Steady-state convection with melting at a boundary
William B. Heard · The Physics of Fluids · 1977
A fluid is heated from below and cooled from above in such a way that a frozen layer overlies a convecting layer. The lower boundary of the fluid is maintained at a constant temperature while the upper boundary of the ice transfers heat to an ambient atmosphere according to Newton’s law of cooling. The Boussinesq equations for convection are used in the mean field approximation and with a one-mode, two-dimensional representation to determine the steady state of the system in the limit of very large Rayleigh number. The fluid-ice interface is corrugated and is a free boundary of the problem. The convective steady states are likely to bifurcate discontinuously from the conductive steady states. Analytical and numerical results are given for the temperature at the upper boundary of the ice, the mean depth of the fluid, and the amplitude of the corrugations.