Appl. Math. Mech. -Engl. Ed., 32(9), 1127–1146 (2011) DOI 10.1007/s10483-011-1487-9 c Shanghai University and Springer-Verlag Berlin Heidelberg 2011

Applied Mathematics and Mechanics (English Edition)

Effect of Hall current on MHD natural convection flow from vertical permeable flat plate with uniform surface heat flux∗ L. K. SAHA1 , S. SIDDIQA2 ,

M. A. HOSSAIN2

(1. Department of Mathematics, University of Dhaka, Dhaka 1000, Bangladesh; 2. Department of Mathematics, COMSATS Institute of Information Technology, Park Road, Chak Shahzad, Islamabad 44000, Pakistan)

Abstract The effect of the Hall current on the magnetohydrodynamic (MHD) natural convection flow from a vertical permeable flat plate with a uniform heat flux is analyzed in the presence of a transverse magnetic field. It is assumed that the induced magnetic field is negligible compared with the imposed magnetic field. The boundary layer equations are reduced to a suitable form by employing the free variable formulation (FVF) and the stream function formulation (SFF). The parabolic equations obtained from FVF are numerically integrated with the help of a straightforward finite difference method. Moreover, the nonsimilar system of equations obtained from SFF is solved by using a local nonsimilarity method, for the whole range of the local transpiration parameter ζ. Consideration is also given to the regions where the local transpiration parameter ζ is small or large enough. However, in these particular regions, solutions are acquired with the aid of a regular perturbation method. The effects of the magnetic field M and the Hall parameter m on the local skin friction coefficient and the local Nusselt number coefficient are graphically shown for smaller values of the Prandtl number P r (= 0.005, 0.01, 0.05). Furthermore, the velocity and temperature profiles are also drawn from various values of the local transpiration parameter ζ. Key words magnetohydrodynamic (MHD) natural convection, Hall current, permeable plate, heat flux Chinese Library Classification O361.3 2010 Mathematics Subject Classification

65F10, 65N55, 76W05

Nomenclature B0 , Cfx , e, g, Grx , J, m, M, κ, τ,

magnetic induction; local skin friction; electronic charge; gravitational acceleration; modified Grashof number; electric current density; Hall parameter; magnetic parameter; thermal conductivity; collision time of electrons with ions;

N ux , p, P r, T, T∞ , x, y, z, u, v, w, V0 , qw ,

local Nusselt number; pressure; Prandtl number; temperature of the fluid; free stream temperature; coordinate directions; velocity components in the x-, y-, and z-directions; transpiration velocity; uniform wall heat flux;

∗ Received Sept. 17, 2010 / Revised May 30, 2011 Corresponding author M. A. HOSSAIN, Professor, Ph. D., E-mail: [email protected]

1128 ω,

L. K. SAHA, S. SIDDIQA, and M. A. HOSSAIN cyclotron frequency of electron.

Greek symbols α, β, ψ, θ,

thermal diffusivity; volumetric expansion coefficient for temperature; stream function; dimensionless temperature function;

ρ, ν, μ, ζ, η,

density; kinematic viscosity; dynamic viscosity; local transpiration parameter; pseudo similarity variable.

∞,

conditions far away from wall.

Subscripts w,

1

conditions at wall;

Introduction

The study of magnetohydrodynamic (MHD) viscous flows with the Hall current has important applications in problems of the Hall accelerators as well as in the flight magnetohydrodynamics. The current trend for the application of magnetohydrodynamics is towards a strong magnetic field (so that the influence of the electromagnetic force is noticeable) and a low density of gas (such as in the space flight and in the nuclear fusion research). Under this condition, the Hall current and ion slip become important. With the above understanding, Sato[1] , Yamanishi[2] , and Sherman and Sutton[3] studied the hydrodynamic flow of a viscous liquid through a straight channel, taking the Hall effect into account. With regards to external hydrodynamic flows, Katagiri[4] discussed the effect of the Hall current on the boundary layer flow past a semi-infinite plate. All of the above authors considered the one-dimensional flow. Pop and Watanabe[5] presented the problem of the free convection flow of a conducting fluid permeated by a transverse magnetic field in the presence of the Hall effects. Considering a uniform magnetic field, the authors reduced the model equations to a system of parabolic partial differential equations, and assumed that ξ (= M x1/2 ) (where M is the magnetic field, and x is the axial distance measured from the leading edge) is the local variable which lies between 0 and 1. Further, Aboeldahab and Elbarbary[6] studied the effect of the Hall current on the MHD free convection flow in the presence of foreign species over a vertical surface, upon which the flow is subjected to a strong external magnetic field. Suction and injection of a different kind of gas through a surface are important in controlling the boundary layer thickness. The rate of the heat transfer has motivated many researchers to investigate its effects on forced and free convection flows. Eichhorn[7] investigated the similarity solution by considering the power-law variations in the plate temperature and transpiration velocity. Further, the free convection flow along a vertical plate with the arbitrary blowing and wall temperature was investigated by Vedhanayagam et al.[8] . The free convection flow over a horizontal plate was investigated by Lin and Yu[9] , in which they considered the temperature and transpiration rates both followed the power-law variations. Moreover, Hossain et al.[10] investigated the natural convection flow from a vertical permeable flat plate with the variable surface temperature, considering the temperature and transpiration rates to follow the powerlaw variation. The study on the effect of the Hall current on the MHD natural convection boundary layer flow past a permeable surface is important from an application point of view in the space flight and the nuclear fusion. Therefore, recently, Saha et al.[11] studied the effect of Hall current on the steady laminar natural convection boundary layer flow of MHD viscous and incompressible fluids from a semi-infinite heated permeable vertical flat plate with an applied magnetic field transversing to it. In this analysis, the transformed boundary layer equations were integrated by employing two distinct methods, namely, the implicit finite difference method together with

Effect of Hall current on MHD natural convection flow

1129

the Keller-box scheme and the local nonsimilarity method for all values of the local transpirax tion parameter ξ = V0 1/4 and for smaller values of the Prandtl number P r (= 0.1, 0.01) that νGrx represent liquid metals. They also obtained the asymptotic solutions to small and sufficently large values of the local transpiration parameter ξ. Moreover, the effects of the Hall current on the unsteady hydromagnetic free convection flow along a porous vertical flat plate with mass diffusion were studied analytically by Hossain and Rashid[12] . In the present investigation, the effect of the Hall current on the MHD natural convection boundary layer flow past a semi-infinite vertical permeable flat plate with a uniform mass flux is studied because of its important application in the space flight and the nuclear fusion. The transformed boundary layer equations are solved near and far from the leading edge with the help of the regular perturbation method. In order to obtain the solution to intermediate locations, the boundary layer equations are first transformed into a convenient form by using the free variable formulation (FVF) and the stream function formulation (SFF). The transformed system of equations obtained via FVF is numerically integrated with the help of the straightforward finite difference method, while the system of equations obtained from SFF is simulated with the local nonsimilarity method. The effects of the pertinent parameters, such as M and m for smaller values of P r (= 0.05, 0.01, 0.005 which are appropriate for liquid metals that are currently used in the nuclear engineering as coolant, e.g., the values of the Prandtl number, for lithium and mercury are found to be 0.05 and 0.01, respectively) on the local skin friction coefficient and the local Nusselt number coefficient at the surface, are discussed in detail. Representative results for the velocity and temperature distributions are also presented for various values of the transpiration parameter ζ.

