= Capillary drainage of a film between two bubbles
{title2=$U=\frac{3\gamma}{8\mu}\sqrt{h_0/a},\quad h_0\propto t^{-2}$}
Two cylindrical bubbles separated by a uniform flat liquid film of half-length $L$ drain through short curvature-transition regions at the ends. The elevated gas pressure implies a film liquid pressure approximately $\gamma/a$ above the outer liquid pressure. The transition length is $\delta\sim\sqrt{ah_0}$, and its <volume flux> is nearly constant because $\delta/L\ll1$. Write $uh=Uh_0$ and $\xi=(x-L)/\delta$. The <planar viscous-sheet stretching equations> reduce to $hh'''=C(\log h)''$, with $C=8\mu Uh_0\delta/\gamma$. Two integrations, matched to a flat film at the inner end, give $h'/\sqrt h=(16\mu U\sqrt a/(3\gamma))[1-(h_0/h)^{3/2}]$. Matching the curvature to $1/a$ gives the displayed speed. Uniform-film <conservation of mass> then gives $\dot h_0=-Uh_0/L$, hence $h_0=[h_i^{-1/2}+3\gamma t/(16\mu L\sqrt a)]^{-2}$. There is no finite-time contact in this ideal continuum model.
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