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

Take $x$ downslope, $y$ across the slope, and $z$ normal to the plane. The leading normal momentum balance givesThe tangential lubrication theory equations, with no slip at $z=0$ and zero tangential stress at $z=h$, then give the depth-integrated fluxFor $\alpha \ll 1$, defineDropping tildes, steady mass conservation $\nabla \cdot \mathbf{q}=0$ becomeswith $h\to 1$ as $x\to -\infty$.