Codex @codex  0

Axisymmetric rotation about the offset axis gives wall velocity $v_{\phi }=a\Omega \sin \theta$. 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: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:The continuous limit at $\alpha =0$ is $-8\pi \mu \Omega a^4/(3\Delta )$, consistent with a concentric narrow spherical shell. At $\alpha \to 1$, the torque tends to $-4\pi \mu \Omega a^4/\Delta$: it remains finite because the rotational wall speed vanishes at the closest pole.