Past exam of the mathematics course of the University of Cambridge / 2026 / iii / Paper 352 / 1 / b / i

Write ${\tau }_d=\rho gH\sin \theta$. For unidirectional velocity $\mathbf{u}=(u(y,z),0,0)$, the nontrivial force balances areFor the power-law fluid,At the free surface $z=H$, $p=p_{\mathrm{atm}}$ and ${\tau }_{xz}=0$. At the bed $z=0$, continuity of tangential velocity and traction gives the local sliding conditionbecause the stated inequalities keep the downhill bed yielded. The velocity and horizontal shear traction are continuous across $y=0$, and the flow tends to uniform states as $y\to \pm \infty$.