= Torque-free rotation of a spherical squirmer
{title2=$\boldsymbol\Omega$}
For a sphere of radius $a$ with prescribed surface slip $u'$ and zero external <torque>, the <Lorentz reciprocal theorem for Stokes flow> gives
$$
\boldsymbol\Omega=-\frac{3}{8\pi a^3}\int_S n\times u'\,dS,
$$
where $n$ points from the sphere into the fluid. Use an auxiliary <rotating sphere in Stokes flow> with arbitrary angular velocity $\widehat\Omega$ and traction $-3\mu\widehat\Omega\times n$. The actual torque-free cross-work vanishes. Substituting the boundary velocity $\Omega\times an+u'$ into the other cross-work leaves $-8\pi\mu a^3\widehat\Omega\cdot\Omega-3\mu\widehat\Omega\cdot\int n\times u'=0$. A possible translation contributes zero because the auxiliary net force is zero.
Back to article page
