Applied Mathematics and Mechanics Computational ...

Appl. Math. Mech. -Engl. Ed., 33(6), 731–748 (2012) DOI 10.1007/s10483-012-1583-7 c

Shanghai University and Springer-Verlag Berlin Heidelberg 2012

Applied Mathematics and Mechanics (English Edition)

Computational study of combined effects of conduction-radiation and hydromagnetics on natural convection flow past magnetized permeable plate∗ M. ASHRAF1 ,

S. ASGHAR2 ,

M. A. HOSSAIN3

(1. Department of Mathematics, COMSATS Institute of Information Technology, Islamabad 44000, Pakistan; 2. Department of Mathematics, King Abdulaziz University, Jeddah 21589, Saudi Arabia; 3. Department of Mathematics, University of Dhaka, Dhaka 1000, Bangladesh)

Abstract The computational study of the combined effects of radiation and hydromagnetics on the natural convection flow of a viscous, incompressible, and electrically conducting fluid past a magnetized permeable vertical plate is presented. The governing non-similar equations are numerically solved by using a finite difference method for all values of the suction parameter ξ and the asymptotic solution for small and large values of ξ. The effects of varying the Prandtl number P r, the magnetic Prandtl number P rm , the magnetic force parameter S, the radiation parameter Rd , and the surface temperature θw on the coefficients of the skin friction, the rate of heat transfer, and the current density are shown graphically and in tables. An attempt is made to examine the effects of the above mentioned physical parameters on the velocity profile, the temperature distribution, and the transverse component of the magnetic field. Key words hydromagnetic, fluctuating, natural convection, magnetized plate, current density, heat transfer Chinese Library Classification O351 2010 Mathematics Subject Classification

76D05

Nomenclature H0 , H∞ , S, f, P rm , Rex , Grx , Cfx , Hx , Hy , N ux ,

reference magnetic field velocity; free stream magnetic field; magnetic field parameter; transformed stream function; magnetic Prandtl number; local Reynolds number; local Grashof number; skin friction; magnetic field along the surface; magnetic field normal to the surface; local Nusselt number;

u, v, T w, T ∞, V0 , Rd , x, y, g,

dimensional axial velocity, m·s−1 ; dimensional normal velocity, m·s−1 ; wall temperature, K; ambient fluid temperature, K; surface mass flux; Plank number (conduction-radiation parameter); axial distance, m; normal distance, m; acceleration due to gravity, m·s−2 .

∗ Received Jun. 15, 2011 / Revised Dec. 28, 2011 Corresponding author M. ASHRAF, Ph. D., E-mail: [email protected]

732

M. ASHRAF, S. ASGHAR, and M. A. HOSSAIN

Greek symbols ψ, φ, ξ, α, µ, η, ν, θ,

fluid stream function, m2 ·s−1 ; transformed stream function for the magnetic field; transpiration parameter; thermal diffusivity, m2 ·s−1 ; dynamical viscosity, kg·m−1 ·s−1 ; similarity transformation; kinematic viscosity, m2 ·s−1 ; dimensionless temperature function;

θw , ρ, σ, σs , γ, β, µ,

surface temperature ratio to the ambient fluid; density of the fluid, kg·m−3 ; electrical conductivity; Stefan-Boltzman constant; magnetic diffusion; coefficient of cubical expansion; magnetic permibility.

∞,

ambient condition.

Subscripts w,

1

wall condition;

Introduction

In many engineering problems, e.g., advanced power plants, nuclear rockets, high speed flights, and re-entry vehicles, the heat transfer by thermal radiation is very important because of the high temperature. In this study, we consider the interaction of radiation with the magnetohydrodynamic (MHD) natural convection flow past a magnetized vertical porous plate. The foundations for the appreciations of the behaviors of viscous, incompressible, and electrically conducting fluids in boundary layers were first given by Greenspan and Carrier[1] in the presence of a symmetrically oriented semi-infinite flat plate. They found the solution by using the Fourier transformation with the asymptotic analysis, and claimed that the velocity gradient at the plate approached zero due to the increase in the applied magnetic field intensity. Davies[2–3] examined the fact that the boundary layer thickness and the drag coefficient diminished steadily as the magnetic force parameter S increased. Gribben[4] considered the study of the two-dimensional MHD boundary layer in steady and incompressible flow under the influence of an external MHD pressure gradient proportional to some powers of the distance along the boundary. Gribben[5] investigated the MHD boundary layer in steady and incompressible flow under the influence of an external magnetic dynamic pressure gradient by using the asymptotic analysis, and found that the skin friction decreased with the increase in the magnetic field. Ramamoorthy[6] numerically investigated that the temperature distribution in the boundary layer was considerably reduced because of the slowing down of the flow by the magnetic field. Tan and Wang[7] showed that the rate of heat transfer decreased with the increases in the magnetic force parameter S and the magnetic field, and the thermal boundary layer thicknesses increased. Hildyard[8] found that the magnetic-field boundary condition used by Gribben was not correct. He obtained the appropriate asymptotic solutions for large and small values of the magnetic Prandtl number P rm . Ingham[9] studied the boundary layer flow on a semi-infinite flat plate placed at zero incidence to a uniform stream of electrically conducting gas with an aligned magnetic field at large distances from the plate. He observed that increasing the magnetic field for a given Mach number or decreasing the Mach number for a given magnetic field thickened the momentum and thermal boundary layer. Glauert[10] investigated the basic steady flow in the case of a uniform field in a magnetized plate. The observation showed that the velocity and magnetic fields were valid for small values of the magnetic field parameter S and for both smaller or larger values of the magnetic Prandtl number P rm . Chawla[11] studied the MHD boundary layer flow past a magnetized plate when a uniform magnetic field in the stream direction was applied, and observed that if the strength of the applied magnetic field exceeded a certain critical value, the boundary layer separation occurred. Simultaneous convection and radiation for the flow of steady and incompressible fluids has been widely studied[12–16] . Chen[17] introduced the radiation effects by assuming that the heat

