The goal of this research program is to construct a general quantitative theory of long-time resonance-driven chaotic transport and mixing in near-integrable autonomous and non-autonomous volume-preserving flows. Specific examples of microscale flows will be used to illustrate the general approach and develop specific tools that can be naturally generalized to a wide class of volume-preserving and Hamiltonian systems. The deterministic theory of resonance processes will be combined with the theory of random walks and theory of stability islands to develop a statistical long-term description of the Lagrangian transport in systems with separatrices and/or resonances. A unique feature of the planned approach is that it will apply both when chaotic advection is the only transport mechanism as well as when chaotic advection competes with thermal or molecular diffusion. This work includes integration of often disconnected methods and techniques used to describe resonant interactions and regular transport into a general transport theory for near-integrable systems, and development of a novel technique to quantify mixing rate, thoroughness, and uniformity for incompressible fluid flows. The PIs' research has direct applications to a wide range of problems in science and engineering, such as the transport of comets and asteroids through the solar system, energy exchange between excitation modes in condensed matter, and motion of charged particles in electromagnetic fields with applications to atmospheric science and magnetic confinement fusion devices. The key application is in the field of microfluidics which promises major advances in drug discovery, medical diagnostics, and national security through its impact on chemical processing and sensor technology. The research program will be tightly integrated with teaching and learning at the undergraduate and graduate levels and will include activities aimed at increased participation of underrepresented groups in research and integration of research advances into the curriculum. The PIs will also seek to extend and establish microfluidics collaborations with the plasma physics community.
As fluids flow, they carry along everything that is floating or dissolved in them. For instance, air flows carry humidity and heat that affect the weather while water solutions and suspensions can transport nutrients needed by plants and animals. While simple flows merely transport their load from point A to point B, complex flows can do much more. Stirring sugar in a cup of coffee quickly and evenly distributes it throughout the cup, making coffee uniformly sweet. This is an example of mixing - the process investigated by a team of scientists at Georgia Tech. Turbulent flows are the most complicated and usually produce very fast mixing. However, many laminar flows can mix almost as well. Numerical algorithms and theoretical models developed by the scientists as a part of this project help describe mixing by laminar flows that occur naturally in simple geometries as well as design flows with desired mixing properties. The design aspect is especially important in such emerging areas as microfluidics, which promises to make medical tests much faster and cheaper. The theoretical description developed by analyzing the models is not limited to the flows arising at the microscale. It also applies to oceanic currents and the atmosphere. Hence, the insights into the mixing behavior of fluid flows which emerged during this investigation may improve our ability to predict, among many other things, the effects of localized events, such as the recent Gulf oil spill, on the environment.
