The goal of this collaborative 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.
The main outcome of the current research program is a general quantitative theory of long-time resonance-driven chaotic transport and mixing in near-integrable autonomous and non-autonomous volume-preserving flows. This was a collaborative effort of two PIs with complementary areas of expertise who have a track record of productive collaboration. The main idea of the project was motivated by the Microfluidics, which promises major advances in drug discovery, medical diagnostics, and national security through its impact on chemical processing and sensor technology. Mixing of fluids at the microscale is one of the most fundamentally difficult problems on the way to practical implementations of microfluidic technology. Our approach provides a better understanding of fundamental principles guiding the development of effective microscale mixing devices. Besides microfluidic problems, we used the wave-particle interaction in Plasma (from the Earth magnetosphere to interstellar clouds) to illustrate the general approach and develop specific calculational tools that can be naturally generalized to a wide class of volume-preserving and Hamiltonian systems. The deterministic theory of transport between the resonance surfaces is be combined with the theory of quasi-random phenomena at the the resonance surfaces to develop a statistical long-term description of the Lagrangian transport in systems with separatrices and/or resonances. We developed methods for computing the size of the mixing domain and the effective rate of mixing. We illustrated that in multiscale systems the overall mixing is governed by mixing on the slowest scale and proposed a method of deriving a single diffusion-type PDE that provides the quantitative description of mixing. The intellectual merit of this work includes (1) the 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 (2) the development of a novel technique to quantify mixing rate, thoroughness, and uniformity for incompressible fluid flows and Hamiltonian plasma systems. The broader impacts of this work include a significant educational component and development of theories and technologies benefiting society at large. Our 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 research program was tightly integrated with teaching and learning at the undergraduate and graduate levels and included activities aimed at increased participation of underrepresented groups in research and integration of research advances into the curriculum. For two graduate students, the results obtained in during the work on the project constituted major parts their Thesis. For undergraduate students, this included both bringing up the research topics from the current proposal into the lectures, but also developing Senior Design projects. For many engineering students, their Senior Design project is the first encounter with a real engineering task, that required a complex approach, from developing a general idea to building something concrete. Thus, developing an interesting and challenging projects is one of the key responsibilities of a professor of engineering both as a mentor and an educator. Students' participation in the research project played a major role in the decision for several of them to pursue advanced degrees. Two of them entered the Graduate program at the Temple University and they are to receive their M.S. degree this year. Based on the discussion with students, the very fact that they participated in the research funded by the NSF was a huge motivation to stay in academia. When told that their designs are used not only for the research, but in the classrooms, students felt elated. For them, it was both the recognition of their labor and a chance to convey their enthusiasm to their peers. The results of the research were broadly disseminated through journal publications, technical presentations at scientific and engineering meetings and conferences both in the US and abroad, colloquia, and seminars at academic institutions.
