 Advection-driven melting front in a porous matrix

For equal phase mass densities, equal sensible heat capacities per unit volume, and negligible thermal diffusion, a porous melting front driven by hot Darcy flux $U$ has speedHere $\Delta \varphi$ is the newly melted volume fraction. Mass conservation preserves $U$ on both sides: the liquid gained by melting fills the newly available pores. The enthalpy increase per unit bulk volume is $\rho c_p\Delta T+\rho L\Delta \varphi$, so its rate of production is balanced by incoming heat flux $\rho c_{p}U\Delta T$.