Mass crystals in vorticity crystals

Abstract

We study the motion of tiny heavy inertial particles advected by a two-dimensional inviscid fluid flow, composed of N identical point vortices regularly placed on a ring and rotating as a solid body, therefore forming a vortex crystal. In the limit of weak particle inertia, we show asymptotically that, in the reference frame of the crystal, inertial particles have N asymptotically stable equilibrium positions located outside the crystal, in agreement with numerical observations by Ravichandran et al. [“Clustering of heavy particles in vortical flows: A selective review,” Sādhanā 42, 597–605 (2017)]. In addition to these “satellite” attracting points, we observe that for N≥3, the center of the ring, though degenerate, is a stable equilibrium position for inertial particles. This creates a kind of cage formed by vortices, where inclusions slowly drift toward the center under the effect of the surrounding vortices. The central attracting point is observed to persist even at larger Stokes numbers, in contrast with the satellite attracting points that vanish when the Stokes number is above some critical value.