Computational study of combined effects of conduction-radiation and hydromagnetics

733

transfer to the gas at the tube wall was proportional to the fourth power of the wall temperature. However, this is physically unrealistic boundary conditions since it neglects the radiation incident on the surface from another surface element. Dussan and Irvine[18] calculated heat transfer by assuming linearized radiation and using an exponential Kernal approximation. A more complete and realistic model that did not require a priori knowledge of the heat transfer coefficient was used by Thorsen[19] and Thorsen and Kachanagom[20] to investigate the effects of radiation on heat transfer for the flow inside circular tubes and used by Liu and Thorsen[21] for the flow between parallel channels. The effects of thermal radiation in different geometries have been discussed by several authors. Chen et al.[22] studied the effects of radiation interaction in the boundary layer flow over a horizontal surface. Arpaci[23] , Cheng and Ozisik[24] , and Sparrow and Cess[25] showed the thermal radiation effects with free convection from a vertical heated plate. The thermal radiation effect with free convection from a heated vertical semi-infinite plate was given by Soundalgekar et al.[26] by using the Colgey-Vincenti equilibrium model. Hossain and Takhar[27] investigated the effect of radiation by using the Rosseland diffusion approximation which led to nonsimilar solutions for the forced and free convection flow of an optically dense viscous incompressible fluid past a heated vertical plate with a uniform free stream velocity and surface temperature. Aboeldahab and Gendy[28] numerically solved the MHD free convection flow of gas past a semi-infinite vertical plate with variable thermophysical properties for a high temperature difference by using the shooting method. Mebine and Adigio[29] analytically demonstrated the thermal radiation effects on the unsteady free convection flow past a vertical porous plate with Newtonian heating by using the Laplace transform technique. Palani and Abbas[30] numerically solved the combined effects of MHD and radiation on free convection flow past an impulsively started isothermal plate with the Rosseland diffusion by using the finite element method. Sparrow and Cess[31] studied the case of uniform suction and blowing through an isothermal vertical wall, and obtained a series solution which was valid near the leading edge. Merkin[32] gave the contribution to the same problem in more details, and obtained asymptotic solutions valid at large distances from the leading edge for both suction and blowing. Clarke[33] numerically solved the boundary layer problem by using the method of matched asymptotic expansions. Merkin[32] described the effects of strong suction and blowing from general body shapes which admited a similarity solution. Vedhanayagam et al.[34] discussed the problem of blowing and wall temperature variations. Clarke and Riley[35] studied the case of a heated isothermal horizontal surface with transpiration in detail. Gupta et al.[36] studied the hydromagnetic steady shear flow along an electrically insulating porous plate. They observed that the velocity at a given point increased with the increase in either the magnetic field or the suction velocity. Bikash and Sharma[37] studied the heat transfer characteristic from a continuous flat surface of an electrically conducting fluid on a non-Newtonian visco-elastic fluid. Zueco and Ahmed[38] carried out the numerical solution of the mixed convection MHD flow past a vertical porous plate. Ali et al.[39] studied the steady MHD stagnation point flow over a stretching sheet. Su and Zheng[40] studied the MHD Falkner-Skan problem by using the Pade Approximation. Shit and Haldar[41] investigated the effect of the thermal radiation on the MHD flow and heat transfer over a non-linear shirking sheet. Ashraf et al.[42] investigated the numerical solutions of hydromagnetic mixed convection flow of a viscous incompressible fluid past a magnetized vertical porous plate. Keeping in view of the above literature survey, we investigate the combined effects of radiation and hydromagnetics on natural convection flow along a magnetized vertical permeable plate. The effects of varying the radiation parameter Rd , the Prandtl number P r, the magnetic Prandtl number P rm , the magnetic force parameter S, and the surface temperature θw on the −3/4 1/4 coefficients of the skin friction GrL x−1/4 Cf , the rate of heat transfer GrL x1/4 N ux , and −3/4 the current density GrL x−1/4 Jw are shown. The effects of the above mentioned parameters on the velocity profile, the temperature distribution, and the transverse component magnetic

734


field are also examined. The numerical solutions for an intermediate range of the transpiration parameter ξ are obtained by using a finite difference method (FDM). The asymptotic solutions are obtained both near and away from the leading edge, and are compared with the numerical solutions obtained by the FDM. The two solutions are found to be in good agreement.

2

Mathematical analysis and governing equations

Here, we consider a steady two-dimensional MHD natural convection flow of an electrically conducting, viscous, and incompressible fluid past a uniformly heated vertical porous plate by including the radiation effects in the energy equation. The diagram illustrating the flow domain and the coordinate system is shown in Fig. 1, where u = 0, v = V0 , T = Tw , Hx = Hw (x), Hy = 0, T = T∞ , u = 0, Hx (∞) = 0, the step length = 0.005 along the x-axis, and the step length = 0.01 with 80 grid points along the y-axis. The x-axis is taken along the surface, and the y-axis is normal to it. In Fig. 1, δM , δT , and δH stand for the momentum, thermal, and magnetic field boundary layer thicknesses, respectively.

Fig. 1

Coordinate system and flow configuration

The momentum, magnetic, and energy flow fields with the influence of the radiation effects are now governed by the following equations: ∂u ∂v + = 0, ∂x ∂y u

(1)

∂u ∂v ∂H x ∂H x ∂2u µ Hx +v =ν 2 + + Hy + gβ(T − T ∞ ), ∂x ∂y ρ ∂x ∂y ∂y

∂H x ∂H y + = 0, ∂x ∂y

(2) (3)

u

∂H x ∂H x ∂u ∂u 1 ∂2H x +v − Hx − Hy = , ∂x ∂y ∂x ∂y γ ∂y2

(4)

u

∂T ∂T ∂2T ∂qr +v =α 2 − , ∂x ∂y ∂y ∂y

(5)

where  ∂T    qr = −Kr ,   ∂y

Kr =

−16σs T 3 , 3αR

∂qr ∂2T = −Kr 2 , ∂y ∂y

u(x, 0) = 0, v(x, 0) = V0 , H x (x, 0) = H w (x),      u(x, ∞) = 0, H x (x, ∞) = 0, T (x, ∞) = 0.

