c Pleiades Publishing, Ltd., 2012. ISSN 1810-2328, Journal of Engineering Thermophysics, 2012, Vol. 21, No. 3, pp. 181–192.
Heat and Mass Transfer of MHD Convective Boundary Layer Flow Past a Stretching Porous Wall Embedded in a Porous Medium S. N. Sahoo* and G. C. Dash Department of Mathematics, Institute of Technical Education and Research, Faculty of Engineering, Siksha‘O’Anusandhan University, Khandagiri, Bhubaneswar-751030, Orissa, India Received June 14, 2011
Abstract—The hydromagnetic convective boundary layer flow past a stretching porous wall embedded in a porous medium with heat and mass transfer in the presence of a heat source and under the influence of a uniform magnetic field is studied. Exact solutions of the basic equations of motion, heat and mass transfer are obtained after reducing them to nonlinear ordinary differential equations. The reduced equations of heat and mass transfer are solved using a confluent hypergeometric function. The effects of the flow parameters such as a suction parameter (N ), magnetic parameter (M ), permeability parameter (Kp ), wall temperature parameter (r), wall concentration parameter (n), and heat source/sink parameter (Q) on the dynamics are discussed. It is observed that the suction parameter appears in the boundary condition ensuring the variable suction at the surface. Transverse component of the velocity increases only when magnetic field strength exceeds certain value, but the thermal boundary layer thickness and concentration distribution increase for all values. Results presented in this paper are in good agreement with the work of the previous author and also in conformity with the established theory. DOI: 10.1134/S1810232812030034
1. INTRODUCTION The study of boundary layer behavior over a continuously moving flat wall has wide applications in technological and manufacturing processes in the industry. These include aerodynamic extrusions of plastic sheets, rolling and extrusion in manufacturing processes, cooling of an infinite metallic plate in a cooled bath, the boundary layer along a liquid film in condensation process, and the boundary layer along the material handing conveyers. In a melt-spinning process an extradition is stretched into a filament when it is drawn from the dye and finally this sheet solidifies through the controlled cooling system. Rajgopal et al. [1] have studied the flow of a viscoelastic fluid over a stretching sheet. Idress et al. [2] have discussed a viscoelastic flow past a stretching sheet in a porous media and heat transfer with an internal heat source. Naseem et al. [3] have investigated the boundary layer flow past a stretching plate with suction, heat and mass transfer, and with variable conductivity. Ray et al. [4] have studied heat transfer in a stagnation point flow toward a stretching sheet. Sanyal et al. [5] have presented an analysis on steady flow and heat transfer of a conducting fluid caused by stretching a porous wall. Mukhopadhayay et al. [6] have investigated study of MHD boundary layer flow over a heated stretching sheet with variable viscosity. Elbashbeshy et al. [7] have studied the flow and heat transfer in a porous medium over a stretching surface with internal heat generation and suction/blowing, but the heat transfer in this flow is analyzed only when the surface is held at constant temperature. Tak et al. [8] have analyzed the flow and heat transfer due to a stretching porous surface in the presence of transverse magnetic field to include heat due to viscous dissipation. Cortell [9] has reported the flow and heat transfer of a fluid through a porous medium over a stretching surface with internal heat generation. Pantokratoras [10] has reinvestigated the MHD boundary layer flow over a heated stretching sheet with variable viscosity. Singh [11] has studied the heat source and radiation effect on a magneto-convection flow of a viscoelastic fluid past a stretching sheet using Kummer’s function. Amkadni et al. [12] have studied *
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the exact solution of magnetohydrodynamics steady-state laminar flow of a viscous incompressible and electrically conducting fluid over a continuous permeable stretching surface. Recently, Anjali Devi et al. [13] have studied viscous dissipation effects on a nonlinear MHD flow in a porous medium over a stretching porous surface. In the present study we have incorporated the effect of source and sink as well as mass transfer aspect to the work of Pantokratoras [10] and considered the medium with uniform porosity. Our study is related to a free convective MHD boundary layer flow of an incompressible viscous fluid past a stretching porous wall embedded in a porous medium in the presence of a transverse magnetic field. 2. FORMULATION AND SOLUTION TO THE PROBLEM We have considered a two dimensional free convective steady laminar flow of an incompressible electrically conducting viscous fluid past a stretching porous wall in the presence of a heat source and a uniform magnetic field in the xy-plane. The x-axis is taken along the wall in the direction of the motion of the flow and the y-axis is perpendicular to it. Let u and v be the components of velocity along x- and y-axes, respectively. It is envisaged that the stretching sheet starts from a thin slit at the origin (0, 0) and the speed of a point on the plate is proportional to its distance from the plate, but the boundary layer approximations still hold. In our discussion, the magnetic Reynolds number is assumed to be small so that the induced magnetic field is negligible in comparison with the applied magnetic field. In addition, the viscous and Joule dissipations are also negligible. Under the above-mentioned assumptions, the steady state boundary layer equations of the flow are given by
u
∂u ∂v + = 0, ∂x ∂y
(1)
∂u ∂2u σB02 υ ∂u +v =υ 2 − u− u. ∂x ∂y ∂y K ρ
(2)
The appropriate boundary conditions for the problem are:
⎫ y=0 : u = Ax, v = −V0 , ⎬ . ⎭ y→∞ :u→0
In order to solve equation (2), we have assumed
u = Axf (η),
√
v = − υAf (η), and η =
(3)
A y. υ
With this choice of u and v, Eq. (1) is identically satisfied. Substituting (4) in Eq. (2), we obtain 2 f (η) + f (η)f (η) − f (η) − M1 f (η) = 0, where M1 =
1 KP
+ M 2 , Kp =
AK υ
(permeability parameter), M 2 =
σB02 Aρ
(4)
(5)
(magnetic parameter).
