Stefan problem

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. The classical Stefan problem aims to describe the evolution of the boundary between two phases of a material undergoing a phase change, for example the melting of a solid, such as ice to water. This is accomplished by solving heat equations in both regions, subject to given boundary and initial conditions. At the interface between the phases (in the classical problem) the temperature is set to the phase change temperature. To close the mathematical system a further equation, the Stefan condition, is required. This is an energy balance which defines the position of the moving interface. Note that this evolving boundary is an unknown (hyper-)surface; hence, Stefan problems are examples of free boundary problems. Analogous problems occur, for example, in the study of porous media flow, mathematical finance and crystal growth from monomer solutions.

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. The classical Stefan problem aims to describe the evolution of the boundary between two phases of a material undergoing a phase change, for example the melting of a solid, such as ice to water. This is accomplished by solving heat equations in both regions, subject to given boundary and initial conditions. At the interface between the phases (in the classical problem) the temperature is set to the phase change temperature. To close the mathematical system a further equation, the Stefan condition, is required. This is an energy balance which defines the position of the moving interface. Note that this evolving boundary is an unknown (hyper-)surface; hence, Stefan problems are examples of free boundary problems. Analogous problems occur, for example, in the study of porous media flow, mathematical finance and crystal growth from monomer solutions. Consider a semi-infinite one-dimensional block of ice initially at melting temperature u=0 for x \in [0;+\infty) . The most well-known form of Stefan problem involves melting via an imposed constant temperature at the left hand boundary, leaving a region [0;s(t)] occupied by water. The melted depth, denoted by s(t) , is an unknown function of time. The Stefan problem is defined by The heat equation: \frac{\partial u}{\partial t} = \frac{\partial^2 u}{\partial x^2}, \quad \forall (x,t) \in [0;s(t)] \times [0;+\infty] A fixed temperature, above the melt temperature, on the left boundary: u(0,t) = 1, \quad \forall t > 0 The interface at the melting temperature is set to u \left(s(t),t \right) = 0 The Stefan condition: \beta \frac{\mathrm{d}}{\mathrm{d}t} s(t) = -\frac{\partial}{\partial x} u \left(s(t), t \right) where \beta is the Stefan number, the ratio of latent to specific sensible heat (where specific indicates it is divided by the mass). Note this definition follows naturally from the nondimensionalisation and is used in many texts however it may also be defined as the inverse of this. The initial temperature distribution: u(x,0) = 0, \; \forall x \geq 0 The initial depth of the melted ice block: s(0) = 0 The Neumann solution, obtained by using self-similar variables, indicates that the position of the boundary is given by s(t) = 2 \lambda \sqrt{\alpha t} where \alpha is diffusivity, and \lambda satisfies the transcendental equation \beta \lambda = \frac{1}{\sqrt{\pi}}\frac{\mathrm{e}^{-\lambda^2}}{\text{erf}(\lambda)}. The temperature in the liquid is then given by T=1-\frac{\text{erf}\left(\frac{x}{2\sqrt{t}}\right)}{\text{erf}(\lambda)}.

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 uses matched asymptotic expansions to prove that Cahn-Hilliard solutions for phase separation problems behave as solutions to a non-linear 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 other than solid-fluid or fluid-fluid. Application of Stefan problem to metal crystallization in electrochemical deposition of metal powders was envisaged by Călușaru The Stefan problem also has a rich inverse theory; in such problems, the melting depth (or curve or hyper-surface) is the known datum and the problem is to find or .

Notes

References

Historical references

. An interesting historical paper on the early days of the theory; a preprint version (in PDF format) 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 (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 (1831¹), has been collected. Sources include scientific journals, symposium or conference proceedings, technical reports and books.

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