Mass Transfer Effects on Unsteady Hydromagnetic Convective Flow past a Vertical Porous Plate in a Porous Medium with Heat Source

The objective of this paper is to analyze the effect of mass transfer on unsteady hydromagnetic free convective flow of a viscous incompressible electrically conducting fluid past an infinite vertical porous plate in presence of constant suction and heat source. The governing equations of the flow field are solved using multi parameter perturbation technique and approximate solutions are obtained for velocity field, temperature field, concentration distribution, skin friction and the rate of heat transfer. The effects of the flow parameters such as Hartmann number M, Grashof number for heat and mass transfer Gr, Gc; permeability parameter Kp, Schmidt number Sc, heat source parameter S, Prandtl number Pr etc. on the flow field are analyzed with the help of figures and tables. It is observed that a growing Hartmann number or Schmidt number retards the mean velocity as well as the transient velocity of the flow field at all points. The effect of increasing Grashof number for heat and mass transfer or heat source parameter is to accelerate both mean and transient velocity of the flow field at all points. The mean velocity of the flow field increases with an increase in permeability parameter while the transient velocity increases for smaller values of Kp (1) and for higher values the effect reverses. A growing Hartmann number decreases the transient temperature of the flow field at all points while a growing permeability parameter or heat source parameter reverses the effect. The Prandtl number increases the transient temperature for small values of Pr (1) and for higher values the effect reverses. The effect of increasing Schmidt number is to reduce the concentration boundary layer thickness of the flow field at all points. The problem has some relevance in the geophysical and astrophysical studies.


INTRODUCTION
The phenomenon of hydromagnetic flow with heat and mass transfer in an electrically conducting fluid past a porous plate embedded in a porous medium has attracted the attention of a good number of investigators because of its varied applications in many engineering problems such as MHD generators, plasma studies, nuclear reactors, oil exploration, geothermal energy extractions and in the boundary layer control in the field of aerodynamics. Heat transfer in laminar flow is important in problems dealing with chemical reactions and in dissociating fluids.
In view of its wide applications, Hasimoto (1957) initiated the boundary layer growth on a flat plate with suction or injection. Soundalgekar (1974) showed the effect of free convection on steady MHD flow of an electrically conducting fluid past a vertical plate. Yamamoto and Iwamura (1976) explained the flow of a viscous fluid with convective acceleration through a porous medium. Mansutti et al. (1993) have discussed the steady flow of a non-Newtonian fluid past a porous plate with suction or injection. Jha (1998) analyzed the effect of applied magnetic field on transient free convective flow in a vertical channel. Chandran and his associates (1998) have discussed the unsteady free convection flow of an electrically conducting fluid with heat flux and accelerated boundary layer motion in presence of a transverse magnetic field. Acharya et al. (1999) have reported the problem of heat and mass transfer over an accelerating surface with heat source in presence of suction and blowing.
The unsteady free convective MHD flow with heat transfer past a semi-infinite vertical porous moving plate with variable suction has been studied by Kim (2000). Singh and Thakur (2002) have given an exact solution of a plane unsteady MHD flow of a non-Newtonian fluid. Sharma and Pareek (2002) explained the behaviour of steady free convective MHD flow past a vertical porous moving surface. Singh and his co-workers (2003) have analyzed the effect of heat and mass transfer in MHD flow of a viscous fluid past a vertical plate under oscillatory suction velocity. Makinde et al. (2003) discussed the unsteady free convective flow with suction on an accelerating porous plate. Sarangi and Jose (2005) studied the unsteady free convective MHD flow and mass transfer past a vertical porous plate with variable temperature. Das and his associates (2006) estimated the mass transfer effects on unsteady flow past an accelerated vertical porous plate with suction employing finite difference analysis. Das et al. (2007) investigated numerically the unsteady free convective MHD flow past an accelerated vertical plate with suction and heat flux. Das and Mitra (2009) discussed the unsteady mixed convective MHD flow and mass transfer past an accelerated infinite vertical plate with suction. Recently, Das and his co-workers (2009) analyzed the effect of mass transfer on MHD flow and heat transfer past a vertical porous plate through a porous medium under oscillatory suction and heat source. More recently, Das et al. (2010) investigated the hydromagnetic convective flow past a vertical porous plate through a porous medium with suction and heat source.
The study of stellar structure on solar surface is connected with mass transfer phenomena. Its origin is attributed to difference in temperature caused by the non-homogeneous production of heat which in many cases can rest not only in the formation of convective currents but also in violent explosions. Mass transfer certainly occurs within the mantle and cores of planets of the size of or larger than the earth. In the present study we therefore, propose to analyze the effect of mass transfer on unsteady free convective flow of a viscous incompressible electrically conducting fluid past an infinite vertical porous plate with constant suction and heat source in presence of a transverse magnetic field. This paper basically highlights the effect of mass transfer on hydromagnetic flow in presence of suction and heat source.

