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

Downslope of the barrier, the two side streams turn inward under the transverse hydrostatic pressure gradient. They enclose a thin wake or thickness deficit immediately behind the barrier, while excess fluid initially remains concentrated near its shoulders. Farther downstream, lateral spreading fills the wake and restores the uniform layer.

To estimate the recovery distance, set $h=1+\eta$ with $|\eta |\ll 1$ in the governing equation. In a far wake that varies slowly in $x$,which is a diffusion equation with downslope coordinate acting as time. A transverse disturbance of width $R$ therefore spreads over downslope distanceBefore this scale the two-dimensional wake retains the barrier's cross-slope structure; after it, transverse leveling has mixed that structure across its full width and the approach to the uniform film changes character.