= Solution
Axisymmetric rotation about the offset axis gives wall <velocity> $v_\varphi=a\Omega\sin\theta$. Its azimuthal <Couette flow> has no pressure-driving divergence, so <pressure> remains constant. The fluid-on-inner-sphere tangential <traction> is $-\mu a\Omega\sin\theta/h$. Multiply it by the moment arm $a\sin\theta$ and integrate:
$$
G_z=-\frac{2\pi\mu\Omega a^4}{\Delta}\int_{-1}^1\frac{1-z^2}{1-\alpha z}\,dz.
$$
Putting $t=\alpha z$ and using the second supplied identity with its generic parameter equal to $\alpha$ gives the <axial rotational resistance of eccentric nested spheres>:
$$
\boxed{G_z=-\frac{2\pi\mu\Omega a^4}{\Delta\alpha^3}\left[2\alpha+(1-\alpha^2)\log\frac{1-\alpha}{1+\alpha}\right].}
$$
The continuous limit at $\alpha=0$ is $-8\pi\mu\Omega a^4/(3\Delta)$, consistent with a concentric narrow spherical shell. At $\alpha\to1$, the <torque> tends to $-4\pi\mu\Omega a^4/\Delta$: it remains finite because the rotational wall speed vanishes at the closest pole.
Back to article page
