Past exam of the mathematics course of the University of Cambridge / 2026 / iii / Paper 352 / 1 / a / Solution

Let $z\in [0,h]$ measure distance through the thin Bingham plastic layer, with a stationary wall at $z=0$ and large-scale velocity $u_b$ at $z=h$. To leading lubrication order the shear stress ${\tau }_b$ is uniform across the layer. If $|{\tau }_b|\le {\tau }_c$, the material is unyielded and the no-slip wall makes the entire layer stationary. After yield,Integrating across the depth gives the Bingham sliding lawEquivalently, for nonzero sliding,