H y (x, 0) = 0,

T (x, 0) = Tw ,

(6)


735

In (6), H w (x) = x1/2 shows the magnetization at the surface. V0 is the surface mass flux which is assumed to be uniform. When the fluid is being withdrawn through the surface, it is negative, while when the fluid is being blown through the surface, it is positive. In the present investigation, we consider that V0 is for the case of the fluid being withdrawn through the surface. Now, for convenience, we introduce the following dependent and independent variables to normalize the boundary layer equations:  1 1 ν ν T − T∞   ,  u = GrL2 u, v = GrL4 v, θ = L L Tw − T∞    H = H0 Gr 12 H , H = H0 Gr 14 H , y = y Gr 14 , x y L x L y L L L L

(7) x x= . L

By substituting (7) into (1)–(6), the dimensionless boundary layer equations and the boundary conditions are given as follows: ∂u ∂v + = 0, ∂x ∂y u

(8)

∂H ∂u ∂u ∂2u ∂Hx x +v = θ + 2 + S Hx + Hy , ∂x ∂y ∂y ∂x ∂y

(9)

∂Hx ∂Hy + = 0, ∂x ∂y

(10)

∂Hx ∂Hx ∂u ∂u 1 ∂ 2 Hx +v − Hx − Hy = , ∂x ∂y ∂x ∂y P rm ∂y 2 ∂θ ∂θ 1 ∂2θ 4 ∂ 3 ∂θ u +v = + (1 + (θ − 1)θ) , w ∂x ∂y P r ∂y 2 3Rd ∂y ∂y ( u(x, 0) = 0, v(x, 0) = V0 , Hx (x, 0) = 1, Hy (x, 0) = 0,

(11)

u

θ(x, 0) = 1,

u(x, ∞) = 0,

Hx (x, ∞) = 0,

(12) (13)

θ(x, ∞) = 0,

where u and v are the dimensionless fluid velocity components in the x- and y-directions, Hx and Hy are the dimensionless x- and y-components of the magnetic field, θ is the dimensionless temperature of the fluid in the boundary layer. Here, P r, S, P rm , Rd , GrL , and θw are the Prandtl number, the magnetic force parameter, the magnetic Prandtl number, the radiation parameter, the Grashof number, and the surface temperature (ratio to wall and ambient fluid temperature) defined as follows: Pr =

ν , α

S=

µH02 , ρL2

P rm =

ν , γ

Rd =

KαR , 3 4σT∞

GrL =

gβ∆T L3 , ν2

θw =

Tw , T∞

where µ, ν, γ, and L are the dynamic fluid viscosity, the kinematic viscosity, the magnetic diffusion, and the characteristic length, respectively.

3

Methods of solutions

We now turn to get the numerical solutions to the problem. For this purpose, we use two methods, i.e., the primitive variable transformation for the FDM and the stream function formulation for the asymptotic series solutions near and away from the leading edge of the plate.

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3.1 Primitive variable transformation To get the set of equations in the convenient form for integration, we define the following parameters of transformation for the dependent and independent variables:   u = x 12 U (ξ, Y ),

1

v = x− 4 (V (ξ, Y ) + ξ),

 H = x 12 φ (ξ, Y ), x 1

1

Hy = x− 4 φ2 (ξ, Y ),

1

Y = x− 4 V (ξ, Y ), θ = θ(ξ, Y ),

1

(14)

ξ = V0 x 4 .

By substituting (14) into (8)–(12) with the boundary conditions (13), we have 1 ∂U 1 ∂U ∂V U +ξ − Y + = 0, 2 ∂ξ 4 ∂Y ∂Y ∂U 1 2 1 ∂U 1 ∂U U + ξU + V − YU −ξ 2 4 ∂ξ 4 ∂Y ∂Y 1 ∂φ ∂2U 1 ∂φ1 1 1 =θ+ + S φ21 + ξφ1 + φ2 − Y φ1 , 2 ∂Y 2 4 ∂ξ 4 ∂Y

(15)

(16)

1 ∂φ1 1 ∂φ1 ∂φ2 1 φ1 + ξ − Y + = 0, 2 4 ∂ξ 4 ∂Y ∂Y ∂φ ∂U 1 ∂φ1 1 ∂φ1 1 ∂U 1 1 ∂ 2 φ1 1 ξU + V − YU −ξ − ξφ1 − φ2 − Y φ1 = , 4 ∂ξ 4 ∂Y ∂Y 4 ∂ξ 4 ∂Y P rm ∂Y 2 ∂θ 1 ∂θ 1 ∂θ ξU + V − YU −ξ 4 ∂ξ 4 ∂Y ∂Y ∂2θ 1 4 4 1 2 ∂θ = 1+ (1 + (θw − 1)θ)3 + ∆ (1 + ∆θ) . Pr 3Rd ∂Y 2 P r Rd ∂Y

(17) (18)

(19)

The appropriate boundary conditions satisfying the above equations are (

U (ξ, 0) = V (ξ, 0) = 0, U (ξ, ∞) = 0,

φ1 (ξ, 0) = 1,

φ1 (ξ, ∞) = 0,

φ2 (ξ, 0) = 0,

θ(ξ, ∞) = 0.

