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On $r=c$, the traction $\mathbf{\sigma }\mathbf{n}$ exerted by the outer fluid on the inner fluid isThe inner fluid exerts the opposite traction on the outer fluid. Its couple is thereforeUsingand the supplied spherical integral,we obtainThus the radius cancels:This is the Torque in rotational Stokes flow between concentric spheres.

When $a\ll b$,the rotational drag torque for a sphere in an effectively unbounded fluid. If $h=b-a\ll a$, thensothe inverse-gap growth expected from a thin Couette flow.

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.