Approximate solution methods for one-dimensional solidification from an incoming fluid

This paper concerns a one-dimensional model for solidification due to incoming supercooled liquid impacting on a substrate that is maintained at a fixed temperature. Using a boundary immobilisation method, and assuming that both the solid and liquid layers remain thin throughout the process, a second-order accurate perturbation expansion is determined. An alternative approximate solution, found using the heat balance integral method, is also described to analyse the problem, and the liquid height and temperatures in the solid and liquid are subsequently found for both approximate solutions. These are then compared with a numerical scheme which solves the full Stefan problem. The perturbation solution is shown to be more accurate, but the HBI method is simpler to implement and avoids complications that arise in the ordering of terms in the perturbation expansion as the difference between the substrate and melting temperature changes.