Attraction of minute particles to invariant regions of volume preserving flows by transients

Direct experimental visualization of the global Hamiltonian progression of two-dimensional Lagrangian flow topologies from integrable to chaotic state

Countless theoretical/numerical studies on transport and mixing in two-dimensional (2D) unsteady flows lean on the assumption that Hamiltonian mechanisms govern the Lagrangian dynamics of passive tracers. However, experimental studies specifically investigating said mechanisms are rare. Moreover, they typically concern local behavior in specific states (usually far away from the integrable state) and generally expose this indirectly by dye visualization. Laboratory experiments explicitly addressing the global Hamiltonian progression of the Lagrangian flow topology entirely from integrable to chaotic state, i.e., the fundamental route to efficient transport by chaotic advection, appear non-existent. This motivates our study on experimental visualization of this progression by direct measurement of Poincaré sections of passive tracer particles in a representative 2D time-periodic flow. This admits (i) accurate replication of the experimental initial conditions, facilitating true one-to-one comparison of simulated and measured behavior, and (ii) direct experimental investigation of the ensuing Lagrangian dynamics. The analysis reveals a close agreement between computations and observations and thus experimentally validates the full global Hamiltonian progression at a great level of detail.

Unexpected trapping of particles at a T junction

A common element in physiological flow networks, as well as most domestic and industrial piping systems, is a T junction that splits the flow into two nearly symmetric streams. It is reasonable to assume that any particles suspended in a fluid that enters the bifurcation will leave it with the fluid. Here we report experimental evidence and a theoretical description of a trapping mechanism for low-density particles in steady and pulsatile flows through T-shaped junctions. This mechanism induces accumulation of particles, which can form stable chains, or give rise to significant growth of bubbles due to coalescence. In particular, low-density material dispersed in the continuous phase fluid interacts with a vortical flow that develops at the T junction. As a result suspended particles can enter the vortices and, for a wide range of common flow conditions, the particles do not leave the bifurcation. Via 3D numerical simulations and a model of the two-phase flow we predict the location of particle accumulation, which is in excellent agreement with experimental data. We identify experimentally, as well as confirm by numerical simulations and a simple force balance, that there is a wide parameter space in which this phenomenon occurs. The trapping effect is expected to be important for the design of particle separation and fractionation devices, as well as used for better understanding of system failures in piping networks relevant to industry and physiology.

Chaotic advection and the emergence of tori in the Küppers-Lortz state

Motivated by the roll-switching behavior observed in rotating Rayleigh-Benard convection, we define a Küppers-Lortz (K-L) state as a volume-preserving flow with periodic roll switching. For an individual roll state, the Lagrangian particle trajectories are periodic. In a system with roll-switching, the particles can exhibit three-dimensional, chaotic motion. We study a simple phenomenological map that models the Lagrangian dynamics in a K-L state. When the roll axes differ by 120 degrees in the plane of rotation, we show that the phase space is dominated by invariant tori if the ratio of switching time to roll turnover time is small. When this parameter approaches zero these tori limit onto the classical hexagonal convection patterns, and, as it gets large, the dynamics becomes fully chaotic and well mixed. For intermediate values, there are interlinked toroidal and poloidal structures separated by chaotic regions. We also compute the exit time distributions and show that the unbounded chaotic orbits are normally diffusive. Although the map presumes instantaneous switching between roll states, we show that the qualitative features of the flow persist when the model has smooth, overlapping time-dependence for the roll amplitudes (the Busse-Heikes model).

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.

Pressure-correlated dispersion of inertial particles in free shear flows

We investigate the relationship between dispersion of inertial particles and the pressure field in free shear flows. A three-dimensional temporally developing particle-laden mixing layer and a three-dimensional spatially developing particle-laden plane jet are studied by means of direct numerical simulation. The incompressible Navier-Stokes equations are accurately solved without any turbulence model. The dispersed inertial particles are traced in the Lagrangian framework. It is found that the instantaneous spatial distribution of inertial particles correlates well with the Laplacian of pressure inverted delta2p , and from the statistical viewpoint, the effect of particle size on the fraction of particle number distributed within different flow zones characterized by inverted delta2p for different particles is negligible when the flow is well developed. The potential explanations and applications for this feature are explored.

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.

Stability of attractors formed by inertial particles in open chaotic flows

Particles having finite mass and size advected in open chaotic flows can form attractors behind structures. Depending on the system parameters, the attractors can be chaotic or nonchaotic. But, how robust are these attractors? In particular, will small, random perturbations destroy the attractors? Here, we address this question by utilizing a prototype flow system: a cylinder in a two-dimensional incompressible flow, behind which the von Kármán vortex street forms. We find that attractors formed by inertial particles behind the cylinder are fragile in that they can be destroyed by small, additive noise. However, the resulting chaotic transient can be superpersistent in the sense that its lifetime obeys an exponential-like scaling law with the noise amplitude, where the exponent in the exponential dependence can be large for small noise. This happens regardless of the nature of the original attractor, chaotic or nonchaotic. We present numerical evidence and a theory to explain this phenomenon. Our finding makes direct experimental observation of superpersistent chaotic transients feasible and it also has implications for problems of current concern such as the transport and trapping of chemically or biologically active particles in large-scale flows.

