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With fast development of microfluidic systems, fluid micro-mixing becomes a very important issue. In this paper, recent developments on various micromixers and their working mechanisms are reviewed, including the external agitation methods applied in active mixing and the channel geometries adopted in passive mixer design. The chaotic mixing and the influences of Re would be mainly discussed. At moderate and high Re, the fluid inertial effects usually facilitate the chaotic mixing. At low Re, generation of chaotic advection becomes more difficult but can still be achieved through fluid manipulations such as stretching and folding. Chaotic mixers can be characterized using dynamical system techniques, such as Poincaré plot, and Lyapunov exponent.

Extensive numerical experiments are performed on tracer dispersion in global reanalysis wind fields. Particle trajectories are computed both along an isobaric (500 hPa) and an isentropic (315 K) surface in a time interval of one year. Besides mean quantities such as advection of the center of mass and growth of tracer clouds, special attention is paid to the evaluation of particle pair separation dynamics. The characteristic behavior for intermediate time scales is Batchelor's dispersion along both surfaces, where the zonal extent of the tracer cloud increases linearly in time. The long-time evolution after 70–80 days exhibits a slower, diffusive dispersion (Taylor regime), in agreement with expectations. Richardson–Obukhov scaling (superdiffusion with an exponent of 3/2) could not be identified in the numerical tests. The results confirm the classical prediction by Batchelor that the initial pair-separation determines subsequent time evolution of tracers. The quantitative dependence on the initial distance differs however from the prediction of the theory.

This paper explores the chaotic properties of an advection system expressed in difference equations form. In the beginning the Aref's blinking vortex system is examined. Then several new lines are explored related to the sink problem (one central sink, two symmetric sinks, eccentric sink and others). Chaotic forms with or without space contraction are presented, analyzed and simulated. Several chaotic objects are formulated especially when special rotation angles or a complex sinus rotation angle are introduced in the rotation-translation difference equations. Very interesting chaotic forms arise when elliptic rotation-translation equations are applied. The simulated chaotic images and attractors express several vortex-like forms resulting in various situations and especially in fluid dynamics.

We model Lagrangian lateral mixing and transport of passive scalars in meandering oceanic jet currents by two-dimensional advection equations with a kinematic stream function with a time-dependent amplitude of a meander imposed. The advection in such a model is known to be chaotic in a wide range of the meander's characteristics. We study chaotic transport in a stochastic layer and show that it is anomalous. The geometry and topology of mixing are examined and shown to be fractal-like. The scattering characteristics (trapping time of advected particles and the number of their rotations around elliptical points) are found to have a hierarchical fractal structure as functions of initial particle's positions. A correspondence between the evolution of material lines in the flow and elements of the fractal and between dynamical and topological measures of the flow are established.

We present the results of experiments on advection-reaction-diffusion processes. Two flows are studied: a blinking vortex flow and a chain of alternating vortices. Mixing in both of these flows has been shown to be chaotic in general. The fluid is composed of the chemicals for the Belousov-Zhabotinsky (BZ) chemical reaction. We investigate the effects of chaotic mixing on the patterns that form in this system. Three experiments are described: (a) pattern formation in the oscillatory BZ reaction; (b) front propagation and mode-locking for the excitable BZ reaction in an oscillating vortex chain; and (c) synchronization of a network of fluid oscillators by superdiffusive transport and Lévy flights. The experiments are complemented by numerical simulations that illustrate the chaotic transport of these flows.

