Igor Mezic

A New Mixing Diagnostic and Gulf Oil Spill Movement

Mixing Chaos Modeling the future movement of chaotic fluids is the basis for predicting the weather and ocean currents. Usually parcels of fluid are traced and geometrical and statistical approaches incorporate parameters for mixing and chaos, as well as the resulting uncertainty. Mezić et al. (p. 486, published online 2 September; see the Perspective by Thiffeault) adapted this approach to consider different mixing and stretching regimes to improve predictions. As a test, they simulated and successfully predicted the spread of oil patches from the Deepwater Horizon oil spill in a model for the Gulf of Mexico. An ocean model can account for the trajectory and fragmentation of the recent Gulf of Mexico oil spill. Chaotic advection has served as the paradigm for mixing in fluid flows with simple time dependence. Its skeletal structure is based on analysis of invariant attracting and repelling manifolds in fluid flows. Here we develop a finite-time theory for two-dimensional incompressible fluid flows with arbitrary time dependence and introduce a new mixing diagnostic based on it. Besides stretching events around attracting and repelling manifolds, this allows us to detect hyperbolic mixing zones. We used the new diagnostic to forecast the spatial location and timing of oil washing ashore in Plaquemines Parish and Grand Isle, Louisiana, and Pensacola, Florida, in May 2010 and the flow of oil toward Panama City Beach, Florida, in June 2010.

Dynamic autoinoculation and the microbial ecology of a deep water hydrocarbon irruption

The irruption of gas and oil into the Gulf of Mexico during the Deepwater Horizon event fed a deep sea bacterial bloom that consumed hydrocarbons in the affected waters, formed a regional oxygen anomaly, and altered the microbiology of the region. In this work, we develop a coupled physical–metabolic model to assess the impact of mixing processes on these deep ocean bacterial communities and their capacity for hydrocarbon and oxygen use. We find that observed biodegradation patterns are well-described by exponential growth of bacteria from seed populations present at low abundance and that current oscillation and mixing processes played a critical role in distributing hydrocarbons and associated bacterial blooms within the northeast Gulf of Mexico. Mixing processes also accelerated hydrocarbon degradation through an autoinoculation effect, where water masses, in which the hydrocarbon irruption had caused blooms, later returned to the spill site with hydrocarbon-degrading bacteria persisting at elevated abundance. Interestingly, although the initial irruption of hydrocarbons fed successive blooms of different bacterial types, subsequent irruptions promoted consistency in the structure of the bacterial community. These results highlight an impact of mixing and circulation processes on biodegradation activity of bacteria during the Deepwater Horizon event and suggest an important role for mixing processes in the microbial ecology of deep ocean environments.

On the integrability and perturbation of three-dimensional fluid flows with symmetry

SummaryThe purpose of this paper is to develop analytical methods for studyingparticle paths in a class of three-dimensional incompressible fluid flows. In this paper we study three-dimensionalvolume preserving vector fields that are invariant under the action of a one-parameter symmetry group whose infinitesimal generator is autonomous and volume-preserving. We show that there exists a coordinate system in which the vector field assumes a simple form. In particular, the evolution of two of the coordinates is governed by a time-dependent, one-degree-of-freedom Hamiltonian system with the evolution of the remaining coordinate being governed by a first-order differential equation that depends only on the other two coordinates and time. The new coordinates depend only on the symmetry group of the vector field. Therefore they arefield-independent. The coordinate transformation is constructive. If the vector field is time-independent, then it possesses an integral of motion. Moreover, we show that the system can be further reduced toaction-angle-angle coordinates. These are analogous to the familiar action-angle variables from Hamiltonian mechanics and are quite useful for perturbative studies of the class of systems we consider. In fact, we show how our coordinate transformation puts us in a position to apply recent extensions of the Kolmogorov-Arnold-Moser (KAM) theorem for three-dimensional, volume-preserving maps as well as three-dimensional versions of Melnikovs method. We discuss the integrability of the class of flows considered, and draw an analogy with Clebsch variables in fluid mechanics.

Uncertainty and sensitivity decomposition of building energy models

As building energy modelling becomes more sophisticated, the amount of user input and the number of parameters used to define the models continue to grow. There are numerous sources of uncertainty in these parameters, especially when the modelling process is being performed before construction and commissioning. Past efforts to perform sensitivity and uncertainty analysis have focused on tens of parameters, while in this work, we increase the size of analysis by two orders of magnitude (by studying the influence of about 1000 parameters). We extend traditional sensitivity analysis in order to decompose the pathway as uncertainty flows through the dynamics, which identifies which internal or intermediate processes transmit the most uncertainty to the final output. We present these results as a method that is applicable to many different modelling tools, and demonstrate its applicability on an example EnergyPlus model.

