Stirring and sedimentation of solid inertial particles in low-Reynolds-number flows has acquired great relevance in multiple environmental, industrial and microfluidic systems, but few detailed numerical studies have focused on chaotically advected experimentally realizable flows. We carry out one-way coupling simulations to study the dynamics of inertial particles in the steady three-dimensional flow in a cylindrical container with exactly counter-rotating lids, which was recently studied by Lackey & Sotiropoulos (Phys. Fluids, vol. 18, 2006, paper no. 053601). We elucidate the rich Lagrangian dynamics of the flow in the vicinity of toroidal invariant regions and show that depending on the Stokes number inertial particles could get trapped for long times in different equilibrium positions inside integrable islands. In the chaotically advected region of the flow the balance between inertia and gravity forces (represented by the settling velocity) can produce a striking fractal sedimentation regime, characterized by a sequence of discrete deposition events of seemingly random number of particles separated by hiatuses of random duration. The resulting staircase-like distribution of the time series of the number of particles in suspension is shown to be a devil's staircase whose fractal dimension is equal to the 0.87 value found in multiple dissipative dynamical systems in nature. Our work sheds new light on the complex mechanisms governing the stirring and deposition of inertial particles and provides new information about the parameters that are relevant in the characterization of particle dynamics in different regions of chaotically advected flows.
Bibliographical noteFunding Information:
This work was supported by NSF grants EAR-0120914 (as part of the National Center for Earth-Surface Dynamics) and EAR-0738726. C. E. has also been partially supported by Fondecyt grant 11080032. Computational resources were provided by the University of Minnesota Supercomputing Institute. We thank the anonymous referee who pointed out equation (5.1) and the insightful connection between sedimentation times and Lagrangian average maps.
- Chaotic advection
- Particle/fluid flows
- Vortex flows