On nonlinear convection in mushy layers: part I. oscillatory modes of convection

We consider the problem of nonlinear convection in horizontal mushy layers during the solidification of binary alloys. We analyse the oscillatory modes of convection in the form of two- and three-dimensional travelling and standing waves. Under a near-eutectic approximation and the limit of large far-field temperature, we determine the solutions to the nonlinear problem by using a perturbation technique, and the stability of the two- and three-dimensional solutions in the form of simple travelling waves, general travelling waves and standing waves is investigated. The results of the stability and the nonlinear analyses indicate preference of supercritical simple travelling rolls over most of the studied range of the parameter values, while supercritical standing rolls can be preferred over some small range of the parameter values. The results of the investigation of the onset of plume convection and chimney formation leading to the occurence of freckles in the alloy crystal indicate that the chimney of the plume can be generated internally or near the lower boundary of the mushy layer. However, due to the oscillatory nature of the flow with respect to time, the initiation of such chimneys, in particular locations in space and time, may either be stopped or be moved away by the oscillatory flow. The roles of a Stefan number, a permeability parameter and a concentration ratio on the flow instability, or otherwise, in both linear and nonlinear regimes are also determined