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The effects of temperature dependent viscosity and thermal conductivity on unsteady MHD convective heat transfer past a semi-infinite vertical porous moving plate with variable suction
Computational Materials Science 40 (2007) 186–192 www.elsevier.com/locate/commatsci
The effects of temperature dependent viscosity and thermal conductivity on unsteady MHD convective heat transfer past a semi-infinite vertical porous moving plate with variable suction M.A. Seddeek a
a,*
, Faiza A. Salama
b
Qassim University, College of Science, Mathematics Department, P.O. Box 237, Buriedah 81999, Saudi Arabia b Department of Mathematics, Faculty of Science, Suez Canal University, Egypt Received 10 May 2006; received in revised form 4 November 2006; accepted 29 November 2006 Available online 30 January 2007
Abstract In this article, we studied the effects of variable viscosity and thermal conductivity on an unsteady two-dimensional laminar flow of a viscous incompressible conducting fluid past a semi-infinite vertical porous moving plate taking into account the effect of a magnetic field in the presence of variable suction. The fluid viscosity is assumed to vary as an inverse linear function of temperature but the thermal conductivity is assumed to vary as a linear function of temperature. It is assumed that the porous plate moves with a constant velocity in the direction of fluid flow, and the free stream velocity follows the exponentially increasing small perturbation law. The governing equations for the flow are transformed into a system of nonlinear ordinary differential equations by perturbation technique and are solved numerically by using the shooting method. The effects of the various parameters on the velocity and temperature profiles as well as the surface skin-friction coefficient are presented graphically. Ó 2006 Elsevier B.V. All rights reserved. Keywords: Variable viscosity; Thermal conductivity; Magnetic field; Variable suction
1. Introduction Natural convection flow and heat transfer in porous moving plate is gaining more attention because of its wide applicability in packed beds, porous insulation, beds of fossil fuels, nuclear waste disposal, resin transfer molding, etc. Most of the existing analytical studies for this problem are based on the constant physical properties of the ambient fluid [1–3]. However, it is known that these properties may change with temperature [4]. To accurately predict the flow and heat transfer rates, it is necessary to take into account this variation of viscosity and thermal conductivity. *
Corresponding author. Permanent address: Department of Mathematics, Faculty of Science, Helwan University, Cairo 11785, Egypt. E-mail address: [email protected] (M.A. Seddeek). 0927-0256/$ - see front matter Ó 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.commatsci.2006.11.012
The influence of variable viscosity on the laminar boundary layer flow and heat transfer due to continuously moving plate is examined by Pop et al. [5]. The free convection flow with variable viscosity and thermal diffusivity along a vertical plate has been studied by Elbashbeshy [6] and Elbashbeshy and Ibrahim [7]. Seddeek [8] studied the effects of magnetic field and variable viscosity on forced non-Darcy flow about a flat plate with variable wall temperature in porous media in the presence of suction and blowing. Also, Ghaly and Seddeek [9] studied the Chebyshev finite difference method for the effects of chemical reaction, heat and mass transfer on laminar flow along a semi-infinite horizontal plate with temperature dependent viscosity. Recently, Seddeek and Salem [10] studied the laminar mixed convection adjacent to vertical continuously stretching sheets with variable viscosity and variable thermal diffusivity.
M.A. Seddeek, F.A. Salama / Computational Materials Science 40 (2007) 186–192
187
Nomenclature a A B0 E f g G k* M n Nu P Pr Re T Tr t U0 u, v V0 x, y
constant depending on the nature of the fluid suction velocity parameter magnetic induction constant depending on the reference state dimensionless stream function gravitational acceleration Grashof number permeability of the porous medium magnetic field parameter dimensionless exponential index Nusselt number pressure Prandtl number Reynolds number temperature constant dimensionless time scale of free stream velocity velocity components in the x and y directions scale of suction velocity distance along and perpendicular to the plate, respectively
The aim of the present work is to consider the case of a semi-infinite moving porous plate in a porous medium with the presence of pressure gradient and constant velocity in the flow direction when the magnetic field is imposed transverse to the plate. In addition, variable viscosity and thermal conductivity effects can become more pronounced in flow and heat transfer over a semi-infinite vertical porous moving plate. Therefore, the purpose of this study is to consider variable viscosity, thermal conductivity and variable suction effects on an unsteady MHD convective heat transfer past a semi-infinite vertical porous moving plate. We have reduced the two-dimensional continuity, momentum and energy equations to a system of nonlinear ordinary differential equations, which are solved numerically by using the shooting method. The effects of various parameters on the flow and heat transfer have been shown graphically.
