Approximate analytical and numerical solutions for a two-dimensional Stefan problem

This paper proposes an approximate analytical and a numerical solution method to a two-dimensional heat conduction problem in which a liquid becomes solidified by heat transfer to a planar mold surface by using a linear perturbation method. It is assumed that the cooling rate is perturbed by a small spatially sinusoidal heat flux at the shell–mold interface. This leads to a corresponding undulation of the solidified shell thickness. Approximate analytical results are obtained for the solid/melt moving interface as a function of time and for the temperature field in the shell. The approximate analytical solution is compared with a numerical solution, and a very good agreement has been found. A limiting analytical solution in which diffusivity of the solidified shell material is assumed to be infinitely large is also obtained, and compared with the numerical predictions to establish the validity of the model and the numerical approach. It is demonstrated that solidified shell materials with higher thermal diffusivities may result in irregular growth of the shell thickness which, generally, causes cracking near the surface.