For two zero-body-force <Stokes flows> $(\mathbf u,\boldsymbol\sigma)$ and $(\widehat{\mathbf u},\widehat{\boldsymbol\sigma})$ in the same domain, the <Lorentz reciprocal theorem for Stokes flow> states
$$
\int_{\partial D}\mathbf u\mathbin\cdot\widehat{\boldsymbol\sigma}\mathbf n\,dS
=\int_{\partial D}\widehat{\mathbf u}\mathbin\cdot\boldsymbol\sigma\mathbf n\,dS.
$$
Apply it first with the auxiliary translating-sphere solution and then with the auxiliary rotating-sphere solution. The swimmer is <force-free> and <torque-free>, while the auxiliary surface tractions are known. The resulting <surface slip velocity> formulas are
$$
\boxed{
\mathbf V=-\frac1{4\pi a^2}\int_{r=a}\mathbf u_s\,dS,
\qquad
\boldsymbol\omega=-\frac3{8\pi a^4}\int_{r=a}\mathbf x\times\mathbf u_s\,dS.}
$$

On $r=a$, the first part of the prescribed slip is
$$
\frac{(\mathbf A\times\mathbf x)\times\mathbf x}{a^2}
=\frac{\mathbf x(\mathbf A\mathbin\cdot\mathbf x)}{a^2}-\mathbf A.
$$
Its surface average is $-2\mathbf A/3$, whereas the $\mathbf B$ term has zero average by <odd function>[oddness]. Thus
$$
\boxed{\mathbf V=\frac23\mathbf A.}
$$
The $\mathbf A$ term contributes no rotation. For the other term, the isotropic second and fourth surface moments give
$$
\int_{r=a}\mathbf x\times\mathbf u_s\,dS
=\frac{8\pi a^3}{15}|\mathbf B|^2\mathbf B,
$$
and hence
$$
\boxed{\boldsymbol\omega=-\frac{|\mathbf B|^2}{5a}\mathbf B.}
$$

Solution

Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 329 / 1 / b / Solution

{wiki}

A squirmer is an idealized spherical <microswimmer> propelled by a prescribed tangential <surface slip velocity>. The <Lorentz reciprocal theorem for Stokes flow> determines its rigid translation and rotation without solving its complete exterior flow.

Squirmer

The surface slip velocity is the fluid velocity relative to the rigid motion of an active particle at its boundary. For a spherical force-free, torque-free swimmer of radius $a$ with slip $\mathbf u_s$,
$$
\mathbf V=-\frac1{4\pi a^2}\int_{r=a}\mathbf u_s\,dS,
\qquad
\boldsymbol\omega=-\frac3{8\pi a^4}\int_{r=a}\mathbf x\times\mathbf u_s\,dS.
$$


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