Past exam of the mathematics course of the University of Cambridge / 2021 / ii / Paper 2 / 39A / a / Solution

The lubrication approximation requires $H/L\ll 1$, small wall slope, and negligible inertia, for example $\rho UH/\mu \ll L/H$. To leading order,With no slip at $z=0,h$,Integration givesIntegrated incompressibility gives $\nabla \cdot \mathbf{q}=0$, so locally $\mathbf{q}=\nabla \wedge (0,0,\psi )$. Solving the flux law for $\nabla p$ and taking its curl yields

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