Interfacial pattern formation in the presence of solidification and thermal convection

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Abstract
An analysis is conducted on the coupling between hydrodynamic behaviors and a solid-liquid interface that forms due to solidification of a portion of a layer of fluid located between almost completely insulating plates. Beginning with a long-wavelength expansion, an evolution equation is derived that includes the effects of the field variables in the liquid phase, which describes the temporal and spatial behavior of the interface. An expression is obtained that shows that increases in the thickness of the solid layer increase the critical value of the Rayleigh number and that deformations in the interface can either decrease or increase this critical value. The interface undergoes a transition from the planar state to a given pattern at the critical value of the Rayleigh number. Analysis of the evolution equation leads to the prediction that the interfacial pattern consists of squares at very small values for the thickness of the solid layer. A critical value of the thickness is found, and for a thickness greater than this value squares and rolls are predicted to coexist as stable interfacial patterns.