Free boundary problem
In mathematics, a free boundary problem is a partial differential equation to be solved for both an unknown function u and an unknown domain Ω. The segment Γ of the boundary of Ω which is not known at the outset of the problem is the free boundary.
The classic example is the melting of ice. Given a block of ice, one can solve the heat equation given appropriate initial and boundary conditions to determine its temperature. But, if in any region the temperature is greater than the melting point of ice, this domain will be occupied by liquid water instead. The boundary formed from the ice/liquid interface is controlled dynamically by the solution of the PDE.
The melting of ice is a Stefan problem for the temperature field T, which is formulated as follows. Consider a medium occupying a region Ω consisting of two phases, phase 1 which is present when T > 0 and phase 2 which is present when T < 0. Let the two phases have thermal diffusivities α1 and α2. For example, the thermal diffusivity of water is 1.4×10−7 m2/s, while the diffusivity of ice is 1.335×10−6 m2/s.
In the regions consisting solely of one phase, the temperature is determined by the heat equation: in the region T > 0,
while in the region T < 0,
This is subject to appropriate conditions on the (known) boundary of Ω; Q represents sources or sinks of heat.
Let Γt be the surface where T = 0 at time t; this surface is the interface between the two phases. Let ν denote the unit outward normal vector to the second (solid) phase. The Stefan condition determines the evolution of the surface Γ by giving an equation governing the velocity V of the free surface in the direction ν, specifically
where L is the latent heat of melting. By T1 we mean the limit of the gradient as x approaches Γt from the region T > 0, and for T2 we mean the limit of the gradient as x approaches Γt from the region T < 0.
In this problem, we know beforehand the whole region Ω but we only know the ice-liquid interface Γ at time t = 0. To solve the Stefan problem we not only have to solve the heat equation in each region, but we must also track the free boundary Γ.
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