Advection of finite-size particles in open flows

Transport, diffusion, and energy studies in the Arnold-Beltrami-Childress map

We study the transport and diffusion properties of passive inertial particles described by a six-dimensional dissipative bailout embedding map. The base map chosen for the study is the three-dimensional incompressible Arnold-Beltrami-Childress (ABC) map chosen as a representation of volume preserving flows. There are two distinct cases: the two-action and the one-action cases, depending on whether two or one of the parameters (A,B,C) exceed 1. The embedded map dynamics is governed by two parameters (α,γ), which quantify the mass density ratio and dissipation, respectively. There are important differences between the aerosol (α<1) and the bubble (α>1) regimes. We have studied the diffusive behavior of the system and constructed the phase diagram in the parameter space by computing the diffusion exponents η. Three classes have been broadly classified-subdiffusive transport (η<1), normal diffusion (η≈1), and superdiffusion (η>1) with η≈2 referred to as the ballistic regime. Correlating the diffusive phase diagram with the phase diagram for dynamical regimes seen earlier, we find that the hyperchaotic bubble regime is largely correlated with normal and superdiffusive behavior. In contrast, in the aerosol regime, ballistic superdiffusion is seen in regions that largely show periodic dynamical behaviors, whereas subdiffusive behavior is seen in both periodic and chaotic regimes. The probability distributions of the diffusion exponents show power-law scaling for both aerosol and bubbles in the superdiffusive regimes. We further study the Poincáre recurrence times statistics of the system. Here, we find that recurrence time distributions show power law regimes due to the existence of partial barriers to transport in the phase space. Moreover, the plot of average particle kinetic energies versus the mass density ratio for the two-action case exhibits a devil's staircase-like structure for higher dissipation values. We explain these results and discuss their implications for realistic systems.

Levitation of heavy particles against gravity in asymptotically downward flows

In the fluid transport of particles, it is generally expected that heavy particles carried by a laminar fluid flow moving downward will also move downward. We establish a theory to show, however, that particles can be dynamically levitated and lifted by interacting vortices in such flows, thereby moving against gravity and the asymptotic direction of the flow, even when they are orders of magnitude denser than the fluid. The particle levitation is rigorously demonstrated for potential flows and supported by simulations for viscous flows. We suggest that this counterintuitive effect has potential implications for the air-transport of water droplets and the lifting of sediments in water.

Memory effects are relevant for chaotic advection of inertial particles

A systematic investigation of the effect of the history force on particle advection is carried out in a paradigmatic model flow of chaotic advection, the von Kármán flow. All investigated properties turn out to heavily depend on the presence of memory when compared to previous studies neglecting this force. We find a weaker tendency for accumulation and for caustics formation. The Lyapunov exponent of transients becomes larger, the escape rates are strongly altered. Attractors are found to be suppressed by the history force, and periodic ones have a very slow, t(-1/2)-type convergence towards the asymptotic form.

Generalization of the JTZ model to open plane wakes

The JTZ model [C. Jung, T. Tel, and E. Ziemniak, Chaos 3, 555 (1993)], as a theoretical model of a plane wake behind a circular cylinder in a narrow channel at a moderate Reynolds number, has previously been employed to analyze phenomena of chaotic scattering. It is extended here to describe an open plane wake without the confined narrow channel by incorporating a double row of shedding vortices into the intermediate and far wake. The extended JTZ model is found in qualitative agreement with both direct numerical simulations and experimental results in describing streamlines and vorticity contours. To further validate its applications to particle transport processes, the interaction between small spherical particles and vortices in an extended JTZ model flow is studied. It is shown that the particle size has significant influences on the features of particle trajectories, which have two characteristic patterns: one is rotating around the vortex centers and the other accumulating in the exterior of vortices. Numerical results based on the extended JTZ model are found in qualitative agreement with experimental ones in the normal range of particle sizes.

Transport and diffusion in the embedding map

We study the transport properties of passive inertial particles in two-dimensional (2D) incompressible flows. Here, the particle dynamics is represented by the four-dimensional dissipative embedding map of the 2D area-preserving standard map which models the incompressible flow. The system is a model for impurity dynamics in a fluid and is characterized by two parameters, the inertia parameter alpha and the dissipation parameter gamma . The aerosol regime, where the particles are denser than the fluid, and the bubble regime, where they are less dense than the fluid, correspond to the parameter regimes alpha>1 and alpha<1 , respectively. Earlier studies of this system show a rich phase diagram with dynamical regimes corresponding to periodic orbits, chaotic structures, and mixed regimes. We obtain the statistical characterizers of transport for this system in these dynamical regimes. These are the recurrence time statistics, the diffusion exponent, and the distribution of jump lengths. The recurrence time distribution shows a power-law tail in the dynamical regimes, where there is preferential concentration of particles in sticky regions of the phase space, and an exponential decay in mixing regimes. The diffusion exponent shows behavior of three types-normal, subdiffusive, and superdiffusive, depending on the parameter regimes. Phase diagrams of the system are constructed to differentiate different types of diffusion behavior, as well as the behavior of the absolute drift. We correlate the dynamical regimes seen for the system at different parameter values with the transport properties observed at these regimes and in the behavior of the transients. This system also shows the existence of a crisis and unstable dimension variability at certain parameter values. The signature of the unstable dimension variability is seen in the statistical characterizers of transport. We discuss the implications of our results for realistic systems.

