Generalized Stefan models accounting for a discontinuous temperature field

Abstract.

We construct a class of generalized Stefan models able to account for a discontinuous temperature field across a nonmaterial interface. The resulting theory introduces a constitutive scalar interfacial field, denoted by $\overline\theta$ and called the equivalent temperature of the interface. A classical procedure, based on the interfacial dissipation inequality, relates the interfacial energy release to the interfacial mass flux and restricts the equivalent temperature of the interface. We show that previously proposed theories are obtained as particular cases when $\overline\theta = \langle\theta\rangle$ or $\overline\theta = \langle\frac{1}{\theta}\rangle^{-1}$ or, more generally, when $\overline\theta = \langle\theta^{r}\rangle\langle\frac{1} {\theta^{1-r}}\rangle^{-1}$ for $0\leq r\leq 1.$ We study in a particular constitutive framework the solidification of an under-cooled liquid and we are able to give a sufficient condition for the existence of travelling wave solutions.