The Binary Alloy Problem: Existence, Uniqueness, and Numerical Approximations

Abstract

In this paper we give an energy formulation of the binary alloy solidification problem. We assume the change in solidification temperature is only the result of the phase diagram; supercooling, surface tension and curvature effects are ignored. Both the mass and heat diffusions are described by partial differential equations which are similar to the enthalpy (energy) formulation for the pure heat diffusion or Stefan problem. For a one space variable problem uniqueness (Theorem 4) and existence (Theorem 5) are established. Also, an implicit time discretization, which has no stability conditions on the time increment, of this formulation is given. This numerical method is shown to converge, as the mesh goes to zero, to the solution of the continuum problem. Numerical examples are given.