 m = 50 m = 10 m = 1.0 (a) 0 2 4 6 0 0.2 0.4 0.6 0.8 η G m = 0, 1.0, 10, 50 (b) 0 2 4 6 0.95 1 1.05 1.1 1.15 1.2 η -H m = 0 m = 0.5 m = 1.0 m = 10 m = 50 (c) 0 2 4 6 8 10 0 0.2 0.4 0.6 0.8 1 η θ m = 0, 0.5, 1.0, 10 (d) 0 2 4 6 8 10 0.625 0.63 0.635 0.64 η θ m = 0, 0.5, 1.0, 10 (e) Figure 4. Variation of (a) radial velocity, (b) tangential velocity, (c) axial velocity, (d) temperature profile, and (e) variable Prandtl number for several values of m . CONVECTIVE HYDROMAGNETIC SLIP FLOW 71 0 2 4 6 0 0.02 0.04 0.06 0.08 0.1 η F γ = 0, 0.2, 0.5, 1 (a) 0 2 4 6 0 0.2 0.4 0.6 0.8 η G γ = 0, 0.2, 0.5, 1 (b) 0 2 4 6 8 10 0.9 1 1.1 1.2 1.3 η -H γ = 0 γ = 0.2 γ = 0.5 (c) 0 2 4 6 8 10 0 0.2 0.4 0.6 0.8 1 η θ γ = 0, 0.2, 0.5, 1 (d) 0 2 4 6 8 10 0.58 0.59 0.6 0.61 0.62 0.63 0.64 0.65 η Pr γ = 0, 0.2, 0.5, 1 Pr =0.64∞ (e) Figure 5. Variation of (a) radial velocity, (b) tangential velocity, (c) axial velocity, (d) temperature profile, and (e) variable Prandtl number for several values of γ . MOHAMMAD M. RAHMAN 72 flow. However it will be swamped out by 2 2 (1 )1 aHa mG m γθ −+ + , the much greater direct proportionality in the factor affecting the tangential velocity in given equation (19). This positive term accelerates the flow for increase in m , explaining the rise in tangential velocity with increase in m . The mechanism by which Hall currents influence hydromagnetic disk flow is therefore via secondary effects and coupling in the momentum equations. Figure 4 (d) reveals that the thickness of the thermal boundary layer decreases as m increases. The variation of the variable Prandtl number for different values of m within the boundary layer is shown in Figure 4(e). It is clearly observed that an increase in m increases Pr within the boundary layer. For very large values of m , increasing the effect of m on Pr is less pronounced due to the fact that 2 1 1 m+ approaches to its limiting value 0 when m → ∞ , and as a consequence resistive effect of the magnetic field on the flow and temperature field is diminished. Figures 5(a)-(d) explain the variation of the nondimensional radial, tangential, axial velocity and temperature profiles for various values of the relative temperature difference parameter γ . From Figure 5(a), we see that due to the existence of the centrifugal force the radial velocity increases and attains its maximum value for all values of γ . It is also observed that the maximum values of the radial velocity are 0.07276499, 0.08221263, 0.09128975 and 0.09893225 for 0γ = , 0.2, 0.5 and 1.0, respectively, and occur at 0.390η = , 0.494, 0.643 and 0.871, respectively. It is seen that the maximum velocity increases by 36% when γ increases from 0 to 1.0. The case 0γ = corresponds to constant property of the working fluid. It is also seen that the smallest maximum value of the radial velocity is found for the case of constant property ( 0γ = ), which contradicts directly the findings of Maleque and Sattar (2005b). From Figure 5(b), it is found that the tangential velocity increases with the increasing values of γ . It can be seen from Figure 5(c) that inward axial velocity decreases with the increase of γ . It is also observed that close to the surface of the disk the effect of γ gives rise to the familiar inflection point profile, which indicates that fluid with variable property on a highly heated surface, may lead to the destabilization of the laminar flow resulting in the development of the viscous sub-layer. Figure 5(d) depicts that temperature profile increases significantly with the increase of γ . Quantitatively, at 8.0η = the value of θ increases by 6026.8% when the value