Amplitude Equation for a Nonlinear Three Dimensional Convective Flow in a Mushy Layer

Abstract

We consider a nonlinear three-dimensional convective flow in a mushy layer. A mushy layer is a partially solidified region formed during solidification of binary alloys. During solidification, fluid flow within the mushy layer can cause vertical chimneys or channels void of solid. These chimneys can generate imperfections in the final form of the solidified alloy. The equations governing the mushy layer system are the continuity equation, heat equation, solute equation, and conservation of momentum which is governed by Darcy's law. A quadratic nonlinear evolution equation satisfied by the amplitude is derived for hexagonal cells. This equation is obtained from the first-order system using the adjoint of the linear system. An explicit solution of the amplitude equation is also presented.