2

Problem formulation

Consider the steady natural convection boundary layer flow of an electrically conducting and viscous incompressible fluid from a semi-infinite heated permeable vertical flat plate maintained with a uniform surface heat flux in the presence of a transverse magnetic field with the effect of the Hall current. The x-axis is along the vertically upward direction, while the y-axis is normal to it. The leading edge of the permeable surface is taken to be coincident with the z-axis. The ambient fluid temperature is maintained at the uniform temperature T∞ . It is assumed that the uniform heat is supplied from the surface of the plate to the fluid, which is maintained throughout the fluid flow at the uniform rate qw . Furthermore, we consider a uniform mass flux V0 through the permeable vertical surface of the plate. The flow configuration and the coordinate system are shown in Fig. 1. It needs to be stated that the effect of the Hall current gives rise to a force in that direction. Hence, the flow becomes three-dimensional. To simplify the problem, we assume that the variations of the flow quantities in the z-direction are absent. This assumption is considered to be valid because the surface is of infinite extent in the z-direction. Although the pattern of electric currents in the extremities in the z-direction is considered to be unrealizable under the assumption, it might be expected to give an insight into the behavior of the Hall current on the flow. The fundamental equations for the steady incompressible MHD flow with the generalized Ohm’s law and Maxwell’s equations, under the assumptions that the fluid is quasi-neutral, and the ion slip and thermoelectric effects can be neglected, are given as follows (see [11]): ∂u ∂v + = 0, ∂x ∂y u

∂u ∂2u σB02 ∂u +v = ν 2 + gβ(T − T∞ ) − (u + mw), ∂x ∂y ∂y ρ(1 + m2 )

(1) (2)

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L. K. SAHA, S. SIDDIQA, and M. A. HOSSAIN

Fig. 1

Flow configuration and coordinate system

u

∂w ∂2w ∂w σB02 +v =ν 2 + (mu − w), ∂x ∂y ∂y ρ(1 + m2 )

(3)

u

∂T ∂2T ∂T +v =α 2. ∂x ∂y ∂y

(4)

The boundary conditions are ⎧ ⎨ u(x, y) = 0, ⎩

u(x, y) = 0,

v(x, y) = −V0 , w(x, y) = 0,

w(x, y) = 0, T (x, y) = T∞

∂T = −qw ∂y at y = ∞. κ

at y = 0,

(5)

Here, u, v, and w are the x, y, and z components of the velocity vector, respectively. p is the pressure, ρ is the density of the fluid, ν is the kinematic coefficient of viscosity, g is the acceleration due to gravity, β is the coefficient of thermal expansion, κ is the thermal conductivity, and T∞ is the ambient fluid temperature. m (= ω 2 τ 2 ) is the Hall parameter, where ω is the cyclotron frequency of the electron, and τ is the collision time of electrons with ions. Deriving the above equations, we assume that (i) the plate is nonconducting, and therefore the component Jy of the current density is zero at the plate and hence zero everywhere, (ii) the applied electric field E = 0 in the case of a short circuit problem, (iii) the magnetic Reynolds number is small, so that the induced magnetic field is negligible in comparison with the applied magnetic field B0 , and (iv) the fluid is isotropic and homogeneous, and has the scalar constant viscosity and electric conductivity. We introduce the following nondimensional dependent and independent variables: ⎧ 2 1 2 ν ν ν 5 5 5 ⎪ ⎪ ⎨ u = L Grx u, v = L Grx v, w = L Grx w, 1 2 y κ ⎪ ⎪ ⎩ y = Grx5 , θ =  Grx5 (T − T∞ ), L Lqw

(6)



4 w in which Grx = gβq κν 2 L is the dimensionless Grashof number that approximates the ratio of the buoyancy force to the viscous force acting on a fluid. Introducing Eq. (6) into the system of Eqs. (1)–(5) and dropping the over bars with brevity yield

∂u ∂v + = 0, ∂x ∂y

(7)

Effect of Hall current on MHD natural convection flow

1131

u

∂u ∂2u ∂u M +v = +θ− (u + mw), 2 ∂x ∂y ∂y 1 + m2

(8)

u

∂w ∂2w M ∂w +v = + (mu − w), ∂x ∂y ∂y 2 1 + m2

(9)

u

∂θ 1 ∂2θ ∂θ +v = . ∂x ∂y P r ∂y 2

(10)

The boundary conditions are ⎧ ⎨ u(x, y) = 0, v(x, y) = −S, w(x, y) = 0, ∂θ = −1 at y = 0, ∂y ⎩ u(x, y) = 0, w(x, y) = 0, θ(x, y) = 0 at y = ∞.

(11)

In Eqs. (7)–(11), M is the magnetic field parameter (or the modified Hartman number) that represents the ratio of the magnetic field strength to the viscous force, P r gives the ratio of the momentum diffusivity to the thermal diffusivity, and S is the transpiration parameter that measures the potential of withdrawal or blowing of the fluid. These parameters are defined as Pr =

ν , α

M=

σB02 L2 − 25 Grx , ρν

S=

1 V0 L Grx − 5 . ν

(12)

Now, we offer the solution methodologies for the problem formulated in the set of Eqs. (7)–(11).

