Past exam of the mathematics course of the University of Cambridge / 2021 / iii / Paper 329 / 3 / b

Treat each short torus segment as a straight cylinder. For translation with velocity $U{\mathbf{e}}_x$, its components normal and tangent to the circular centreline are $U\cos \theta$ and $-U\sin \theta$. Integrating the local resistance per unit length around $ds=Rd\theta$ gives

Rotation at angular velocity $\Omega {\mathbf{e}}_z$ gives the everywhere tangential speed $\Omega R$. Only the axial-cylinder resistance $2\pi \mu {\epsilon }^{-1/2}$ contributes. Multiplying the force by its moment arm $R$ and integrating around the centreline gives