On Mixed Oscillatory and Steady Modes of Nonlinear Convection in Mushy Layers

We consider the problem of mixed oscillatory and steady modes of nonlinear compositional convection in horizontal mushy layers during the solidification of binary alloys. Under a near-eutectic approximation and the limit of large far-field temperature, we determine a number of two- and three-dimensional weakly nonlinear mixed solutions, and the stability of these solutions with respect to arbitrary three-dimensional disturbances is then investigated. The present investigation is an extension of the problem of mixed oscillatory and steady modes of convection, which was investigated by Riahi (J Fluid Mech 517: 71–101, 2004), where some calculated results were inaccurate due to the presence of a singular point in the equation for the linear frequency. Here we resolve the problem and find some significant new results. In particular, over a wide range of the parameter values, we find that the properties of the preferred and stable solution in the form of particular subcritical mixed standing and steady hexagons appeared to be now in much better agreement with the available experimental results (Tai et al., Nature 359:406–408, 1992) than the one reported in Riahi (J Fluid Mech 517:71–101, 2004). We also determined a number of new types of preferred supercritical solutions, which can be preferred over particular values of the parameters and at relatively higher values of the amplitude of convection.