2. Mathematical formulation Consider a two-dimensional unsteady flow of a laminar, incompressible fluid past a semi-infinite vertical porous moving plate embedded in a porous medium and subjected to a transverse magnetic field (see the sketch of the physical model). We assume that the fluid properties are isotropic and constant, except for the fluid viscosity l, which is assumed to vary as an inverse linear function of temperature T, in the form
Greek symbols a fluid thermal conductivity b thermal conductivity parameter b1 the coefficient of volumetric expansion e scale constant (1) r electrical conductivity l fluid viscosity m kinematic viscosity q density h dimensionless temperature hr variation viscosity parameter Superscripts / differentiation with respect to y dimensional properties * Subscripts P plate w wall conditions 1 free stream conditions
1 1 ¼ ð1 þ dðT T 1 ÞÞ l l1
or
1 ¼ EðT T r Þ; l
where E¼
d l1
and
1 Tr ¼ T1 ; d
Here l1 and T1 are the fluid free stream dynamic viscosity and the fluid free stream temperature, E and Tr are the constants and their values depend on the reference state and thermal property of the fluid, i.e. d. In general, E < 0 for fluid such as liquids and E > 0 for gases. Also, we assume that the fluid thermal conductivity, a, is assumed to vary as a linear function of temperature in the form [11] a ¼ a0 ð1 þ aðT T 1 ÞÞ; where a0 is the thermal diffusivity at temperature Tw and a is a constant depending on the nature of the fluid. In general a > 0 for fluids such as water and air, while a < 0 for fluids such as lubrication oils. This form can be rewritten in the non-dimensional form: a ¼ a0 ð1 þ bhÞ; where b = a(Tw T1) is the thermal conductivity parameter and Tw is the value of the plate temperature. The range of variations of b can be taken as follows: for air 0 6 b 6 6, for water 0 6 b 6 0.12 and for lubrication oils 0.1 6 b 6 0.
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V ¼ V 0 ð1 þ eAen t Þ;
* P
U ,x
ð6Þ
where A is a real positive constant, e and eA are small and less than unity, and V0 is a scale of suction velocity which has non-zero positive constant. Outside the boundary layer, Eq. (2) gives
U *∞ , T∞
B0 Porous medium
0
0
It is assumed that there is no applied voltage, which implies the absence of an electric field. The transversely applied magnetic field and magnetic Reynolds number are very small and hence the induced magnetic field is negligible [12]. Viscous and Darcy’s resistance terms are taken into account with constant permeability of the porous medium. Under the usual boundary layer approximation, the governing equation for this problem can be written as follows:
ð1Þ
u rB20 u ; k ð2Þ ð3Þ
subject to the boundary conditions
u !