3

Solution methodologies

Here, we propose the methods used to investigate the model defined in Eqs. (7)–(11). First, we adopt two formulations, namely, FVF and SFF, which reduce the boundary layer equations into the convenient form. The equations of FVF are integrated by employing the straightforward finite difference method, and the equations of SFF are simulated with the help of the local nonsimilarity method. Second, the solutions near and far from the surface of the plate are obtained by the regular perturbation method. 3.1 Free variable formulation FVF is initiated in Eqs. (7)–(11) before employing the numerical scheme for the solution to the problem. For this reason, consider the following continuous transformations: ⎧ ⎨ u = x 35 U (ζ, Y ), v = x− 15 (V (ζ, Y ) − ζ), w = x 35 W (ζ, Y ), (13) ⎩ θ = x 15 θ(ζ, Y ), Y = x− 15 y, ζ = Sx− 15 , where Y is the self similarity variable, and ζ is the streamwise distribution of the transpiration velocity. By introducing the above formulations, we get the following parabolic partial differential equations:  3 1  ∂U 1 ∂U ∂V U + V −ζ − Y + ζ + = 0, (14) 5 5 ∂Y 5 ∂ζ ∂Y  ∂U 3 2  1 ∂U ∂2U 1 M U + V −ζ − YU + ζU = +θ− ζ 2 (U + mW ), 2 5 5 ∂Y 5 ∂ζ ∂Y 1 + m2

(15)

  ∂W 3 1 1 ∂W ∂2W M UW + V − ζ − Y U + ζU = + ζ 2 (mU − W ), 5 5 ∂Y 5 ∂ζ ∂Y 2 1 + m2

(16)

  ∂θ 1 1 ∂θ 1 ∂2θ 1 Uθ + V − ζ − Y U + ζU = . 5 5 ∂Y 5 ∂ζ P r ∂Y 2

(17)

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L. K. SAHA, S. SIDDIQA, and M. A. HOSSAIN

The boundary conditions are ⎧ ⎨ U (ζ, Y ) = V (ζ, Y ) = W (ζ, Y ) = 0, ∂θ = −1 at Y = 0, ∂Y ⎩ U (ζ, Y ) = 0, W (ζ, Y ) = 0, θ(ζ, Y ) = 0 at Y = ∞.

(18)

Now, Eqs. (14)–(17) subject to the boundary conditions (18) are discretized by a centraldifference scheme for the diffusion terms and a backward-difference scheme for the convective terms. The resulting implicit tridiagonal algebraic system is obtained as follows: For the x-momentum equation, we have A1 Ui−1,j + B1 Ui,j + C1 Ui+1,j = D1 , where

(19)

 Yi Ui,j  ΔY , A1 = 1 + Vi,j − ζj − 5 2 3 1 ζ  M j B1 = −2 − (ΔY )2 + Ui,j − ΔY ζ 2, 5 5 Δζ 1 + m2  Yi Ui,j  ΔY , C1 = 1 − Vi,j − ζj − 5 2  1 U U mM ζj2 i,j i,j−1 D1 = − ζj + θi,j − (ΔY )2 . W i,j 5 Δζ 1 + m2

For the z-momentum equation, we have A2 Wi−1,j + B2 Wi,j + C2 Wi+1,j = D2 , where

(20)

 Yi Ui,j  ΔY , A2 = 1 + Vi,j − ζj − 5 2 3 1 ζ  M j B2 = −2 − (ΔY )2 + Ui,j − ΔY ζ2, 5 5 Δζ 1 + m2  Yi Ui,j  ΔY , C2 = 1 − Vi,j − ζj − 5 2  1 U W mM ζj2 i,j i,j−1 D2 = − ζj (ΔY )2 . U − i,j 5 Δζ 1 + m2

Finally, for the energy equation, we have A3 θi−1,j + B3 θi,j + C3 θi+1,j = D3 , where

 1 Yi Ui,j  ΔY + Vi,j − ζj − , Pr 5 2  1 ζj  2 B3 = − − (ΔY )2 1 + Ui,j , Pr 5 Δζ  1 Yi Ui,j  ΔY + Vi,j − ζj − , C3 = Pr 5 2 A3 =

1 Ui,j θi,j−1 D3 = − (ΔY )2 ζj . 5 Δζ

(21)

Effect of Hall current on MHD natural convection flow

1133

The implicit tridiagonal algebraic system of Eqs. (19)–(21) is solved using the Gaussian elimination method for the unknowns U , W , and θ independently. In the computation, the continuity equation is used to directly measure the normal velocity vector V from the following expression:  1  Vi,j = Vi−1,j − (Vi,j − ζj ) − Yi (Ui,j − Ui−1,j ) 5 3 ΔY ζj ΔY (Ui,j + Ui−1,j ) − (Ui,j − Ui,j−1 ). − (22) 5 2 5Δζ The computation starts at ζ = 0.0, and then it marches up to ζ = 100.0. At every ζ station, the computations are iterated unless the difference of the results of two successive iterations becomes to be less than or equal to 10−5 . The Y and ζ grids are set at 0.01 and 0.025, and the maximum value of Y is taken to be 40. These values are chosen in the present computation after running several tests for the convergence. Recently, this method has been successfully used by Siddiqa et al.[13] to investigate the high Prandtl number effects on the natural convection flow over an inclined flat plate with the internal heat generation and the variable viscosity. Once the quantities U and θ and their derivatives are evaluated at each ζ step, we can calculate the local −3/5 −1/5 skin friction coefficient Cfx Grx and the local Nusselt number coefficient N ux Grx , which are significant from the engineering point of view. The expressions for these physical quantities are shown below ⎧  ∂U  − 35 ⎪ ⎪ C Gr = , x fx ⎨ ∂Y Y =0 (23) 1 ⎪ 1 ⎪ − 5 ⎩ N ux Grx = . θ Y =0 Furthermore, we introduce SFF, which is also used to solve the boundary layer problem in the intermediate region. 3.2 Stream function formulation The straightforward introduction of the transformation in terms of the stream function given in (27) reduces the set of Eqs. (7)–(11) into the following nonsimilar partial differential equations:  4 3 2 M 1   ∂f  2   ∂f f  + f f  − f  + ζf  + θ − ζ f − f , (24) ζ (f + mg) = 5 5 1 + m2 5 ∂ζ ∂ζ  4 3 M 1   ∂g 2   ∂f g  + f g  − f  g + ζg  − ζ f − g , (25) ζ (g − mf ) = 5 5 1 + m2 5 ∂ζ ∂ζ 1  ∂θ ∂f  1  4  1  , (26) θ + f θ − f θ + ζθ = ζ f  − θ Pr 5 5 5 ∂ζ ∂ζ where the functions f, g, and θ are defined as ⎧ 4 ψ(x, y) = x 5 (f (ζ, η) + ζ), ⎪ ⎪ ⎪ ⎪ 3 ⎪ ⎪ 5 ⎪ ⎨ w(x, y) = x g(ζ, η), θ(x, y) = θ(ζ, η), ⎪ ⎪ ⎪ ζ = Sx− 15 , ⎪ ⎪ ⎪ ⎪ ⎩ 1 η = x− 5 y,

(27)

and ψ is the stream function that satisfies the continuity equation given in (7) and defined as u=

∂ψ , ∂y

v=−

∂ψ . ∂x

(28)

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L. K. SAHA, S. SIDDIQA, and M. A. HOSSAIN

In Eqs. (24)–(26), primes denote the differentiation of the functions with respect to the pseudosimilarity variable η. In addition, ζ is the locally varying transpiration parameter that may also be interpreted as a scaled streamwise variable. The boundary conditions appropriate for the system of Eqs. (24)–(26) are 

f (ζ, 0) = f  (ζ, 0) = g(ζ, 0) = 0,

θ (ζ, 0) = −1,

f  (ζ, ∞) = g(ζ, ∞) = θ (ζ, ∞) = 0.

