= Premelted-film lifting of a disk
{title2=$U=\frac{A}{6\pi\mu a^2}\left(1-\frac{2WT_m}{\pi a^2\rho L\Delta T}\right)$}
In an equal-density, no-slip <lubrication approximation>, repulsive <disjoining pressure> $A/(6\pi d^3)$ sustains a <premelting> film of thickness $d$ beneath a rigid disk of radius $a$. With imposed radial undercooling $\Delta T(1-r^2/a^2)$, local <thermomolecular pressure> is $P_T=\rho L\Delta T(1-r^2/a^2)/T_m$, and $d^3=A/(6\pi P_T)$. A constant vertical lifting rate $U$ requires inward volume flux $q_r=-Ur/2$. Integrating the resulting <lubrication pressure> and disk <force balance> gives the displayed rate, where $W$ is the apparent load, $\mu$ is liquid <dynamic viscosity>, $L$ is specific <latent heat>, and $T_m$ is the absolute bulk melting <temperature>. Positive lifting requires $W<\pi a^2\rho L\Delta T/(2T_m)$. For fixed positive load, the rate rises from zero at that threshold towards $A/(6\pi\mu a^2)$ as undercooling increases. That plateau is an asymptotic limit within this constitutive model, not a claim that continuum-film physics persists at arbitrarily low <temperatures>.
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