Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 329 / 1 / Solution

Put $\delta =H/L\ll 1$. In a thin gap, the streamwise Stokes flow balance isso $P\sim \mu UL/H^2$. The characteristic shear stress is $\tau \sim \mu U/H$, and thereforeThis is the pressure-dominance scaling of lubrication theory.

Let the centre displacement be $\lambda \Delta$ in the downward direction. To first order in $\Delta /a$, projecting this displacement onto the radial direction at polar angle $\theta$ changes the concentric gap $\Delta$ by $-\lambda \Delta \cos \theta$. Hence