 Superheated two-phase Stefan similarity

For equal phase thermal properties, a roof undercooling $\Delta T$ and liquid superheat $\Theta$, the planar Stefan problem has interface $a=2\lambda \sqrt{\kappa t}$. The solid error function profile and liquid complementary-error-function profile give $S\lambda =e^{-{\lambda }^2}[1/\mathrm{erf}\lambda -(\Theta /\Delta T)/\mathrm{erfc}\lambda ]/\sqrt{\pi }$. Here $S=L/(c_p\Delta T)$ is the latent-to-sensible heat ratio. Liquid heat supplied to the interface reduces crust growth.