Effect of permeability on steady flow in a dendrite layer

We consider the problem of nonlinear steady convective flow in a horizontal dendrite layer during alloy solidification. We analyze the effect of permeability of the layer on the stationary modes of convection in the form of two-dimensional rolls and threedimensional patterns. Under a near-eutectic approximation and the limit of large far-field temperature, we determine the twoand three-dimensional solutions to the weakly nonlinear problem by using a perturbation technique, and the stability of these solutions are investigated with respect to arbitrary three-dimensional disturbances. An inverse form of the permeability function introduces two non-negative non-dimensional parameters K1 and K2 that are significant in the present problem. The results of the analyses in particular range of values of the magnitude |ε| of the amplitude of convection indicate, in particular, that the effects of K1 and K2 on the flow pattern are destabilizing and stabilizing, respectively, and different types of flow pattern can be stable for particular values of these parameters. For sufficiently small and non-zero values of |ε| and K1, the steady flow pattern in the form of subcritical hexagons can stable. For |ε| beyond some value and depending on the values of the parameters of the problem, supercritical rolls, squares or rectangles can be stable. For K1=0.0, the only stable flow pattern is that due to steady rolls.