Particle segregation by Stokes number for small neutrally buoyant spheres in a fluid

It is a commonly observed phenomenon that spherical particles with inertia in an incompressible fluid do not behave as ideal tracers. Due to the inertia of the particle, the planar dynamics are described in a four-dimensional phase space and thus can differ considerably from the ideal tracer dynamics. Using finite-time Lyapunov exponents, we compute the sensitivity of the final position of a particle with respect to its initial velocity, relative to the fluid, and thus partition the relative velocity subspace at each point in configuration space. The computations are done at every point in the relative velocity subspace, thus giving a sensitivity field. The Stokes number, being a measure of the independence of the particle from the underlying fluid flow, acts as a parameter in determining the variation in these partitions. We demonstrate how this partition framework can be used to segregate particles by Stokes number in a fluid. The fluid model used for demonstration is a two-dimensional cellular flow.

Moving finite-size particles in a flow: a physical example of pitchfork bifurcations of tori

The motion of small, spherical particles of finite size in fluid flows at low Reynolds numbers is described by the strongly nonlinear Maxey-Riley equations. Due to the Stokes drag, the particle motion is dissipative, giving rise to the possibility of attractors in phase space. We investigate the case of an infinite cellular flow field with time-periodic forcing. The dynamics of this system are studied in a part of the parameter space. We focus particularly on the size of the particles, whose variations are most important in active physical processes, for example, for aggregation and fragmentation of particles. Depending on their size the particles will settle on different attractors in phase space in the long-term limit, corresponding to periodic, quasiperiodic, or chaotic motion. One of the invariant sets that can be observed in a large part of this parameter region is a quasiperiodic motion in the form of a torus. We identify some of the bifurcations that these tori undergo, as particle size and mass ratio relative to the fluid are varied. In this way we provide a physical example for sub- and supercritical pitchfork bifurcations of tori.

Can aerosols be trapped in open flows?

The fate of aerosols in open flows is relevant in a variety of physical contexts. Previous results are consistent with the assumption that such finite-size particles always escape in open chaotic advection. Here we show that a different behavior is possible. We analyze the dynamics of aerosols both in the absence and presence of gravitational effects, and both when the dynamics of the fluid particles is hyperbolic and nonhyperbolic. Permanent trapping of aerosols much heavier than the advecting fluid is shown to occur in all these cases. This phenomenon is determined by the occurrence of multiple vortices in the flow and is predicted to happen for realistic particle-fluid density ratios.

Clustering, chaos, and crisis in a bailout embedding map

We study the dynamics of inertial particles in two-dimensional incompressible flows. The particle dynamics is modeled by four-dimensional dissipative bailout embedding maps of the base flow which is represented by 2-d area preserving maps. The phase diagram of the embedded map is rich and interesting both in the aerosol regime, where the density of the particle is larger than that of the base flow, as well as the bubble regime, where the particle density is less than that of the base flow. The embedding map shows three types of dynamic behavior, periodic orbits, chaotic structures, and mixed regions. Thus, the embedding map can target periodic orbits as well as chaotic structures in both the aerosol and bubble regimes at certain values of the dissipation parameter. The bifurcation diagram of the 4-d map is useful for the identification of regimes where such structures can be found. An attractor merging and widening crisis is seen for a special region for the aerosols. At the crisis, two period-10 attractors merge and widen simultaneously into a single chaotic attractor. Crisis induced intermittency is seen at some points in the phase diagram. The characteristic times before bursts at the crisis show power-law behavior as functions of the dissipation parameter. Although the bifurcation diagram for the bubbles looks similar to that of aerosols, no such crisis regime is seen for the bubbles. Our results can have implications for the dynamics of impurities in diverse application contexts.

Coexistence of inertial competitors in chaotic flows

We investigate the dynamics of inertial particles immersed in open chaotic flows. We consider the generic problem of competition between different species, e.g., phytoplankton populations in oceans. The strong influence from inertial effects is shown to result in the persistence of different species even in cases when the passively advected species cannot coexist. Multispecies coexistence in the ocean can be explained by the fact that the unstable manifold is different for each advected competitor of different size.

Finite-size effects on open chaotic advection

We study the effects of finite-sizeness on small, neutrally buoyant, spherical particles advected by open chaotic flows. We show that, when observed in the configuration or physical space, the advected finite-size particles disperse about the unstable manifold of the chaotic saddle that governs the passive advection. Using a discrete-time system for the dynamics, we obtain an expression predicting the dispersion of the finite-size particles in terms of their Stokes parameter at the onset of the finite-size induced dispersion. We test our theory in a system derived from a flow and find remarkable agreement between our expression and the numerically measured dispersion.

Continuum description of finite-size particles advected by external flows: the effect of collisions

The equation of the density field of an assembly of macroscopic particles advected by a hydro-dynamic flow is derived from the microscopic description of the system. This equation allows one to recognize the role and the relative importance of the different microscopic processes implicit in the model: the driving of the external flow, the inertia of the particles, and the collisions among them. The validity of the density description is confirmed by comparisons of numerical studies of the continuum equation with direct simulation Monte Carlo simulations of hard disks advected by a chaotic flow. We show that the collisions have two competing roles: a dispersinglike effect and a clustering effect (even for elastic collisions). An unexpected feature is also observed in the system: the presence of collisions can reverse the effect of inertia, so that grains with lower inertia are more clusterized.