Heat-balance correction to a premelted-film growth model ($\rho L\dot{h}=K_s(\Delta T-\delta T)/h-K_l\delta T/a$)

Combine the full quasi-steady Stefan condition above with $\delta T=T_m{p}_T/(\rho L)$, $a=(A/(6\pi p_T){)}^{1/3}$ and the Darcy flow law to obtain an implicit pressure-growth system. Even when latent heat is negligible, a conductivity ratio $K_s/K_l=r$ changes the leading molecular driving pressure by $r^{3/4}$. Thus a reduced equal-conductivity formula requires an explicit material and rate approximation.