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

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,