 One-sided kinetic solidification dispersion

Use undercooling $\Delta T=T_m-T_{\infty }$, solidification attachment mobility $\mathcal{M}$, velocity scale $U_0=\mathcal{M}\Delta T$, length $\ell =\kappa /U_0$ and time $\kappa /U_0^2$. Let $S=L/(c_p\Delta T)$ be the latent-to-sensible heat ratio, $\theta =(T-T_{\infty })/\Delta T$, and $\Gamma ={\gamma }_T/(\Delta T\ell )$. In the one-sided liquid-diffusion model, a travelling planar front has $\theta =S{e}^{-V\xi }$ and $V=1-S$. For a corrugation $\zeta e^{\sigma t+i\alpha y}$, the decaying thermal perturbation is $A_1{e}^{-p\xi }$, with $p^2-Vp=\sigma +{\alpha }^2$ and the branch $p=[V+\sqrt{V^2+4(\sigma +{\alpha }^2)}]/2$. The displaced kinetic condition gives $A_1=(SV-\sigma -\Gamma {\alpha }^2)\zeta$. The displaced Stefan condition gives $S\sigma \zeta =p{A}_1-S{V}^2\zeta$. Eliminating $A_1$ proves the displayed dispersion relation. Neglect of solid-side perturbation heat flux is part of this model.