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The conservation of mass equation is $h_t+(uh{)}_x=0$. The flat region changes on time scale $L/U$, whereas a transition has advection time $\delta /U$. Hence $h_t=O(U{h}_0/L)$, and its flux change across the transition is $O(U{h}_0\delta /L)$, small compared with the entering flux $U{h}_0$. To leading order the transition is quasisteady and

Eliminate $u=q/h$ in the longitudinal equation. Then $h{u}_x=-q(\log h{)}_x$, soUse $\xi =(x-L)/\delta$, with $\delta =\sqrt{a{h}_0}$, and use a prime for differentiation with respect to $\xi$. This gives the third-order equationHere $h$ remains a dimensional thickness; only the longitudinal coordinate is scaled.

Match at the inner end to $h\to h_0$, $h^{\prime }\to 0$, $h^{\prime \prime }\to 0$. Since $(h{h}^{\prime \prime }-h^{\prime 2}/2{)}^{\prime }=h{h}^{\prime \prime \prime }$, the first integral isDivide by $h^{3/2}$ to obtain $(h^{\prime }/\sqrt{h}{)}^{\prime }=B{h}^{\prime }/h^{5/2}$. Its second integral, using the flat-film limit to fix the constant, isLet $A=16\mu U\sqrt{a}/(3\gamma )>0$. In the overlap $h\gg h_0$, the equation gives $h^{\prime }\sim A\sqrt{h}$, hence $h^{\prime \prime }\sim A^2/2$. The curvature of each physical interface is $h_{xx}/2\sim A^2/(4a{h}_0)$. Matching to the outer bubble curvature $1/a$ therefore requires $A=2\sqrt{h_0}$ and fixes the numerical coefficient:

Finally the uniform-film evolution becomes ${\dot{h}}_0=-3\gamma h_0^{3/2}/(8\mu L\sqrt{a})$. If $h_0(0)=h_i$ at the start of the drainage regime, its solution isThus capillary drainage of a film between two bubbles gives an algebraic $t^{-2}$ decrease, and the two bubbles do not touch in finite time in this ideal model. As $h_0$ decreases, $h_0/\delta$ and $\delta /L$ become still smaller, so the geometric scale separation improves. At molecular thicknesses, disjoining pressure and changes in the assumed stress-free boundary condition can invalidate the clean continuum-sheet description; the algebraic law alone makes no claim about that physical endpoint.

In the short end transition, the full thickness changes by $O(h_0)$ while the curvature of each interface changes to $O(1/a)$. Since each interface has height $h/2$, the curvature scale is $O(h_0/{\delta }^2)$, with order-one factors irrelevant to this estimate. ThereforeThis is the geometric scale of the capillary drainage of a film between two bubbles. The assumed regime additionally requires $h_0\ll a$ and $\sqrt{a{h}_0}\ll L$, so the transition is long compared with its thickness and short compared with the flat sheet length.