Boundary layer ¯ows in a vertical porous enclosure induced by opposing buoyancy forces A. Amahmid a , M. Hasnaoui a , M. Mamou b , P. Vasseur b, * a De Âpartement de Physique, Faculte  des Sciences Semlalia, Marrakech, B.P. S-15, Maroc b Ecole Polytechnique, C.P. 6079, Suc. `Centre-Ville', Montreal, P.Q., Canada H3C 3A7 Received 3 August 1998; received in revised form 4 January 1999 Abstract Double diusive natural convection induced in a vertical porous layer subject to opposing horizontal gradients of heat and mass is studied analytically and numerically using the Darcy model with the Boussinesq approximation. The governing parameters for the problem are the thermal Rayleigh number, R T , the Lewis number, Le, the buoyancy ratio, N and the aspect ratio, A, of the enclosure. The analytical solution is developed on the basis of the parallel ¯ow approximation. A numerical study is performed to validate the results of the analytical predictions. It is demonstrated in this study that there exists a domain in (Le, N ) plane where, at large R T , boundary layer pro®les are obtained for the velocity and density but not for the temperature and concentration. The boundary layer regimes obtained for this domain (N < 0) are extremely dierent from those found in the previous studies for the case of aiding buoyancy forces (N>0). # 1999 Elsevier Science Ltd. All rights reserved. 1. Introduction Double diusive natural convection induced in a ¯uid-saturated porous medium is widely encountered in nature and technological processes. The engineering applications of the problem are of importance in many situations like the migration of moisture through air contained in ®brous insulations, food processing, con- taminant transport in ground water, electrochemical processes, etc. While considerable research has been carried out on thermally driven natural convection in porous media (see for instance Nield and Bejan [1]), relatively less work has been done on natural convec- tion due to combined buoyancy forces. The aim of the present study is to discuss a boundary layer solution for free convection, within a rectangular cavity, due to opposing buoyancy eects (N < 0). Past studies on double diusive convection in a ver- tical porous enclosure indicate that the resulting ¯ows can be very dierent from those driven by the tempera- ture ®eld solely, especially when the buoyancy forces are opposing each other. For instance, for the case of opposing and equal buoyancy forces (N=  1) the rest state, in a vertical cavity with constant temperature and concentration on the vertical walls, is an exact sol- ution of the problem. The stability of this solution was investigated by Charrier-Mojtabi et al. [2] on the basis of the linear stability theory. The critical Rayleigh number, for the onset of supercritical convection, was derived by these authors in terms of the aspect ratio A of the cavity and the Lewis number Le. Their results were extended by Mamou et al. [3] to consider the case of an inclined cavity. It was demonstrated that for values of Lewis number around unity, overstability is possible, provided that the normalized porosity of the International Journal of Heat and Mass Transfer 42 (1999) 3599±3608 0017-9310/99/$ - see front matter # 1999 Elsevier Science Ltd. All rights reserved. PII: S0017-9310(99)00019-8 www.elsevier.com/locate/ijhmt * Corresponding author. Tel.: +1-514-340-4711; fax: +1- 514-340-5917. E-mail address: vasseur@meca.polymtl.ca (P. Vasseur)