Squirmer

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.

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.

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.

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 ${\mathbf{u}}_s$,