Multiple solutions for double diffusive convection in a shallow porous cavity with vertical fluxes of heat and mass

The Darcy model with the Boussinesq approximation is used to study double-diffusive natural convection in a shallow porous cavity. The horizontal walls are subject to uniform fluxes of heat and mass, while the side vertical walls are exposed to a constant heat flux of intensity aq′, where a is a real number. Results are presented for −20⩽RT⩽50, −20⩽RS⩽20, 5⩽Le⩽10, 4⩽A⩽8 and −0.7⩽a⩽0.7, where RT,RS, Le and A correspond to thermal Rayleigh number, solutal Rayleigh number, Lewis number and aspect ratio of the enclosure, respectively. In the limit of a shallow enclosure (A≫1) an asymptotic analytical solution for the stream function and temperature and concentration fields is obtained by using a parallel flow assumption in the core region of the cavity and an integral form of the energy and the constituent equations. In the absence of side heating (a=0), the solution takes the form of a standard Bénard bifurcation. The asymmetry brought by the side heating (a≠0) to the bifurcation is investigated. For high enough Rayleigh numbers, multiple steady states near the threshold of convection are found. These states represent flows in opposite directions. In the range of the governing parameters considered in the present study, a good agreement is observed between the analytical predictions and the numerical simulations of the full governing equations.