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## Physics of Mixing |

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It took the fluid mechanics community until the early 1980's to appreciate that flow kinematics is a very difficult subject! In part this was due to an almost exclusive focus on the Eulerian representation of the problem. Viewing flow kinematics from the Lagrangian point of view shows that this problem can, in fact, encompass the dynamical systems theory of non-autonomous systems with up to three degrees of freedom. In particular, flow kinematics can be either regular or chaotic. Flow kinematics provides a theory of fluid mixing in the idealized limit where the substance to be mixed advects as a passive scalar. This is often a good approximation but deviations from it are also interesting and important. The lecture will give a bit of historical survey and perspective on where we are with the dynamical systems approach, where we have been, and where we may be going, emphasizing those aspects that appear to have lasting relevance.
I will review two topological approaches to stirring and mixing. The first involves constructing systems such that the fluid motion is topologically complex, usually by imposing a specific motion of rods. I will discuss optimization strategies that can be implemented. The second is diagnostic, where flow characteristics are deduced from observations of periodic or random orbits and their topological properties.
Decomposition and Diagnostics" I will discuss an approach to understanding passive scalar mixing in fluids using properties of an infinite-dimensional operator, called the Koopman operator. Particular attention is given to three-dimensional, time-dependent flows. The analysis leads to development of a method for visualization of invariant sets where 2-dimensional surfaces of section are obtained in three-dimensional, time-dependent flows, providing a generalization to the Poincare section concept. An application to a three-dimensional, time-periodic, divergence-free flow with swirl, will be given. I also discuss a decomposition, based on spectral properties of the Koopman operator, named the Koopman Mode Decomposition, that allows detection of purely time-periodic Eulerian components of flows with possibly more complex time-dependence, and discuss the differences and advantages over the commonly used Proper Orthogonal Decomposition. This new decomposition enables extension of chaotic advection concepts to flows with complex time dependence. Finally, I will discuss a new way of finding mixing and stretching zones in two-dimensional flows with complex time-dependence, and review a recent application to the Gulf Oil Spill surface oil spread forecast.
Open flows are characterised by the presence of inflow and outflow, and the consequent exchange of matter between the system of interest and the outside. Many important natural and man-made open flows display a chaotic dynamics, with the result that the advection dynamics is governed by structures in phases space that have a fractal geometry. This leads to a number of consequences to the dynamical properties of particles advected by the flow, and results in anomalous kinetic properties for chemical reactions or biological processes taking place in the flow. In this talk I will review the main results of this field, and present some recent developments.
Mixing and heat exchange are two of the fundamental operations for industrial transformation and are key determinants of the quality of many manufactured products. In practice for industry the physics of mixing means chaotic advection in laminar flows, and chaotic advection becomes commercially important when it is used to cost effectively augment and control industrial transport processes or to enable the creation of new products. Why use chaotic advection for laminar flow industrial processes? As the physics of mixing is neither widely taught in the engineering curriculum, nor widely used in industry, there is often a risk-aversion barrier that takes a long time to overcome; the question of 'why chaos' comes up frequently in trying to commercialize chaotic mixing processes. In this talk I will describe three examples of how our group is using chaotic advection to add value to a commercial operation. Each example is at a different point on the time path to general use. The first example is a device for continuous mixing and/or heat exchange. It gives better or equivalent quality results while reducing energy use by 60-96% and is starting to be used in the food industry. The second example is a process for intensifying extraction of heat or minerals from the ground and/or for confinement of fluid bodies in the ground. These processes use transient switching of the pressure applied at different wells to produce periodic reorientation of subsurface flows and have shown up to a 100% increase in extraction rate. Negotiations with a consortium of companies are underway to fund larger scale laboratory experiments and field trials. The third example is a device that uses a steady flow produced by a steady boundary motion to generate globally chaotic advection; we have also derived an analytic expression for the velocity field. Preliminary discussions are underway to use this device for a biomedical suspension to which current mixing methods impart too many bubbles.
