Unidirectional solidification with a mushy layer. The influence of weak convection

Motivated by industrial and geophysical solidification problems such as segregation in metallic castings and expulsion of brine from growing sea ice, we present and solve a model for steady convection in a mushy layer of a binary mixture. A non-linear problem of one-dimensional convection is studied. The model under consideration describes the main properties of the convective heat and mass transport. The concentration field in the liquid, solid and mushy phases, as well as the rate of solidification and mushy layer thickness are found analytically as functions of all thermophysical parameters. The role of convective transfer is detailed in the analysis. The domain of applicability of exact analytical solutions is found.