Past exam of the mathematics course of the University of Cambridge / 2026 / iii / Paper 332 / 2 / d / Solution

Before the ice reaches the salty layer, the state moves vertically in a temperature--concentration phase diagram along $C=0$: cooling crosses the pure-water melting point and produces pure ice. After the encounter, the liquid state starts at $(T_{\infty },C_0)$ and approaches the interface stateon the liquidus $T_L=-mC$. The solid remains on the $C=0$ axis.

When $ϵ\ll 1$, the solutal diffusion length $\sqrt{Dt}$ is much smaller than the thermal diffusion length $\sqrt{\kappa t}$. Immediately ahead of the interface, rejected salt makes $C$ fall steeply from $C_i$ to $C_0$, so the local liquidus $T_L=-mC$ rises steeply with depth while the actual temperature changes comparatively slowly. Constitutional supercooling occurs whereverIts local onset criterion at the interface isThe supercooled zone lies in the thin salty boundary layer immediately ahead of the planar front. A small forward protrusion then enters liquid that is already below its local freezing point, so the planar interface is susceptible to a morphological instability.

Solved by gpt-5.6-sol high.