Self-ordering of microscopic particles in time-periodic flows

Small macroscopic particles advected by fluid flows are generally believed to follow the surrounding fluid when their Stokes number, $St \ll 1$. We show that even when particles are as small as $St \sim 10^{-6}$, the inertia effects may lead to spontaneous self-organization of particles into dynamic coherent structures. The arising structures are typically spiral curves. They are a collective phenomenon observed when an ensemble of particles is evolved. While the structures are robust for a range of control parameters, they become sensitive quickly dissolve and particles mix with the fluid outside this range. We explain the structures' formation by the dynamical effect of phase-locking. It occurs for particle turn-over motions in vortical time-periodic flows. We show that this mechanism is responsible for the surprising assembly of particles into rotating spirals that was discovered experimentally in thermocapillary flows more than a decade ago and has remained unexplained until now. In our exposition we lean on the results of our numerical simulations, which reproduce the effect in physically realistic regimes. We expect that similarly to phase-locking in dynamical systems, this effect is subtle but generic and may cause localization and ordering of particles in time-periodic flows that abound in nature and applications.