In mathematics and its applications, particularly to phase transitions in matter, a Stefan problem is a particular kind of boundary value problem for a system of partial differential equations (PDE), in which the boundary between the phases can move with time.
This is accomplished by solving heat equations in both regions, subject to given boundary and initial conditions.
Analogous problems occur, for example, in the study of porous media flow, mathematical finance and crystal growth from monomer solutions.
[2] However, some 60 years earlier, in 1831, an equivalent problem, concerning the formation of the Earth's crust, had been studied by Lamé and Clapeyron.
The underlying PDEs are not valid at the phase change interfaces; therefore, an additional condition—the Stefan condition—is needed to obtain closure.
The regularity of the equation has been studied mainly by Luis Caffarelli[4][5] and further refined by work of Alessio Figalli, Xavier Ros-Oton and Joaquim Serra[6][7] The one-phase Stefan problem is based on an assumption that one of the material phases may be neglected.
This is a mathematically convenient approximation, which simplifies analysis whilst still demonstrating the essential ideas behind the process.
A further standard simplification is to work in non-dimensional format, such that the temperature at the interface may be set to zero and far-field values to
The most well-known form of Stefan problem involves melting via an imposed constant temperature at the left hand boundary, leaving a region
Also, Stefan problems can be applied to describe phase transformations other than solid-fluid or fluid-fluid.
In fact the liquid layer is often in motion, thus requiring advection or convection terms in the heat equation.
In the solidification of supercooled melts an analysis where the phase change temperature depends on the interface velocity may be found in Font et al.[15] Nanoscale solidification, with variable phase change temperature and energy/density effects are modelled in.