In this talk I will review the dynamics of 3D volume preserving maps, as representative of the motion of fluid parcels in 3D unsteady (time-periodic) incompressible flows. I will also show some realizations of this type of flows where several aspects of the former dynamics are present. Finally, I will briefly present some features of the behaviour of imperfectly passive particles in the same class of flows.
The fields of application of microfluidics are diverse, but chemistry and biology certainly hold particular places. In a microfluidic device dedicated to a chemical or biological application, the fundamental task consists in fostering biochemical reactions in the microfluidic system, which obviously requests fluid mixing to be achieved on-chip. It has been recognized early that mixing in miniaturized systems is uneasy even though scaling laws indicate that diffusive times decrease as the square of the system size. This is why, over the last ten years, all sorts of micromixers were invented, and now they are classified in terms of Reynolds and Peclet numbers. In this period, chaotic mixing ideas developed in the eighties proved to be particulary relevant and efficient. In the talk, I will try to summarize the microhistory of this domain, stress underlying physical ideas, and attempt to make some guess regarding its future evolution.
The principles of scalar mixing in randomly stirred media are discussed, aiming at describing the overall concentration distribution of the mixture, its shape, and rate of deformation as the mixture evolves towards uniformity. We emphasize the impact of the individual trajectories of the particles, as well as the interaction rules between the particles, on the mixture evolution and fine structure. The discussion is illustrated by several experiments, both at high, and low Reynolds number; it also suggests a new numerical method for scalar mixing.
A new mathematical model for a meandering oceanic current is suggested. It based upon modification of the Von Kármán vortex street model. The suggested modification allows one to approximate observed patterns in the meandering Gulf Stream. This ocean current is characterized by the following: 1) an eastward-propagating meandering jet; 2) regions of recirculating fluid below and above meander crests and troughs (seen in a coordinate system co-moving with the meanders; 3) regions of westward-propagating fluid near recirculation regions on either side of the jet. The inclusion of eddies above the recirculation regions and the jet enhance transport and mixing across the jet. Calculations show that more than a half of the circular area above the crest point of the third meander may contain warm fluid from a central area of the jet. To study mixing across the jet we examine deformation of this circular area back in time, so we can determine from which part of the jet that area is composed. Contour line tracking method conserving all topological properties in 2-D flows is used for this procedure.
Granular flow in a container rotated sequentially about two axes is a realistic system in which to investigate kinematic structures of 3D volume-preserving maps. It is shown that the dynamics can be restricted to 2D invariant surfaces, or be fully-3D, depending on the rates of rotation. For the former, KAM-like tubes and manifolds are constructed by stacking up their 1D versions from a sequence of invariant surfaces. By requiring that the KAM-like tubes be “as small as possible,” optimal angles of rotation are found. These predictions are confirmed by mixing simulations.
The behavior of mixing in time-interleaved sequential micromixers is illustrated by means of simulations and theoretical aspects. A critical comparison with the "Nguyen model" (Microfluidic Nanofludic 1:373 (2005)) is exposed. The "anomalous" scaling law exhibited by the mixing boundary layer is illustrated and characterized. Finally practical implications are discussed.
We define the concept of a transitory dynamical system and present a new method for quantifying transport between coherent structures in exact volume preserving transitory flows. Our methods require little Lagrangian information and rely on knowing only particular heteroclinic orbits at the intersection of stable and unstable manifolds in such systems. We illustrate the application of this theory with a 2D rotating double gyre and a 3D transitory ABC flow. Jemil Znaien “Coherent structures in a time-periodic viscous mixing flow” Periodically driven laminar flows occur in many industrial processes e.g. micro-mixer in lab-on-a-chip devices. The present study is motivated by better understanding fundamental transport phenomena in 3D viscous time-periodic flows. Both three-dimensional Particle Tracking Velocimetry measurements and numerical simulations are performed to investigate the 3D advection of a passive scalar in a lid-driven cylindrical cavity flow: one endwall sets the fluid in motion via time-periodic repetition of a sequence of n piecewise steady translations. We concentrate on the formation and interaction of coherent structures due to fluid inertia, which play an important role in 3D mixing by geometrically determining the tracer transport. The disintegration of these structures by fluid inertia reflects an essentially 3D route to chaos. Data from tracking experiments of small particles will be compared with predictions from numerical simulations on transport of passive tracers.
