= Large-concentration stagnant-mush similarity
{title2=$\Omega^2=1+\mathcal S/\mathcal C$}
For $\mathcal C=C_0/\Delta C\gg1$ and <inverse Stefan number> $\mathcal S=O(\mathcal C)$, a <stagnant mushy-layer model> has nearly constant effective <heat capacity>, giving $\Omega^2\theta_t=\kappa\theta_{zz}$ with $\Omega^2=1+\mathcal S/\mathcal C$. With negligible liquid <solutal diffusivity>, roof value $\theta=-1$, positive liquid superheat $\theta_\infty$, and edge $h=2\lambda\sqrt{\kappa t}$, the mush profile is $\theta=-1+\operatorname{erf}(\Omega\eta)/\operatorname{erf}(\Omega\lambda)$. Matching <heat flux> requires $\Omega e^{\lambda^2}\operatorname{erfc}\lambda=\theta_\infty e^{\Omega^2\lambda^2}\operatorname{erf}(\Omega\lambda)$. The square on $\Omega$ in the exponential is essential. The <porosity> $\mathcal C/(\mathcal C-\theta)$ records the leading composition balance; the linear thermal profile is asymptotic, not exact at finite $\mathcal C$.
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