Stabilizing an adverse density difference in the presence of phase change

Abstract

Given two phases in equilibrium in a porous solid, the heavy phase lying above the light phase in a gravitational field, we stabilize this adverse density arrangement by heating from below and derive a formula for how steep the temperature gradient must be to do this. The input temperature gradient has two effects on the stability of our system. Its effect on the heat convection is destabilizing, its effect on the heat conduction at the surface is stabilizing. By directing our attention to the case of zero growth rate, we obtain the critical value of the input temperature gradient as it depends on the permeability of the porous solid, the density difference across the surface, the distance between the planes bounding our system, and the physical properties. Our problem makes connections to the Bénard problem where it has two, one, or no critical points, and to the Rayleigh–Taylor problem where it has no critical points.