of γ increases from 0 to 1.0. Thus the thickness of the thermal boundary layer increases markedly with the increase of γ which is a direct contradiction to the findings of Maleque and Sattar (2005b), and Osalusi and Sibanda (2006). Studying a limited set of parameter values such as 0γ = , 0.5 (Maleque and Sattar, 2005b) and 0γ = , 0.01 (Osalusi and Sibanda, 2006) and considering Prandtl number as constant within the boundary layer, they concluded that an increase in γ does not change the thickness of the thermal boundary layer. Figure 5(e) shows that variable Prandtl number Pr decreases very rapidly within the boundary layer for the increase of γ . For 0γ = variable Prandtl number Pr equals the ambient Prandtl number Pr∞ . For a fixed value of γ ( 0)> , Pr increases as η increases and for η → ∞ , i.e. outside the boundary layer, it converges to its ambient value Pr∞ . From this figure it is also clear that at the surface of the disk (at 0η = ), 0γ = , 0.2, 0.5, and 1 corresponds to Pr 0.64= , 0.625, 0.607, 0.585 when other parameter values are fixed. Thus the effects of Pr on the velocity and temperature functions give the reverse effect of γ on them. CONVECTIVE HYDROMAGNETIC SLIP FLOW 73 0 2 4 6 0 0.02 0.04 0.06 0.08 0.1 η F ε = 0, 0.2, 1.0, 4.0, 8.0 (a) 0 2 4 6 0 0.2 0.4 0.6 0.8 1 η G ε = 0, 0.2, 1.0, 4.0, 8.0 (b) 0 2 4 6 0.8 0.9 1 1.1 1.2 η -H ε = 0 ε = 0.2 ε = 1.0 ε = 4 ε = 8 (c) 0 2 4 6 8 10 0 0.2 0.4 0.6 0.8 1 η θ ε = 0, 1.0, 4.0, 8.0 (d) 0 2 4 6 8 10 0.625 0.63 0.635 0.64 η Pr ε = 0, 1.0, 4.0, 8.0 (e) Figure 6. Variation of (a) radial velocity, (b) tangential velocity, (c) axial velocity, (d) temperature profile, and (e) variable Prandtl number for several values of ε . MOHAMMAD M. RAHMAN 74 0 2 4 6 0 0.02 0.04 0.06 0.08 0.1 η F ε = 0, 0.2, 1.0, 4.0, 8.0 (a) 0 2 4 6 0 0.2 0.4 0.6 0.8 1 η G ε = 0, 0.2, 1.0, 4.0, 8.0 (b) 0 2 4 6 0.8 0.9 1 1.1 1.2 η -H ε = 0 ε = 0.2 ε = 1.0 ε = 4 ε = 8 (c) 0 2 4 6 8 10 0 0.2 0.4 0.6 0.8 1 η θ ε = 0, 1.0, 4.0, 8.0 (d) 0 2 4 6 8 10 0.625 0.63 0.635 0.64 η Pr ε = 0, 1.0, 4.0, 8.0 (e) Figure 7. Variation of (a) radial velocity, (b) tangential velocity, (c) axial velocity, (d) temperature profile, and (e) variable Prandtl number for several values of ε . CONVECTIVE HYDROMAGNETIC SLIP FLOW 75 In Figures 6(a)-(d) we displayed velocity and temperature profiles for various values of the slip factor (ε ). Here 0ε = represents no-slip condition at the surface of the disk. From Figure 6(a) we see that the radial boundary layer decreases very rapidly with the increase of the slip factor. The thickness of the radial boundary layer is higher for no-slip flow compared to the slip flow. Fig. 6(a) further indicates that for large values of ε i.e. ε → ∞ , the rotating disk does not cause rotation of the fluid particles. Because in this range of ε the flow becomes entirely potential, there will be no motion in the fluid. This can be further explained as follows: the centrifugal force acting on the rotating disk (as like a centrifugal fan) will throw out the fluid that sticks to it. On the other hand, the flow in the axial direction will come forward to compensate for this thrown fluid. But increasing the slip on the surface of the disk reduces the amount of fluid that can stick on it; as a consequence the efficiency of the rotating disk is reduced substantially and is unable to transfer its circumferential momentum to the fluid particles. A reduction in the circumferential velocity results in a reduction in the centrifugal force which in turn decreases the inward axial velocity substantially as can be seen from Figure 6(c). From Figure 6(d) we see that the