The problem is named after Josef Stefan (Jožef Stefan), the Slovenian physicist who introduced the general class of such problems around 1890, in relation to problems of ice formation.[1] 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 (orinterfaces) 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 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 to the dimensionless Stefan problem consists of finding u and s such that [2]
Applications
Apart from modelling melting of solids, Stefan problem is also used as a model for the asymptotic behaviour (in time) of more complex problems: for example, Pego[3] 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.[4] 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.[5]
The Stefan problem also has a rich inverse theory: in such problems, the meting depth (orcurveorhypersurface) s is the known datum and the problem is to find uorf.[6]
^R. L. Pego. (1989). Front Migration in the Nonlinear Cahn-Hilliard Equation. Proc. R. Soc. Lond. A.,422:261–278.
^Vermolen, F. J.; Gharasoo, M. G.; Zitha, P. L. J.; Bruining, J. (2009). "Numerical Solutions of Some Diffuse Interface Problems: The Cahn–Hilliard Equation and the Model of Thomas and Windle". International Journal for Multiscale Computational Engineering. 7 (6): 523–543. doi:10.1615/IntJMultCompEng.v7.i6.40.
^Alvarenga HD, Van de Putter T, Van Steenberge N, Sietsma J, Terryn H (Apr 2009). "Influence of Carbide Morphology and Microstructure on the Kinetics of Superficial Decarburization of C-Mn Steels". Metall Mater Trans A. 46: 123. Bibcode:2015MMTA...46..123A. doi:10.1007/s11661-014-2600-y.
^(Kirsh 1996) harv error: no target: CITEREFKirsh1996 (help).
Kamenomostskaya, S. L. (1958), "On Stefan Problem", Nauchnye Doklady Vysshey Shkoly, Fiziko-Matematicheskie Nauki (in Russian), 1 (1): 60–62, Zbl0143.13901. The earlier account of the research of the author on the Stefan problem.
Tarzia, Domingo Alberto (July 2000), "A Bibliography on Moving-Free Boundary Problems for the Heat-Diffusion Equation. The Stefan and Related Problems", MAT, Series A: Conferencias, seminarios y trabajos de matemática., 2: 1–297, ISSN1515-4904, MR1802028, Zbl0963.35207. 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.
![{\displaystyle {\begin{aligned}{\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}},\\[6pt]-{\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}},&&\\[6pt]u{\big (}s(t),t{\big )}&=0,&&t>0,&&{\text{the Dirichlet condition at the water-ice interface: setting melting/freezing temperature}},\\[6pt]{\frac {\mathrm {d} s}{\mathrm {d} t}}&=-{\frac {\partial u}{\partial x}}{\big (}s(t),t{\big )},&&t>0,&&{\text{Stefan condition}},\\[6pt]u(x,0)&=0,&&x\geq 0,&&{\text{initial temperature distribution}},\\[6pt]s(0)&=0,&&&&{\text{initial depth of the melted ice block}}.\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7829b74aa06fe0e02f0bc6add23fe36b79e09830)
