Stefan problem

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. The classical Stefan problem aims to describe the temperature distribution in a homogeneous medium undergoing a phase change, for example ice passing to water: this is accomplished by solving the heat equation imposing the initial temperature distribution on the whole medium, and a particular boundary condition, the Stefan condition, on the evolving boundary between its two phases. Note that this evolving boundary is an unknown (hyper-)surface: hence, Stefan problems are examples of free boundary problems.

Historical note

The problem 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.

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 (or interfaces) 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 flux discontinuity 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 u ≡ 0 for x ∈ [0, +∞). Heat flux of f(t) is introduced at the left boundary of the domain causing the block to melt down leaving an interval [0, s(t)] occupied by water. The melted depth of the ice block, denoted by s(t), is an unknown function of time; the solution of the Stefan problem consists of finding u and s such that

$\begin{align} \frac{\partial u}{\partial t} &= \frac{\partial^2 u}{\partial x^2} &&\text{in } \{(x,t): 0 < x < s(t), t>0\}, && \text{the heat equation},\\ -\frac{\partial u}{\partial x}(0, t) &= f(t), && t>0, &&\text{the Neumann condition at the left end of the domain describing the inlet heat flux}, &&\\ u\big(s(t),t\big) &= 0, && t>0, &&\text{the Dirichlet condition at the water-ice interface: setting melting/freezing temperature},\\ \frac{\mathrm{d}s}{\mathrm{d}t} &= -\frac{\partial u}{\partial x}\big(s(t), t\big), && t>0, &&\text{Stefan condition},\\ u(x,0) &= 0, && x\geq 0, &&\text{initial temperature distribution},\\ s(0) &= 0, && &&\text{initial depth of the melted ice block}. \end{align}$

Applications

Apart from the modelling of melting of solids, Stefan problems are also used as models for the asymptotic behaviour with respect to time 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. Also 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 (or curve or hypersurface) s is the known datum and the problem is to find u or f.

See also

• Free boundary problem
• Moving boundary problem
• Olga Arsenievna Oleinik
• Shoshana Kamin
• Stefan's equation

References

Historical references

• . An interesting historical paper on the early days of the theory: a preprint version (in PDF format) is available here [1].

Scientific and general references

• . Contains an extensive bibliography, 460 items of which deal with the Stefan and other free boundary problems, updated to 1982.
• .  – via De Gruyter (subscription required) 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 (M–FBP) for the heat-diffusion equation (H–DE), 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 (1831), has been collected. Sources include scientific journals, symposium or conference proceedings, technical reports and books.