Sorting of small inertial particles in vortical, 3D fluid flows

Vortical flows have been shown to support attractors of buoyant particles in the form of limit cycles. The position of the attractor generally depends on particle characteristics such as size and buoyancy. A continuous range of characteristics gives rise to a range of limit cycles, together which can form a surface, volume, or some other geometric object in space. Particles will be sorted over this object according to their characteristics. We present some examples of these objects using a simple model of a three-dimensional ocean eddy as a background fluid flow. Here, the limit cycles consist of circular horizontal orbits, whose location depends on the size and density of the particles, as reflected in Stokes number and buoyancy parameters, respectively. If just particle size is varied, the attractors combine to form a surface on which particles are sorted by size. When both parameters are varied, the particles are sorted over a volume. We compute these objects using a general form of the Maxey-Riley-Gatignol (MRG) equations and compare solutions to those based on two common approximations: the MRG equations without the history term and the slow manifold reduction. Also discussed are rates of attraction and basins of attraction, and it is found that although larger particles are attracted most rapidly, they tend to have smaller basins of attraction. The history force is found to be important in most cases. Neglect of this force allows particles to become less constrained to follow fluid trajectories, freeing them to approach an attractor more rapidly.