U 1
t
T ¼ T w þ eðT w T 1 Þen ¼ U 0 ð1 þ ee
n t
Þ;
T ! T1
o2 u h0 ou ; dy 2 h hr oy " 2 # oh 1 o2 h oh nt oh ð1 þ eAe Þ ¼ ð1 þ bhÞ 2 þ b ; ot oy Pr dy oy þ Gh þ
at y ¼ 0; as y ! 1;
ð4Þ ð5Þ
where up is velocity of the plate and e the scalar constant (1). From the continuity of Eq. (1), it is clear that the suction velocity normal to the plate is a function of time only and we shall take it in the form
ð9Þ ð10Þ
where hr is variable viscosity parameter. The boundary conditions become
u ! U 1;
oT o oT oT þ v ¼ a ot oy oy oy
u ¼ up ;
ou ou dU 1 1 ð1 þ eAent Þ ¼ þ Mþ ðU 1 uÞ ot oy K dt
u ¼ U p;
þ qgb1 ðT T 1 Þ mq
0
In view of Eqs. (6)–(8), the governing equations (2) and (3) reduce to the following non-dimensional form:
The sketch of the physical model
ov ¼ 0; oy ou op o ou ou ¼ þ v þ l q ot oy ox oy oy
ð7Þ
The boundary layers Eqs. (1)–(3), along with the boundary conditions (4) and (5), are solved using the perturbation technique as the similarity solution is not possible. First we convert the partial differential equations into non-similar form making use of the following transformation [13]: 9 > u ¼ Uu 0 ; v ¼ Vv 0 ; y ¼ V 0my ; > > > > > > up t V 20 U 1 > > U 1 ¼ U0 ; U p ¼ U0 ; t ¼ m ; = ð8Þ k V2 1 > > h ¼ TTwT ; n ¼ nV 2v ; k ¼ v2 0 ; > T 1 > 0 > > > 2m > dB mb1 gðT w T 1 Þ > m 0 ; : Pr ¼ a0 ; M ¼ qV 2 ; G ¼ 2 U V
g
v*
dp dU 1 m ¼ q þ q U 1 þ rB20 U 1 : k dx dt
h ¼ 1 þ eent h!0
at y ¼ 0; as y ! 1:
) ð11Þ
3. The method of solution In order to reduce the above system of partial differential equations to a system of ordinary differential equations in dimensionless form, we may represent the velocity and temperature as u ¼ f0 ðyÞ þ eent f1 ðyÞ þ Oðe2 Þ þ ;
ð12Þ
h ¼ g0 ðyÞ þ eent g1 ðyÞ þ Oðe2 Þ þ :
ð13Þ
Substituting in (9) and (10) and by comparing the harmonic and non-harmonic terms, neglecting those terms of order e2, we get for (f0, g0) and (f1, g1) the equations
M.A. Seddeek, F.A. Salama / Computational Materials Science 40 (2007) 186–192
1 1 f 0 g0 ¼ 0; f000 þ f00 þ M þ ð1 f0 Þ þ Gg0 þ k hr g0 0 0
ð14Þ
1 f100 þ f10 nf1 Af 00 þ n M þ ð1 f1 Þ k 1 1 þ Gg1 ðf 0 g0 þ f10 g00 Þ þ f00 g00 g01 ¼ 0; ðg0 hr Þ 0 1 ðg0 hr Þ2
ð15Þ ð16Þ
g000 ð1 þ bg0 Þ þ Prg00 þ bg02 0
¼ 0; g1 g00 00 0 0 0 g1 ð1 þ bg0 Þ þ b 2g0 g1 ð1 þ g0 Þ þ Prðg01 ng1 Ag00 Þ ¼ 0: 1 þ bg0
ð17Þ
The boundary conditions can be written as f0 ¼ U p ; f 1 ¼ 0; g0 ¼ 1; g1 ¼ 1 f0 ¼ 1; f 1 ¼ 1; g0 ! 0; g1 ! 0
at y ¼ 0; as y ¼ 1:
ð18Þ
Fig. 1. Effects of hr on velocity profiles.
The governing boundary layer equations (14)–(17) subject to the boundary condition (18) are solved numerically. The shooting method for linear equations is based on replacing the boundary value problem by two initial value problems, and the solution of the boundary value problem is a linear combination of the two initial value problems. Of special importance for this flow and heat transfer situation is the heat transfer coefficient in terms of the Nusselt number as follows: sw ou sw ¼ ¼ ; qU 0 V 0 oy y¼0 oT =oy jw oh 1 ; NuRex ¼ ; Nu ¼ x oy y¼0 Tw T1 where Rex ¼ V m0 x is the Reynolds number. Fig. 2. Effects of b on velocity profile.
4. Results and discussion In the previous sections, we have formulated and solved the problem of the influence of variable viscosity and thermal conductivity on an unsteady MHD convection heat transfer past a semi-infinite vertical porous moving plate. In the numerical computation, the Prandtl number, Pr = 0.733 which corresponds to air, and various values of the material parameters are used. In addition, the boundary condition y ! 1 is approximated by ymax = 6, which is sufficiently large for the velocity to approach the relevant stream velocity. From Fig. 1 it can be seen that increase in the value of the viscosity-variation parameter hr leads to an increase in the velocity profile near the surface of the plate. An increase in the value of viscosityvariation parameter hr also leads to an increase in the viscosity near the surface of the plate and this approaches unity value at the outer-edge of the boundary layer for every value of the viscosity-variation parameter considered here. Figs. 2 and 3 show that the dimensionless velocity u
Fig. 3. Effects of b on temperature profiles.
