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

The long-wave approximation makes $u$ independent of $z$ at leading order. Mass conservation and symmetry about $z=0$ then giveFor an incompressible Newtonian fluid,The leading normal-stress balance on either surface isSince ${\sigma }_{xx}-{\sigma }_{zz}=4\mu u_x$, it follows that

For a slice of length $\delta x$, the axial forces are the integrated normal stresses $2h{\sigma }_{xx}$ on its vertical ends, ambient pressure on the varying end height, and the horizontal components of surface tension on its two sloping faces. Expanding their difference to first order in $\delta x$ cancels the uniform $p_a$ terms and the lower-order capillary terms, leavingThe kinematic condition on $z=h(x,t)$ is $h_t+u{h}_x=w$. Substituting $w=-h{u}_x$ gives the second required relation,

The gas pressure exceeds the distant liquid pressure by the capillary pressure $\gamma /a$ associated with each cylindrical bubble. In the flat film the interface curvature is nearly zero, so its liquid pressure is close to the gas pressure and therefore exceeds the external liquid pressure by approximately $\gamma /a$. This pressure excess drives liquid out through the two transition regions.

Inside the flat region $h=h_0(t)$, so $h_{xxx}=0$. The extensional-force equation gives $(h_0{u}_x{)}_x=0$, hence $u_x$ is independent of $x$. Symmetry gives $u(0,t)=0$ and define $u(L,t)=U(t)$, soThe mass-conservation equation becomesIt contains no $x$ dependence, so an initially uniform film remains uniform within the flat region.

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

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

Insert the velocity from part d into the uniform-film equation:If $h_i=h_0(0)$, integration givesThusThe film thins algebraically and does not reach zero thickness in finite time within this continuum model; rupture would require physics omitted here, such as intermolecular forces.