The corresponding boundary conditions are:
⎫ : f (η) = 1, f (η) = N, ⎬ , ⎭ η → ∞ : f (η) → 0
η=0
where N =
√V0 υA
(6)
is the suction parameter.
In view of boundary conditions (6), the solution of equation (5) by Gupta and Gupta [14] is: f (η) = α1 + α2 e−αη ,
2 1 1 1 2 + 4(1 + M ) . , α = − , and α = N N + where α1 = α −M 2 1 α α 2 JOURNAL OF ENGINEERING THERMOPHYSICS Vol. 21 No. 3 2012
(7)
HEAT AND MASS TRANSFER OF MHD CONVECTIVE BOUNDARY LAYER
Hence, the exact solution shown in (7) can be expressed as 1 2 f (η) = α − M1 − e−αη . α
and its nondimensional form is: The skin-friction is given by τ ∗ = μ ∂u ∂y
183
(8)
y=0
τ = f (0) = −α.
(9)
2.1. Heat Transfer Analysis The equation of energy is given by u
∂T κ ∂2T ∂T +v = − Q (T − T∞ ). ∂x ∂y ρCp ∂y 2
(10)
The appropriate boundary conditions are:
⎫ y = 0 : T = Tw , ⎬ , y → ∞ : T → T∞ ⎭
where Q =
Qu x
(11)
(volumetric rate of heat absorption), Tw = T∞ + A1 xr (wall temperature).
To solve Eq. (10), we have introduced the following nondimensional temperature θ defined as θ=
T − T∞ . Tw − T∞
The temperature T , following Nield and Bejan [15, p. 107], can be expressed as T (y) = T∞ + A1 xr θ(η).
(12)
In view of Eqs. (4) and (12), Eq. (10) reduces to θ (η) + Pr f (η)θ (η) − Pr (r + Q)f (η)θ(η) = 0.
(13)
The boundary conditions (11) transformed to
⎫ : θ = 1, ⎬ . η→∞ : θ→0⎭ η=0
(14)
In order to obtain a solution of Eq. (13), we introduce a new variable ξ defined as ξ=−
Pr e−αη . α2
(15)
Hence, with the help of Eqs. (8) and (15), Eq. (13) transformed to
M1 dθ d2 θ + (r + Q)θ = 0. ξ 2 + 1 − Pr 1 − 2 − ξ dξ α dξ The boundary conditions (14) now reduced to ξ=
− αP2r
ξ=0
⎫ : θ = 1, ⎬ . : θ=0 ⎭
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(16)
(17)
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The solution of Eq. (16) in view of the boundary conditions (17) in terms of confluent hypergeometric function (or Kummer’s function) is:
M1 M1 M1 F Pr 1 − 2 − r − Q ; Pr 1 − 2 + 1 ; ξ Pr 1− 2 α α2 α α
. θ(ξ) = − ξ (18) M1 M1 Pr Pr F Pr 1 − 2 − r − Q ; Pr 1 − 2 + 1 ; − 2 α α α Hence, the expression for θ(η) is:
M1 M1 Pr −αη F Pr 1 − − r − Q ; Pr 1 − 2 + 1 ; − 2 e M α2 α α −αPr 1− 21 η α
. θ(η) = e M1 M1 Pr F Pr 1 − 2 − r − Q ; Pr 1 − 2 + 1 ; − 2 α α α
The nondimensional heat transfer in terms of Nusselt number (Nu ) at the wall is given by M
Pr 1 − 21 − r − Q dθ Pr M1 α = −αPr 1 − 2 + Nu = M1 dη η=0 α α Pr 1 − 2 + 1 α
M1 M1 Pr F Pr 1 − 2 − r − Q + 1 ; Pr 1 − 2 + 2 ; − 2 α α α
. × M1 M1 Pr F Pr 1 − 2 − r − Q ; Pr 1 − 2 + 1 ; − 2 α α α
(19)
(20)
2.2. Mass Transfer Analysis The equation for species concentration is given by u
∂C ∂2C ∂C +v =D 2. ∂x ∂y ∂y
(21)
The appropriate boundary conditions are:
⎫ y = 0 : C = Cw , ⎬ , y → ∞ : C → C∞ ⎭
(22)
where Cw = C∞ + A2 xn . To solve Eq. (21), we have introduced the following nondimensional concentration h defined as h=
C − C∞ . Cw − C∞
The concentration C can be expressed as C(y) = C∞ + A2 xn h(η).