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and the dimensionless temperature h increase as the thermal conductivity parameter b increases. This is because as b increases the thermal conductivity of the fluid increases. This increase in the fluid thermal conductivity increases the fluid temperature and accordingly its velocity. Figs. 4 and 5 show that the dimensionless velocity u and the dimensionless temperature h increase as the exponential index n increases. In Fig. 6, the velocity is plotted for several values of M. As seen in this figure, for a given value of M, the velocity profiles decrease monotonically with an increase in y. For small and large values of y, the effect of M is rather insignificant. Only when 0.5 < y < 2.5 is the value of the velocity reduced significantly depending on M. Fig. 7 shows the effects of the permeability of the porous medium parameter k on the velocity distribution. As shown, the velocity is increasing with the increasing dimensionless porous medium parameter. The effect of
Fig. 6. Effects of M on velocity profiles.
Fig. 4. Effects of n on velocity profiles.
Fig. 7. Effects of k on velocity profiles.
Fig. 5. Effects of n on temperature profiles.
the dimensionless porous medium parameter K becomes smaller as K increases. Physically, this result can be achieved when the holes of the porous medium are very large so that the resistance of the medium may be neglected. Fig. 8 shows the effect of the Grashof number G on the velocity distribution. As shown, the velocity is increasing with increasing G. In addition, the curves show that the peak value of velocity increases rapidly near the wall of the plate as Grashof number increases, and then decays to the relevant free stream velocity. The velocity profiles for different values of plate moving velocity in the direction of fluid flow Up are described in Fig. 9. Although we have different initial plate moving velocities, the velocity decays to the constant value for the given material parameter. Fig. 10 depicts the variation of the surface skin friction with the suction velocity parameter A for various values of plate moving velocity Up. As shown, the surface skin
M.A. Seddeek, F.A. Salama / Computational Materials Science 40 (2007) 186–192
191
Fig. 11. Effects of n on surface skin friction.
Fig. 8. Effects of G on velocity profiles.
friction decreases by increasing the plate moving velocity Up. The variation of the surface skin friction with the suction velocity parameter A for different values of the dimensionless exponential index n is shown in Fig. 11. From this figure, it can be seen that the surface skin friction increases as the dimensionless exponential index n is increased. 5. Conclusion
Fig. 9. Effects of Up on velocity profiles.
The problem of unsteady, laminar and hydromagnetic heat transfer past a semi-infinite vertical porous moving plate with variable suction has been studied. The fluid viscosity is assumed to varies as an inverse linear function of temperature but the thermal conductivity is assumed to varies as a linear function of temperature. The porous plate was assumed to move with a constant velocity in the direction of the fluid flow. The governing equations were developed and transformed into a system of nonlinear ordinary differential equations by perturbation technique and are solved numerically by using the shooting method, the surface skin-friction coefficient as well as the details of velocity and temperature fields are presented for various values of parameters of the problem. The numerical results indicate that the velocity increases with the increase in variable viscosity, thermal conductivity, the exponential index, porous medium, Grashof number and plate moving velocity, but it decreases as the magnetic field parameter increases. Also the temperature increases as the variable thermal conductivity and the exponential index increase. The surface skin friction decreases as the plate moving velocity increases, but it increases as the exponential index parameter increases. It is hoped that the present work will serve as a vehicle for understanding more complex problems involving the various physical effects investigated in the present problem. References
Fig. 10. Effects of Up on surface skin friction.
[1] A.A. Raptis, Int. J. Energy Res. 10 (1986) 97. [2] A.A. Raptis, N. Kafousias, Int. J. Energy Res. 6 (1982) 241.
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[3] V.M. Soundalgekar, Proc. Roy. Soc. London A 333 (1973) 25. [4] I.A. Hassanien, ZAAM. Z. Angew. Math. Mech. 79 (11) (1999) 786. [5] I. Pop, R.S. Gorla, M. Rashidi, Int. J. Eng. Sci. 30 (1997) 1. [6] E.M.A. Elbashbeshy, Int. J. Eng. Sci. 38 (2000) 207. [7] E.M.A. Elbashbeshy, F.N. Ibrahim, J. Phys. D 26 (1993) 2137. [8] M.A. Seddeek, J. Appl. Mech. Tech. Phys. 43 (1) (2002) 13.