thermal boundary layer increases as slip factor ε increases. Figure 6(e) shows a decreasing effect of ε on the variable Prandtl number throughout the boundary layer. In Table 5 we present skin-friction in radial and tangential directions and rate of heat transfer for various values of the pertinent parameters for a fixed value of Pr . It can be seen that skin-friction in the radial direction decreases while skin-friction in the tangential direction increases with the increase of the suction parameter ( 0)sw < . On the other hand, the rate of heat transfer increases with the increase of the suction parameter. An opposite effect is observed for the case of injection ( 0)sw > . Table 5 also shows that skin-friction in the radial direction increases for all increasing values of the Hartmann number except in the range of 0 0.707Ha≤ < (not precisely determined). In this range of Ha , radial skin-friction decreases as Ha increases. Tangential skin-friction increases while the rate of heat transfer decreases for all increasing values of the Hartmann number. The effects of the Hall current parameter on the radial and tangential skin-frictions and the rate of heat transfer can be seen from Table 5. Skin-friction in the radial direction increases within the range of 0 1m≤ ≤ . Outside of this range of m an opposite behavior is observed. Tangential skin-friction decreases when m increases within the range of 0 1m≤ ≤ , and outside this range of m tangential skin-friction increases with the further increase of m . The rate of heat transfer increases with the increase of m for some cm m< . But for the existence of strong Hall current i.e. cm m> the rate of heat transfer decreases with the further increase of m . The effect of increasing Eckert number Ec has a decreasing effect on the radial skin-friction and on the rate of heat transfer whereas it has a very minor increasing effect on the tangential skin-friction as can be seen from Table 5. The variation of the radial and tangential skin-frictions and the rate of heat transfer for some selected values of the slip factor ε are shown in Table 5. From here we see that skin-friction in both directions decreases with the increase of the slip factor. The largest skin-friction is found for the case of no-slip at the surface. On the other hand the rate of heat transfer increases with the increase of slip factor within the range of 0 1ε≤ ≤ . But outside of this range of ε , the rate of heat transfer decreases with the further increase of the slip factor. Thus the rate of heat transfer can be strongly controlled by controlling the slip on the disk. MOHAMMAD M. RAHMAN 76 Table 5. Numerical values of (0)F′ , (0)G′− and (0)θ′− for various values of sw , 2Ha , m , Ec , and ε with 0.2γ = , Pr 0.625= , 0.7a = , 0.83b = , 1.0d = − . sw 2Ha m Ec ε (0)F′ (0)G′− (0)θ′− 0.5 0.5 0.5 0.2 0.2 0.30626574 0.57918115 0.05772346 0.0 0.5 0.5 0.2 0.2 0.28341282 0.69042730 0.16076663 -0.5 0.5 0.5 0.2 0.2 0.24952322 0.81807709 0.30083407 -1.0 0.5 0.5 0.2 0.2 0.20853272 0.96222468 0.47057060 -2.0 0.5 0.5 0.2 0.2 0.12995186 1.28487051 0.86659713 -1.0 0.0 0.5 0.2 0.2 0.21477746 0.78866948 0.49905518 -1.0 0.5 0.5 0.2 0.2 0.20853272 0.96222468 0.47057060 -1.0 0.8 0.5 0.2 0.2 0.20907598 1.04165765 0.45885733 -1.0 1.0 0.5 0.2 0.2 0.21011307 1.08825472 0.45234655 -1.0 0.5 0.0 0.2 0.2 0.13956413 0.95761163 0.43868718 -1.0 0.5 0.5 0.2 0.2 0.20853272 0.96222468 0.47057060 -1.0 0.5 1.0 0.2 0.2 0.24406352 0.93179389 0.49107437 -1.0 0.5 10 0.2 0.2 0.23051412 0.80677029 0.50468008 -1.0 0.5 50 0.2 0.2 0.21822032 0.79212583 0.50044457 -1.0 0.5 0.5 0.0 0.2 0.20858805 0.96208379 0.53127404 -1.0 