Let a representative fixed volume have mixture <mass density> $\bar\rho=\rho_s\varphi+\rho_l(1-\varphi)$. Since the crystal framework is rigid and stationary, the liquid mass flux is $\rho_l\mathbf u$, where $\mathbf u$ is <Darcy velocity>. <Mass conservation> gives
$$
\partial_t\bar\rho+\nabla\cdot(\rho_l\mathbf u)=0.
$$
The constant phase densities therefore imply the <phase-change volume source in a rigid mush>
$$
\boxed{\nabla\cdot\mathbf u=(1-r)\varphi_t,\qquad r=\rho_s/\rho_l.}
$$
For ice less dense than brine, increasing <solid fraction> creates an expansion flow; when the phase densities agree, this source vanishes.

Solution

Past exam of the mathematics course of the University of Cambridge / 2019 / iii / Paper 332 / 2 / a / Solution

For constant phase <mass densities> $\rho_s,\rho_l$, a fixed solid framework has mixture density $\bar\rho=\rho_s\varphi+\rho_l(1-\varphi)$. <Mass conservation> gives
$$
\partial_t\bar\rho+\rho_l\nabla\cdot\mathbf u=0,\qquad \nabla\cdot\mathbf u=(1-r)\varphi_t,\qquad r=\rho_s/\rho_l,
$$
where $\mathbf u$ is <Darcy flux>. Equal phase densities give a divergence-free flux. Unequal densities require expansion or contraction flow during freezing or melting.

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 Incoming links: Phase-change volume source in a rigid mush