= One-sided kinetic solidification dispersion
{title2=$\sigma(1+S/p)=SV-\Gamma\alpha^2-SV^2/p$}
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 $A_1e^{-p\xi}$, 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 $S\sigma\zeta=pA_1-SV^2\zeta$. Eliminating $A_1$ proves the displayed <dispersion relation>. Neglect of solid-side perturbation heat flux is part of this model.
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