= Solution
Impose the <eutectic temperature> at $\zeta=0$ in the <large-concentration thermal profile of a pulled mush>:
$$
-1=-\frac{\theta_\infty}{\Omega}(e^{r\Omega H}-1).
$$
Therefore
$$
\boxed{\frac{Vh}{\kappa}=H=\frac1{r\Omega}\log\left(1+\frac\Omega{\theta_\infty}\right),\qquad \Omega=1+\frac S{r\mathcal C}.}
$$
The thickness is proportional to <thermal diffusivity> divided by pulling speed. Latent-heat release modifies the exponential decay rate through $\Omega$, while the far-liquid temperature fixes how far the profile can remain below the <liquidus>.
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