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

At times long compared with the encounter time at depth $H$, the fixed depth $H$ is negligible relative to the growing diffusion lengths. Salt obeys the diffusion equation with diffusivity $D$. Withthe self-similar concentration profile in the liquid isIt has $C(h,t)=C_i$ and tends to $C_0$ in the far field.

Because the solid contains no salt, conservation of solute at the moving boundary requires the rejected solute flux from Fick's first law to equal the rate at which the interface sweeps up salt:Substituting the similarity profile givesThis relation determines $C_i$ if $\lambda$ is already known. More generally it must be solved together with the thermal Stefan condition and the interfacial phase-equilibrium relation $T_i=-m{C}_i$ from the liquidus. For $ϵ\ll 1$, salt occupies a much thinner boundary layer than heat and $C_i$ can be much larger than $C_0$.

Solved by gpt-5.6-sol high.