(23)
In view of Eqs. (4) and (23), Eq. (21) reduces to h + Sc f h − nSc f h = 0,
(24)
υ D.
where Sc = The boundary conditions (22) transformed to
⎫ η = 0 : h = 1, ⎬ . η→∞ : h→0⎭
(25)
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In order to obtain a solution of Eq. (24), we introduce a new variable ξ defined as ξ=−
Sc e−αη . α2
(26)
Hence, with the help of Eqs. (8) and (26), Eq. (24) transformed to
M1 dh d2 h + nh = 0. ξ 2 + 1 − Sc 1 − 2 − ξ dξ α dξ
(27)
The boundary conditions (25) now reduced to
⎫ Sc ξ = − 2 : h = 1, ⎬ α . , ⎭ ξ=0 : h=0
(28)
The solution of Eq. (27) in view of the boundary conditions (28) in terms of confluent hypergeometric function (or Kummer’s function) is:
M M 1 1 Sc 1− M21 F Sc 1 − 2 − n ; Sc 1 − 2 + 1 ; ξ α α2 α α
. (29) h(ξ) = − ξ M1 M1 Sc Sc F Sc 1 − 2 − n ; Sc 1 − 2 + 1 ; − 2 α α α Hence, the expression for h(η) is:
M1 M1 Sc −αη F Sc 1 − − n ; Sc 1 − 2 + 1 ; − 2 e M α2 α α −αSc 1− 21 η α
. h(η) = e M1 M1 Sc F Sc 1 − 2 − n ; Sc 1 − 2 + 1 ; − 2 α α α
(30)
The dimensionless concentration gradient in terms of Sherwood number (Sh ) at the wall is given by M1
Sc 1 − 2 − n dh Sc M1 α = −αSc 1 − 2 + Sh = M1 dη η=0 α α Sc 1 − 2 + 1 α (31)
M1 M1 Sc F Sc 1 − 2 − n + 1 ; Sc 1 − 2 + 2 ; − 2 α α α
. × M1 M1 Sc F Sc 1 − 2 − n ; Sc 1 − 2 + 1 ; − 2 α α α 3. RESULTS AND DISCUSSION The problem of the steady laminar flow with heat and mass transfer of a viscous incompressible electrically conducting fluid past a stretching porous wall embedded in a porous medium in the presence of a uniform transverse magnetic field and heat source/sink has been considered. The solutions for velocity, heat transfer, and mass transfer are given in Eqs. (8), (19), and (30), respectively. The effects of various flow parameters on longitudinal velocity, transverse velocity, temperature field, and concentration distribution have been studied analytically and presented with the help of Figs. 1–5. Further, the effects of the flow parameters on skin friction, Nusselt number, and Sherwood number have been discussed with the help of Tables 1, 2, and 3, respectively. For numerical computation of temperature distribution, the values of Prandtl number are chosen to be 0.71 and 7.0, which represent air and water, respectively. For the concentration distribution, the values of Schmidt number are chosen in such a way that they represent the diffusing chemical species of most common interest in air. For example, in air, due to the presence of H2 and H2 O, the values of Sc are 0.22 and 0.60, respectively. JOURNAL OF ENGINEERING THERMOPHYSICS Vol. 21
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Fig. 1. Effect of N , M , and Kp on nondimensional transverse velocity profile.
