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Critical Stokes number for the capture of inertial particles by recirculation cells in 2D quasi-steady flows

Inertial particles are often observed to be trapped, temporarily or permanently, by recirculation cells which are ubiquitous in natural or industrial flows. In the limit of small particle inertia, determining the conditions of trapping is a challenging task, as it requires a large number of numerical simulations or experiments to test various particle sizes or densities. Here, we investigate this phenomenon analytically and numerically in the case of heavy particles (e.g. aerosols) at low Reynolds number, to derive a trapping criterion that can be used both in analytical and numerical velocity fields. The resulting criterion allows to predict the characteristics of trapped particles as soon as single-phase simulations of the flow are performed. Our analysis is valid for two-dimensional particle-laden flows in the vertical plane, in the limit where the particle inertia, the free-fall terminal velocity, and the flow unsteadiness can be treated as perturbations. The weak unsteadiness of the flow generally induces a chaotic tangle near heteroclinic or homoclinic cycles if any, leading to the apparent diffusion of fluid elements through the boundary of the cell. The critical particle Stokes number Stc below which aerosols also enter and exit the cell in a complex manner has been derived analytically, in terms of the flow characteristics. It involves the non-dimensional curvature-weighted integral of the squared velocity of the steady fluid flow along the dividing streamline of the recirculation cell. When the flow is unsteady and St > Stc, a regular motion takes place due to gravity and centrifugal effects, like in the steady case. Particles driven towards the interior of the cell are trapped permanently. In contrast, when the flow is unsteady and St < Stc, particles wander in a chaotic manner in the vicinity of the border of the cell, and can escape the cell.