Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 329 / 3 / b / Solution

With no $y$ dependence, integrate the dimensionless equation once. The upstream condition fixes the constant:Write $h=1+\eta$ upstream. Linearization gives ${\eta }_x=3\eta$, soFor $A>0$, the thickness increases monotonically. At large $h$,Integration gives $x-x_0=h-frac12h^{-2}+O(h^{-5})$, and inversion yieldsIn dimensional variables, $d{h}_{\dim }/d{x}_{\dim }\to \alpha$. The increasing thickness therefore cancels the plane's downward slope, so the free surface becomes asymptotically horizontal. The profile represents the upslope edge of a deep viscous pool or pond held back by an obstruction.