[9] A.Y. Ghaly, M.A. Seddeek, Chaos, Solitons Fract. 19 (2004) 61. [10] M.A. Seddeek, A.M. Salem, Heat Mass Transfer 41 (2005) 1048– 1055. [11] J.C. Slattery, Momentum, Energy and Mass Transfer in Continua, McGraw Hill, New York, 1972. [12] K.A. Helmy, ZAMM 78 (4) (1998) 255. [13] Y.J. Kim, Int. J. Eng. Sci. 38 (2000) 833.
The effects of temperature dependent viscosity and thermal conductivity on unsteady MHD convective heat transfer past a semi-infinite vertical porous moving plate with variable suction M.A. Seddeek a
a,*
, Faiza A. Salama
b
Qassim University, College of Science, Mathematics Department, P.O. Box 237, Buriedah 81999, Saudi Arabia b Department of Mathematics, Faculty of Science, Suez Canal University, Egypt Received 10 May 2006; received in revised form 4 November 2006; accepted 29 November 2006 Available online 30 January 2007
Abstract In this article, we studied the effects of variable viscosity and thermal conductivity on an unsteady two-dimensional laminar flow of a viscous incompressible conducting fluid past a semi-infinite vertical porous moving plate taking into account the effect of a magnetic field in the presence of variable suction. The fluid viscosity is assumed to vary as an inverse linear function of temperature but the thermal conductivity is assumed to vary as a linear function of temperature. It is assumed that the porous plate moves with a constant velocity in the direction of fluid flow, and the free stream velocity follows the exponentially increasing small perturbation law. The governing equations for the flow are transformed into a system of nonlinear ordinary differential equations by perturbation technique and are solved numerically by using the shooting method. The effects of the various parameters on the velocity and temperature profiles as well as the surface skin-friction coefficient are presented graphically. Ó 2006 Elsevier B.V. All rights reserved. Keywords: Variable viscosity; Thermal conductivity; Magnetic field; Variable suction
1. Introduction Natural convection flow and heat transfer in porous moving plate is gaining more attention because of its wide applicability in packed beds, porous insulation, beds of fossil fuels, nuclear waste disposal, resin transfer molding, etc. Most of the existing analytical studies for this problem are based on the constant physical properties of the ambient fluid [1–3]. However, it is known that these properties may change with temperature [4]. To accurately predict the flow and heat transfer rates, it is necessary to take into account this variation of viscosity and thermal conductivity. *
Corresponding author. Permanent address: Department of Mathematics, Faculty of Science, Helwan University, Cairo 11785, Egypt. E-mail address: [email protected] (M.A. Seddeek). 0927-0256/$ - see front matter Ó 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.commatsci.2006.11.012
The influence of variable viscosity on the laminar boundary layer flow and heat transfer due to continuously moving plate is examined by Pop et al. [5]. The free convection flow with variable viscosity and thermal diffusivity along a vertical plate has been studied by Elbashbeshy [6] and Elbashbeshy and Ibrahim [7]. Seddeek [8] studied the effects of magnetic field and variable viscosity on forced non-Darcy flow about a flat plate with variable wall temperature in porous media in the presence of suction and blowing. Also, Ghaly and Seddeek [9] studied the Chebyshev finite difference method for the effects of chemical reaction, heat and mass transfer on laminar flow along a semi-infinite horizontal plate with temperature dependent viscosity. Recently, Seddeek and Salem [10] studied the laminar mixed convection adjacent to vertical continuously stretching sheets with variable viscosity and variable thermal diffusivity.
