On thermodynamically consistent Stefan problems with variable surface energy

TL;DR

This paper studies a thermodynamically consistent two-phase Stefan problem with temperature-dependent surface tension, analyzing the existence, stability, and long-term behavior of solutions, including conditions for stability of multiple spheres.

Contribution

It introduces a thermodynamically consistent Stefan problem with variable surface energy and analyzes solution stability and convergence to equilibrium.

Findings

Solutions exist globally if no singularities occur

Multiple spheres are unstable with small surface heat capacity

Spheres are stable with large surface heat capacity when no kinetic undercooling

Abstract

A thermodynamically consistent two-phase Stefan problem with temperature-dependent surface tension and with or without kinetic undercooling is studied. It is shown that these problems generate local semiflows in well-defined state manifolds. If a solution does not exhibit singularities, it is proved that it exists globally in time and converges towards an equilibrium of the problem. In addition, stability and instability of equilibria is studied. In particular, it is shown that multiple spheres of the same radius are unstable if surface heat capacity is small; however, if kinetic undercooling is absent, they are stable if surface heat capacity is sufficiently large.