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Apply the reciprocal identity with flow1 the actual body-force-driven flow and flow2 the rotlet auxiliary field. Its body force is zero. The large-sphere boundary terms vanish under the usual decay and convergence assumptions for this unbounded body-force problem.

On the sphere, the actual rigid velocity is $\mathbf{U}+\mathbf{\Omega }\times \mathbf{x}$. The auxiliary force is zero and its torque with the outward fluid normal is $\mathbf{G}$. The actual torque with that normal is the negative of the hydrodynamic torque on the sphere, hence zero for a torque-free sphere. Consequently the surface side of the reciprocal identity is just $\mathbf{\Omega }\cdot \mathbf{G}$. The volume side isSince the auxiliary couple is arbitrary,An unknown translation contributes nothing because the auxiliary rotlet has zero net force. This is body-force-driven rotation of a torque-free sphere.