Density-change flow in a pulled mushy layer

If the solid skeleton translates with velocity $-V{\mathbf{e}}_z$ and the fully solid material has density ${\rho }_s$, steady mass conservation fixes Darcy flux toThe liquid's laboratory velocity is therefore ${\mathbf{v}}_l=-rV{\mathbf{e}}_z$ wherever liquid exists. This upward relative flux for $r<1$ supplies the expansion needed when liquid becomes a less dense solid.

With pure solid crystals, negligible solute diffusion, and the density-change flow in a pulled mushy layer, steady solute balance gives $(1-\phi )C=C_0$. If $\theta =(T-T_0)/\Delta T$, $\mathcal{C}=m{C}_0/\Delta T$, and the liquidus is $T-T_0=-m(C-C_0)$, then