Kinematic reversibility of Stokes flow

Linearity of the Stokes equations implies that reversing all imposed forces and boundary velocities reverses the entire velocity field and retraces particle paths.

For a smooth passing encounter of a force-driven sphere with a fixed sphere, isotropy and reflection symmetry imply $\mathbf{U}=[m_{\perp }(R)I+(m_{\parallel }(R)-m_{\perp }(R))\mathbf{nn}]\mathbf{F}$. Along a force axis $X$, $U_X$ is even and the transverse velocity is odd in $X$. Uniqueness then makes the trajectory fore-aft symmetric and restores the incoming impact parameter downstream. This consequence of kinematic reversibility of Stokes flow remains valid with lubrication resistance, provided the particles remain smooth, noncontacting and subject only to the stated forces. Surface roughness and contact can invalidate these assumptions.

A force-free swimmer with a reciprocal shape stroke cannot achieve net locomotion in an inertia-free Newtonian fluid. A single real shape coordinate that retraces an interval gives a reciprocal cycle. The conclusion concerns free swimming; externally imposed translations can instead force and pump the fluid.

For spheres linked by a length $\ell (t)$, axial mobility and $F_1+F_2=0$ imply $\dot{X}=K(\ell )\dot{\ell }$ for their midpoint. The displacement over a periodic stroke is $\oint K(\ell )d\ell =0$. Unequal radii permit instantaneous midpoint motion but not net translation. This is an explicit one-shape-coordinate example of the scallop theorem.

If spatial reflection leaves a Stokes configuration and forcing equivalent to flow reversal while preserving one candidate velocity component, uniqueness forces that component to vanish.