= 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> to
$$
\mathbf u=(1-r)V(1-\varphi)\mathbf e_z,\qquad r=\rho_s/\rho_l.
$$
The 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.
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