Capillary drainage of a film between two bubbles ($U=\frac{3\gamma }{8\mu }\sqrt{h_0/a},h_0\propto t^{-2}$)

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 $\delta \sim \sqrt{a{h}_0}$, and its volume flux is nearly constant because $\delta /L\ll 1$. Write $uh=U{h}_0$ and $\xi =(x-L)/\delta$. The planar viscous-sheet stretching equations reduce to $h{h}^{\prime \prime \prime }=C(\log h{)}^{\prime \prime }$, with $C=8\mu U{h}_0\delta /\gamma$. Two integrations, matched to a flat film at the inner end, give $h^{\prime }/\sqrt{h}=(16\mu U\sqrt{a}/(3\gamma ))[1-(h_0/h{)}^{3/2}]$. Matching the curvature to $1/a$ gives the displayed speed. Uniform-film conservation of mass then gives ${\dot{h}}_0=-U{h}_0/L$, 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.