In this presentation I will discuss the effect of a single particle on the dynamics of mixing. First, a number of examples are presented in a square cavity. It is shown that even a regular periodic motion of a single particle can induce chaotic advection around the particle, as a result of the perturbation of the flow introduced by the freely rotating solid particle. This perturbation is of a hyperbolic nature. In fact, stretching and folding of the fluid elements are guaranteed by the occurrence of the hyperbolic flow perturbation centered at the particle and by the rotation of the freely suspended particle, respectively. In the second part of the talk, chaotic mixing induced by break-up and reformation of a magnetic chain under the influence of a rotating magnetic field, is studied.
Left-right symmetry breaking is critical to vertebrate embryonic development; in many species this process begins with the cilia-driven flow in a structure termed the ‘node’. A field of primary ‘whirling’ cilia, tilted towards the posterior, transport morphogen-containing vesicles towards the left, initiating left-right asymmetric development. Theoretical models based on the point-force stokeslet and point-torque rotlet singularities explain how rotation and surface-tilt produce directional flow. Simulations predict that vesicles released within one cilium length of the epithelium are generally transported to the left via a ‘loopy’ drift motion, sometimes involving highly unpredictable detours around leftward cilia. A range of phenomena including changing shape of the node during the few relevant days of development, missing cilia, stationary cilia and randomness impact on mixing and vesicle transport will also be discussed.
We have studied the mixing of viscous fluids in 2-D closed and open flows, where stirring rods create chaotic advection. The analysis of dye concentration fields reveals how some regions submitted to low stretching, such as the vicinity of fixed walls, slow down the speed of mixing in the entire chaotic region. Models derived from the baker's map are shown to reproduce most observed mechanisms.
In certain (2+1)-dimensional dynamical systems, the braiding of periodic orbits provides a framework for analyzing chaos in the system through application of the Thurston-Nielsen classification theorem. Periodic orbits generated by the dynamics can behave as physical obstructions that ‘stir’ the surrounding domain and serve as the basis for this topological analysis. We give evidence that, even in the absence of periodic orbits, almost-cyclic regions identified using a transfer operator approach can reveal an underlying structure that enables topological analysis of chaos in the domain.
I present a methodology to build Lagrangian descriptors that are valid for arbitrary time dependent flows. They succeed in depicting a time dependent phase portrait for general non-periodic flows detecting simultaneously, invariant manifolds, hyperbolic and non-hyperbolic flow regions. Applications in realistic geophysical flows will be reported.
We present a quantitative long-term description of resonant mixing in 3-D near-integrable flows. As a model problem we use the flow in the annulus between two coaxial elliptic counter-rotating cylinders. We illustrate that such resonance phenomena as scattering on resonance and capture into resonance create mixing by causing the jumps of adiabatic invariants. We calculate the width of the mixing domain and estimate the rate of the chaotic advection. We illustrate that the resulting mixing can be described in terms of a single diffusion-type PDE.
The talk presents the results of investigation of the microfluidics mixing processes in a rectangular cavity flow induced by electro-osmotic excitation. Analytical solution is presented for the velocity field in the cavity under various electric potential distributions. The location of the periodic points in the flow are accurately established and the structure of stable and unstable manifolds is discussed. The optimal form of excitation is suggested in order to obtain most effective mixing regime in the cavity. The regular and chaotic regions are identified under various condition of excitation.