θ(ξ, 0) = 1,

(20)

Now, we first discretize (15)–(19) and the boundary conditions (20) by using the FDM and central difference for convective terms, out of which we get a system of tri-diagonal algebraic equations as follows: U − U 1 U Vi+1,j − Vi−1,j 1 i,j i,j−1 i+1,j − Ui−1,j Ui,j + ξi − Yj + = 0, (21) 2 ∆ξ 4 2∆Y 2∆Y U − U U U 1 2 1 1 i,j i,j−1 i+1,j − Ui−1,j i+1,j − Ui−1,j Ui,j + ξUi,j + Vi,j − Yj Ui,j − ξi 2 4 ∆ξ 4 2∆Y 2∆Y Ui+1,j − 2Ui,j + Ui−1,j 1 1 φ1i,j−1 − φ1i,j = θi,j + + S φ21i,j + ξi φ1i,j 2 ∆Y 2 4 ∆ξ 1 φ1i+1,j − φ1i−1,j + φ2i,j − Yj φ1i,j , (22) 4 2∆Y 1 1 φ1i,j−1 − φ1i,j 1 φ1i+1,j − φ1i−1,j φ2i+1,j − φ2i−1,j φ1i,j + ξi − Yj + = 0, (23) 2 4 ∆ξ 4 2∆Y 2∆Y


737

φ φ φ 1 1 1i,j−1 − φ1i,j 1i+1,j − φ1i−1,j 1i+1,j − φ1i−1,j ξi Ui,j + Vi,j − Yj Ui,j − ξi 4 ∆ξ 4 2∆Y 2∆Y 1 1 Ui+1,j − Ui−1,j − ξi φ1i,j Ui,j−1 − Ui,j ∆ξ − φ2i,j − Yj φ1i,j 4 4 2∆Y 1 φ1i+1,j − 2φ1i,j + φ1i−1,j = , (24) P rm ∆Y 2 θ θ θ 1 1 i,j−1 − θ i,j i+1,j − θ i−1,j i+1,j − θ i−1,j ξi Ui,j + Vi,j − Yj Ui,j − ξi 4 ∆ξ 4 2∆Y 2∆Y θ1i+1,j − 2θ1i,j + θ 1i−1,j 1 4 1+ (1 + (θw − 1)θ)3 . (25) = Pr 3Rd ∆Y 2 The discretized boundary conditions are (

U (ξ, 0) = V (ξ, 0) = 0, U (ξ, ∞) = 0,

φ1 (ξ, 0) = 0,

φ1 (ξ, ∞) = 0,

φ′1 (ξ, 0) = 0,

θ(ξ, 0) = 1,

θ(ξ, ∞) = 0.

(26)

These tri-diagonal equations are then solved by the Gaussian elimination technique. The computation is started at X = 0, and then marches the downstream implicitly. Here, we taken ∆ξ = 0.005 and ∆Y = 0.01 for the ξ and Y grids and the maximum value of y as 80 in the following computation. It takes 28.9 seconds CPU run time with the error tolerance ε = 0.000 01 for the convergence of the obtained numerical solutions. Throughout, the solutions are obtained for smaller values of the Prandtl number P r, the magnetic Prandtl number P rm , the magnetic force parameter S, the radiation parameter Rd , and the surface temperature θw , which are appropriate for liquid metal often used in nuclear cooling systems. Finally, the solutions are obtained for different values of pertinent physical parameters, namely, the magnetic field parameter S, the magnetic Prandtl number P rm , the Prandtl number P r, the radiation parameter Rd , and the surface θw . The values of P r taken here are so small that they are appropriate for liquid metal, often used as coolant in the nuclear devices. The results −3/4 are obtained in the coefficients of the skin friction GrL x−1/4 Cf , the rate of heat transfer 1/4 −3/4 GrL x1/4 N ux , and the current density GrL x−1/4 Jw defined in (27). The effects of different physical parameters are also obtained in the forms of the velocity, the temperature, and the transverse component of the magnetic field as shown graphically in Figs. 6–10. Once we know the solutions to (15)–(19), we can measure the physical quantities such as the coefficients of the skin friction, the rate of heat transfer, and the current density from the relation given below, which are important from the application point of view, from the following dimensionless expressions:  ∂u −3/4 −1/4   Gr x C = ,  f L   ∂Y Y =0    4 3 ∂θ 1/4 GrL x1/4 N ux = − 1 + θ ,  3Rd w ∂Y Y =0    ∂φ   1   GrL−3/4 x−1/4 Jw = . ∂Y Y =0

(27)

In the following section, the solutions to small and large local transpiration parameters ξ are obtained.

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3.2 Stream function formulation To get the numerical solutions for small and large local transpiration parameters for the steady state equations, we re-define the flow variables as follows:  3 1 1 1 1 1 ∂f   f (Y ) − Y f ′ (Y ) + ξ +ξ ,  Y = x− 4 y, u = x 2 f ′ (Y ), v = −x− 4   4 4 4 ∂ξ     1 ∂θ 1 ′ 1 ′ (28) θ = x−1 ξ − Y θ , θy = x− 4 θ ,  x 4 ∂ξ 4     3  1 1 1 1 1 ∂φ   φ(Y ) − Y φ′ (Y ) + ξ , ξ = V0 x 4 ,  Hx = x 2 φ′ (Y ), Hy = −x− 4 4 4 4 ∂ξ

which reduces the set of equations to

3 3 1 1 f ′′′ + f f ′′ − f ′2 + θ − S φφ′′ − φ′2 + ξf ′′ 4 2 4 2 ′ ′ 1 ∂f ∂f ∂φ ∂φ = ξ f′ − f ′′ − S φ′ − φ′′ , 4 ∂ξ ∂ξ ∂ξ ∂ξ 1 ′′′ 3 ′′ 3 ′′ 1 ∂φ′ ∂f ′ ∂φ ∂f φ + f φ − f φ + ξφ′′ = ξ f ′ − φ′ + f ′′ − φ′′ , P rm 4 4 4 ∂ξ ∂ξ ∂ξ ∂ξ 1 4 3 ′ 1 ∂θ ′ ′ ′ ′ ∂f 1+ (1 + (θw − 1)θ)3 θ + f θ + ξθ = ξ f ′ −θ . Pr 3Rd 4 4 ∂ξ ∂ξ The boundary conditions satisfying the above equations are ( f (ξ, 0) = f ′ (ξ, 0) = 0, φ(ξ, 0) = 0, φ′ (ξ, 0) = 1, f ′ (ξ, ∞) = 0,

φ′ (ξ, ∞) = 0,

θ(ξ, 0) = 1,

θ(ξ, ∞) = 0.

