= Small-diffusivity constitutional-supercooling threshold
{title2=$\theta_-=\theta_+(1+2\epsilon)+O(\epsilon^2)$}
For a <saline Stefan problem> with common thermal properties, $\epsilon=\sqrt{D/\kappa}\ll1$, and fixed <latent-to-sensible heat ratio>, set $T_0=-mC_0$, $\theta_-=T_0-T_{-\infty}$ and $\theta_+=T_\infty-T_0>0$. A finite-$\lambda$ <similarity solution> has $T_i=(T_\infty+T_{-\infty})/2+O(\epsilon)$. At the onset of <constitutional supercooling>, $\lambda=O(\epsilon)$ and the error in this mean is $O(\epsilon^2)$. The <salt-rejection function for a saline Stefan front> then gives $m\lambda C_i=(\theta_--\theta_+)/(2\sqrt\pi)+O(\epsilon^2)$, whereas $\sqrt{Dt}\,T_{l,x}(a^+)=\epsilon\theta_+/\sqrt\pi+O(\epsilon^2)$. Comparing the actual <temperature gradient> to the <liquidus> <gradient> gives the displayed onset, with supercooling on the side of greater $\theta_-$. For a freezing front at $a\ge0$ and $D<\kappa$, the ratio of these two <gradients> decreases into the liquid, so their interface comparison also decides whether a supercooled interval occurs anywhere ahead. The formula is an asymptotic criterion; finite-$\epsilon$ onset requires the full <Stefan condition> and salt balance.
Back to article page