M.A. Seddeek, F.A. Salama / Computational Materials Science 40 (2007) 186–192
187
Nomenclature a A B0 E f g G k* M n Nu P Pr Re T Tr t U0 u, v V0 x, y
constant depending on the nature of the fluid suction velocity parameter magnetic induction constant depending on the reference state dimensionless stream function gravitational acceleration Grashof number permeability of the porous medium magnetic field parameter dimensionless exponential index Nusselt number pressure Prandtl number Reynolds number temperature constant dimensionless time scale of free stream velocity velocity components in the x and y directions scale of suction velocity distance along and perpendicular to the plate, respectively
The aim of the present work is to consider the case of a semi-infinite moving porous plate in a porous medium with the presence of pressure gradient and constant velocity in the flow direction when the magnetic field is imposed transverse to the plate. In addition, variable viscosity and thermal conductivity effects can become more pronounced in flow and heat transfer over a semi-infinite vertical porous moving plate. Therefore, the purpose of this study is to consider variable viscosity, thermal conductivity and variable suction effects on an unsteady MHD convective heat transfer past a semi-infinite vertical porous moving plate. We have reduced the two-dimensional continuity, momentum and energy equations to a system of nonlinear ordinary differential equations, which are solved numerically by using the shooting method. The effects of various parameters on the flow and heat transfer have been shown graphically.
2. Mathematical formulation Consider a two-dimensional unsteady flow of a laminar, incompressible fluid past a semi-infinite vertical porous moving plate embedded in a porous medium and subjected to a transverse magnetic field (see the sketch of the physical model). We assume that the fluid properties are isotropic and constant, except for the fluid viscosity l, which is assumed to vary as an inverse linear function of temperature T, in the form
Greek symbols a fluid thermal conductivity b thermal conductivity parameter b1 the coefficient of volumetric expansion e scale constant (1) r electrical conductivity l fluid viscosity m kinematic viscosity q density h dimensionless temperature hr variation viscosity parameter Superscripts / differentiation with respect to y dimensional properties * Subscripts P plate w wall conditions 1 free stream conditions
1 1 ¼ ð1 þ dðT T 1 ÞÞ l l1
or
1 ¼ EðT T r Þ; l
where E¼
d l1
and
1 Tr ¼ T1 ; d
Here l1 and T1 are the fluid free stream dynamic viscosity and the fluid free stream temperature, E and Tr are the constants and their values depend on the reference state and thermal property of the fluid, i.e. d. In general, E < 0 for fluid such as liquids and E > 0 for gases. Also, we assume that the fluid thermal conductivity, a, is assumed to vary as a linear function of temperature in the form [11] a ¼ a0 ð1 þ aðT T 1 ÞÞ; where a0 is the thermal diffusivity at temperature Tw and a is a constant depending on the nature of the fluid. In general a > 0 for fluids such as water and air, while a < 0 for fluids such as lubrication oils. This form can be rewritten in the non-dimensional form: a ¼ a0 ð1 þ bhÞ; where b = a(Tw T1) is the thermal conductivity parameter and Tw is the value of the plate temperature. The range of variations of b can be taken as follows: for air 0 6 b 6 6, for water 0 6 b 6 0.12 and for lubrication oils 0.1 6 b 6 0.
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M.A. Seddeek, F.A. Salama / Computational Materials Science 40 (2007) 186–192
V ¼ V 0 ð1 þ eAen t Þ;
* P
U ,x
ð6Þ
where A is a real positive constant, e and eA are small and less than unity, and V0 is a scale of suction velocity which has non-zero positive constant. Outside the boundary layer, Eq. (2) gives
U *∞ , T∞
B0 Porous medium
0
0
It is assumed that there is no applied voltage, which implies the absence of an electric field. The transversely applied magnetic field and magnetic Reynolds number are very small and hence the induced magnetic field is negligible [12]. Viscous and Darcy’s resistance terms are taken into account with constant permeability of the porous medium. Under the usual boundary layer approximation, the governing equation for this problem can be written as follows:
ð1Þ
u rB20 u ; k ð2Þ ð3Þ
subject to the boundary conditions
u !
