Particle trapping in vortex crystals

We study the motion of inertial particles in two-dimensional inviscid and viscous vortex crystals, where vortices are placed at the tips of a regular polygon, and determine the conditions under which long-term trapping of particles can occur. In crystals with a central vortex, we find that trapping points in addition to those found in a previous analysis may exist, finding furthermore that particle trajectories may approach a limit cycle that can be described by means of asymptotic methods. We show that these newly discovered fixed points persist temporarily in the presence of viscosity, and study the different transitions following viscous vortex mergers. We find that for moderate Reynolds numbers, the annular merger of the vortex crystals forms an annular vortex layer that can keep particles trapped within a disk centred at the origin. For larger Reynolds numbers, pairwise vortex merger leads to crystals with a smaller number of vortices, with particles clustering at the fixed points of the newly formed vortex crystal.