(29) (30) (31)

(32)

3.2.1 Small ξ Since ξ is small (ξ ≪ 1) near the leading edge, we can expand all the depending functions in powers of ξ. Accordingly, we consider that f (ξ, Y ) =

∞ X

ξ i fi (Y ),

φ(ξ, Y ) =

i=0

∞ X

ξ i φi (Y ),

θ(ξ, Y ) =

∞ X

ξ i θi (Y ).

(33)

i=0

i=0

Substituting the above equations into (29)–(31) and taking the term up to O(ξ), the following sets of equations can be obtained: (i) O(ξ 0 ) 3 3 1 1 f0′′′ + f0 f0′′ − f0′2 + θ0 − S φ0 φ′′0 − φ′2 = 0, 4 2 4 2 0 1 ′′′ 3 3 φ + f0 φ′′0 − f0′′ φ0 = 0, P rm 0 4 4 3 (1 + α(1 + ∆θ0 )3 )θ0′′ + 3α∆(1 + ∆θ0 )2 θ0′2 + P rf0 θ0′ = 0. 4 The boundary conditions are ( f0 (0) = f0′ (0) = 0, f0′ (∞)

= 0,

φ′0 (∞)

φ0 (0) = 0, = 0,

φ′0 (0) = 1,

θ0 (∞) = 0.

θ0 (0) = 1,

(34) (35) (36)

(37)


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(ii) O(ξ 1 ) 3 5 f1′′′ + (f0 f1′′ − Sφ0 φ′′1 ) + (f0′′ f1 − Sφ′′0 φ1 ) − (f0′ f1′ − Sφ′0 φ′1 ) + θ1 + f0′′ = 0, 4 4 1 ′′′ 3 5 3 5 1 1 φ1 + f0 φ′′1 + f1 φ′′0 − f0′′ φ1 − f0′′ φ1 − f0′ φ′1 + f1′ φ′0 + φ′′0 = 0, P rm 4 4 4 4 4 4 (1 + α(1 + ∆θ0 )3 )θ1′′ + 3α∆(1 + ∆θ0 )2 (θ1 θ0′′ + 2θ0′ θ1′ ) 3 1 + 6α∆2 θ1 (1 + ∆θ0 )θ0′2 + P r f0 θ1′ + f1 θ0′ − f0′ θ1 + θ0′ = 0. 4 4

The related order boundary conditions are ( f1 (0) = f1′ (0) = 0, φ1 (0) = 0, f1′ (∞)

φ′1 (∞)

= 0,

= 0,

φ′1 (0) = 0,

(38) (39)

(40)

θ1 (0) = 0,

(41)

θ1 (∞) = 0.

Equations (31)–(40) are nonlinear coupled equations. The solutions to these equations can be obtained by the Nactsheim-Swigert iteration technique together with the six-order implicit Runge-Kutta-Butcher initial value solver. We can calculate the values of the coefficients of the skin friction, the rate of heat transfer, and the current density at the surface in the region near the leading edge against ξ from the following expressions: −3/4 −1/4

Cf = f ′′ (0),

(42)

Jw = φ′′ (0), 4 3 ′ 1/4 GrL x1/4 N ux = − 1 + θ θ (0). 3Rd w

(43)

GrL

x

−3/4 −1/4

GrL

x

(44)

The results obtained with the help of (42)–(44) are given in Tables 1–3 for small values of ξ. Here and after here, † and †† stand for small and large values of ξ, respectively. It is found that as the surface temperature θw increases, the coefficients of the skin friction and the rate of heat transfer increase in the downstream regime, while the coefficient of the current density changes slightly throughout the entire regime. From this physical phenomena, we can observe that with the increase in the surface temperature θw , the ambient fluid temperature in the domain of the fluid flow decreases. By the Fourier law of heat transfer, the flow direction of heat transfer is towards down the stream regime, where we can see the changes in the coefficients of the skin friction and the rate of heat transfer with good margin, and the coefficient of the current density is dominated. From these tables, we can also claim that the results obtained by both the two methods are in excellent agreement. Table 1

ξ 0.0 0.1 0.5 1.0 2.0 4.0 8.0 10.0

−3/4

Numerical values of GrL x−1/4 Cfx obtained for θw = 0.5, 1.5, 2.5 against ξ with P rm = 0.1, Rd = 2 0.0, S = 0.2, and P r = 0.1 by two methods θw = 0.5

θw = 1.5

θw = 2.5

FDM

Asymptotic

FDM

Asymptotic

FDM

Asymptotic

1.929 58 2.017 64 2.458 30 2.970 60 3.663 65 3.226 90 1.296 61 1.011 53

1.920 21† 1.942 59† 2.054 52† 2.944 31† – – 1.265 62†† 1.010 00††

1.937 40 2.026 28 2.471 16 2.990 06 3.701 20 3.317 10 1.359 90 1.060 63

1.928 33† 1.952 15† 2.071 27† 2.922 01† – – 1.316 01†† 1.010 00††

1.944 80 2.034 28 2.483 07 3.008 07 3.735 92 3.400 79 1.422 97 1.109 71

1.971 00† 2.249 32† 2.510 93† 3.083 75† – – 1.412 65†† 1.010 00††

740 Table 2

ξ

M. ASHRAF, S. ASGHAR, and M. A. HOSSAIN 1/4

Numerical values of GrL x1/4 N ux obtained for θw = 0.5, 1.5, 2.5 against ξ with P rm = 0.1, Rd = 20.0, S = 0.2, and P r = 0.1 by two methods θw = 0.5

