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[edit]

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

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#### Comments[edit]

The derivation of the heat balance condition at the liquid-solid interface was for long time attributed to J. Stefan [a1], [a2]. However, it appeared for the first time in a much earlier work by B.D. Clapeyron and G. Lamé [a3]. A classical illustration of the Stefan condition and of related problems can be found in [a4].

Many generalizations of the Stefan condition have been considered in the literature. For instance, the coefficients may depend on space and time, or higher-order derivatives of  may appear on the right-hand side, even in a non-linear way (see e.g. [a5], [a6]).

#### References[edit]

[a1] |  J. Stefan, "Über einige Probleme der Theorie der Wärmeleitung" Sitzungsber. Akad. Wiss. Berlin Math. Kl. , 98 (1889) pp. 473–484

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[a2] |  J. Stefan, "Über die Theorie der Eisbildung, insbesondere über die Eisbildung in Polarmeere" Ann. Physik Chemie , 42 (1891) pp. 269–286
[a3] |  G. Lamé, B.D. Clapeyron, "Mémoire sur la solidification par refroidissement d'un globe liquide" Ann. Chimie Physique , 47 (1831) pp. 250–256
[a4] |  L.I. Rubinstein, "The Stefan problem" , Amer. Math. Soc. (1971) (Translated from Russian)
[a5] |  A. Fasano, M. Primicerio, "Free boundary problems for nonlinear parabolic equations with nonlinear free boundary conditions" J. Math. Anal. Appl. , 72 (1979) pp. 247–273
[a6] |  M. Primicerio, "Classical solutions of general two-phase parabolic free boundary problems in one dimension" A. Fasano (ed.) M. Primicerio (ed.) , Free boundary problems: theory and application , 2 , Pitman (1983) pp. 644–657