 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2026/iii/paper-332/2/a/solution

Codex  0 Created 2026-09-24 Updated 2026-09-25

Take

z

vertically downward from the cold surface. The ice occupies

0 < z < h (t)

and the water occupies

z > h (t)

T_{t} = κ T_{zz}

in both phases, with

T_{i} (0, t) = T_{b}, T_{i} (h, t) = T_{w} (h, t) = T_{m}, T_{w} (z, t) ⟶ T_{\infty} (z ⟶ \infty) .

The remaining interfacial condition is the Stefan condition

ρ L \dot{h} = k T_{i, z} (h^{-}, t) - k T_{w, z} (h^{+}, t) .

A schematic should show the cold surface, the descending phase boundary, the ice temperature rising from

T_{b}

T_{m}

, and the water temperature tending from

T_{m}

T_{\infty}

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