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For $\mathcal{C}\gg 1$ and latent-to-sensible heat ratio $S\gg 1$ 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 ${\theta }^{\prime \prime }+r\Omega {\theta }^{\prime }=0$ in the mush and ${\theta }^{\prime \prime }+r{\theta }^{\prime }=0$ in the liquid. Matching temperature and heat flux at $\zeta =H$, with $\theta (0)=-1$ and liquid temperature ${\theta }_{\infty }>0$ far above, givesThe latent heat modifies the mush's effective thermal decay rate and therefore its thickness.

Use $\zeta =Vz/\kappa$, $H=Vh/\kappa$, and $S=L/(c_p\Delta T)$, the latent-to-sensible heat ratio. Transforming the heat equation to the fixed apparatus frame and substituting Darcy flux gives the exact steady mush equationFor $\mathcal{C}\gg 1$, the solid fraction is $\phi =-\theta /\mathcal{C}+O({\mathcal{C}}^{-2})$. Since $S/\mathcal{C}=O(1)$, the latent term remains leading order, whereas the correction $(1-r)\phi {\theta }_{\zeta }$ is small. Define $\Omega =1+S/(r\mathcal{C})$. The large-concentration thermal profile of a pulled mush then obeysWith $\theta (H)=0$ and $\theta \to {\theta }_{\infty }$ in the liquid,The thermal conductivity agrees on both sides and the solid fraction vanishes at the mush–liquid interface, so there is no jump in latent production there: continuity of heat flux gives ${\theta }_m^{\prime }(H)={\theta }_l^{\prime }(H)=r{\theta }_{\infty }$. ThusThe boundary value ${\theta }_m(0)=-1$ determines $H$, as calculated next. The crystal fraction is obtained from the previous part's formula, with the temperature field understood to this leading asymptotic accuracy.

Melting raises permeability of a porous medium. An advancing protrusion therefore offers a less resistive path to hot liquid, attracting larger Darcy flux and receiving more heat. The resulting faster melting reinforces the protrusion: this is a reactive infiltration instability. A permeability decrease would reverse this feedback, while zero contrast gives no growth in this ideal model.

For positive contrast, the growth rate is proportional to transverse wavenumber, so the model has no finite fastest-growing wavelength. The latent-to-sensible heat ratio slows the front but supplies no short-wave cutoff. Finite thermal transport and a resolved melting zone can introduce a length scale; pore geometry eventually limits the continuum description. If interface curvature changes the equilibrium melting temperature through the Gibbs--Thomson relation, that can also oppose short-wave corrugations. The modified dispersion relation must include such effects before a preferred wavelength can be calculated; a length scale alone does not guarantee that every added mechanism selects one.