Figure 1 shows the effect of suction parameter (N ), magnetic parameter (M ), and permeability parameter (Kp ) on transverse velocity. From boundary condition (6) it is clear that the transverse velocity is equal to the suction parameter on the surface of the plate. It is evident from curves VIII and IX that for higher values of the suction parameter, the transverse velocity increases at all points of the flow. Due to the presence of magnetic field as well as porous matrix, the transverse velocity decreases. Hence, it is concluded that Lorentz force acts adversely reducing the transverse component of velocity. On careful observation we see from curves I and II that there is no change in the transverse velocity when M changes from 0 to 0.5. But when M changes from 0.5 to 1.0 and more, there is a significant increase in the transverse velocity. Hence, magnetic field is effective to modify the flow field when it exceeds certain limit. Figure 2 depicts the effect of the suction parameter (N ), magnetic parameter (M ), and permeability parameter (Kp ) on the longitudinal velocity. It is clear that due to an increase in the magnetic parameter, there is a fall in the longitudinal velocity because of the retarding effect of the magnetic force. Further, the velocity boundary layer thickness is reduced due to the effect of magnetic field. An increase in the suction parameter decreases the longitudinal velocity at all points. Again, it is marked that in the absence of porous matrix (Kp = 50.0) the longitudinal velocity increases as there is no resistance offered by the porous matrix along the flow direction. Figure 3 depicts the effect of suction parameter (N ), magnetic parameter (M ), and Prandtl number (Pr ) on the temperature field. From the figure it is clear that the temperature distribution is a two layer character, one for water and the other for air. Temperature is reduced significantly in the case of water. It is observed that an increase in the suction parameter decreases the temperature field, whereas the reverse effect is observed in the presence of magnetic field. Thus, it may be concluded that onset of Lorentz force due to interaction of conducting fluid and magnetic field enhances the temperature at all points leading to an increase in the thermal boundary layer thickness. Further, an increase in Prandtl number decreases the temperature field exhibiting a multilayer character. This is consistent with the fact that the thermal boundary layer thickness decreases with increasing Prandtl number. JOURNAL OF ENGINEERING THERMOPHYSICS Vol. 21 No. 3 2012
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Fig. 2. Effect of N , M , and Kp on nondimensional longitudinal velocity profile.
Figure 4 shows the effect of permeability parameter (Kp ), wall temperature parameter (r), heat source parameter (Q), and Prandtl number (Pr ) on the temperature field. It is observed that in the case of air (Pr = 0.71), an increase in the wall temperature parameter and the heat source parameter leads to an increase in the temperature of the fluid. In the case of water (Pr = 7.0), an increase in the wall temperature parameter and heat source parameter does not affect the temperature distribution. In both cases, in the absence of porous matrix, the temperature of the fluid decreases. Curves V and XI represent the case of without porous matrix in which the temperature attains the minimum. Again, it is observed that an increase in Prandtl number reduces the temperature of the fluid. Figure 5 displays the effect of the suction parameter (N ), magnetic parameter (M ), permeability parameter (Kp ), and wall concentration parameter (n) on the concentration profile. It is seen that heavier species diffuses more slowly than the lighter one, leading to a higher concentration level in the case of lighter species. An important parameter is wall concentration parameter (n). From C = Cw + Axn , it is clear that n = 0 represents a linear x-dependent variation of concentration on the wall and n = 0 represents independent of x. Comparing almost coincident curves I and V, it is marked that no significant change in concentration distribution occurs due to variable wall concentration. Further, a fall of concentration is marked in the layers away from the plate. Moreover, an increase in concentration is marked with greater magnetic field strength and in the presence of porous matrix, but reverse effect is observed in the case of suction. To conclude, by reducing magnetic field strength with heavier diffusing species, equilibrium is attained far away from the plate, whereas due to resistance of porous matrix and for reduced suction the process becomes slow. From Table 1, it is to note that all the entries are negative. It is seen that magnetic, permeability, and JOURNAL OF ENGINEERING THERMOPHYSICS Vol. 21
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Fig. 3. Effect of N and M on temperature profile when Kp = r = Q = 0.5.
