= 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 speed
$$
V=\frac{U}{1+S\Delta\phi},\qquad S=\frac{L}{c_p\Delta T}.
$$
Here $\Delta\phi$ 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 $\rho c_p\Delta T+\rho L\Delta\phi$, so its rate of production is balanced by incoming heat flux $\rho c_pU\Delta T$.
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