(29)

Now, we apply the local nonsimilary method on the set of Eqs. (24)–(26) subject to the boundary conditions given in Eq. (29). This method embodies two essential features: First, the nonsimilar solution at any specific streamwise location is found (i.e., each solution is locally autonomous). Second, the local solutions are found from differential equations, which are numerically solved by well-established techniques, such as the forward integration (e.g., the Runge-Kutta scheme) in conjunction with a shooting procedure to determine the unknown boundary conditions at the wall. The method also allows some degree of self-checking for the accuracy of numerical results. In the local nonsimilarity method, all terms in the transformed conservation equations are retained with the ζ derivatives distinguished by the introduction of the new functions f1 = ∂f ∂ζ ,

∂θ g1 = ∂g ∂ζ , and θ1 = ∂ζ . These represent three additional unknown functions. Therefore, it is necessary to deduce three further equations to determine f1 , g1 , and θ1 . This is accomplished by creating the subsidiary equations by the differentiation of the transformed conservation equations and boundary conditions (i.e., the system of equations for f , g, and θ) with respect ∂g1 ∂θ1 1 to ζ. The subsidiary equations for f1 , g1 , and θ1 contain the terms ∂f ∂ζ , ∂ζ and ∂ζ , and their η derivatives. If these terms are neglected, the system of equations for f , g, θ, f1 , g1 , and θ1 reduces to a system of ordinary differential equations that provides locally autonomous solutions in the streamwise direction. This form of the local nonsimilarity method is referred to as the second level of truncation, because approximations are made by dropping terms in the second level equation (the f , g, and θ equations are the first-level equations). The procedure described above in the formulation of the local nonsimilarity method can result in a large number of ordinary differential equations that may require the simultaneous solutions. For example, at the second level of truncation, there are six equations involving f , g, θ, f1 , g1 , and θ1 . It is expected that the accuracy of the local nonsimilarity method results depends on the truncation level. Minkowycz and Sparrow[14] initially developed the local nonsimilary method. Later this method was applied by many investigators, i.e., Hossain et al.[10] and Chen[15] in order to solve various nonsimilar boundary layer problems. Below, we give only the equations valid up to the second level of truncation. The first-order truncation is

4 3 2 M 1 f  + f f  − f  + ζf  + θ − ζ 2 (f  + mg) = ζ(f1  f  − f  f1 ), 5 5 1 + m2 5 4 3 M 1 ζ 2 (g − mf  ) = ζ(f  g1 − g  f1 ), g  + f g  − f  g + ζg  − 5 5 1 + m2 5 1 1  4  1  θ + f θ − f θ + ζθ = ζ(f  θ1 − θ f1 ) Pr 5 5 5

(30) (31) (32)

along with the boundary conditions 

f (ζ, 0) = f  (ζ, 0) = g(ζ, 0) = 0,

θ (ζ, 0) = −1,

f  (ζ, ∞) = g(ζ, ∞) = θ (ζ, ∞) = 0.

(33)

Effect of Hall current on MHD natural convection flow

1135

The second-order truncation is 4 7 2M ζ f1 + f f1 − f  f1 + f1 f  + f  + ζf1 + θ1 − (f  + mg) 5 5 1 + m2 −

M ζ2 1 2 (f1 + mg1 ) = ζ(f1 − f1 f1 ), 2 1+m 5

(34)

4 4 3 2M ζ(g − mf  ) g1 + f  g1 − f  g1 + f1 g  − f1  g + g  + ζg1 − 5 5 5 1 + m2 −

1 M ζ 2 (g1 − mf1 ) = ζ(f1 g1 − g1 f1 ), 1 + m2 5

(35)

1  4  1  2 1 θ1 + f θ1 − f1θ − f  θ1 + f1 θ + θ + ζθ1 = ζ(f  θ1 − θ1 f1 ). Pr 5 5 5 5

(36)

The associated boundary conditions are 

f1 (ζ, 0) = f1 (ζ, 0) = g1 (ζ, 0) = 0,

θ1 (ζ, 0) = −1,

f1 (ζ, ∞) = g1 (ζ, ∞) = θ1 (ζ, ∞) = 0.

(37)

It can be seen that the system of Eqs. (30)–(33) and (34)–(37) forms the coupled nonlinear system of ordinary differential equations taking ζ as a parameter. These equations are solved numerically, employing the Nachtsheim-Swigert[16] iteration technique. Here, the solutions to these equations are obtained up to the second level of truncation in terms of the local skin friction coefficient and the local Nusselt number coefficient, which can be expressed as follows: ⎧ − 35  ⎪ ⎨ Cfx Grx = f (ζ, 0), 1 ⎪ ⎩ N ux Grx− 5 =

1 . θ(ζ, 0)

(38)

The numerical results obtained here are discussed in view of several physical parameters, namely, the magnetic field parameter M , the Hall parameter m, and the Prandtl number P r with the values of ζ starting from 0.0 to 100.0. In Section 3.3, solutions to considerably small values of the local transpiration parameter ζ are obtained by employing the regular perturbation method. 3.3 Regular perturbation solutions (RPS) for small ζ Since, near the leading edeg ζ is sufficiently small, we expand the functions f , g, and θ given in Eqs. (24)–(26) along with the boundary conditions (29) in the powers of ζ as follows: ⎧ ∞ ⎪ ⎪ ⎪ f (η, ζ) = ζ i fi (η), ⎪ ⎪ ⎪ ⎪ i=0 ⎪ ⎪ ⎪ ⎪ ∞ ⎨ ζ i gi (η), g(η, ζ) = ⎪ ⎪ i=0 ⎪ ⎪ ⎪ ⎪ ∞ ⎪ ⎪ ⎪ ⎪ ζ i θi (η). θ(η, ζ) = ⎪ ⎩

(39)

i=0

Substituting Eq. (39) into Eqs. (24)–(26) along with the boundary conditions (29), and equating the coefficients of like powers of ζ, consequently, we get the following system of equations.

