Past exam of the mathematics course of the University of Cambridge / 2020 / ii / Paper 2 / 38B / iii / Solution

Let the supplied volume per unit width be $V(t)$. The governing balance gives the scalesEliminating $x$ yieldsFor the head height to approach a nonzero constant $h_{\infty }$, the total supplied volume must therefore grow linearly:This is the constant-head supply for a gravity-driven thin film.