~) Pergamon Int. J. Heat Mass TransJer. Vol. 38, No. 8, pp. 1517 1525, 1995 Copyright ~ 1995 Elsevier Science Ltd Printed in Great Britain. All rights reserved 0017 9310/95 $9.50+0.00 0017-9310(94)00340-8 Conjugate mixed convection on a vertical surface in a porous medium I. POP Faculty of Mathematics, University of Cluj, R-3400 Cluj, CP 253, Romania and D. LESNIC and D. B, INGHAM Department of Applied Mathematical Studies, University of Leeds, Leeds LS2 9JT, U.K. (Received 19 January 1994) Abstract--The aim of this paper is to present a detailed analysis of the problem of steady conjugate mixed- convection flow along a vertical finite fiat plate which is embedded in a porous medium under the boundary- layer approximation, The problem then reduces to a parabolic partial differential equation which involves only the buoyancy parameter, 2. The cases of both aiding (2 > 0) and opposing (2 < 0) flows are considered. Full numerical and asymptotic solutions are obtained over a wide range of values of 2 and the results for the temperature profiles on the plate and in the convective fluid are presented. It is found that, unlike all other problems previously investigated, in both a porous and a non-porous medium and for all inclinations of the plate, unseparated flows can be obtained in this conjugate situation even when there is an opposing flow when 2 1> - 1. Further, when 2 is very large and negative, predictions of the separation point of the boundary layer from the plate are also reported. 1. INTRODUCTION The problems which occur in conjugate free and forced convection from vertical and horizontal sur- faces in a viscous fluid have been the concern of researchers for more than 30 years (see, for example, the recent review article by Martynenko and Soko- vishin [1]). However, very little research work has been performed on the corresponding porous medium configuration [2, 3]. In the mixed-convection region it is important to study such problems because fre- quently the boundary layer separates, which has a considerable effect on the heat transfer characteristics. In this paper we present an analysis of the problem of conjugate mixed convection from a vertical finite flat plate which is embedded in a porous medium. We propose new non-dimensional co-ordinates which are such that the conjugation parameter is scaled from the governing equations. Thereby, the problem depends only on one parameter, namely the buoyancy parameter, )~, which is the ratio of the Rayleigh to the Peclet number, Ra/Pe. Both the situation when the flow and buoyancy force are in the same direction, which is referred to as assisting flow, and that when they are in opposite directions, which is referred to as opposing flow, are discussed. It is worth mentioning that, in all cases previously investigated, in both a porous medium and a non-porous medium, it has always been found that, when there is an opposing flow, no matter what the inclination of the plate, the boundary layer separates at some distance along the plate. However, in the present conjugate problem the flow does not separate for 2 >/ -1, i.e. there is a regime in which the flow does not separate even though the flow is opposing in nature. When 2 < - 1 the flow does separate and the separation point has been determined from the full numerical solution. It is found that, as one would have physically predicted, the smaller the value of 2 the sooner flow separates. When 2 is very large and negative an estimate of the distance along the plate where the flow separates is presented. Further, in this paper we predict the tem- perature fields both in the boundary layer adjacent to the plate and at the solid-fluid interface, which are determined by the common solution of the energy equations for the fluid and the solid, respectively. 2. GOVERNING EQUATIONS The co-ordinate system and flow variables are shown in a schematic diagram (Fig. 1). The model is based on a vertical rectangular plate, of length l and thickness b, which is embedded in a porous medium and over which the fluid flows with an undistorted uniform speed U~. The outside surface of the plate is maintained at a constant temperature To, while the ambient fluid is at a uniform temperature T~, where To > T~ (aiding flow) or To < T~ (opposing flow). We assume that the boundary-layer approximation holds in the convective fluid and that the plate is thin relative to its length, i.e. b/l<< 1, so that the axial 1517