Effect of mass transfer on the free convective flow through a porous medium with variable permeability in slip flow regime with couple stresses

Abstract

This paper examines the problem of mass transfer on MHD unsteady free convective flow of a polar fluid through a porous medium of variable permeability bounded by an infinite horizontal porous plate in slip flow regime. The permeability of the porous medium decreases exponentially with time about a constant mean. Using perturbation technique the expressions for the velocity distribution, mean angular velocity of rotation of particles, concentration distribution and skin friction are obtained. The effects of permeability parameter K0, Magnetic parameter M, Prandtl number Pr , Schimdt number Sc ,and Grashof number Gr ,Modified Grashof number Gm entering into the problem on velocity, temperature distribution and concentration distribution are shown graphically and discussed numerically. It can be observed that this velocity decreases with the increase in M, Pr, Sc and increases with the increase in K0, Gr, Gm. Temperature and Concentration decreases with the increase in the value of Pr and Sc respectively. INTRODUCTION The convection problem in a porous medium has important application in geothermal reservoirs and geothermal energy extractions. It is obvious that in order to utilize the geothermal energy to maximum, one should have a complete and precise knowledge about the amount of perturbations needed to generate convection current in geothermal fluids. Also the knowledge of the quantity of perturbation is essential to entreat convection current in mineral fluids that is found in the earth’s crust helps one to utilize minimal energy to extract the minerals. Cheng and Lau[1] and Cheng and Teckchandani [2] obtained numerical solutions for the convective flow in a porous medium bounded by two isothermal parallel plates in the presence of withdrawl of the fluid. All the above mentioned studies treat the permeability and the conductivity or thermal resistance of the medium as constant or neglect the effect of porosity. Lai and Kulacki [6] studied Coupled heat and mass transfer by natural convection from vertical surfaces in porous media. Lin and Wu[7] discussed Coupled heat and mass transfer by laminar natural convection from vertical plate. Malasetty and Gaikwad[8] investigated effect of cross diffusion on double diffusion convection in the presence of horizontal gradients. Yan et. al [11] have discussed Simultaneous heat and mass transfer in laminar mixed convection flows between vertical parallel plate with asymmetric heating. Yan[12] studied Turbulent mixed convection heat and mass transfer past a vertical porous plate. Mazumdar and Deka[9] investigated MHD flow past an impulsively started infinite vertical plate in presence of thermal radiation. In geothermal region situation may arise when the flow becomes unsteady and sliping at the boundary may take place as well. In such situation of slip flow, ordinary continuum approach fails to yield satisfactory results. Many authors have solved problems taking slip conditions at the boundary. Jain and Teneja [3] solved the problem of magnetopolar flow through a porous medium in slip flow regime. Khandelwal et al [5] have studied effect of couple stresses on the flow through a porous medium with variable permeability in slip flow regime. Varshney et. al.[10] have discussed effect of heat transfer on the flow through a porous medium with variable permeability in slip flow regime with couple stress. Our problem under study is an extension of the problem of Varshney et. al.[10] with mass transfer. The effects of different parameter entering into the problem viz. Permeability parameter Ko, Magnetic Am. J. Sci. Ind. Res., 2011, 2(3): 469-477 470 parameter M, Prandtl number Pr , Schimdt number Sc ,and Grashof number Gr ,Modified Grashof number Gm entering into the problem on velocity, temperature distribution and concentration distribution are shown graphically and discussed numerically FORMULATION AND SOLUTION OF THE PROBLEM Let us consider the unsteady flow of a polar fluid through a porous medium of variable permeability in slip flow regime past an infinite horizontal porous plate in presence of a transverse magnetic field. The external velocity of the fluid is assumed as nt 0 U [1 e ] − + e . The flow is chosen to be at small magnetic Reynolds number which enables us to neglect the induced magnetic field. The x-axis is taken along the plate and y-axis normal to it. Under these conditions the equations which govern the flow are: v 0 y ∂ = ∂ (1)