U 1
t
T ¼ T w þ eðT w T 1 Þen ¼ U 0 ð1 þ ee
n t
Þ;
T ! T1
o2 u h0 ou ; dy 2 h hr oy " 2 # oh 1 o2 h oh nt oh ð1 þ eAe Þ ¼ ð1 þ bhÞ 2 þ b ; ot oy Pr dy oy þ Gh þ
at y ¼ 0; as y ! 1;
ð4Þ ð5Þ
where up is velocity of the plate and e the scalar constant (1). From the continuity of Eq. (1), it is clear that the suction velocity normal to the plate is a function of time only and we shall take it in the form
ð9Þ ð10Þ
where hr is variable viscosity parameter. The boundary conditions become
u ! U 1;
oT o oT oT þ v ¼ a ot oy oy oy
u ¼ up ;
ou ou dU 1 1 ð1 þ eAent Þ ¼ þ Mþ ðU 1 uÞ ot oy K dt
u ¼ U p;
þ qgb1 ðT T 1 Þ mq
0
In view of Eqs. (6)–(8), the governing equations (2) and (3) reduce to the following non-dimensional form:
The sketch of the physical model
ov ¼ 0; oy ou op o ou ou ¼ þ v þ l q ot oy ox oy oy
ð7Þ
The boundary layers Eqs. (1)–(3), along with the boundary conditions (4) and (5), are solved using the perturbation technique as the similarity solution is not possible. First we convert the partial differential equations into non-similar form making use of the following transformation [13]: 9 > u ¼ Uu 0 ; v ¼ Vv 0 ; y ¼ V 0my ; > > > > > > up t V 20 U 1 > > U 1 ¼ U0 ; U p ¼ U0 ; t ¼ m ; = ð8Þ k V2 1 > > h ¼ TTwT ; n ¼ nV 2v ; k ¼ v2 0 ; > T 1 > 0 > > > 2m > dB mb1 gðT w T 1 Þ > m 0 ; : Pr ¼ a0 ; M ¼ qV 2 ; G ¼ 2 U V
g
v*
dp dU 1 m ¼ q þ q U 1 þ rB20 U 1 : k dx dt
h ¼ 1 þ eent h!0
at y ¼ 0; as y ! 1:
) ð11Þ
3. The method of solution In order to reduce the above system of partial differential equations to a system of ordinary differential equations in dimensionless form, we may represent the velocity and temperature as u ¼ f0 ðyÞ þ eent f1 ðyÞ þ Oðe2 Þ þ ;
ð12Þ
h ¼ g0 ðyÞ þ eent g1 ðyÞ þ Oðe2 Þ þ :
ð13Þ
Substituting in (9) and (10) and by comparing the harmonic and non-harmonic terms, neglecting those terms of order e2, we get for (f0, g0) and (f1, g1) the equations
M.A. Seddeek, F.A. Salama / Computational Materials Science 40 (2007) 186–192
1 1 f 0 g0 ¼ 0; f000 þ f00 þ M þ ð1 f0 Þ þ Gg0 þ k hr g0 0 0
ð14Þ
1 f100 þ f10 nf1 Af 00 þ n M þ ð1 f1 Þ k 1 1 þ Gg1 ðf 0 g0 þ f10 g00 Þ þ f00 g00 g01 ¼ 0; ðg0 hr Þ 0 1 ðg0 hr Þ2
ð15Þ ð16Þ
g000 ð1 þ bg0 Þ þ Prg00 þ bg02 0
¼ 0; g1 g00 00 0 0 0 g1 ð1 þ bg0 Þ þ b 2g0 g1 ð1 þ g0 Þ þ Prðg01 ng1 Ag00 Þ ¼ 0: 1 þ bg0
ð17Þ
The boundary conditions can be written as f0 ¼ U p ; f 1 ¼ 0; g0 ¼ 1; g1 ¼ 1 f0 ¼ 1; f 1 ¼ 1; g0 ! 0; g1 ! 0
at y ¼ 0; as y ¼ 1:
ð18Þ
Fig. 1. Effects of hr on velocity profiles.
The governing boundary layer equations (14)–(17) subject to the boundary condition (18) are solved numerically. The shooting method for linear equations is based on replacing the boundary value problem by two initial value problems, and the solution of the boundary value problem is a linear combination of the two initial value problems. Of special importance for this flow and heat transfer situation is the heat transfer coefficient in terms of the Nusselt number as follows: sw ou sw ¼ ¼ ; qU 0 V 0 oy y¼0 oT =oy jw oh 1 ; NuRex ¼ ; Nu ¼ x oy y¼0 Tw T1 where Rex ¼ V m0 x is the Reynolds number. Fig. 2. Effects of b on velocity profile.