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For the zeroth-order problem, we have 4 3 2 f0 + f0 f0 − f0 + θ0 = 0, 5 5 4 3 g0 + f0 g0 − f0 g0 = 0, 5 5 1  4 1 θ0 + f0 θ0 − f0 θ0 = 0. Pr 5 5 The corresponding boundary conditions are  f0 = f0 = g0 = 0, f0 → 0,

g0 → 0,

θ0 = −1 at η = 0, θ0 → 0,

at η → ∞.

(40) (41) (42)

(43)

Similarly, the first-order problem is found to be 4 7 f1 + f0 f1 + f0 f1 − f0 f1 + f0 + θ1 = 0, 5 5 4 3 4 g1 + f0 g1 + f1 g0 − f1 g0 − f0 g1 = 0, 5 5 5 1  4 2 1 θ1 + f0 θ1 + f1 θ0 − f0 θ1 − f1 θ0 = 0 Pr 5 5 5 with the boundary conditions  f1 = f1 = g1 = 0, f1 → 0,

g1 → 0,

θ1 = 0 at η = 0, θ1 → 0,

at η → ∞.

(44) (45) (46)

(47)

For the higher-order equations, i.e., for i  2, the corresponding equations are obtained from  f i + fi−1 + θi −

 gi + gi−1 −

i   4 r 3 r   M   + frfi−r − + fr fi−r , (f + mg ) = i−2 i−2 2 1+m 5 5 5 5 r=0

i   4 r M 3 r    + g + f , (g − mf ) = f − g i−2 i−2 r i−r r i−r 1 + m2 5 5 5 5 r=0

i   4 r  1  1 r    θi + θi−1 + θr fi−r + fr θi−r . = − Pr 5 5 5 5 r=0

The boundary conditions are  fi = fi = gi = 0, fi → 0,

gi → 0,

θi = 0 θi → 0,

at η = 0, at η → ∞.

(48)

(49)

(50)

(51)

In the above equations, f0 , g0 , and θ0 are the well-known free convection similarity functions for the flow around a constant temperature semi-infinite vertical plate, and the pairs of equations for f1 , g1 , and θ1 and fi , gi , and θi (i = 2, 3, · · · ) are effectively the first- and higher-order corrections to the flow due to the interaction of transpiration of the fluid through the surface in the presence of the Hall effect. Furthermore, Eqs. (48)–(51) (for each i  2) are linear but coupled, and may be found by pairwise sequential solutions. These pairs of equations have been integrated by using an implicit Runge-Kutta initial value solver introduced by Butcher[17]

Effect of Hall current on MHD natural convection flow

1137

in connection with the iterative scheme of Nachtsheim-Swigert[16]. In view of various physical parameters, the numerical results obtained near the leading edge of the plate are discussed in −3/5 terms of the local skin friction coefficient Cfx Grx and the local Nusselt number coefficient −1/5 N ux Grx , which are significant from the engineering point of view and can be computed from the following relations: − 35

= f  (ζ, 0) = f0 + ζf1 + ζ 2 f2 + · · · , 1 1 −1 = . N ux Grx 5 = θ(ζ, 0) θ0 + ζθ1 + ζ 2 θ2 + · · ·

Cfx Grx

(52) (53)

It is worthy mentioning that in the absence of the Hall effect, the underlying model reduces to the one discussed by Wilks and Hunt[18] . The present numerical results are compared with those in Ref. [18] in Table 1. Wilks and Hunt[18] aimed to investigate the influence of the uniform surface heat flux on the boundary layer flow. They tackled the problem numerically by using the Keller box method for the entire regions where the regular perturbation method was initiated to handle the problem in the neighborhood of the plate. Moreover, they adopted the inverse coordinate expansion technique for the deliberately large values of the locally varying parameter. Numerical values of the wall shear stress f0 (0) and the rate of heat transfer θ0 (0) thus obtained are shown in Table 1 and compared with those of Wilks and Hunt[18] for several values of P r. The comparison shows excellent compatibility between the present analysis and the work done by Wilks and Hunt[18] . Table 1 Pr

Numerical values of coefficients of skin friction and rate of heat transfer Wilks and Hunt[18]

f0

θ0 (0)

1.50

1.091 7

1.00

1.374 4

0.72

Present f0 

θ0 (0)

−1.975 7

1.091 1

−1.976 2

−1.872 8

1.373 9

−1.873 0

1.655 3

−1.799 4

1.653 7

−1.799 8

0.50

2.033 1

−1.727 6

2.030 6

−1.728 0

0.20

3.390 0

−1.586 7

3.380 6

−1.587 3

0.10

4.958 5

−1.510 9

4.931 6

−1.511 9

0.05

7.208 5

−1.455 4

7.132 3

−1.546 8

0.02

11.718 5

−1.404 7

11.717 2

−1.404 7

0.01

16.829 7

−1.378 6

16.825 2

−1.378 6

Further, discussion is carried out on the numerical results obtained in terms of the local skin −3/5 −1/5 friction coefficient Cfx Grx and the local Nusselt number coefficient N ux Grx against small ζ ∈ [0.0, 1.5]. These results are shown in Figs. 2–4 and compared with the results obtained for the entire values of the local transpiration parameter ζ. The comparison validates the numerical result acquired in the upstream region (near the leading edge of the plate). Now, we focus on the downstream region, where the local transpiration parameter ζ takes the large values. 3.4 Asymptotic solution (ASS) for large ζ Here, attention is given to the solution to Eqs. (24)–(26) along with (29) when the locally varying variable ζ is large. The order of the magnitude analysis of various terms in Eqs. (24)– (26) shows that the largest ones are f  and ζf  in Eqs. (24), g  and ζf  in Eq. (25), and θ and ζθ in Eq. (26). In the respective equations, both the terms have to be balanced in the magnitude. The only way to do this is to assume that η is small, and hence the derivatives are large. Given that θ = O(1) as ζ → ∞, it is essential to find an appropriate scaling for f and η. By balancing f  , θ, and ζf  terms in Eq. (24), it is found that η = O(ζ) and f = O(ζ −4 ) as ζ → ∞. Therefore, the following transformations may be introduced:

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L. K. SAHA, S. SIDDIQA, and M. A. HOSSAIN

Fig. 2

Local skin friction and Nusselt number with different M and ζ where P r = 0.005 and m = 2.0

Fig. 3

Local skin friction and Nusselt number with different P r and ζ where M = 0.5 and m = 2.0

Fig. 4

Local skin friction and Nusselt number with different m and ζ where M = 0.5 and P r=0.005

Effect of Hall current on MHD natural convection flow

f = ζ −4 f (η),

η = ζη,

g = ζ −3 g(η),

θ = ζ −1 θ(η).

