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.
Historical note The problem is named after Josef Stefan (Jožef Stefan), the Slovenian physicist who introduced the general class of such problems around 1890 in a series of four papers concerning the freezing of the ground and the formation of sea ice. 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. Stefan's problem admits a similarity solution, this is often termed the Neumann solution, which was allegedly presented in a series of lectures in the early 1860s. A comprehensive description of the history of Stefan problems may be found in Rubinstein.
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 PDEs are not valid at the 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 either side of the phase boundary, and is usually derived from a physical constraint. In problems of heat transfer with phase change, for instance, conservation of energy dictates that the discontinuity of heat flux at the boundary must be accounted for by the rate of latent heat release (which is proportional to the local velocity of the interface). The regularity of the equation has been studied mainly by Luis Caffarelli and further refined by work of Alessio Figalli, Xavier Ros-Oton and Joaquim Serra
Mathematical formulation
The one-dimensional one-phase Stefan problem The one-phase Stefan problem is based on an assumption that one of the material phases may be neglected. Typically this is achieved by assuming that a phase is at the phase change temperature and hence any variation from this leads to a change of phase. 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 + 1 {\displaystyle +1} or − 1 {\displaystyle -1} . Consider a semi-infinite one-dimensional block of ice initially at melting temperature u = 0 {\displaystyle u=0} for x ∈ [ 0 ; + ∞ ) {\displaystyle 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 ) ] {\displaystyle [0;s(t)]} occupied by water. The melted depth, denoted by s ( t ) {\displaystyle s(t)} , is an unknown function of time. The Stefan problem is defined by
The heat equation: ∂ u ∂ t = ∂ 2 u ∂ x 2 , ∀ ( x , t ) ∈ [ 0 ; s ( t ) ] × [ 0 ; + ∞ ] {\displaystyle {\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 , ∀ t > 0 {\displaystyle u(0,t)=1,\quad \forall t>0}
The interface at the melting temperature is set to u ( s ( t ) , t ) = 0 {\displaystyle u\left(s(t),t\right)=0}
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