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

LetFlux conservation gives $u=U/H$. Substitution into the extensional-force equation yields the third-order equationSinceone integration, using $H\to 1$ and its derivatives tending to zero on the flat-film side, givesPut $q(H)=H_{\xi }$. After division by $q$, this becomesThe integrating factor $H^{-1/2}$ givesA second integration and $q(1)=0$ therefore give

On the bubble side, matching to its cylindrical shape gives $H\sim {\xi }^2/2$, so $H^{-1/2}{H}_{\xi }\to \sqrt{2}$. Taking $H\to \infty$ in the integrated equation yields $8V/3=\sqrt{2}$. Hence