Slip Effect on the MHD Three Dimensional Free Convective Flow of a Viscous Fluid Through a Porous Medium with Periodic Permeability

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Published on International Journal of Biology, Physics & Matematics
ISSN: 2721-3757, Volume 1, Issue 1, page 148 – 154
Publication Date: 25 October 2018

Mehfooz Hussain Shah, M. A. Rana, Nosheen Zareen Khan, & M. Shoaib
Department of Mathematics & Statistics, Riphah International University

M. Shoaib
COMSATS Institute of Information Technology
Kamra Road, Post code 43600, Attock, Pakistan

Abstract
The purpose of this work is to study the three dimensional free convective flow and heat transfer through a porous medium with periodic permeability. The porous medium is bounded by an infinite vertical porous plate. Assuming the free stream velocity to be uniform, approximate solutions are obtained for the flow field, the temperature field, the skin friction and the rate of heat transfer. Effects of various non-dimensional parameters emerging in the model are visualized graphically.

Keywords: Free Convective Flow, Porous Medium, Periodic Permeability, Perturbation Method, Analytic Solutions.

1. Introduction
In recent years, natural convection flows through a porous medium have gained the attention of many workers due to their applications in science and engineering. For example, in agriculture engineering to study seepage of water in river beds, the underground water resources, etc. Because of such important applications, Singh and Kumar [1] studied unsteady two-dimensional free convective flow through a porous medium bounded by infinite vertical porous plates, Singh and Kumar [2] have studied unsteady two-dimensional free convective flow through a porous medium.
Most of the workers considered only the two-dimensional flows by taking permeability of the porous medium either constant or time dependent. Many workers [3-4] analyzed the effects of such transverse permeability distribution on the flow of an incompressible fluid through a porous medium. Further, three dimensional flow through porous medium with heat transfer taking periodic permeability into account is investigated by Singh and Sharma [5]; Jain et al. [6] presented the influence of periodic permeability and temperature of free convective three dimensional flow of a viscous fluid through porous medium with slip effects. Recently, Reddy et al. [7] have extended the work of Singh and Sharma [5] by adding MHD effects. They noted that velocity of the fluid reduces due heat sink and magnetic field.
The study of Magnetohydrodynamic (MHD) flows with heat transfer is very important because of its various industrial applications, for example, cooling of nuclear reactor, power generation system, etc. Mamta Gohal and Namrata Naraniya [8] analyzed three dimensional free convective Couette flow with transpiration cooling, Ahmed Sahim [9] studied the MHD and chemical influences on unsteady flow of an incompressible electrically conducting viscous fluid over a vertical porous semi-infinite porous plate in a slip-flow regime.

2. Mathematical Analysis
Steady flow of a viscous incompressible fluid through porous medium is considered. The medium is bounded by an infinite vertical porous plate in x^* z^*-plane with x^*-axis along the plate in upward direction. The y^*-axis is taken perpendicular to the plane of the plate and directed into the fluid flow. The suction through the plate is assumed to be constant while free stream velocity U is considered to be uniform. The permeability is taken of the form ……… where k_o^*denotes the mean permeability, ϵ(<<1) is the amplitude of the permeability variation, and l is the wavelength of permeability distribution. Due to permeability of the term (3.1), the flow becomes three-dimensional. The velocity components along x^*-,y^*-,z^*- axes are u^*,v^*,w^* respectively while T^* denotes the temperature. Moreover, density depending on temperature whiles other fluid properties are considered to be constant ………