Large-concentration stagnant-mush similarity (${\Omega }^2=1+\mathcal{S}/\mathcal{C}$)

For $\mathcal{C}=C_0/\Delta C\gg 1$ 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 $\theta =-1+\mathrm{erf}(\Omega \eta )/\mathrm{erf}(\Omega \lambda )$. Matching heat flux requires $\Omega e^{{\lambda }^2}\mathrm{erfc}\lambda ={\theta }_{\infty }{e}^{{\Omega }^2{\lambda }^2}\mathrm{erf}(\Omega \lambda )$. 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 $\mathcal{C}$.