On weakly nonlinear convection in mushy layers during solidification of alloys

We consider the problem of weakly nonlinear buoyant convection in horizontal mushy layers with permeable mush-liquid interface during the solidification of binary alloys. We analyse the effects of several parameters of the problem on the stationary modes of convection in the form of either a hexagonal pattern or a non-hexagonal pattern such as rolls, rectangles and squares. No assumption is made on the thickness of the mushy layer, and a number of simplifying assumptions made in previous theoretical investigations of the problem are relaxed here in order to study the problem based on a more realistic model. Using both analytical and numerical methods, we determine the steady solutions for the nonlinear problem in a range of the Rayleigh number R near its critical value. Both the nonlinear basic state and variable permeability of the present problem favour hexagon-pattern convection. The results of the analyses and computations indicate that depending on the range of values of the parameters, bifurcation to hexagonal or non-hexagonal convection can be either supercritical or subcritical. However, among all the computed solutions in the particular range of values of the parameters that are most relevant to those of the experiments, only convection in the form of down-hexagons with downflow at the cell centres and upflow at the cell boundaries, was found to be realizable, in the sense that its amplitude increases with R.