Natural convection in a shallow cavity containing two superposed layers of immiscible binary liquids

Abstract

This paper reports an analytical study of the natural convection in a shallow rectangular cavity containing two superposed layers of binary immiscible fluids. A uniform heat flux is applied to the horizontal walls of the enclosure while the vertical walls are impermeable and adiabatic. The solutal buoyancy forces are assumed to be induced either by the imposition of constant fluxes of mass on the horizontal walls (double-diffusive convection) or by a temperature gradient (Soret effects). An analytical solution of the steady form of the governing equations is derived using a parallel flow approximation. The critical Rayleigh numbers for the onset of critical, $${R_{\rm TC}^{\rm sup}}$$ , or subcritical, $${R_{\rm TC}^{\rm sub}}$$ , convection are predicted by the present theory. For finite amplitude convection the present model yields explicitly the flow patterns, Nusselt and Sherwood numbers in terms of the governing parameters of the problem. In the limit of a single fluid layer the present theory is found to be in agreement with the results reported in the literature.