1139

(54)

Equations (24)–(26) with the introduction of transformation given in Eq. (54) become  ∂f  ∂f  M 1 − f  , (f  + mg) = ζ −4 f  2 1+m 5 ∂ζ ∂ζ  ∂g ∂f  M 1 − g , g  + g − (g − mf  ) = ζ −4 f  2 1+m 5 ∂ζ ∂ζ  ∂θ 1  ∂f  1 − θ θ + θ = ζ −4 f  Pr 5 ∂ζ ∂ζ f  + f  + θ −

together with the boundary conditions  f (ζ, 0) = f  (ζ, 0) = g(ζ, 0) = 0,

θ  (ζ, 0) = −1,

f (ζ, ∞) = g(ζ, ∞) = θ(ζ, ∞) = 0.

(55) (56) (57)

(58)

For the sufficiently large ζ, we can neglect the right-hand side of these equations. Hence, the equations take the forms  M  f + mg = 0, 2 1+m  M

  g +g − g − mf  = 0, 2 1+m 1  θ + θ = 0, Pr f  + f  + θ −

which satisfy the following boundary conditions:  f (0) = f  (0) = g(0) = 0,

θ (0) = −1,

f (∞) = g(∞) = θ(∞) = 0.

(59) (60) (61)

(62)

One can easily obtain the solution to Eq. (61) with the boundary conditions given in Eq. (62). The solution is found to be θ(η) =

1 −P rη e . Pr

(63)

Now, Eqs. (59)–(60) can be written as follows: V  + V  −

M (1 − im) 1 −P rη e V =− , 1 + m2 Pr

(64)

where V represents the complex function whose real part is the function f , and the imaginary part is the function g, i.e., V = f  + ig. The solution to Eq. (64) with the boundary conditions in Eq. (62) can be written as V (η) =

1 (e−hη − e−P rη ), P r(P r(P r − 1) − M1 )

(65)

Pr − h , P r(P r(P r − 1) − M1 )

(66)

with V (0) =

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L. K. SAHA, S. SIDDIQA, and M. A. HOSSAIN

and V  (0) =

(P r − A1 )(P r(P r − 1) − A0 ) − B1 B0 P r((P r(P r − 1) − A0 )2 + B02 ) −i

(P r − A1 )B0 + (P r(P r − 1) − A0 )B1 . P r((P r(P r − 1) − A0 )2 + B02 )

(67)

Now, we can obtain the expression for f at the surface of the plate, by taking the real part of the function V . Thus, we have f  (0) =

(P r − A1 )(P r(P r − 1) − A0 ) − B1 B0 , P r((P r(P r − 1) − A0 )2 + B02 )

(68)

where M Mm , B0 = , M1 = A0 − iB0 , 2 1+m 1 + m2 1 1 1 A = √ (((1 + 4A0 )2 + 16A20 m2 ) 2 + (1 + 4A0 )) 2 , 2 1 1 1 B = √ (((1 + 4A0 )2 + 16A20 m2 ) 2 − (1 + 4A0 )) 2 , 2 1+A B , B1 = , h = A1 + iB1 . A1 = 2 2 A0 =

(69) (70) (71) (72)

Thus, Eqs. (63)–(72) enable us to calculate the various flow characteristics, such as the local −3/5 and the rate of the heat transfer in terms of the local Nusselt skin friction coefficient Cfx Grx −1/5 number coefficient N ux Grx , which are, respectively, given below ⎧ 3 (P r − A1 )(P r(P r − 1) − A0 ) − B1 B0 ⎪ ⎨ Cfx Grx− 5 = ζ −2 , P r((P r(P r − 1) − A0 )2 + B02 ) (73) ⎪ ⎩ − 15 N ux Grx = ζP r. From the above solutions, it can be seen that for the considerably large values of the local tran−3/5 spiration parameter ζ, the value of the local skin friction coefficient Cfx Grx is approximate −1/5 2 to be 1/P r ζ, and the local Nusselt number coefficient N ux Grx ≈ P rζ. The asymptotic values of the local skin friction coefficient and the local Nusselt number coefficient thus obtained for different values of the pertinent parameters are shown and compared with the numerical results obtained for all values of ζ in Figs. 2–4. The comparison shows excellent agreement between the analytical and numerical analyses for large and all values of the local transpiration parameter ζ.

4

Results and discussion

In the present work, we investigate the problem of the laminar natural convective boundary layer flow above a semi-infinite vertical permeable flat plate with the uniform heat flux. Here, we consider the uniformly applied magnetic field normal to the surface of the plate. In addition, the Hall current effects are also taken into the consideration. FVF and SFF are initiated to convert the boundary layer equations into the nonlinear partial differential equations, which are fully solved with the straightforward finite difference numerical scheme in connection with the Gaussian elimination method and the local nonsimilarity method, respectively, for the whole range of ζ. However, in the case of the upstream region, where ζ is assumed to be small enough,

Effect of Hall current on MHD natural convection flow

1141

the regular perturbation method is adopted. Moreover, the asymptotic analysis is taken into account to acquire the solution away from the leading edge of the plate, where ζ is sufficiently −3/5 large. The results are presented in terms of the local skin friction coefficient Cfx Grx and −1/5 the local Nusselt number coefficient N ux Grx for small P r. Concern has also been shown towards the velocity as well as the temperature distribution for various physical parameters involved in controlling the flow field. −3/5 The numerical values of the local skin friction coefficient Cfx Grx and the local Nusselt −1/5 number coefficient N ux Grx , against the transpiration parameter ζ, ranging from 0.0 to 100.0 for the values of the magnetic field parameter M = 0.1, the Hall parameter m = 1.0, and the Prandtl number P r=0.1, have been shown in Table 2. On the one hand, it is observed that for the increasing values of the transpiration parameter ζ, the values of the local skin-friction coefficient tend to increase near the leading edge, and then diminish slowly. On the other hand, the local Nusselt number coefficient enhances rapidly. This comparison clearly validates that the solutions to the small and large values of ζ are in excellent agreement with the straightforward finite difference solutions as well as the local nonsimilarity solutions, which ensures that the numerical scheme used here is quite accurate for the underlying problem. It is also seen that the straightforward finite difference method gives the solutions for all values of ζ. However, the local nonsimilarity method cannot give solutions for very large values of ζ. Table 2

Numerical values of local skin friction coefficient and local Nusselt number coefficient against transpiration parameter ζ for P r=0.1, M = 0.1, and m = 1.0 3/5

