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A spherical squirmer can rotate without disturbing the surrounding fluid if its prescribed surface slip exactly cancels the velocity of rigid rotation: $u^{\prime }=-\Omega \times an$. The laboratory-frame boundary velocity is then zero. Uniqueness of Stokes flow with decay at infinity gives identically zero exterior velocity. In particular $u^{\prime }=b_0\sin \theta e_{\varphi }$ gives $\Omega =-(b_0/a)e_z$. The cancellation concerns the total fluid boundary velocity, not the solid material's angular velocity.

A spherical squirmer with tangential surface slip velocity $(I-nn)(A+Bn)$, where $B$ is a symmetric second-rank tensor and a traceless second-rank tensor, translates with $U=-2A/3$ and does not rotate in an unbounded quiescent fluid. With $S=x\cdot Bx$ its exterior Stokes flow isThis follows from the Unscaled Papkovich–Neuber representation using harmonic functions as potentials $\Phi =-a^{2}Bx/(2r^3)$ and $\chi =a^{3}A\cdot x/(3r^3)+a^{4}S/(2r^5)$. Matching the surface coefficients proves the formula. The two modes generate a potential dipole and a stresslet, respectively.