Past exam of the mathematics course of the University of Cambridge / 2017 / ib / Paper 1 / 17D / a / Solution

Choose $x$ downslope along the rigid plane and $z$ normal to it into the fluid, with the wall at $z=0$ and the flat free surface at $z=h$. The gravitational body force per unit volume has components $(\rho g\sin \alpha ,-\rho g\cos \alpha )$. The wind's tangential traction on the top surface is $-S$ in the $x$ direction; the atmospheric normal traction is $-p_{\mathrm{atm}}$ in the $z$ direction. At the wall there is no-slip boundary condition, and the wall supplies the opposing normal force and a tangential traction whose sign depends on the wind strength.

The velocity is parallel to the plane, $(u(z),0)$. Its velocity profile is a concave quadratic, with $u(0)=0$ and negative slope at the free surface when $S>0$. The diagram uses moderate wind, for which the flow is still everywhere downslope. Stronger wind can reverse the upper portion, or the entire moving layer; the later cases quantify this.