On inward solidifying cylinders and spheres initially not at their fusion temperature

A boundary fixing series technique, previously used on melting and freezing problems, initially at their fusion temperature, is generalized to initially subcooled, melting cylinders and spheres with constant surface temperatures. For planar geometry the method gives the known exact solution. For these non-linear Stefan problems the boundary fixing series technique involves introducing a transformation which fixes three points, the origin, the moving boundary and the outer surface of the cylinder or sphere. For these geometries this transformation is necessarily singular and this singularity arises in the approximations for the temperature profiles but not in the approximation for the motion of the moving boundary. Numerical comparisons are made with the enthalpy method and perturbation solutions for the cylinder. Two terms of the method generate accurate analytic estimates for the boundary motion, which in contrast to the perturbation solution are valid over the full range of Stefan numbers. Numerical results indicate that the singularity in the temperature profiles is overcome to a certain extent by further transformations and one term of the series, with the singularity removed, provides a simple analytic estimate which posseses the essential qualitative features of the numerical solution.