Periodic knolls and valleys: coexistence of solid and liquid states in granular suspensions

We report the spontaneous emergence of a doubly periodic train of sedimented knolls in a dense suspension. These solidified knolls rise out of, and coexist alongside, a sea of freely flowing liquid in a slowly rotating horizontal bottle. We apply a variable viscosity model that permits simultaneous analysis of fluidlike and solidlike behaviors that are ubiquitous in a variety of sedimenting flows. The model generates qualitative agreement with experiments, and produces new insights into mechanisms by which sedimented structures form.

Universality in active chaos

Many examples of chemical and biological processes take place in large-scale environmental flows. Such flows generate filamental patterns which are often fractal due to the presence of chaos in the underlying advection dynamics. In such processes, hydrodynamical stirring strongly couples into the reactivity of the advected species and might thus make the traditional treatment of the problem through partial differential equations difficult. Here we present a simple approach for the activity in inhomogeneously stirred flows. We show that the fractal patterns serving as skeletons and catalysts lead to a rate equation with a universal form that is independent of the flow, of the particle properties, and of the details of the active process. One aspect of the universality of our approach is that it also applies to reactions among particles of finite size (so-called inertial particles).

Superpersistent chaotic transients in physical space: advective dynamics of inertial particles in open chaotic flows under noise

Superpersistent chaotic transients are characterized by an exponential-like scaling law for their lifetimes where the exponent in the exponential dependence diverges as a parameter approaches a critical value. So far this type of transient chaos has been illustrated exclusively in the phase space of dynamical systems. Here we report the phenomenon of noise-induced superpersistent transients in physical space and explain the associated scaling law based on the solutions to a class of stochastic differential equations. The context of our study is advective dynamics of inertial particles in open chaotic flows. Our finding makes direct experimental observation of superpersistent chaotic transients feasible. It also has implications to problems of current concern such as the transport and trapping of chemically or biologically active particles in large-scale flows.

Uniform resonant chaotic mixing in fluid flows

Laminar flows can produce particle trajectories that are chaotic, with nearby tracers separating exponentially in time. For time-periodic, two-dimensional flows and steady three-dimensional (3D) flows, enhancements in mixing due to chaotic advection are typically limited by impenetrable transport barriers that form at the boundaries between ordered and chaotic mixing regions. However, for time-dependent 3D flows, it has been proposed theoretically that completely uniform mixing is possible through a resonant mechanism called singularity-induced diffusion; this is thought to be the case even if the time-dependent and 3D perturbations are infinitesimally small. It is important to establish the conditions for which uniform mixing is possible and whether or not those conditions are met in flows that typically occur in nature. Here we report experimental and numerical studies of mixing in a laminar vortex flow that is weakly 3D and weakly time-periodic. The system is an oscillating horizontal vortex chain (produced by a magnetohydrodynamic technique) with a weak vertical secondary flow that is forced spontaneously by Ekman pumping--a mechanism common in vortical flows with rigid boundaries, occurring in many geophysical, industrial and biophysical flows. We observe completely uniform mixing, as predicted by singularity-induced diffusion, but only for oscillation periods close to typical circulation times.

Advection of finite-size particles in open flows

It is known that small, spherical particles with inertia do not follow the local velocity field of the flow. Here we investigate the motion of such particles and particle ensembles immersed in open, unsteady flows which, in the case of ideal pointlike tracers, generate chaotic Lagrangian trajectories. Due to the extra force terms in the equations of motion (such as Stokes drag, added mass) the inertial tracer trajectories become described by a high-dimensional (2d+1, with d being the flow's dimension) chaotic dynamics, which can drastically differ from the (d+1)-dimensional ideal tracer dynamics. As a consequence, we find parameter regimes (in terms of density and size), where long-term tracer trapping can occur for the inertial particle, even for flows in which no ideal, pointlike passive tracers can be trapped. These studies are performed in a model of a two-dimensional channel flow past a cylindrical obstacle. Since the Lagrangian tracer dynamics is sensitive to the particle density and size parameters, a simple geometric setup in such flows could be used as a (low-density) particle mixture segregator.

Selective sensitivity of open chaotic flows on inertial tracer advection: catching particles with a stick

We investigate the effects of finite size and inertia of a small spherical particle immersed in an open unsteady flow which, for ideal tracers, generates transiently chaotic trajectories. The inertia effects may strongly modify the chaotic motion to the point that attractors may appear in the configuration space. These studies are performed in a model of the two-dimensional flow past a cylindrical obstacle. The relevance to modeling efforts of biological pathogen transport in large-scale flows is discussed. Since the tracer dynamics is sensitive to the particle inertia and size, simple geometric setups in such flows could be used as a particle mixture segregator separating and trapping particles.