Soret-driven convection coupled to the morphology of a solid–liquid interface

Abstract

The importance of Soret-driven convection in solidification processes is illustrated through an analysis of the coupling between convective currents and the deformations in a solid–liquid interface. The interface appears on freezing an upper portion of a layer of a dilute binary fluid. The presence of the interface and the amount of the solid strongly affect the stability properties of the liquid mixture and, in turn, the state of the liquid determines the patterns formed on the interface. Marginal stability curves are constructed by plotting the critical Rayleigh number and the critical wave number against the thickness of the solid layer. At small values of a positive separation ratio, increases in the thickness of the solid layer are found to have a destabilizing effect, the static state becomes unstable at smaller values of the Rayleigh number, whereas at larger values for the separation ratio this effect disappears. As the separation ratio is increased, a change of slope at zero thickness in the solid layer, from negative to positive, occurs in the plots of the critical wave number. For the case of a sufficiently large separation ratio, a weakly nonlinear analysis leads to the prediction of a bifurcation diagram that is characterized by two regions of bistability. A stable stationary convective structure consisting of down-hexagons appears through a subcritical bifurcation and for a small range of Rayleigh numbers coexists with the static state. Squares become stable at higher Rayleigh numbers and for a small parameter range coexist with down-hexagons. These stationary structures, down-hexagons and squares, are imprinted as patterns on the solid–liquid interface through the action of the convective currents. It is deduced that down-hexagons appear as a result of the coupling of the convective motion in the melt with the deformations in the interface and do not occur in the absence of solidification.