The transport properties of a two-dimensional, inviscid incompressible vortex flow using dynamical systems techniques are examined. The flow field induced by the interaction of a bottom topography with a background flow in a 2-layer fluid is considered. Using the concept of background currents, the dynamically consistent stream function of this flow is constructed. If the background flow is steady, then the trajectories of fluid particles coincide with streamlines. The flow field consists of a vortex region (VR) with closed streamlines bounded by a separatrix and a flowing region (FR) with streamlines going to infinity. In the presence of the oscillating perturbation the picture changes dramatically; fluid particles are entrained and detrained from the VR and chaotic particle motion occurs. We focus on how the change in frequency of the perturbation affects the transport and distribution of passive tracers. It's carried out experiment that allows us to estimate the number of fluid particles, initially uniformly filled whole vortex region, leaving the VR. The evolution particle number shows that when the perturbation frequency increases the mean transport decreases. However, the dependence of particle number leaved the VR on perturbation frequency is complicated. It has both global maximum and local extremums. The analyse of evolution Poincaré map from the perturbation frequency showed that local extremums are related to the disappearance and overlap of resonance bands. It is suggested that the disappearance of resonance bands related to the limited of dependence of fluid particle travel time in the VR on the distance from tracer location to vortex center at absent of perturbation.

When a shallow layer of inviscid fluid flows over a substrate, the fluid particle trajectories are, to leading order in the layer thickness, geodesics on the two-dimensional curved space of the substrate. Since the two-dimensional geodesic equation is a two degree-of-freedom autonomous Hamiltonian system, it can exhibit chaos, depending on the shape of the substrate. We find chaotic behaviour for a range of substrates.

The distributions of “time of flight” (time spent by a single fluid particle between two crossings of the Poincaré section, also called first-return time in dynamical systems community) are investigated numerically for 3D stationary chaotic mixers. Above all, we study the large tails of those distributions, and show that two types of behaviors are encountered. In the case of slipping walls, as expected, we obtain an exponential decay, which, however, does not scale with the Lyapunov exponent. Using a simple model, we suggest that this decay is related to the negative eigenvalues of the fixed points of the flow. Hence long time of flight correspond to particles that are stuck for a while in the vicinity of stable manifolds of fixed points. This property is then applied to the rate of decay of scalar energy in the case of diffusing species with global chaos. More precisely, we suggest that mixing in 3D-flows is governed by regions of slow rate of strain rather than by the Lyapunov exponent. When no-slip walls are considered, as predicted by the model, the behavior is radically different, with a very large tail following a power law with an exponent close to -3. The model suggests that long times of flights correspond to particles stuck for a while in the vicinity of solid walls.

We study the transport and diffusion of a scalar in a two-dimensional flow generated by the co-rotation of two cylinders and perturbed periodically by an oscillating cylinder. The flow is both stratified with a linear density gradient using salted water, and viscous in order to prevent Ekman pumping and centrifugal instabilities. The evolution of a blob of scalar injected close to the stagnation point is observed. First, the distance between the stable and the unstable manifolds is measured as the distance between the maximum and the minimum of the dye undulating pattern, and is compared successfully to the Melnikov function. Second, the diffusion of the stretched filament close to the hyperbolic point has been measured and validated theoretically. Thirdly, the fractal structure created by the flow and its homogenization by diffusion has been observed qualitatively at late stages.

The transport of passive tracers inside a two-dimensional differentially heated cavity is investigated numerically in the oscillatory regime, both in the vicinity and far from the corresponding Hopf bifurcation. Differential transport at low Rayleigh number is essentially linked to the exchange of fluid between two symmetric homoclinic tangles. A hierarchy of models is developed for the transient homogenisation process, the third order model reproducing well the observed behaviour. The vertical transport is efficient only at higher values of the Rayleigh number once all invariant tori have resonated. Signatures of non-hyperbolicity are shown by monitoring the variance of the coarse-grained concentration vs. time.

Transport of water masses across oceanic jet currents is considered in kinematic and dynamic models of chaotic advection of passive scalars. From the point of view of dynamical systems theory it is a problem of breakdown of the last transport barrier with changing control parameters. We find numerically two scenarios for the breakdown of the transport barrier in a kinematic model of a western boundary meandering jet current like the Gulf Stream in the Atlantic ocean and the Kuroshio in the Pacific ocean and discuss causes for such a breakdown. The cross-frontal chaotic transport is shown to occur as well in a dynamic model of chaotic advection in the Antarctic circumpolar current with two Rossby waves. In both the models the breakdown of the transport barrier may occur at comparatively small values of the amplitudes of perturbations.