Cellular interfacial patterns in the Rayleigh-Bénard problem with phase change

Abstract

The coupling between the flow field due to free convection with the morphology of a solid-liquid interface, formed upon the freezing of a thin layer of a monocomponent liquid from above, is investigated by means of a double expansion in the small amplitude of motion and a heat transfer Biot number. The latter is defined as the ratio of the thermal conductivity of the plates to that of the liquid. The presence of the interface and the thickness of the solidified portion of the layer affect the stability of the static liquid layer located below the interface. It is found that increases in the thickness of the solid delay the onset of convection and lead to a decrease in the critical wavelength. We show that the bifurcation from the conductive state is either subcritical or supercritical, depending on the strength of the coupling between the temperature fluctuations in the liquid and the interface deformation. A weakly nonlinear stability analysis on the regular and stationary interfacial patterns is undertaken. It reveals a supercritical square pattern when the coupling is weak and subcritical up-hexagons otherwise.