Stefan problem

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In mathematics and its applications, particularly to phase transitions in matter, a Stefan problem (also Stefan task) is a particular kind of boundary value problem for a partial differential equation (PDE), adapted to the case in which a phase boundary can move with time. Hence, Stefan problems are examples of moving boundary problems.

It is named after Jožef Stefan, the Slovene physicist who introduced the general class of such problems around 1890, in relation to problems of ice formation. This question had been considered earlier, in 1831, by Lamé and Clapeyron.

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 for 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.