θw = 1.5

θw = 2.5

FDM

Asymptotic

FDM

Asymptotic

FDM

Asymptotic

0.0 0.1 0.5 1.0 2.0 4.0 8.0 10.0

0.151 53 0.155 06 0.172 47 0.194 13 0.240 00 0.367 08 0.775 46 0.972 61

0.145 48† 0.153 74† 0.165 05† 0.194 19† – – 0.808 24†† 1.000 00††

0.149 48 0.152 84 0.169 39 0.189 92 0.233 21 0.352 33 0.739 07 0.927 21

0.145 65† 0.154 09† 0.164 30† 0.186 55† – – 0.781 87†† 0.956 72††

0.147 58 0.150 79 0.166 57 0.186 08 0.227 05 0.339 06 0.705 94 0.885 84

0.142 28† 0.150 88† 0.163 91† 0.185 20† – – 0.756 31†† 0.943 34††

Table 3

Numerical values of GrL x−1/4 Jw obtained for θw = 0.5, 1.5, 2.5 against ξ with P rm = 0.1, Rd = 20.0, S = 0.2, and P r = 0.1 by two methods

ξ 0.0 0.1 0.5 1.0 2.0 4.0 8.0 10.0

−3/4

θw = 0.5

θw = 1.5

θw = 2.5

FDM

Asymptotic

FDM

Asymptotic

FDM

Asymptotic

0.012 03 0.015 48 0.039 25 0.080 08 0.178 98 0.390 06 0.797 67 0.998 29

0.015 69† 0.017 07† 0.034 00† 0.080 16† – – 0.800 00†† 1.000 00††

0.011 19 0.014 61 0.038 27 0.079 03 0.177 91 0.389 19 0.797 56 0.998 21

0.017 50† 0.017 08† 0.032 99† 0.075 39† – – 0.800 00†† 1.000 00††

0.010 40 0.013 80 0.037 35 0.078 05 0.176 92 0.388 36 0.797 33 0.998 12

0.012 06† 0.010 76† 0.031 94† 0.078 27† – – 0.800 00†† 1.000 00 ††

3.2.2 Large ξ Now, attention is given in finding the solution to (29)–(31) along with the boundary conditions (32) when ξ is large. The order of the magnitude analysis of various terms in these equations shows that the largest terms are f ′′′ and ξf ′′ in (29), φ′′′ and ξφ′′ in (30), and θ′′ and ξθ′ in (31). In the respective equations, both terms have to be balanced in the magnitude, and the only way to do this is to assume that η is small so that its derivative is large. It is essential to find the appropriate scaling for f , φ, θ, and η. On balancing f ′′′ and ξf ′′ in (29), φ′′′ and ξφ′′ in (30), and θ′′ and ξθ′ in (31), it is found that η = O(ξ −1 ), f = O(ξ −3 ), and φ = O(ξ −3 ). Therefore, the following transformations can be introduced: ( Y = ξ −1 η, f (ξ, Y ) = ξ −3 F (ξ, η), (45) φ(ξ, Y ) = ξ −3 Φ(ξ, η), θ(ξ, Y ) = Θ(ξ, η). Then, by using (45) in (29)–(31), we can obtain the following set of equations: ∂F ′ ∂Φ′ 1 ∂F ∂Φ − F ′′ −S Φ − Φ′′ , F ′′′ + F ′′ + Θ = ξ −3 F ′ 4 ∂ξ ∂ξ ∂ξ ∂ξ ∂Φ′ 1 1 ∂F ′ ∂Φ ∂F Φ′′′ + Φ′′ = ξ −3 F ′ − Φ′ + F ′′ − Φ′′ , P rm 4 ∂ξ ∂ξ ∂ξ ∂ξ ′ ∂Θ 1 4 1 ∂F 1+ (1 + (θw − 1)Θ)3 Θ′ + Θ′ = ξ −3 F ′ − Θ′ . Pr 3Rd 4 ∂ξ ∂ξ

The boundary equations satisfying the above equations are ( F (ξ, 0) = F ′ (ξ, 0) = 0, Φ(ξ, 0) = 0, Φ′ (ξ, 0) = 1, ′

F (ξ, ∞) = 0,

′

Φ (ξ, ∞) = 0,

Θ(ξ, ∞) = 0.

Θ(ξ, 0) = 1,

(46) (47) (48)

(49)


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Now, expanding the functions F , Φ, and Θ in powers of ξ −3 by substituting (50) into (45)–(48), F (ξ, η) =

1 X

ξ

−3m

Fm (η),

Φ(ξ, η) =

m=0

1 X

ξ

−3m

Φm (η),

Θ(ξ, η) =

m=0

1 X

ξ −3m Θm (η),

(50)

m=0

and equating the coefficients of the equal powers of ξ from both sides, we have F0′′′ + F0′′ + Θ0 = 0,

(51)

Φ′′′ 0

(52)

+

P rm Φ′′0

= 0,

′ (1 + α(1 + ∆Θ0 )3 )Θ′′0 + 3α∆(1 + ∆Θ0 )2 Θ′2 0 + P rΘ0 = 0.

The boundary conditions are ( F0 (0) = F0′ (0) = 0, F0′ (∞)

= 0,

Φ0 = 0,

Φ0 (∞) = 0,

Φ′0 (0) = 1,

Θ0 (0) = 1,

Θ0 (∞) = 0,

(53)

(54)

from which we can see that F1′′′ + F1′′ + Θ1 = 0,

(55)

′′ Φ′′′ 1 + P rm Φ1 = 0, 3

(1 + α(1 + ∆Θ0 )

)Θ′′1

(56) 2

+ 3α∆(1 + ∆Θ0 )

(Θ1 Θ′′0

′ + 6α∆2 Θ1 (1 + ∆Θ0 )Θ′2 0 + P rΘ1 = 0, ( F1 (0) = F1′ (0) = 0, Φ1 = 0, Φ′1 (0) = 0,

F1′ (∞)

= 0,

Φ1 (∞) = 0,

+

2Θ′0 Θ′1 ) (57)

Θ1 (0) = 0,

Θ1 (∞) = 0.

