Past exam of the mathematics course of the University of Cambridge / 2017 / iii / Paper 332 / 2 / i

Let $E(\lambda )=\sqrt{\pi }\lambda e^{{\lambda }^2}\mathrm{erfc}(\lambda )$. The salt equation in the saline Stefan problem gives $C_i-C_0=E(\lambda )C_i$. Since the complementary error function is positive on the real axis and $C_i>0$, the signs of $C_i-C_0$ and $\lambda$ agree. Thus freezing is equivalent to $C_i>C_0$, or $T_i<T_0=-m{C}_0$. Using the leading thermal balance,at leading order away from a vanishing temperature difference. In fact the exact onset is also ${\theta }_{-}={\theta }_{+}$: at $\lambda =0$ the salt balance forces $C_i=C_0$, and the Stefan condition then forces $T_{-\infty }+T_{\infty }=2T_0$. The solution with $\lambda =0$ has a stationary interface and balancing conductive heat fluxes, despite nontrivial diffusion of the initial temperature discontinuity. For completeness, the exact freezing criterion follows without relying on that local crossing. By the salt-rejection function for a saline Stefan front, $T_i(\lambda )$ decreases continuously from $T_0$ to $-\infty$ as positive $\lambda$ increases. If $T_{-\infty }<T_0$, let ${\lambda }_s$ be where $T_i=T_{-\infty }$. On $[0,{\lambda }_s]$, the solid-side thermal term decreases, the liquid-side term increases, and the latent-heat term increases: the factors $e^{-{ϵ}^2{\lambda }^2}/\mathrm{erfc}(-ϵ\lambda )$ and $e^{-{ϵ}^2{\lambda }^2}/\mathrm{erfc}(ϵ\lambda )$ decrease and increase respectively. These monotonicities follow from the positive integral representation of $e^{v^2}\mathrm{erfc}v$. The thermal residual starts at $({\theta }_{-}-{\theta }_{+})/\sqrt{\pi }$ and is negative at ${\lambda }_s$, so a positive root exists uniquely precisely when ${\theta }_{-}>{\theta }_{+}$. Beyond ${\lambda }_s$ both thermal contributions to freezing are negative, so no further root is possible. If $T_{-\infty }\ge T_0$, the same sign argument excludes every positive root.