Mass crystals and cage effect 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 forming a 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. (Sadhana 42, 2017). In addition to these "satellite" attracting points, we observe that for $N \ge 3$ the center of the ring, though degenerate, is a stable equilibrium position for inertial particles. This creates a kind of cage effect, where inclusions slowly drift towards the center under the effect of the surrounding vortices. This cage effect 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.