(58)

The solution obtained by these equations enables us to calculate the solutions to different parameters for large values of ξ from the following expressions: −3/4 −1/4

GrL

x

−3/4 −1/4

Cf = F ′′ (0),

Jw = Φ′′ (0), 4 3 ′ 1/4 GrL x1/4 N ux = − 1 + θ Θ (0). 3Rd w GrL

x

(59) (60) (61)

The results obtained by (59)–(61) are given in Tables 1–3 for large values of ξ, and compared with the solutions obtained by the FDM, showing to be in good agreement.

4

Results and discussion

Equations (21)–(25) along with the boundary conditions (26) are numerically solved for all values of the transpiration parameter ξ by using the FDM. Equations (29)–(31) and the boundary conditions (32) sufficiently near to the plate and away from the plate are solved by using the asymptotic series solutions. Later, to test the accuracy, the results obtained by the FDM are compared with the results obtained by the asymptotic series solution. They are found to be in excellent agreement. We shall now give a brief discussion on the effects of the Prandtl number P r, the magnetic force parameter S, the magnetic parameter P rm , the radiation parameter −3/4 Rd , and the surface temperature θw on the coefficients of the skin friction GrL x−1/4 Cf , the 1/4 −3/4 rate of heat transfer GrL x1/4 N ux , and the current density GrL x−1/4 Jw in Section 4.1. The detail of the velocity profile, the temperature distribution, and the transverse component of the magnetic field for varying parameters P r, P rm , S, Rd , and θw for different values of the transpiration parameter ξ is given in Section 4.2.

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4.1

Effect of physical parameters on skin friction, magnetic intensity, and rate of heat transfer Figures 2(a)–2(c) illustrate the influence of different values of the radiation parameter Rd on the coefficients of the skin friction, the rate of heat transfer, and the current density at the surface. From these figures, it is shown that the coefficient of the skin friction decreases and the coefficient of the rate of heat transfer increases at very gross margin, while the coefficient of KαR the current density increases very slightly up to ξ = 100 . The relationship of Rd = 4σT 3 means ∞ that with the increase in Rd , the product of the thermal diffusivity and roseland absorption coefficient increase, and the ambient fluid temperature decreases, which slow down the motion of the fluid. Therefore, the coefficient of the skin friction decreases in the downstream regime very grossly, and the coefficient of the rate of heat transfer increases due to natural convection. The variation in Rd therefore generally has no inhibiting effect on the development of the current density.

Fig. 2

Coefficients at surface against ξ for Rd = 1.0, 2.5, 5.0, 10.0 with P rm = 0.1, P r = 0.1, S = 0.1, and θw = 1.1

It is shown in Figs. 3(a)–3(c) that with the increase in the magnetic force parameter S, the coefficient of the skin friction increases actively in the middle range of the surface of the plate, while the coefficients rate of heat transfer and the current density increase at very low margin near the surface of the plate and remain the same after ξ = 100 . The reason is that with the increase in the magnetic force parameter S, the magnetic energy increases, which extracts the kinetic energy of the fluid and results the increases in the coefficients of the skin friction, the rate of heat transfer, and the current density.

Fig. 3

Coefficients at surface against ξ for S = 0.0, 0.3, 0.6, 0.9 with P rm = 0.1, Rd = 1.0, P r = 0.1, and θw = 1.1


743

Figures 4(a)–4(c) illustrates the response of different values of the magnetic Prandtl number P rm on the coefficients of the skin friction, the rate of heat transfer, and the current density. It can be seen that the coefficient of the skin friction decreases slightly in the downstream regime and the coefficient of the rate of heat transfer remains unchanged. We can also see that the coefficient of the current density decreases very actively, having its maximum response for P rm = 0.001 and its exactly zero value for P rm = 0.1. These happen because with the increase in the magnetic Prandtl number P rm , the induced current within the boundary layer tends to spread away from the surface, which results in thickening the momentum and the magnetic field boundary layer thickness. However, in the case of heat transfer, this factor is not dominant in the flow domain. With this reason, the coefficients of the skin friction and the current density decrease, while the coefficient of the rate of heat transfer remains unchanged.

Fig. 4

Coefficients at surface against ξ for P rm = 0.001, 0.010, 0.050, 0.100 with Rd = 1.0, P r = 0.1, S = 0.1, and θw = 1.1

Finally, in Figs. 5(a)–5(c), we show the effects of the variation of the Prandtl number P r on the physical quantities such as the coefficients of the skin friction, the rate of heat transfer, and the current density at the surface. We have examined that the coefficient of the skin friction decreases moderately near the surface but grossly decreases in the down stream regime, the coefficient of the rate of heat transfer increases very slowly in the upstream regime but this margin extends in the downstream regime, and the coefficient of the current density increases slightly in the upstream regime. The reason is that the increase in the value of P r corresponds to the increase in the kinematic viscosity of the fluid and reduces the thermal diffusion. It is very interesting to note that the increase in the kinematic viscosity leads to the increase in the momentum boundary layer thickness and the decrease in the thermal diffusion

Fig. 5

Coefficient of at surface against ξ for P r = 0.01, 0.05, 0.08, 0.10 with Rd = 10.0, P rm = 0.1, S = 0.1, and θw = 1.1

