On the formulation of hyperbolic Stefan problems

Abstract

Abstract. The study of phase change processes governed by hyperbolic heat transfer is at an embryonic stage. We raise here some of the relevant questions and make some remarks on the formulation and qualitative behavior of hyperbolic Stefan problems. In particular, we correct an error in the interface condition appearing in two earlier studies, and present an explicit solution to a simple one-phase problem and study its behavior. Finally we describe an enthalpy (weak) formulation for a two-phase problem and report on a few numerical experiments based on it.

A One-Phase Hyperbolic Stefan Problem in Multi-Dimensional Space

“…Otherwise, the problem can't be well-posed. Were (3-11) violated, the solution would be nonunique, just as in the problem discussed in [15]; see also [9] for explanation.…”

“…The formulation of the phase-change problem for the hyperbolic heat transfer model (1-4) or (1-5) depends upon the conditions imposed on the phase-change interface S(t), which are determined from physical consideration. There are the classical conditions assuming continuous temperature across the phase-change interface, see [2,15]. A different condition was proposed in [4,16], where the temperature is supposed to sustain a jump across the phase-change interface.…”

The well-posedness of a hyperbolic Stefan problem

“…In [6] was given a formulation of the hyperbolic Stefan problem based upon the traditional assumption that the temperatures on the two sides of the phase change boundary are prescribed and equal. Also, an explicit solution was given in [6] where the phase change front propagates faster than sound speed and consequently is physically unacceptable.…”

“…Also, an explicit solution was given in [6] where the phase change front propagates faster than sound speed and consequently is physically unacceptable.…”

“…In this paper, we study the Stefan problem for the hyperbolic heat model in the classical formulation of [6]. For both the one-phase and the two-phase problem, we get the local and global solution.…”