Wedge solidification with differing types of boundary conditions

Solidification, in the large Stefan number limit, is considered within a wedge of fluid, initially at fusion temperature. The wedge walls have different boundary conditions, one is Neumann and the other Dirichlet. The large Stefan number limit allows a reduction to complex variables, and ultimately a solution utilizing conformal mappings and a connection with Fuchsian differential equations. The physical example is of interest as the one dimensional solutions corresponding to either Dirichlet or Neumann problems propagate at different velocities, and the connection between these near the corner region can lead to changes in sign of the curvature of the front profile. This manifests itself in the mathematics as an additional singularity in the Fuchsian equation and this enriches the problem over that usually encountered; there are now four singular points rather than three, and unknown parameters appear that must be determined.

Co-authored with H.V. Hoang, DAMTP, University of Cambridge

IMA J. Appl. Math, 67, 509--524, 2002