 Torque-free rotation of a spherical squirmer

For a sphere of radius $a$ with prescribed surface slip $u^{\prime }$ and zero external torque, the Lorentz reciprocal theorem for Stokes flow giveswhere $n$ points from the sphere into the fluid. Use an auxiliary rotating sphere in Stokes flow with arbitrary angular velocity $\hat{\Omega }$ and traction $-3\mu \hat{\Omega }\times n$. The actual torque-free cross-work vanishes. Substituting the boundary velocity $\Omega \times an+u^{\prime }$ into the other cross-work leaves $-8\pi \mu a^3\hat{\Omega }\cdot \Omega -3\mu \hat{\Omega }\cdot \int n\times u^{\prime }=0$. A possible translation contributes zero because the auxiliary net force is zero.