ISSN 1009 - 3095 Joumal of Zhejiang University SCIENCE V.4, No.2, P. 181 - 185, Mar. - Apr., 2003 http://www, periodicals, com. cn; http://www, zju. edu. cn/English http://www, zjupress, com; http://lib, zju. edu. cn/eindex, htm; jzus @ zju. edu. cn 181 Three dimensional Couette flow and heat transfer through a porous medium with variable permeability CHAUDHARY R. C.1, SHARMA Pawan Kumar2 ( Department of Mathematics, University of Rajasthan, Jaipur 302004, India ) E-mail : 1 ramacharanchaudhary @ rediffmail, com; 2 pawkumar 20 @yahoo, co. in Received Sept. 5,2002 ; revision accepted Dec. 6,2002 Abstract: This paper reports research on the effects of variations in injection velocity and permeability on the heat transfer and flow through a highly porous medium between two horizontal parallel plates situated at constant distance with constant suction by the upper plate. Due to this type of variation in injection velocity and in permeability the flow becomes three dimensional. The governing equations are solved by adopting complex variable notations to obtain the expressions for the velocity and temperature field. The skin-friction along the main flow direction and rate of heat transfer are discussed with the help of graphs. Key words: Couette flow, Porous medium, Heat transfer, Permeability injection/suction. Document code: A CLC number:O29:TBll INTRODUCTION The flow of a viscous liquid over and through a porous medium has been the subject of intens- ive studies in recent years because of its natural occurrence in the movement of water and oil in- side the earth and the flow of rivers through po- rous banks. It has applications in many engineer- ing and biomedical problems. In view of the geo- physical applications the flows through porous medium, a series of investigations conducted by Raptis et al. (1981a; 1981b; 1982) into the steady flow past a vertical wall. Raptis (1983) and Raptis et al. (1985) further studied the un- steady free convective flow through highly porous medium bounded by an infinite porous plate. Recently Singh et. al (1983) studied a free con- vective flow through a porous medium with peri- odic permeability variation. Assuming periodic permeability variation Singh et al. (1995a) stud- ied the oscillatory flow through a porous medi- um. Singh et. al (1995b) also studied the effect of transverse periodic variation of the permeabili- ty on the steady flow and heat transfer. Singh (1999) studied three dimensional couette flow with transpiration cooling. Also Singh et al. (2000) studied the three dimensional fluctuating flow and heat transfer through a porous medium with variable permeability. However, to the best of the author's knowledge, the applications of the variations of injection velocity and perme- ability on three dimensional Couette flow has not yet appeared in the literature. Therefore the ob- ject of this paper is to investigate the simulta- neous effects of injection and permeability varia- tions on three dimensional Couette flow and heat transfer through porous variable permeability me- dium. GOVERNING EQUATIONS An unsteady Couette flow of a viscous incom- pressible fluid between two parallel flat highly porous plates is considered. The upper plate in uniform motion U is subjected to a constant suc- tion V 0 and the lower to a variation of injection velocity of the form: V* (z* ,t* ) = V0[1 + r - oo*t*)~ (1) Also the permeability variations of the porous medium is of the form k* (z* ,t* ) = ko /[1 + ecos(Trz /d - t )1 (2) Where k 0 * is the mean permeability of the me- dium, co* is the frequency of fluctuations, t* is