Travelling solidification front in a pure supercooled melt ($T_l=T_{\infty }+\Delta T{e}^{-V\xi /\kappa },S=1$)

For equal thermal properties and bounded travelling profiles on two deep half-spaces, the solid temperature is $T_m$ and the liquid profile is exponential. The Stefan condition requires $L=c_p\Delta T$. This balances the latent heat of solidification with warming of the initially supercooled liquid.

Retaining perturbation thermal diffusion givesThe displaced interfacial temperature and Stefan condition give the displayed implicit dispersion relation. In the capillarity-free limit its growing branch is exactly $\sigma =V\alpha$, which checks the limits of the small-wavelength thermal approximation for a solidification front.

Approximating perturbation temperatures by harmonic fields decaying as $e^{\pm \alpha \xi }$, while retaining displacement through the base gradient, produces this reduced dispersion relation. The Gibbs--Thomson relation supplies $\Gamma =\gamma T_m/(\rho L)$. This short-wave reduction suppresses perturbation advection and time dependence; its constant term and a threshold at $\alpha \kappa /V=O(1)$ are not controlled next-order results of the full diffusion problem.

The reduced dimensionless growth rate $g(q)=-1+q-2q^3/R_c$ has a positive maximum precisely when $R_c>27/2$. Its maximizing $q=\sqrt{R_c/6}$ follows by differentiation. At equality $q=3/2$, outside a strict $q\gg 1$ short-wave regime. The inequality is an exact criterion for the reduced cubic, not a quantitative onset theorem for full thermal stability of a travelling solidification front.