Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 329 / 2 / c / Solution

Matching the flat film to a cylindrical interface gives the local parabolic lubrication gapThe thickness changes by $O(h_0)$ across the transition, soIn the extensional-force equation, viscous and capillary terms have scalesTheir balance gives

The transition adjusts on time $\delta /U$, whereas the flat film drains on time $L/U$. Since $\delta \ll L$, the transition is quasi-steady. Its thin-film mass flux therefore satisfies