Stefan condition

A condition describing the law of motion of the boundary between two different phases of matter and expressed as a law of energy conservation under phase transformation. For example, the boundary between the solid and liquid phases of matter in a solidifying process (or a melting process) can be described in the one-dimensional case by a function $\xi=\xi(t)$, connected to the temperature distribution $u(x,t)$ by means of the Stefan condition:

$$\lambda\rho_1\frac{d\xi}{dt}=k_1\frac{\partial u(\xi(t)-0,t)}{\partial x}-k_2\frac{\partial u(\xi(t)+0,t)}{\partial x},\quad t>0$$

(for the significance of the symbols, see Stefan problem).

The mass

$$\rho_1\Delta\xi=\rho_1[\xi(t+\Delta t)-\xi(t)]$$

solidifies (or melts) in the course of time $\Delta t$. The amount of heat $\lambda\rho_1\Delta\xi$ thus required is equal to the difference between the amounts of heat passing through the boundaries $\xi(t)$ and $\xi(t+\Delta t)$:

$$\lambda\rho_1\Delta\xi=\left[k_1\frac{\partial u(\xi(t)-0,t)}{\partial t}-k_2\frac{\partial u(\xi(t+\Delta t)+0,t+\Delta t)}{\partial x}\right]\Delta t.$$

Hence, when $\Delta t\to0$, the Stefan condition is obtained. Moreover, the temperature on the boundary between the two phases $\xi=\xi(t)$ is assumed to be continuous and its value is taken equal to the known temperature of melting.

Similar conditions on unknown boundaries which arise in studies on certain other processes and which follow from conservation laws are also called Stefan conditions (see Differential equation, partial, free boundaries).

References

 [1] A.N. Tikhonov, A.A. Samarskii, "Equations of mathematical physics" , Pergamon (1963) (Translated from Russian)