Cytoplasmic streaming circulates the contents of large eukaryotic cells, often with complex flow geometries. A largely unanswered question is the significance of these flows for molecular transport and mixing. Motivated by “rotational streaming” in Characean algae, we solve the advection-diffusion dynamics of flow in a cylinder with bidirectional helical forcing at the wall. A circulatory flow transverse to the cylinder’s long axis, akin to Dean vortices at finite Reynolds numbers, arises from the chiral geometry. Strongly enhanced lateral transport and longitudinal homogenization occur if the transverse Peclet number is sufficiently large, with scaling laws arising from boundary layers.
We study the stochastic dynamics of an array of rotors on a substrate that are coupled by hydrodynamic interaction at zero Reynolds number. The rotors that are modeled by an effective rigid body, are driven by an internal torque and exerts an active force on the surrounding fluid. The long-ranged nature of the hydrodynamic interaction between the rotors causes a rich pattern of dynamical behaviors including phase ordering and turbulent spiral waves. The model provides a novel example of coupled oscillators with long-range interaction. Our results suggest strategies for designing controllable microfluidic mixers using the emergent behavior of hydrodynamically coupled active components.
Ever since the idea of chaotic advection was envisaged by Arnold and Henon, it was understood that many fluid flows would not mix perfectly because of the presence of long thin invariant tori. Nonetheless it is possible to make examples which do mix perfectly and even robustly so. I will survey several types of such example. A good outcome of this workshop would be to get at least one of them into a form where it could be implemented physically.
Heating or cooling of highly viscous fluids, characterized by high Prandtl numbers, can be efficiently achieved using the methods developed in the few last decades in the field of chaotic advection. Indeed, for these fluids the characteristics time of diffusion is very high compared to convection one. We propose a heating/cooling process composed by a tank and two vertical rods. These three elements are heated and rotate around their own axes. One of the particularities of this situation is the fact that the scalar (temperature) source is located at the walls, and the mixing process must extract the energy from them and homogenize the temperature inside the rest of fluid. The unsteady flow and heat transfer inside the mixer (in a 2D plan) were numerically modeled. Different stirring protocols were studied, depending on the type of the temporal modulation of the angular velocity, its period and on the respective direction of rotation of the walls. In the particular case of alternated rotations, “strange” eigenmodes patterns of temperature were detected in this self-similar flow. The mixing of fluids with different rheological behaviors (shear-thinning, shear-thickening and Newtonian) was studied. The differences in the mixing induced by the type of the thermal boundary condition (constant wall temperature or prescribed wall heat flux) were highlighted with their implications on the mixing strategy. Finally, the impact of the temperature-dependence of the fluid viscosity on the mixing efficiency was pointed out.
The interaction between convective and molecular transport results in different physical phenomenologies depending on whether the system is bounded and closed or unbounded. In closed impermeable systems, it controls the rate at which the equilibrium state is reached. In periodic unbounded media, this local interaction determines the large (Darcy) scale dynamics of average transport and dispersion properties. This lecture discusses how these different physics contexts give rise to the same mathematical paradigm, namely the advection-diffusion equation on the n-dimensional torus. Physical examples ranging from the dynamics of mixing-controlled reactions in Stokes flows to recent microfluidic separation devices are addressed.
The understanding of fluid mixing has been transformed as a result of dynamical systems theory, its fingerprint being stretching and folding, the hallmark of deterministic chaos. Here we introduce a different mixing mechanism - cutting and shuffling - that has a theoretical foundation in a relatively new area of mathematics called piecewise isometries (PWIs). The mixing properties of PWIs that arise from the study of the kinematics of flows are fundamentally different from the stretching and folding mechanism of the familiar chaotic advection. As a physical example we show experimentally that PWIs capture essential aspects of mixing of granular materials in a three-dimensional tumbler. Simulations connect the PWI theory with experiments and demonstrate how the combination of cutting and shuffling (the PWI framework) with stretching due to advection contribute to increasing interfacial boundaries leading to effective mixing.
We study the planktonic biological activity in the wake of an island which is close to an upwelling region providing nutrients for the growth of the plankton. In particular we show that mesoscale vortices act as incubators for plankton growth leading to localized plankton blooms within vortices as well as to dominance patterns of species competing for the same nutrients. [Back] |