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

Axisymmetric mass conservation in the spherical gap givesSince $h_t=-V\cos \theta$ and regularity requires $q\sin \theta =0$ at $\theta =0$, integration gives

The leading pressure-driven lubrication flux isSubstitution of $h$ and $q$ givesand therefore

Take downward as the positive vertical direction. The constant pressure contributes no resultant, while the pressure force on the inner sphere is opposite its outward normal. ThusWith $t=\lambda \cos \theta$ and the supplied integral,The sign is upward for $V>0$, so this is a drag force. As $\lambda \to 0$, the bracket is $4{\lambda }^3/3+O({\lambda }^5)$ and $F_z\to -8\pi \mu a^{4}V/{\Delta }^3$.