Classical thermodynamics, heat transfer

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• ConceptCritical point

  A critical point of a real function is a point where the derivative is zero or does not exist. All interior local extrema of the function occur at critical points. The node critical point gives the precise definition.

• ConceptHeat equation

  The heat equation $\partial_t u = \Delta u$ describes how temperature, or the concentration of a diffusing substance, evolves in time. Here $\Delta$ is the Laplacian in the space variables. It is the model parabolic partial differential equation. Its solutions become smooth at once, spread information at infinite speed, and decay toward equilibrium. It is linked to probability through the Wiener process, and to geometry through the spectrum of the Laplacian.

• ProblemStefan problem

  The one-dimensional one-phase Stefan problem describes a solid at its melting temperature $0$ that occupies the half-line $x > 0$ and melts when heat enters at $x = 0$. Let $a > 0$ be the thermal diffusivity of the liquid, $k > 0$ its thermal conductivity and $L > 0$ the latent heat per unit volume. Let $g \colon [0, \infty) \to [0, \infty)$ be a given boundary temperature. Find a function $s \colon [0, \infty) \to [0, \infty)$ with $s(0) = 0$, the position of the interface, and a temperature $u(x, t)$ defined for $0 \le x \le s(t)$, such that: