Dust trapping in inviscid vortex pairs

Abstract

The motion of tiny heavy particles transported in a co-rotating vortex pair, with or without particle inertia and sedimentation, is investigated. The dynamics of non-inertial sedimenting particles is shown to be chaotic, under the combined effect of gravity and of the circular displacement of the vortices. This phenomenon is very sensitive to particle inertia, if any. By using nearly hamiltonian dynamical system theory for the particle motion equation written in the rotating reference frame, one can show that small inertia terms of the particle motion equation strongly modify the Melnikov function of the homoclinic trajectories and heteroclinic cycles of the unperturbed system, as soon as the particle response time is of the order of the settling time (Froude number of order unity). The critical Froude number above which chaotic motion vanishes and a regular centrifugation takes place is obtained from this Melnikov analysis and compared to numerical simulations. Particles with a finite inertia, and in the absence of gravity, are not necessarily centrifugated away from the vortex system. Indeed, these particles can have various equilibrium positions in the rotating reference frame, like the Lagrange points of celestial mechanics, according to whether their Stokes number is smaller or larger than some critical value. An analytical stability analysis reveals that two of these points are stable attracting points, so that permanent trapping can occur for inertial particles injected in an isolated co-rotating vortex pair. Particle trapping is observed to persist when viscosity, and therefore vortex coalescence, is taken into account. Numerical experiments at large but finite Reynolds number show that particles can indeed be trapped temporarily during vortex roll-up, and are eventually centrifugated away once vortex coalescence occurs.