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 magnitude
$$
\frac{\mu\Omega a\sin\theta}{h(\theta)}.
$$
Its moment arm about the vertical axis is $a\sin\theta$, and $dS=2\pi a^2\sin\theta\,d\theta$. Consequently
$$
G_z=-\frac{2\pi\mu\Omega a^4}{\Delta}
\int_0^\pi\frac{\sin^3\theta}{1-\lambda\cos\theta}\,d\theta.
$$
Putting $t=\lambda\cos\theta$ and using the supplied integral gives
$$
\boxed{
G_z=-\frac{2\pi\mu\Omega a^4}{\lambda^3\Delta}
\left[
2\lambda+(1-\lambda^2)
\log\left(\frac{1-\lambda}{1+\lambda}\right)
\right]}.
$$
This <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>.

Solution

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

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2024/iii/paper-329/1/ii/solution