 Axial rotational resistance of eccentric nested spheres

Rotation about the offset axis has azimuthal Couette flow and no pressure variation at leading lubrication order. Integrating the tangential traction gives torque $-{\mathcal{R}}_{\Omega }\Omega$ on the inner sphere. The concentric limit is $8\pi \mu a^4/(3\Delta )$ and the contact limit is finite, $4\pi \mu a^4/\Delta$, because the wall velocity vanishes at the closest pole.