Symmetry-breaking strategies for particle manipulation in Stokes flow

The meaningful manipulation of particles in Stokes flow by hydrodynamic forces, i.e., predictable crossing of streamlines, faces a dilemma: the small magnitude of such effects requires repeated displacements through periodic driving, yet the time-reversibility of Stokes flow precludes the use of straightforward oscillatory flow strategies. Breaking reciprocal symmetry does, however, lead to systematic displacements as in the case of the well-known scallop theorem. Here, we instead focus on rigorously quantifying the effects of breaking geometric symmetry of flows and/or particles. Analyzing dynamical systems modeling particles in 2D vortical Stokes flows in a channel, we find that the breaking of at least two symmetries is necessary to steer the particles onto attractive limiting trajectories (independent of initial conditions): For spherical particles, whose displacements are effected by interaction with channel walls, vortex symmetries both along and across the channel have to be broken. For elongated particles, modeled as rigid dumbbells, the asymmetric particle shape allows for attractive limit cycles even without wall interactions, but attractive limit cycles are again only reached for fully symmetry-broken vortices, while the breaking of only one symmetry results in quasi-periodic motion, whose trajectories depend on initial conditions. Awareness of these limitations influences the design of microfluidic devices for particle accumulation or capture, and also informs valid strategies for Deterministic Lateral Displacement, where the spatial symmetry of obstacle geometry plays a similar role.