Structure, Transport, and Chaos in Volume-Preserving Dynamics

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The persistence of quasiperiodic motion on codimension-one tori in nearly-integrable volume-preserving maps is explained by KAM theory. However, the robustness of these tori and the existence of remnants upon destruction are understood only in two-dimensions. The PI proposes to study tori of three-dimensional maps, and to generalize the residue criterion discovered by Greene and the anti-integrable limit discovered by Aubry. Studies will include symmetry reduction, invariance, and the loss of integrability for general, structure-preserving maps and flows. An important application is the optimization of mixing in open duct flows used in the continuous blending of materials. Our current understanding of the mixing process is, for the most part, limited to flows that are in essence two-dimensional and either closed or recycling. Transport in three-dimensional systems can be quantified by the flux through the destroyed structures, computed using a generalized action based on Lagrangian forms, thereby obtaining accurate and computationally efficient volume fluxes. The PI and students will use the concept of transitory dynamics to quantify and optimize transport in open flows. The extension to episodic and more general time-dependence will clarify the definition of Lagrangian coherent structures in aperiodic dynamics.The complexity of patterns obtained by mixing a passive scalar in a fluid can be observed by anyone pouring cream into hot coffee. That this process is not fully understood is perhaps less obvious. If the flow is sufficiently turbulent then mixing is rapid and uniformity is not hard to achieve. If, however, the flow is slow, on a small scale, or viscous, then mixing is much more difficult. Yet, such processes are important to many applications including the development of micrometer scale bioreactors and effective mixing of polymer and granular materials. A predictive theory for laminar mixing would also contribute to the understanding of climate modeling and pollution dispersal in the atmosphere as well as nutrient dispersal and spawning efficiencies for sea life. Mixing in laminar flows proceeds by stretching and folding due to chaotic motion that gives rise to fine-scale structure where diffusion is effective. Any measure of mixing requires quantification of chaos and its concomitant transport. Chaotic motion in incompressible fluids has some similarities to that in conservative dynamics. The later models are used to predict the lifetime of particles in accelerators, obtain rates for simple chemical reactions, calculate confinement times in plasma fusion devices, understand the spectra of highly excited atomic systems, and design efficient spacecraft trajectories. For chaotic dynamics, prediction of specific trajectories is difficult; nevertheless, chaos can be profitably utilized, for example, to improve efficiency of spacecraft trajectories, by judiciously applying small course corrections, or to enhance the lifetimes of particles in confinement devices and the rates of chemical reactions. In this study, chaos will be used to optimize mixing with the goal of obtaining practical designs for open, three-dimensional, mixing devices.