4. Results and discussion In the previous sections, we have formulated and solved the problem of the influence of variable viscosity and thermal conductivity on an unsteady MHD convection heat transfer past a semi-infinite vertical porous moving plate. In the numerical computation, the Prandtl number, Pr = 0.733 which corresponds to air, and various values of the material parameters are used. In addition, the boundary condition y ! 1 is approximated by ymax = 6, which is sufficiently large for the velocity to approach the relevant stream velocity. From Fig. 1 it can be seen that increase in the value of the viscosity-variation parameter hr leads to an increase in the velocity profile near the surface of the plate. An increase in the value of viscosityvariation parameter hr also leads to an increase in the viscosity near the surface of the plate and this approaches unity value at the outer-edge of the boundary layer for every value of the viscosity-variation parameter considered here. Figs. 2 and 3 show that the dimensionless velocity u
Fig. 3. Effects of b on temperature profiles.
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and the dimensionless temperature h increase as the thermal conductivity parameter b increases. This is because as b increases the thermal conductivity of the fluid increases. This increase in the fluid thermal conductivity increases the fluid temperature and accordingly its velocity. Figs. 4 and 5 show that the dimensionless velocity u and the dimensionless temperature h increase as the exponential index n increases. In Fig. 6, the velocity is plotted for several values of M. As seen in this figure, for a given value of M, the velocity profiles decrease monotonically with an increase in y. For small and large values of y, the effect of M is rather insignificant. Only when 0.5 < y < 2.5 is the value of the velocity reduced significantly depending on M. Fig. 7 shows the effects of the permeability of the porous medium parameter k on the velocity distribution. As shown, the velocity is increasing with the increasing dimensionless porous medium parameter. The effect of
Fig. 6. Effects of M on velocity profiles.
Fig. 4. Effects of n on velocity profiles.
Fig. 7. Effects of k on velocity profiles.
Fig. 5. Effects of n on temperature profiles.
the dimensionless porous medium parameter K becomes smaller as K increases. Physically, this result can be achieved when the holes of the porous medium are very large so that the resistance of the medium may be neglected. Fig. 8 shows the effect of the Grashof number G on the velocity distribution. As shown, the velocity is increasing with increasing G. In addition, the curves show that the peak value of velocity increases rapidly near the wall of the plate as Grashof number increases, and then decays to the relevant free stream velocity. The velocity profiles for different values of plate moving velocity in the direction of fluid flow Up are described in Fig. 9. Although we have different initial plate moving velocities, the velocity decays to the constant value for the given material parameter. Fig. 10 depicts the variation of the surface skin friction with the suction velocity parameter A for various values of plate moving velocity Up. As shown, the surface skin
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Fig. 11. Effects of n on surface skin friction.
Fig. 8. Effects of G on velocity profiles.
friction decreases by increasing the plate moving velocity Up. The variation of the surface skin friction with the suction velocity parameter A for different values of the dimensionless exponential index n is shown in Fig. 11. From this figure, it can be seen that the surface skin friction increases as the dimensionless exponential index n is increased. 5. Conclusion
Fig. 9. Effects of Up on velocity profiles.
The problem of unsteady, laminar and hydromagnetic heat transfer past a semi-infinite vertical porous moving plate with variable suction has been studied. The fluid viscosity is assumed to varies as an inverse linear function of temperature but the thermal conductivity is assumed to varies as a linear function of temperature. The porous plate was assumed to move with a constant velocity in the direction of the fluid flow. The governing equations were developed and transformed into a system of nonlinear ordinary differential equations by perturbation technique and are solved numerically by using the shooting method, the surface skin-friction coefficient as well as the details of velocity and temperature fields are presented for various values of parameters of the problem. The numerical results indicate that the velocity increases with the increase in variable viscosity, thermal conductivity, the exponential index, porous medium, Grashof number and plate moving velocity, but it decreases as the magnetic field parameter increases. Also the temperature increases as the variable thermal conductivity and the exponential index increase. The surface skin friction decreases as the plate moving velocity increases, but it increases as the exponential index parameter increases. It is hoped that the present work will serve as a vehicle for understanding more complex problems involving the various physical effects investigated in the present problem. References
Fig. 10. Effects of Up on surface skin friction.
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