suction parameters decrease the tangential stress at the surface. This implies that all the forcing forces reduce the frictional resistance at the surface. From Table 2, it is observed that all the entries of Nusselt number are negative. The parameters related to magnetic field, suction, wall temperature, and heat source decrease the rate of heat transfer at the surface of the wall. Again, it is seen that in the absence of porous matrix the rate of heat transfer increases. From Table 3, it is also observed that all the entries of Sherwood number are negative. The rate of mass transfer decreases due to the suction parameter, porous matrix, and wall concentration parameter for Schmidt number 0.22 and 0.60. But the magnetic parameter enhances the rate of mass transfer for all the two values of Schmidt number. 4. CONCLUSIONS A theoretical study of steady laminar flow with heat and mass transfer of an incompressible electrically conducting viscous fluid past a stretching porous wall embedded in a porous medium in the presence of heat source/sink and a uniform magnetic field is presented. Some of the important findings of the problem are listed below. JOURNAL OF ENGINEERING THERMOPHYSICS Vol. 21 No. 3 2012
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Fig. 4. Effect of Kp , r, Q, and Pr on temperature profile when M = 0.5 and N = 0.4.
Table 1. Effects of various parameters on skin-friction (τ ) M
Kp
N
τ
0.5
1.0
0.4
−1.628286
1.0
1.0
0.4
−1.943560
0.5
10.0
0.4
−1.219804
0.5
1.0
0.8
−1.869694
(i) The effect of magnetic field is to reduce the nondimensional transverse velocity, nondimensional longitudinal velocity, and skin friction, but the reverse effect is observed in the case of temperature and concentration distribution. (ii) Onset of Lorentz force due to interaction of conducting fluid and magnetic field enhances the temperature at all points, leading to increase in the thermal boundary layer thickness. (iii) The thermal boundary layer thickness decreases with increasing Prandtl number. (iv) Heavier diffusing species reduces concentration exhibiting a multilayer character. (v) The rate of heat transfer increases in the absence of porous matrix. (vi) Presence of magnetic field enhances the rate of mass transfer. JOURNAL OF ENGINEERING THERMOPHYSICS Vol. 21
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Fig. 5. Effect of N , M , Kp , and n on concentration profile.
NOTATIONS x, y—coordinate system u, v—components of velocity along the x- and y-direction A, B—positive constants B0 —uniform magnetic field Cp —specific heat at constant pressure K—permeability of the medium Kp —permeability parameter C—concentration κ—thermal conductivity D—diffusivity coefficient M —magnetic parameter N —suction parameter Pr —Prandtl number Q—heat source parameter Q —volumetric rate of heat absorption T —temperature T∞ —temperature far away from the wall Tw —wall temperature JOURNAL OF ENGINEERING THERMOPHYSICS Vol. 21 No. 3 2012
HEAT AND MASS TRANSFER OF MHD CONVECTIVE BOUNDARY LAYER Table 2. Effects of various parameters on Nusselt number (Nu ) M
Kp
N
r
Q
Pr
Nu
0.5
0.5
0.4
0.5
0.5
0.71
−1.058257
0.5
0.5
0.8
0.5
0.5
0.71
−1.164332
1.0
0.5
0.4
0.5
0.5
0.71
−1.075673
0.5
1.0
0.4
0.5
0.5
0.71
−1.037502
0.5
0.5
0.4
1.0
0.5
0.71
−3.118766
0.5
0.5
0.4
0.5
1.0
0.71
−3.118766
0.0
0.5
0.4
0.5
0.5
0.71
−1.058257
0.5
100.0
0.4
0.5
0.5
0.71
−1.011216
0.5
0.5
0.4
0.5
−0.5
0.71
−0.535787
0.5
0.5
0.4
0.5
0.5
7.00
−4.426128
0.5
0.5
0.8
0.5
0.5
7.00
−6.833032
1.0
0.5
0.4
0.5
0.5
7.00
−4.445090
0.5
1.0
0.4
0.5
0.5
7.00
−4.375855
0.5
0.5
0.4
1.0
0.5
7.00
−4.836488
0.5
0.5
0.4
0.5
1.0
7.00
−4.836488
0.0
0.5
0.4
0.5
0.5
7.00
−4.426128
0.5
100.0
0.4