ζ 0.000 00 0.100 17 0.201 34 0.410 75 0.509 84 0.601 37 0.809 41 1.012 24 1.206 30 2.014 27 4.021 86 8.028 49 10.017 87 12.002 58 15.116 10 18.103 24 20.009 94 25.190 30 30.161 86 40.314 01 50.237 10 60.146 53 70.583 94

Cfx /Grx FVF 3.114 77 3.172 74 3.231 53 3.348 62 3.403 76 3.451 17 3.551 31 3.642 07 3.715 44 3.882 53 3.092 39 0.945 85 0.609 05 0.425 13 0.269 40 0.188 46 0.154 60 0.097 52 0.068 25 0.039 07 0.025 51 0.018 06 0.013 32

1/5

= f  (ζ, 0) RPS & ASS 3.114 78s 3.172 85s 3.230 74s 3.347 12s 3.400 08s 3.447 53s 3.549 19s 3.638 50s 3.713 21s − − − 0.589 16l 0.410 43l 0.258 77l 0.180 42l 0.147 67l 0.093 18l 0.064 99l 0.036 38l 0.023 43l 0.016 34l 0.011 87l

N ux /Grx LNS 3.114 00 3.171 71 3.229 59 3.346 10 3.399 15 3.446 69 3.548 58 3.638 13 3.713 07 3.882 79 3.068 35 0.948 08 0.609 51 0.424 59 0.267 71 0.186 70 0.152 81 0.096 42 − − − − −

= 1/θ(ζ, 0)

FVF

RPS & ASS 0.264 04s 0.267 77s 0.271 55s 0.279 39s 0.283 12s 0.286 57s 0.294 45s 0.301 29s 0.310 67s − − − 1.001 79l 1.200 26l 1.511 61l 1.810 32l 2.000 99l 2.519 03l 3.016 19l 4.031 40l 5.023 71l 6.014 65l 7.058 39l

0.264 04 0.267 97 0.271 75 0.279 61 0.283 35 0.286 81 0.294 70 0.302 46 0.309 93 0.341 88 0.439 32 0.803 66 1.001 36 1.198 36 1.506 66 1.801 45 1.989 12 2.496 77 2.981 39 3.963 72 4.913 15 5.851 27 6.828 73

LNS 0.264 11 0.267 90 0.271 68 0.279 53 0.283 26 0.286 71 0.294 58 0.302 30 0.309 73 0.341 36 0.440 45 0.803 38 1.001 94 1.200 45 1.511 82 1.810 33 2.001 00 2.519 07 − − − − −

s and l denote the small and large values of ζ, respectively. −3/5

−1/5

4.1 Effect of pertinent physical parameters on Cf x Grx and N ux Grx −3/5 The local skin friction coefficient Cfx Grx and the local Nusselt number coefficient −1/5 N ux Grx against the transpiration parameter ζ for different values of the magnetic parameter M (=0.0, 0.1, 1.0, 10.0) are displayed in Figs. 2(a) and 2(b), respectively, for the case of P r=0.005 and m=2.0. From these figures, we see that the excellent agreement lies between the results obtained for small, large, and all values of the local transpiration parameter ζ. Fig-

1142

L. K. SAHA, S. SIDDIQA, and M. A. HOSSAIN

ures 2(a) and 2(b) also show that, when the magnetic parameter increases, both the coefficients of the local skin friction and the local Nusselt number decrease. This is expected since the intense amount of the magnetic field inside the boundary layer literally increases the Lorentz force that significantly opposes the flow in the reverse direction. Thus, the magnetic field acts as the retarding force that causes the coefficients of the local skin friction and the local Nusselt number to decrease significantly. It is observed in Fig. 2(a) that the curves for M = 0.0, 0.1, and 1.0 attain the local maxima, and after that the skin friction decreases and settles down to the corresponding asymptotic values as ζ increases. The respective maximum values for M =0.0, 0.1, and 1.0 are 44.432 34, 20.557 01, and 11.182 55, respectively, and they occur at ζ = 20.328, 6.689 94, and 1.855 18, respectively. However, for the stronger magnetic field, the maxima do not occur in the case of M = 10.0. In addition, it is inferred from Fig. 2(b) that the local Nusselt number coefficient for M = 0.1, 1.0, and 10.0 attains the local minima near the leading edge of the plate, and then slowly achieves the asymptotic values as ζ increases. The corresponding minimum values are 0.084 48, 0.059 00, and 0.047 00 which take place at ζ = 12.164 6, 5.890 11, and 2.975 54, respectively. In the case of the heat transfer, it is observed that the local minima do not exist for the weaker magnetic field. Moreover, the momentum and thermal boundary layer thickness decrease for the case of the strong magnetic field applied normal to the surface of the plate. −3/5 Furthermore, the effects of the local skin friction coefficient Cfx Grx and the local Nusselt −1/5 number coefficient N ux Grx , against the transpiration parameter ζ for different values of the Prandtl number P r = 0.05, 0.01, and 0.005 are displayed in Figs. 3(a) and 3(b), respectively. The other physical quantities, namely, the magnetic field parameter M is chosen to be 0.5, and the Hall parameter m takes the value 2.0. One can observe that the local skin friction coefficient diminishes while the rate of the heat transfer enhances for the increasing values of P r. Physically, it is possible because an increase in the value of P r leads to the decrease in the thermal conductivity of the fluid, which conclusively lowers the magnitude of the frictional force between the viscous layers. Conclusively, the skin friction at the surface of the plate decreases as the local Nusselt number increases within the boundary layer. Likewise, we find that the thickness of the thermal boundary layer increases as P r tends to increase. In addition, the local maximum occurs for each curve in Fig. 3(a) for P r = 0.005, 0.01, and 0.05, and the respective maximum values are 12.953 62, 10.072 92, and 5.134 848, which appear at ζ = 2.96, 3.62, and 2.12, respectively. The effects of the varying Hall parameter m (= 0.0, 1.5, and 3.0) on the local skin friction −3/5 −1/5 coefficient Cfx Grx and the local Nusselt number coefficient N ux Grx , against the transpiration parameter ζ, are displayed in Figs. 4(a) and 4(b), respectively, where P r = 0.005 and M = 0.5. These figures depict that the coefficients in the local skin friction and the local Nusselt number enhance considerably due to the increase of the Hall parameter m. Unlike the magnetic field M , the Hall effect supports the fluid flow inside the boundary layer, and ultimately gives rise to both the skin friction and the rate of the heat transfer. It can also be observed in Fig. 4(a) that for each value of m, there exists a local maximum in the local skin friction coefficient near the leading edge, and then its value decreases asymptotically as ζ increases. For m = 0.0, 1.5, and 3.0, the maximum values occur at ζ = 1.91, 2.52, and 3.47, and they are 10.458 08, 12.170 27, and 14.36, respectively. 4.2 Effect of pertinent physical parameters on velocity and temperature profiles The numerical values of the dimensionless velocity profile f  (ζ, η) and the dimensionless temperature profile θ(ζ, η), obtained with the effect of different physical parameters, are presented for the low values of P r. The effects of the local transpiration parameter ζ on the velocity and temperature distributions are depicted in Figs. 5(a) and 5(b), respectively. In these figures, ζ takes the values of 3.0, 6.0, and 8.0, while the other physical parameters are P r = 0.005, M = 0.5, and m =

