A time-dependent formulation of the mushy-zone free-boundary problem

Abstract

We present time-dependent governing equations and boundary conditions for the mushy-zone free-boundary problem that are valid in an arbitrary frame of reference. The model for time-evolving mushy zones is more complicated than in the steady case because the interface velocity $\bm{w}$ can be distinct from both the velocity of the dendrites $\bm{v}$ and the fluid velocity $\bm{u}$. We consider the limit of negligible solutal diffusivity, where there are four types of boundary condition at the mush–liquid interface, depending on both the direction of flow across the interface and the direction of the interface motion relative to the solid phase. We illustrate these boundary conditions by examining a family of one-dimensional problems in which a binary material is chilled from a fixed cold point in the laboratory frame of reference while fluid is pumped through the resulting mushy layer at a rate $Q$ and the mushy layer itself is translated at a rate $V$. This allows us to exhibit three of the four types of mushy-layer interfaces. We show that the fourth type cannot occur in this scenario.