Body-force-driven rotation of a torque-free sphere ($\mathbf{\Omega }=\frac{1}{8\pi \mu }{\int }_{r>a}\frac{\mathbf{x}\times \mathbf{f}}{r^3}dV$)

Use a rotlet as the auxiliary Stokes flow, with a rigid sphere at its center. Its exterior-domain boundary torque is the imposed auxiliary couple. In the Lorentz reciprocal theorem, the actual sphere's zero torque removes the other surface term, while the volume term is the rotlet dotted with the body force. Varying the auxiliary couple gives the displayed angular velocity, assuming the integrals and the far boundary limit converge.