Pseudo-steady-state solutions for solidification in a wedge

Polubarinova-Kochina's analytical differential equation method is used to determine the pseudo-steady-state solution to problems involving the freezing (solidification) of wedges of liquid which are initially at their fusion temperature. In particular: we consider four distinct problems for wedges which are: freezing with the same constant boundary temperature, freezing with the same constant boundary heat fluxes, freezing with distinct constant boundary temperatures and freezing with distinct constant fluxes at the boundaries. For the last two problems, a Heun's differential equation with an unknown singularity is derived, which in both cases admits a particularly elegant simple solution for the special case when the wedge angle is pi. The moving boundaries obtained are shown pictorially.