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6 Free Boundary Problems and Phase Transitions
102

Fig. 6.8. Penitentes

and μ solves a transcendental equation:
μ
μ
2
2
μ exp(μ ) exp(−z )dz = .
2L
0
√
Note that the thickness of the ice layer behaves like t. Stefan found coincidence
of this theoretical result with the experimental data available to him.
There is a convenient reformulation of the Stefan problem in terms of a de-
generate parabolic equation, making use of the enthalpy formulation of heat
flow. The physical enthalpy e is related to the temperature θ by

θ = β(e) , (6.20)
where
L
β(e) = e + , for e< 0
2
L L
β(e) = 0, for − <e< (6.21)
2 2
L
β(e) = e − , for e> 0
2