| The behavior of convective instabilities in solidifying media is of interest in applications spanning materials processing and geophysics. Under a broad set of circumstances, constitutional supercooling occurs, resulting in the formation of a layer of dendritic growth called a mushy layer. The mushy layer has both solid and fluid components. We consider the problem of nonlinear steady buoyant convection in horizontal mushy layers during the solidification of binary alloys. Both cases of zero vertical volume flux and constant pressure at the upper mush-liquid interface were investigated. We analyze the effects of several parameters of the problem on the stationary modes of convection in the form of either hexagonal cells or nonhexagonal cells 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 nonlinear analyses are relaxed here in order to study a richer set of phenomena. Using both analytical and computational methods, we determine the steady solutions to the weakly nonlinear problem by using a perturbation technique for both constant and variable permeability, referred to as passive and reactive mushy layer cases, respectively. Both the nonlinear basic state and the reactive mushy zone of the present problem favor hexagon-pattern convection. The results of the analyses and computations indicate in particular that, depending on the range of values of the parameters, bifurcation to nonhexagonal convection can be either supercritical or subcritical, while bifurcation to hexagon-pattern convection, corresponding to the smallest value of the Rayleigh number, is subcritical. For reactive mushy layers, subcritical down-hexagons with downflow at the cell centers and upflow at the cell boundaries, which have been observed in related experiments, were predicted. Supercritical nonhexagons were also predicted in particular ranges of values of the parameters. |
