The problem is named after Josef Stefan, the Slovenian 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.
Premises to the mathematical description
From a mathematical point of view, the phases are merely regions in which the solutions of the underlying PDE are continuous and differentiable up to the order of the PDE. In physical problems such solutions represent properties of the medium for each phase. The moving boundaries are infinitesimally thin surfaces that separate adjacent phases; therefore, the solutions 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 fluxdiscontinuity at the interface.
Mathematical formulation
The one-dimensional one-phase Stefan problem
Consider a semi-infinite one-dimensional block of ice initially at melting temperature for. Heat flux of is introduced at the left boundary of the domain causing the block to melt down leaving an interval occupied by water. The melted depth of the ice block, denoted by, is an unknown function of time; the solution to the dimensionless Stefan problem consists of finding and such that
Applications
Apart from modelling melting of solids, Stefan problem is also used as a model for the asymptotic behaviour of more complex problems. For example, Pego uses matched asymptotic expansions to prove that Cahn-Hilliard solutions for phase separation problems behave as solutions to a nonlinear Stefan problem at an intermediate time scale. Additionally, the solution of the Cahn–Hilliard equation for a binary mixture is reasonably comparable with the solution of a Stefan problem. In this comparison, the Stefan problem was solved using a front-tracking, moving-mesh method with homogeneous Neumann boundary conditions at the outer boundary. Also, Stefan problems can be applied to describe phase transformations. The Stefan problem also has a rich inverse theory; in such problems, the meting depth is the known datum and the problem is to find or.
Historical references
. An interesting historical paper on the early days of the theory; a preprint version is available here .
Scientific and general references
. Contains an extensive bibliography, 460 items of which deal with the Stefan and other free boundary problems, updated to 1982.
. An important monograph from one of the leading contributors to the field, describing his proof of the existence of a classical solution to the multidimensional Stefan problem and surveying its historical development.
. The paper containing Olga Oleinik's proof of the existence and uniqueness of a generalized solution for the three-dimensional Stefan problem, based on previous researches of her pupil S.L. Kamenomostskaya.
. The earlier account of the research of the author on the Stefan problem.
. In this paper the author proves the existence and uniqueness of a generalized solution for the three-dimensional Stefan problem, later improved by her master Olga Oleinik.
. A comprehensive reference, written by one of the leading contributors to the theory, updated up to 1962–1963 and containing a bibliography of 201 items.
. The impressive personal bibliography of the author on moving and free boundary problems for the heat-diffusion equation, containing about 5900 references to works appeared on approximately 884 different kinds of publications. Its declared objective is trying to give a comprehensive account of the existing western mathematical–physical–engineering literature on this research field. Almost all the material on the subject, published after the historical and first paper of Lamé–Clapeyron, has been collected. Sources include scientific journals, symposium or conference proceedings, technical reports and books.
