Stefan Problem - Premises To The Mathematical Description

Premises To The Mathematical Description

From a mathematical point of view, the phases are merely regions in which the coefficients of the underlying PDE are continuous and differentiable up to the order of the PDE. In physical problems such coefficients represent properties of the medium for each phase. The moving boundaries (or interfaces) are infinitesimally thin surfaces that separate adjacent phases; therefore, the coefficients of the underlying PDE and its derivatives may suffer discontinuities across interfaces.

The underlying PDE is not valid at phase change interfaces; therefore, an additional condition—the Stefan condition—is needed to obtain closure. The Stefan condition expresses the local velocity of a moving boundary, as a function of quantities evaluated at both sides of the phase boundary, and is usually derived from a physical constraint. In problems of heat transfer with phase change, for instance, the physical constraint is that of conservation of energy, and the local velocity of the interface depends on the heat flux discontinuity at the interface.