A steady state phase change problem

Abstract

In a previous paper [4] the solution to the Stefan problem for a one-dimensional semi-infinite slab with constant boundary and initial conditions was shown to be given by the limit of solutions to a nonlinear parabolic equation for the internal energy. In this paper we obtain the same result for the Stefan problem in a bounded two- or three-dimensional domain, with constant boundary conditions. This result further justifies the application of the methods of [3], [4] to the Stefan problem in higher dimensions. In Section 1 the problem to be solved is stated, and a simple solution given. In Sections 2, 3 this solution is shown to be obtainable from a limit of solutions to a related problem for the specific internal energy, as well as a solution to a related problem in the calculus of variations. 1. Notation and Statement of the Problem. Let ? be a bounded region of the x, y plane having a smooth boundary F and consisting of material which undergoes a change of phase, from Phase to Phase II, at the critical temperature TC (see Fig. 1); our results apply as well for a three-dimensional region. (Phases I and II can represent frozen and melted states of the material.) Let H be the latent heat of the material which is lost in the transition from Phase II to Phase I, cl, K1 and C2, K2 the specific heat and conductivity of Phase I and Phase II material, respectively, and Ki = Ki/cip, i = 1, 2, where p is the density of Phase I and II material, which we assume to be the same.