Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 329 / 1 / c / ii

Write $\mathbf{\Omega }={\mathbf{G}}_1/(8\pi \mu a^3)$. The incident rotlet of sphere 1 at sphere 2 isand it is harmonic away from sphere 1. Since sphere 2 is force-free, Faxén's first law therefore givesThe vorticity of the rotlet isBecause ${\mathbf{G}}_2=0$, Faxén's rotational law gives

The symmetric rate-of-strain tensor of the incident rotlet at sphere 2 isAfter translation and rotation have matched the uniform and antisymmetric parts of the incident flow, the leading perturbation from sphere 2 is the stresslet part of the supplied straining-sphere solution:Part b gives its vorticity asAt the centre of sphere 1, $\mathbf{x}=-\mathbf{R}$, soApplying Faxén's rotational law to sphere 1 produces half this ambient vorticity and proves