Convection of a binary fluid saturating a shallow porous cavity subjected to cross heat fluxes

Abstract

In this work, natural convection in a differentially heated binary mixture is studied analytically and numerically. The fluid is subjected to the Soret effect and is contained in a shallow rectangular porous cavity. All four faces are exposed to uniform heat fluxes, opposite faces being heated and cooled, respectively. Analytical solutions for the stream function, temperature and concentration fields are obtained using a parallel flow assumption in the core region of the cavity and an integral form of the energy and constituent equations. Numerical confirmation of the analytical predictions is also obtained. Results are presented first in the presence of a vertical temperature gradient (a = 0) for which the solution takes the form of a standard Bénard bifurcation. For this situation, steady bifurcations are either pitchfork or subcritical, depending on the separation parameter ϕ and Lewis number Le. The imperfection brought by a horizontal temperature gradient (a≠0) to the bifurcation is then investigated. Both the nonlinear analytical model and the numerical solution indicate that, depending on a, ϕ and Le, the onset of motion occurs through subcritical bifurcations. The existence of transcritical bifurcations is also demonstrated. The special case where the buoyancy forces induced by the thermal and solutal forces are opposing and of equal intensity (ϕ =-1) is also discussed. For this particular situation, the supercritical Rayleigh number for the onset of convection is predicted on the basis of a linear stability analysis. Multiple steady states near the threshold of convection are found.