User:Compsonheir/Stefan problem

In applied mathematics, the Stefan problem constitutes the determination of the temperature distribution in a medium consisting of more than one phase, for example ice and liquid water. While in principle the heat equation suffices to determine the temperature within each phase, one must also ascertain the location of the ice-liquid interface. Note that this evolving boundary is an unknown (hyper-)surface: hence, the Stefan problem is a free boundary problem.

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.

Mathematical formulation

Consider a substance consisting of two phases which have densities ρ₁, ρ₂, heat capacities c₁, c₂ and thermal conductivities k₁, k₂. Let T be the temperature in the medium, and suppose that phase 1 exists for T < T_m while phase 2 is present for T > T_m, where T_m is the melting temperature. With a source Q, the temperature T satisfies the partial differential equation

\rho _{1}c_{1}{\frac {\partial T}{\partial t}}=k_{1}\nabla ^{2}T+Q

in the region where T < T_m,

\rho _{2}c_{2}{\frac {\partial T}{\partial t}}=k_{2}\nabla ^{2}T+Q

in the region where T > T_m. However, there is an additional equation determining the location of the interface between the two phases. Let V be the velocity of the phase boundary; we adopt the convention that V is positive when the phase boundary is moving towards phase 2 and negative when moving towards phase 1. Then V is determined by the equation

(\rho _{2}L+(\rho _{2}c_{2}-\rho _{1}c_{1})T_{m})V=k_{1}|\nabla T_{1}|-k_{2}|\nabla T_{2}|

.

Here the subscript ₁ is a shorthand for

\nabla T_{1}(x)=\lim _{\stackrel {y\to x}{T(y)<T_{m}}}\nabla T(y)

and vice versa for ₂.

Derivation

The Stefan condition

(L+(\rho _{2}c_{2}-\rho _{1}c_{1})T_{m})V\cdot n=(k_{2}\nabla T_{2}-k_{1}\nabla T_{1})\cdot n

is the key component of this problem. It can be derived in a similar fashion to the Rankine-Hugoniot conditions for conservation laws, which we do below.

First, we note that the internal energy of the material is

E=\left\{{\begin{array}{ll}\rho _{1}c_{1}T&T<T_{m}\\\rho _{2}c_{2}T+\rho _{2}L&T>T_{m}\end{array}}\right.,

where L is the latent heat of melting. The differential equations and Stefan condition are consequences of the fundamental relation

\partial _{t}E=\nabla \cdot (k\nabla T)+Q,

where k is the thermal conductivity and Q represents diabatic heating.

Unfortunately, the differential equation for the internal energy does not make sense in a neighborhood of the region where the substance is melting: the internal energy is discontinuous due to the additional latent heat necessary to melt a solid, and the thermal conductivity is discontinuous too.

To remedy this situation, we interpret the equation for the internal energy as a conservation law by integrating the PDE over a small box in space-time containing a point on the surface

\Sigma =\{(x,t):T(x,t)=T_{m}\},\,

using the divergence theorem and then taking the limit as the size of this box shrinks to zero. For the sake of an unambiguous definition we take

n=\lim _{x\to \Sigma _{t}}{\frac {\nabla T}{|\nabla T|}}.\,

The vector n points in the same direction whether the limit is taken from the solid or liquid side of the phase boundary, since the gradient always points in the direction of increasing T. Now, note that the jump of E going from the solid to liquid phase is