Past exam of the mathematics course of the University of Cambridge / 2026 / iii / Paper 329 / 3 / a / Solution

Let $y$ measure distance normal to the plane. The normal momentum balance and capillary pressure condition giveThe downslope lubrication equation isApply no slip $u(0)=-U$ and zero tangential stress $u_y(h)=0$. Integration gives the fluxThe thin-film equation $h_t+q_x=0$ is therefore

Long-wave information near a uniform film propagates with the kinematic speedWhen $c>0$, disturbances travel from $x=-\infty$ into the domain, so $h\to h_0$ is an admissible upstream boundary condition. When $c<0$, information travels toward $x=-\infty$ from the pool, so the same condition cannot independently be imposed there.