Analytical and numerical solution to the convection problem in a shallow cavity filled with two immiscible superposed fluids

摘要

Natural convection, in a shallow horizontal cavity filled with two layers immiscible fluids, is studied both numeric and analytic. The horizontal walls of the enclosure are heated by the bottom and cooled by the upper by a uniform heat fluxes while the verticals one are subject to heat fluxes of variable intensity. The analytical solution, based on the parallel flow approximation, is found to be in good agreement with a numerical solution. The existence of both natural and antinatural flows is demonstrated. Indeed, the critical Rayleigh number for the onset of motion is dependent on the side heating intensity. In the case where the upper fluid is more viscous than the bottom one, multiple solutions are possible as well as the a steady states in a range of ratio of the viscosity of the fluids. The fact that one of the two fluids is more heat conductive, affect not only heat transfer generated but also the intensity of flows and the existence of the third cell where we found several solutions. The present model yields explicitly the flow patterns and Nusselt numbers in terms of the governing parameters of the problem. (C) 2014 Elsevier Masson SAS. All rights reserved.