0.5
0.5
7.00
−4.230008
0.5
0.5
0.4
0.5
−0.5
7.00
−3.612946
Table 3. Effects of various parameters on Sherwood number (Sh ) M
Kp
N
n
Sc
Sh
0.5
0.5
0.4
0.5
0.22
−0.282235
1.0
0.5
0.4
0.5
0.22
−0.266061
0.5
1.0
0.4
0.5
0.22
−0.304490
0.5
0.5
0.8
0.5
0.22
−0.341275
0.5
0.5
0.4
1.0
0.22
−0.712057
0.0
0.5
0.4
0.5
0.22
−0.282235
0.5
100.0
0.4
0.5
0.22
−0.338350
0.5
0.5
0.4
0.5
0.60
−0.632114
1.0
0.5
0.4
0.5
0.60
−0.610696
0.5
1.0
0.4
0.5
0.60
−0.658505
0.5
0.5
0.8
0.5
0.60
−0.803027
0.5
0.5
0.4
1.0
0.60
−0.982853
0.0
0.5
0.4
0.5
0.60
−0.632114
0.5
100.0
0.4
0.5
0.60
−0.691839
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V0 —constant suction velocity U —characteristic velocity Cw —species concentration at the wall C∞ —species concentration away from the wall Sc —Schmidt number r—wall temperature parameter n—wall concentration parameter
Greek Symbols υ—coefficient of kinematic viscosity ρ—density of the fluid σ—electrical conductivity of the medium REFERENCES 1. Rajgopal, K.R., Na, T.Y., and Gupta, A.S., Flow of a Visco-Elastic Fluid over a Stretching Sheet, Rheol. Acta, 1984, vol. 23, pp. 213–221. 2. Idress, M.K. and Abel, M.S., Visco-Elastic Flow past a Stretching Sheet in Porous Media and Heat Transfer with Internal Heat Source, Ind. J. Th. Phys., 1996, vol. 44, pp. 233–244. 3. Naseem, A. and Khan, N., Boundary Layer Flow past a Stretching Plate with Suction and Heat Transfer with Variable Conductivity, Int. J. Eng. Mat. Sci., 2000, vol. 7, pp. 51–53. 4. Ray Mahapatra, T. and Gupta, A.S., Heat Transfer in Stagnation Point Flow towards a Stretching Sheet, Heat Mass Transfer, 2002, vol. 38, pp. 517–521. 5. Sanyal, D.C. and Dasgupta, S., Steady Flow and Heat Transfer of a Conducting Fluid Caused by Stretching a Porous Wall, Ind. J. Th. Phys., 2003, vol. 51, pp. 47–58. 6. Mukhopadhayay, S., Layek, G.C., and Samad, S.A., Study of MHD Boundary Layer Flow over a Heated Stretching Sheet with Variable Viscosity, Int. J. Heat Mass Transfer, 2005, vol. 48, pp. 4460–4466. 7. Elbashbeshy, E.M.A. and Bazid, M.A.A., Heat Transfer in Porous Medium over a Stretching Surface with Internal Heat Generation Suction or Injection, Appl. Math. Comp., 2004, vol. 158, pp. 799–807. 8. Tak, S.S. and Lodha, A., Flow and Heat Transfer Due to a Stretching Porous Surface in Presence of Transverse Magnetic Field, Acta Ciencia Indica, 2005, vol. 31 M, no. 3, pp. 657–663. 9. Cortell, R., Flow and Heat Transfer of a Fluid through a Porous Medium over a Stretching Surface with Internal Heat Generation/Absorption and Suction/Blowing, Fluid Dyn. Res., 2005, vol. 37, pp. 231–245. 10. Pantokratoras, A., Study of MHD Boundary Layer Flow over a Heated Stretching Sheet with Variable Viscosity: A Numerical Investigation, Int. J. Thermal Sci., 2008, vol. 51, pp. 104–110. 11. Singh, A.K., Heat Source and Radiation Effects on Magneto-Convection Flow of a Viscoelastic Fluid past a Stretching Sheet: Analysis with Kummer’s Function, Int. Comm. Heat Mass Transfer, 2008, vol. 35, pp. 637–642. 12. Amkadni, M., Azzaouzia, A., and Hammouch, Z., On the Exact Solutions of Laminar MHD Flow over a Stretching Flat Plate, Comm. Nonlinear Sci. Num. Simulation, 2008, vol. 13(2), pp. 359–364. 13. Anjali Devi, S.P. and Ganga, B., Viscous Dissipation Effects on Nonlinear MHD Flow in a Porous Medium over a Stretching Porous Surface, Int. J. Appl. Math. Mech., 2009, vol. 5(7), pp. 45–59. 14. Gupta, P.S. and Gupta, A.S., Heat and Mass Transfer on a Stretching Sheet with Suction or Blowing, Cnadian J. Chem. Eng., 1977, vol. 55, pp. 744–746. 15. Nield, D.A. and Bejan, A., Convection in Porous Media, New York: Springer-Verlag, 1999.
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