Pressure-limited spherical freezing in a porous medium ($\Lambda =k\gamma \mu ϵ/(\Pi {\rho }_{s}L)$)

A growing ice-filled sphere expels liquid because ice is less dense than water. With porosity $\varphi$, $ϵ=1-{\rho }_s/\rho$, and pressure-induced freezing-point depression $T_i=T_m-\gamma p_i$, the Darcy velocity gives $p_i=\mu \varphi ϵa\dot{a}/\Pi$. Quasi-steady spherical thermal conduction gives $\varphi {\rho }_{s}La\dot{a}=k(T_i-T_{\infty })$. Hence $a\dot{a}=k(T_m-T_{\infty })/[\varphi {\rho }_{s}L(1+\Lambda )]$, where $\Lambda =k\gamma \mu ϵ/(\Pi {\rho }_{s}L)$. The interface approaches the far-field temperature at low permeability of a porous medium, while its pressure tends to a finite equilibrium value.