Past exam of the mathematics course of the University of Cambridge / 2006 / iii / Paper 78 / 4 / Solution

Let the tangential velocity scale be $U$, gap scale $H$ and along-gap length $L\gg H$. The tangential Stokes equation balances $P/L$ with $\mu U/H^2$, while wall shear is $\tau \sim \mu U/H$. ThereforeThis is lubrication pressure dominates shear stress; pressure force projections may nevertheless be small in a particular direction or may give no torque on a spherical surface.

Write the signed centre displacement along the downward axis as $d=\alpha \Delta$. A point $an$ on the inner sphere has distance $|an+d{e}_z|=a+d\cos \theta +O({\Delta }^2/a)$ from the outer centre. The leading normal gap to radius $a+\Delta$ is consequentlyThe local-curvature corrections are smaller by $\Delta /a$. This approximation also follows by solving the outer-sphere intersection along the ray $n$ from the inner centre and expanding its positive root.