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

Rotation about the vertical axis gives the inner surface the azimuthal speed $\Omega a\sin \theta$. The leading Couette flow shear traction is opposing and has magnitudeIts moment arm about the vertical axis is $a\sin \theta$, and $dS=2\pi a^2\sin \theta d\theta$. ConsequentlyPutting $t=\lambda \cos \theta$ and using the supplied integral givesThis viscous shear torque opposes the rotation. Its concentric limit is $-8\pi \mu \Omega a^4/(3\Delta )$, agreeing with the thin-gap limit of Torque in rotational Stokes flow between concentric spheres.