= Large-concentration thermal profile of a pulled mush
For $\mathcal C\gg1$ and <latent-to-sensible heat ratio> $S\gg1$ with $S/\mathcal C=O(1)$, put $\Omega=1+S/(r\mathcal C)$, $\zeta=Vz/\kappa$, and $H=Vh/\kappa$. The steady <heat equation> becomes $\theta''+r\Omega\theta'=0$ in the mush and $\theta''+r\theta'=0$ in the liquid. Matching temperature and <heat flux> at $\zeta=H$, with $\theta(0)=-1$ and liquid temperature $\theta_\infty>0$ far above, gives
$$
\theta_m=\frac{\theta_\infty}{\Omega}-(1+\theta_\infty/\Omega)e^{-r\Omega\zeta},\qquad \theta_l=\theta_\infty[1-e^{-r(\zeta-H)}],\qquad H=\frac1{r\Omega}\log(1+\Omega/\theta_\infty).
$$
The latent heat modifies the mush's effective thermal decay rate and therefore its thickness.
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