0.5 0.5 0.2 0.2 0.20853272 0.96222468 0.47057060 -1.0 0.5 0.5 0.4 0.2 0.20847731 0.96236597 0.40974692 -1.0 0.5 0.5 0.8 0.2 0.20836700 0.96264871 0.28795509 -1.0 0.5 0.5 1.0 0.2 0.20831208 0.96279014 0.22698731 -1.0 0.5 0.5 0.2 0.0 0.36777423 1.17657272 0.43706560 -1.0 0.5 0.5 0.2 0.2 0.20853272 0.96222468 0.47057060 -1.0 0.5 0.5 0.2 1.0 0.05117315 0.53881089 0.48991829 -1.0 0.5 0.5 0.2 4.0 0.00578668 0.20435375 0.47419596 -1.0 0.5 0.5 0.2 8.0 0.00158190 0.11227042 0.46583227 Finally, the significance of the relative temperature difference (γ ) on the rate of heat transfer for both variable Prandtl number ( PrV ) and constant Prandtl number ( PrC ) is tabulated in Table 6. From this table we see that in both cases the rate of heat transfer from the surface of the disk to the fluid decreases for all increasing values of γ . We also see that rate of heat transfer for the variable property case is lower than the constant property case and the relative error between them increases significantly with the increase of γ . Therefore, consideration of Prandtl number as constant within the boundary layer for variable property is unrealistic. It is also mentionable that for our studied parameter values the relationship between the relative temperature difference parameter and the variable Prandtl number is an inverse relationship. So, the effect of Pr on the radial and tangential skin-frictions and on the rate of heat transfer is just the reverse of the effect of γ on them. CONVECTIVE HYDROMAGNETIC SLIP FLOW 77 Table 6. Numerical values of (0)θ′− for various values of γ for 1.0sw = − , 2 0.5Ha = , 0.5m = , 0.2Ec = , and 0.2ε = with 0.7a = , 0.83b = , 1.0d = − . (0)θ′− γ PrV PrC Pr Pr Error = 100 Pr V C V − × 0.0 0.64904702 0.64904702 0.0% 0.2 0.47057060 0.47627008 1.2% 0.5 0.31906666 0.32769165 2.7% 1.0 0.19338897 0.20237407 4.6% 3.0 0.05065792 0.05674936 12.0% 5.0 0.01661496 0.02066049 24.3% 8. Conclusions In this study we experiment numerically on the effects of Hall current, viscous dissipation and Joule heating on hydromagnetic slip flow over a porous rotating disk taking into account the variable properties of the fluid. We illustrate the flow and heat transfer characteristics in terms of non-dimensional velocity and temperature profiles and tabulate skin-friction and rate of heat transfer, and show how the flow fields are influenced by the material parameters entering into the problem. As a result of computations the following conclusions can be drawn: 1. Suction stabilizes the boundary layer’s growth. 2. Slip factor significantly controls the flow and heat transfer characteristics. 3. Increasing slip factor forces decrease of the Prandtl number within the boundary layer. 4. Hall parameter markedly controls the radial and axial flows. For strong Hall current (large 1)m > flow along these directions decreases. 5. Hall current strongly controls the rate of heat transfer from the disk to the fluid. Very strong Hall current may reduce the heat transfer rate. 6. Hall current increases variable Prandtl number within the boundary layer. 7. The resistive effect of an applied magnetic field (Lorentz force) on the velocity and temperature profiles is apparent. 8. Increasing viscous dissipation parameter (or Eckert number) decreases the rate of heat transfer from the disk to the fluid. 9. The rate of heat transfer in a fluid of constant property is higher than in a fluid of variable property. 10. The thickness of the thermal boundary layer is lower for a fluid of constant property than for a corresponding fluid of variable property. 11. 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