Full thermal stability of a travelling solidification front ($\sigma =(V-2\kappa \Gamma {\alpha }^2/\Delta T)Q-V^2/(2\kappa )$)

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.