Singular Perturbation Theory for Melting or Freezing in Finite Domains Initially Not at the Fusion Temperature

Abstract

This paper treats the problem of the inward solidification at large Stefan number 1/ε, ε = CP(Ti − Tf)/L, of a finite slab which is initially at an arbitrary temperature Ti above the melting point. The face at which the heat is removed is maintained at a constant temperature below fusion while the opposite face is either (a) insulated or (b) kept at the initial temperature. Perturbation series solutions in ε are obtained for both the short-time scale characterizing the transient diffusion in the liquid phase and the long-time scale characterizing the interface motion. The asymptotic matching of the two series solutions shows that to O(ε1/2) the short-time series solution for interface motion for the insulated Case (a) is uniformly valid for all time. A singular perturbation theory is, however, required for the isothermal Case (b) since the interface motion is affected to this order by the inhomogeneous temperature distribution in the liquid phase.