Phase-change volume source in a rigid mush

For constant phase mass densities ${\rho }_s,{\rho }_l$, a fixed solid framework has mixture density $\bar{\rho }={\rho }_s\phi +{\rho }_l(1-\phi )$. Mass conservation giveswhere $\mathbf{u}$ is Darcy flux. Equal phase densities give a divergence-free flux. Unequal densities require expansion or contraction flow during freezing or melting.