Mush–Chimney Convection

Mush–chimney convection occurs in a system consisting of a layer of mixed solid and liquid bounded by liquid above and solid below. The solid phase of the mush is composed of the denser constituent of the alloy, making the residual liquid compositionally buoyant as solidification proceeds. The liquid is in steady, nonlinear convection, consisting of a downward flow in the mush and an upward flow in a chimney, free of solid, which formed spontaneously in the mush as a result of the interaction between flow and structure. A well-posed continuum mathematical model of this system is developed. A novel feature of this analysis is the development of an equation that determines the chimney radius. Material diffusion is an essential ingredient in this equation, but is negligible elsewhere. In the parameter range considered, the problem is well posed without the necessity of specifying the condition of marginal phase equilibrium. A remarkably simple analytic solution is obtained for the convective flow in the chimney.