Melnikov-Based Dynamical Analysis of Microcantilevers in Scanning Probe Microscopy

We study the dynamical behavior of a microcantilever-sample system that forms the basis for the operation of atomic force microscopes (AFM). We model the microcantilever by a single mode approximation. The interaction between the sample and the cantilever is modeled by a Lennard--Jones potential which consists of a short-range repulsive potential and a long-range van der Waals (vdW) attractive potential. We analyze the dynamics of the cantilever sample system when the cantilever is subjected to a sinusoidal forcing. Using the Melnikov method, the region in the space of physical parameters where chaotic motion is present is determined. In addition, using a proportional and derivative controller, we compute the Melnikov function in terms of the parameters of the controller. Using this relation, controllers can be designed to selectively change the regime of dynamical interaction.

Control of mixing in fluid flow: a maximum entropy approach

In many technological processes a fundamental stage involves the mixing of two or more fluids. As a result, the design of optimal mixing protocols is a problem of both fundamental and practical importance. In this paper, the authors formulate a prototypical mixing problem in a control framework, where the objective is to determine the sequence of fluid flows that will maximize entropy. By developing the appropriate ergodic-theoretic tools for the determination of entropy of periodic sequences, they derive the form of the protocol which maximizes entropy among all of the possible periodic sequences composed of two shear flows orthogonal to each other. The authors discuss the relevance of their results in the interpretation of previous studies of mixing protocols.

Chaotic mixing in a bounded three-dimensional flow

Even though the first theoretical example of chaotic advection was a three-dimensional flow (Henon 1966), the number of theoretical studies addressing chaos and mixing in three-dimensional flows is small. One problem is that an experimentally tractable three-dimensional system that allows detailed experimental and computational investigation had not been available. A prototypical, bounded, three-dimensional, moderate-Reynolds-number flow is presented; this system lends itself to detailed experimental observation and allows high-precision computational inspection of geometrical and dynamical effects. The flow structure, captured by means of cuts with a laser sheet (experimental Poincare section), is visualized via continuously injected fluorescent dye streams, and reveals detailed chaotic structures and chains of high-period islands. Numerical experiments are performed and compared with particle image velocimetry (PIV) and flow visualization results. Predictions of existing theories for chaotic advection in three-dimensional volume-preserving flows are tested. The ratio of two frequencies of particle motion – the frequency of motion around the vertical axis and the frequency of recirculation in the plane containing the axis – is identified as the crucial parameter. Using this parameter, the number of islands in the chain can be predicted. The same parameter – using as a base-case the integrable motion – allows the identification of operating conditions where small perturbations lead to nearly complete mixing.

A method for visualization of invariant sets of dynamical systems based on the ergodic partition

We provide an algorithm for visualization of invariant sets of dynamical systems with a smooth invariant measure. The algorithm is based on a constructive proof of the ergodic partition theorem for automorphisms of compact metric spaces. The ergodic partition of a compact metric space A, under the dynamics of a continuous automorphism T, is shown to be the product of measurable partitions of the space induced by the time averages of a set of functions on A. The numerical algorithm consists of computing the time averages of a chosen set of functions and partitioning the phase space into their level sets. The method is applied to the three-dimensional ABC map for which the dynamics was visualized by other methods in Feingold et al. [J. Stat. Phys. 50, 529 (1988)]. (c) 1999 American Institute of Physics.

Mixing in the shear superposition micromixer: three-dimensional analysis.

In this paper, we analyse mixing in an active chaotic advection micromixer. The micromixer consists of a main rectangular channel and three cross–stream secondary channels that provide ability for time–dependent actuation of the flow stream in the direction orthogonal to the main stream. Three–dimensional motion in the mixer is studied. Numerical simulations and modelling of the flow are pursued in order to understand the experiments. It is shown that for some values of parameters a simple model can be derived that clearly represents the flow nature. Particle image velocimetry measurements of the flow are compared with numerical simulations and the analytical model. A measure for mixing, the mixing variance coefficient (MVC), is analysed. It is shown that mixing is substantially improved with multiple side channels with oscillatory flows, whose frequencies are increasing downstream. The optimization of MVC results for single side–channel mixing is presented. It is shown that dependence of MVC on frequency is not monotone, and a local minimum is found. Residence time distributions derived from the analytical model are analysed. It is shown that, while the average Lagrangian velocity profile is flattened over the steady flow, Taylor–dispersion effects are still present for the current micromixer configuration.

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.