 Solution

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.