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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 $\rho c_{p}U\Delta T$.

A planar advection-driven melting front in a porous matrix with permeabilities $k$ ahead and $k+\Delta k$ behind has the ideal sharp-front growth rateThis follows by matching harmonic pressure perturbations and normal Darcy flux, then converting flux to front speed by the enthalpy balance. It predicts no finite fastest-growing wavelength when $\Delta k>0$.

The heat capacity at constant pressure is $C_p=(\delta Q/\partial T{)}_p=T(\partial S/\partial T{)}_p$ for reversible heat input. The enthalpy is $H=U+pV$, so $dH=TdS+Vdp$. Along an isobar this gives $\boxed{C_p=(\partial H/\partial T{)}_p}$.