FORMULATION OF THE PROBLEM
Consider the unsteady free convective mass transfer flow of a viscous incompressible electrically conducting fluid past an infinite vertical porous plate in presence of constant suction and heat source and transverse magnetic field. Let the x-axis be taken in vertically upward direction along the plate and y-axis normal to it. The physical sketch and geometry of the problem is shown in Fig. 1. Neglecting the induced magnetic field and the Joulean heat dissipation and applying Boussinesq's approximation the governing equations of the flow field are given by: Continuity equation: Momentum equation: Concentration equation: The boundary conditions of the problem are: Introducing the following non-dimensional variables and parameters, The corresponding boundary conditions are:

Skin Friction
The skin friction at the wall is given by

Heat Flux
The heat flux at the wall in terms of Nusselt number is given by

RESULTS AND DISCUSSIONS
The effect of mass transfer on unsteady free convective flow of a viscous incompressible electrically conducting fluid past an infinite vertical porous plate with constant suction and heat source in presence of a transverse magnetic field has been studied. The governing equations of the flow field are solved employing multi parameter perturbation technique and approximate solutions are obtained for velocity field, temperature field, concentration distribution, skin friction and rate of heat transfer. The effects of the pertinent parameters on the flow field are analyzed and discussed with the help of velocity profiles (Figs. 2-13), temperature profiles (Figs. 14-17), concentration distribution (Fig. 18)

Velocity Field
The velocity of the flow field is found to change more or less with the variation of the flow parameters.

Temperature Field
The temperature of the flow field suffers a substantial change with the variation of the flow parameters such as Prandtl number Pr, Hartmann number M, heat source parameter S and permeability parameter Kp. These variations are shown in Figs. 14-17. The temperature profiles are in good agreement with those of Das et al. (2010). Figure 14 depicts the effect of Prandtl number on the temperature field keeping other parameters of the flow field constant. It is interesting to observe that for lower value of Pr (1), it enhances the transient temperature while for higher values the effect reverses. Figure 15 shows the effect of magnetic parameter on the temperature field. The effect of Hartmann number is to retard the temperature of the flow field at all points. Curve with M=0 corresponds to the non-MHD flow. This shows that in absence of magnetic field the temperature first rises near the plate and thereafter, it falls. In other curves there is a decrease in temperature at all points. This shows the dominating effect of the magnetic field due to the action of the Lorentz force in the flow field. The effect of heat source parameter on the temperature field is presented in Fig. 16. The heat source parameter is found to enhance the temperature of the flow field at all points. In Fig. 17, we analyze the effect of permeability parameter on the temperature field. A growing permeability parameter is found to increase the temperature of the flow field at all points.

Concentration Distribution
The variation in the concentration boundary layer of the flow field is shown in Fig. 18 due to the change in the Schmidt number Sc. Curves with Sc=0.22, 0.30, 0.6 and 0.78 respectively, represent the concentration distribution in presence of H2, He, H2O vapour and NH3 in the flow field. Comparing the curves of the said figure it is observed that a growing Schmidt number decreases the concentration boundary layer thickness of the flow field at all points.

Skin Friction
The values of skin friction at the wall against Kp for different values of Hartmann number M and heat source parameter S are entered in Tables 1 and 2 respectively. From Table 1, it is observed that a growing Hartmann number M reduces the skin friction at the wall for a fixed value of the permeability parameter due to the action of Lorentz force in the flow field. It is further observed from Table 2 that both permeability parameter Kp and heat source parameter S enhance the skin friction at the wall. Our observation for skin friction agrees with those of Das et al. (2010).

Rate of Heat Transfer
The rate of heat transfer at the wall varies with the variation of Prandtl number Pr, Hartmann number M, permeability parameter Kp. These variations are entered in the Tables 3-5. From Table 3, we observe that a growing Prandtl number or permeability parameter increase the magnitude of the rate of heat transfer at the wall. Further, it is observed from Table 4 that an increase in Hartmann number reduces its value for a given value of Prandtl number. Again from Table 5, we see that for a given value of permeability parameter it enhances the magnitude of rate of heat transfer for small values of M and for higher values the effect reverses due to the magnetic pull of the Lorentz force acting on the flow field. These variations agree with those of Das et al. (2010) with a little deviation for higher value of M.

CONCLUSIONS
We summarize below the following results of physical interest on the velocity, temperature and the concentration distribution of the flow field and also on the wall shear stress and rate of heat transfer at the wall.