Nonlinear evolution analysis and stability for convection in a mushy layer with permeable interface

We consider the problem of two- and three-dimensional nonlinear buoyant flows in horizontal mushy layers during the solidification of binary alloys. We study the nonlinear evolution of such flow based on a recently developed realistic model for the mushy layer with permeable interface. The evolution approach is based on a Landau type equation for the amplitude of the secondary nonlinear solution, which can be in the form of rolls, squares, rectangles or hexagons. Using both analytical and computational methods, we calculate the solutions to the evolution equation near the onset of motion for both subcritical and supercritical regimes and determine the stable solutions. We find, in particular, that for several investigated cases with different parameter regimes, secondary solution in the form of subcritical down-hexagons or supercritical up-hexagons can be stable. However, the preferred solution for smallest values of the Rayleigh number and the amplitude of motion is in the form of subcritical down-hexagons. This result appears to agree with the experimental observation on the form of the convective flow near the onset of motion.