= Body-force-driven rotation of a torque-free sphere
{title2=$\boldsymbol\Omega=\frac1{8\pi\mu}\int_{r>a}\frac{\mathbf x\times\mathbf f}{r^3}dV$}
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.
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