Particle grouping in oscillating flows

An equation describing the dynamics of spherical particles in an oscillating Stokesian flow in the frame of reference moving with the phase velocity of the wave, and only taking into account the contribution of the drag force, is simplified in two limiting cases. Firstly, the case when Stokes numbers are small is considered. Secondly, the analysis focuses on the case when the initial location of the particles is close to the location where the particles are grouped (their velocities and accelerations in the wave frame of reference are equal to zero), $x_{\lim }$. This is followed by an analysis of the dynamics of non-Stokesian particles. In all cases, the analytical results are validated against the results of numerical solution of the equation of particle motion. Three types of trajectories are predicted when particles approach $x_{\lim }$: the trajectories describing the monotonic approach to $x_{\lim }$, the trajectories describing the approach to $x_{\lim }$ with oscillations and trajectories repelled from $x_{\lim }$. These are identified with stable nodes, stable foci and saddles. The trajectories in the zone between stable nodes and foci are identified as stable stars. Using Dulac's criterion, it is pointed out that none of the particle trajectories in the position–velocity plane can be closed. This result is illustrated by the trajectories calculated using the numerical solution of the equation for particle dynamics for various parameter values.