Paper presented at the 9th International Conference on Heat Transfer, Fluid Mechanics and Thermodynamics, Malta, 16-18 July, 2012.
The solidiﬁcation of a pure liquid phasechange material in the presence of natural convection is a commonly recurring problem in natural science and technology. The numerical solution of this Stefan problem is made diﬃcult by the fact that there is initially no solid phase; hence, the classical 1D Neumann similarity solution is often used for the purposes of initiating a computation. However, if the solid and liquid phases have diﬀerent densities at the solidiﬁcation temperature, this solution is not valid. This paper considers the limit of the coupled heat and momentum equations for small times, and ﬁnds that it is not possible to solve the corresponding problem, when the densities are diﬀerent, without introducing a singularity into the liquid velocity and pressure. The solution to a non-classical Stefan problem, where cooling is due to a constant heat ﬂux, is also considered, and is found to be free from such singularities