Perturbation and stability analyses of flow in a mushy layer with permeable interface

We consider the problem of weakly nonlinear convective flow stability in a horizontal mushy layer with permeable interface. No assumption is made on the thickness of the mushy layer, and a number of simplifying assumptions made in previous related nonlinear analyses are lifted here in order to determine the results based on a more realistic model. Using perturbation and stability analyses and some numerical calculations, we determine the steady stable solutions to the weakly nonlinear problem under certain range of the parameter values where such analyses are valid. We find, in particular, that depending on the range of values of the parameters, the convective flow in the form of down-hexagons, with down-flow at the cells’ centers and up-flow at the cells’ boundaries, up-hexagons, with up-flow at the cells’ centers and down-flow at the cells’ boundaries, rectangles, squares and rolls can be possible.