= Premelted-film particle pushing
{title2=$V=\frac{A}{9\pi\mu a^2}\frac{c+2}{(1-c)(c+3)}$}
For a sphere partially below a planar melting isotherm, write $c=\cos\theta_0$ and depth $a(\cos\theta-c)$. Local <premelting> gives film thickness $d^3=K/(\cos\theta-c)$. Conservation in the thin spherical film gives flux $Va\sin\theta/2$ and <pressure> $p=3\mu Va^2(\cos\theta-c)^2/K$. Integrating vertical repulsive and hydraulic stresses over the wetted cap yields the displayed speed. The unknown thermodynamic constant $K$ cancels, but a narrow rim where the film ceases to be thin is excluded from the leading approximation.
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