Effect of Hall current on MHD natural convection flow

1143

2.0. One can observe that the velocity of the fluid decreases, while the temperature increases considerably within the boundary layer. Further, it can be seen that for ζ > 0, there exists the suction/withdrawal of the fluid, because the velocity of the fluid decreases. This phenomenon is expected because the suction pulls the fluid toward the wall, and the buoyancy force acts as the pulling force. Thus, inside the boundary layer region, the velocity reduces, and the temperature enhances. It is worthy mentioning that the momentum boundary layer thickness reduces, while the thermal boundary layer thickness increases slightly. The computational experiment shows that the asymptotic temperature profile is achieved for ζ > 10.0.

Fig. 5

Velocity and temperature profiles with different ζ and η where P r=0.005, M = 0.5, and m = 2.0

Moreover, the dimensionless velocity and the temperature profiles against η for distinct values of P r are shown in Figs. 6(a) and 6(b), respectively. Here, P r takes the values of 0.005, 0.01, and 0.05, while other parameters are ζ=4.0, M =0.5, and m=2.0. It can be seen from these figures that both the velocity and temperature profiles decrease as P r increases, which is expected. We see that the velocity and temperature profiles diffuse quickly and achieve their asymptotic values for the large values of P r, and consequently the thermal as well as momentum boundary layer thickness decreases. This phenomenon occurs because of some definite value of the kinematic viscosity, for which the thermal diffusivity of the fluid decreases. As a result, the velocity and the temperature of the fluid reduce.

Fig. 6

Velocity and temperature profiles with different P r and η where ζ = 4.0, M = 0.5, and m = 2.0

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L. K. SAHA, S. SIDDIQA, and M. A. HOSSAIN

Furthermore, the influence of the magnetic field on the dimensionless velocity f  (ζ, η) and the temperature θ(ζ, η) against η is plotted in Fig. 7 for M = 0.0, 1.0, and 2.0, P r = 0.005, ζ = 4.0, and m = 2.0. This figure demonstrates that the velocity of the fluid diminishes, whilst the temperature distribution enhances within the boundary layer with the magnetic parameter M rising from 0 to 2. The magnetic field acts like an opposing force or a resistive force. It has the tendency to slow down the motion of the fluid. This characteristic is observed in Fig. 7(a) because the velocity is reduced owing to the increase in the strength of the magnetic field, which is applied normal to the flow direction. It is also noticed that the thermal as well as momentum boundary layer thickness increases due to the intensification of the magnetic field parameter M.

Fig. 7

Velocity and temperature profiles with different M and η where P r=0.005, ζ=4.0, and m = 2.0

Last, we draw our attention toward the influence of the Hall parameter m on the velocity f  (ζ, η) and the temperature θ(ζ, η) of the fluid. In Fig. 8, the Hall parameter m is assumed to be 0.0, 1.0, and 2.0, while other quantities are chosen to be P r=0.005, ζ=4.0, and M =0.5. It can be seen that the velocity of the fluid increases, while the temperature falls down for the increasing values of the Hall parameter m. Moreover, the momentum boundary layer thickness increases slightly, while the thickness of the thermal boundary layer tends to decrease. Besides, it is observed that as compared with the magnetic field, the Hall current effects oppose the motion of the fluid.

Fig. 8

Velocity and temperature profiles with different m and η where P r=0.005, M =0.5, and ζ = 4.0

Effect of Hall current on MHD natural convection flow

5

1145

Conclusions

The present investigation deals with the effects of the Hall current on the MHD natural convection flow along a vertical permeable flat plate with a uniform surface heat flux. The governing boundary layer equations are simulated employing four distinct methodologies, namely, the extended series solution method for the smaller values of ζ, the asymptotic solutions for the large values of ζ, the local nonsimilarity method with the second level of truncation, and the straightforward finite difference method for all ζ ∈ [0, 100]. The simulated results are expressed −3/5 in terms of the local skin friction coefficient Cfx Grx and the local Nusselt number coefficient −1/5 N ux Grx . Later, we discuss the influence of various physical parameters on the velocity and temperature profiles of the fluid within the boundary layer. From the present investigation, the following results can be concluded: (i) An increase in the magnetic parameter M serves to the decrease of both the local skin friction coefficient and the local Nusselt number coefficient. (ii) Owing to an increase in the Hall parameter m, the coefficients of the local skin friction and the local Nusselt number are enhanced, but the effect of m on the rate of the heat transfer is very small. (iii) An increase in the transpiration parameter ζ leads to a decrease in the velocity of the fluid, but the temperature profile rises effectively and achieves its asymptotic profile for ζ > 10.0. (iv) It is also observed that the velocity distribution decreases, while the temperature gets strengthen inside the boundary layer for the increasing values of the magnetic parameter M . However, the opposite effects are found in the case when the Hall parameter tends to rise.

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[13] Siddiqa, S., Asghar, S., and Hossain, M. A. Natural convection flow over an inclined flat plate with internal heat generation and variable viscosity. Mathematical and Computer Modelling, 52, 1739–1751 (2010) [14] Minkowycz, W. J. and Sparrow, E. M. Numerical solution scheme for local nonsimilarity boundary layer analysis. Numerical Heat Transfer, 1, 69–85 (1978) [15] Chen, T. S. Parabolic system: local nonsimilarity method. Handbook of Numerical Heat Transfer (eds. Minkowycz, W. J., Sparrow, E. M., Schneider, G. E., and Pletcher, R. H.), Wiley, New York (1988) [16] Nachtsheim, P. R. and Swigert, P. Satisfaction of the Asymptotic Boundary Conditions in Numerical Solution of the System of Nonlinear Equations of Boundary Layer Type, NASA TN D-3004, New York (1965) [17] Butcher, J. C. Implicit Runge-Kutta process. Math. Comp., 18, 50–55 (1964) [18] Wilks, G. and Hunt, R. Magnetohydrodynamic free convection flow about a semi-infinite plate at whose surface the heat flux is uniform. ZAMP, 35, 34–49 (1984)