744


which gives a thin thermal boundary layer thickness. These are responsible for the aforementioned phenomena. 4.2 Effects of physical parameters on velocity, temperature, and transverse component of magnetic field The velocity, temperature, and transverse component of the magnetic field distributions obtained by the FDM for various values of the transpiration parameter ξ are displayed in Figs. 6–10. The aim of these figures is to display how the profile vary in ξ. The transpiration parameter ξ in the present investigation is taken as positive for suction. It is shown that the values of the velocity, temperature, and transverse component of the magnetic field decrease in the magnitude as ξ increases. This phenomena establish the very strong reason that the suction slows down the motion of the fluid in the down stream regime and the values of the aforementioned physical quantities decrease. Thus, the numerical results in Figs. 6(a)–6(c) indicate that the momentum, thermal, and magnetic field boundary layer thicknesses decrease as ξ = 1.0, 3.0, 5.0, 8.0, 10.0 increases for Rd = 1.0, 10.0, P r = 0.1, P rm = 0.1, S = 0.1, and θw = 1.1. It can be seen that with the increase in the radiation parameter Rd in the fluid, the values of the velocity, temperature, and transverse component of the magnetic field decrease, which lead to the decreases in the momentum, thermal, and magnetic field boundary layer thicknesses. The variation in the magnetic force parameter S = 0.0, 0.4 for the case of suction increases the values of the momentum and magnetic field profiles, and there is no change seen in the temperature distribution, which is expected because the direction of the magnetic field is in favor of the flow, as shown in Figs. 7(a)–7(b).

Fig. 6

Magnetic field for ξ = 1.0, 3.0, 5.0, 8.0, 10.0 against Y for Rd = 1.0, 10.0 with P rm = 0.1, S = 0.1, P r = 0.1, and θw = 1.1

Fig. 7

Magnetic field for ξ = 1.0, 3.0, 5.0, 8.0, 10.0 against Y for S = 0.0, 0.4 with P rm = 0.1, Rd = 1.0, P r = 0.1, and θw = 1.1


745

In Figs. 8(a)–8(b), we can see the effect of the Prandtl number P r = 0.01, 0.1 by keeping other parameters fixed. It is noted that the values of the velocity decrease slightly, the temperature distribution decreases and is separated into regions, and the transverse component of magnetic field remains the same. This is because that the role of P r in the magnetic field equation (24) is not very prominent. The effects of varying the magnetic Prandtl number P rm = 0.01, 0.1 for P r = 0.1, S = 0.1, Rd = 1.0, and θw = 1.1 on the velocity, temperature, and transverse component of the magnetic field are shown in Figs. 9(a)–9(b). It is clear from these figures that with the increase in the magnetic Prandtl number P rm , the value of the velocity profile increases slightly, the temperature distribution remains the same, and the transverse component of the magnetic field decreases drastically and is separated into two parts in the flow domain. Whereas from Figs. 10(a)–10(b), we can see that with the increase in the surface temperature θw , the values of the velocity and temperature distributions increase, which results the increases in the momentum and thermal boundary layer thicknesses. The transverse component of the magnetic field increases for ξ = 1.0, and remains the same for other values of the transpiration parameter ξ.

Fig. 8

Magnetic field for ξ = 1.0, 3.0, 5.0, 8.0, 10.0 against Y for P r = 0.01, 0.10 with Rd = 1.0, S = 0.1, P rm = 0.1, and θw = 1.1

Fig. 9

Magnetic field for ξ = 1.0, 3.0, 5.0, 8.0, 10.0 against Y for P rm = 0.01, 0.10 with P r = 0.1, S = 0.1, Rd = 1.0, and θw = 1.1

5

Conclusions

In summing up what has been discussed above, we remarke the effects of different physical parameters such as the radiation parameter Rd , the magnetic force parameter S, the Prandtl

746

Fig. 10


Magnetic field for ξ = 1.0, 3.0, 5.0, 8.0, 10.0 against Y for θw = 0.5, 1.1 with P r = 0.1, S = 0.1, Rd = 1.0, and P rm = 0.1

number P r, the magnetic Prandtl number P rm , and the surface temperature θw on the coef−3/4 1/4 ficients of the skin friction GrL x−1/4 Cf , the rate of heat transfer GrL x1/4 N ux , and the −3/4 current density GrL x−1/4 Jw as follows. (i) The coefficient of the skin friction decreases and the coefficient of the rate of heat transfer increases in the downstream regime, while the coefficient of the current density increases slightly up to the value of ξ=1 with the increase in the radiation parameter Rd . (ii) The increase in the magnetic force parameter S increases the coefficient of the skin friction actively in the middle range of the flow domain, while increases the coefficient of the rate of heat transfer and current density at a very low difference near the surface of the plate. (iii) With varying the magnetic Prandtl number P rm , the coefficient of the skin friction decreases slightly in the downstream regime, the coefficient of the rate of heat transfer remains unchanged, while the coefficient of the current density decreases very actively. (iv) With the increase in the Prandtl number P r, the coefficient of the skin friction decreases, while the coefficients of the rate of heat transfer and the current density increase. (v) With the increase in the surface temperature θw , the coefficients of the skin friction and the rate of heat transfer increase, and the coefficient of the current density changes slightly. (vi) The velocity, temperature, and transverse component of the magnetic field distributions decrease in the magnitude as the transpiration parameter ξ increases. The numerical results indicate that the momentum, thermal, and magnetic field boundary layer thicknesses decrease for the case of suction for two different values of the radiation parameter Rd . (vii) The variation in the magnetic force parameter S for the case of suction leads to the decrease in the value of the momentum and magnetic field profile with no change in the temperature distribution. (viii) With the increase in the Prandtl number P r, the values of the velocity profile decreases slightly, the temperature distribution decreases and is separated into two regions in the flow domain, and there is no change seen in the transverse component of the magnetic field. (ix) With the increase in the magnetic Prandtl number P rm , the values of the velocity profile increase slightly, and the temperature distribution remains unchanged, while the transverse component of the magnetic field decreases and is separated into two regions in the flow domain. (x) With the increase in the surface temperature θw , the values of the velocity and temperature distributions increase. Thus, the momentum and thermal boundary layer thicknesses increase. The transverse component of the magnetic field increases very slowly only for ξ = 1.0. Acknowledgements

The authors are grateful to the reviewers for their valuable suggestions to improve this article. One of the author M. ASHRAF is also grateful to Department of Mathematics of COMSATS Institute of Information Technology for the award of research fellewship.


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