Put $\delta=H/L\ll1$. In a thin gap, the streamwise <Stokes flow> balance is
$$
\frac PL\sim\mu\frac U{H^2},
$$
so $P\sim\mu UL/H^2$. The characteristic <shear stress> is $\tau\sim\mu U/H$, and therefore
$$
\boxed{\frac\tau P\sim\frac HL=\delta\ll1}.
$$
This is the <lubrication pressure dominates shear stress>[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
$$
\boxed{h(\theta)=\Delta(1-\lambda\cos\theta)}.
$$

Solution

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

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2024/iii/paper-329/1/solution