The broad objectives of this work are to develop the theory of topological chaos and to advance the application of this theory to fluid stirring. The main focus is on fluid systems for which a unique top-level periodic orbit of pseudo-Anosov type has been "built in." This work is accomplished using an integrated program of theoretical, numerical, and experimental research. In particular, the investigators (1) advance the rigorous mathematical theory underlying topological chaos; (2) use the theory to develop an understanding of the fundamental mechanisms of mixing in several canonical fluid flows of practical interest; and (3) leverage these developments to advance current laminar mixing technology. The mathematical/theoretical and numerical/experimental components of this project are mutually supporting. The theory provides ideas and a basic framework for the design and analysis of fluid mixing, while the fluid applications provide evidence for or against the posed conjectures and suggest ideas for additional development of the theory. The intellectual merit of this work includes its extensions of the Thurston-Nielsen theory, its contributions to the fundamental understanding of fluid stirring and the role of topological methods in global predictions of chaos, and its detailed analysis of fluid systems of interest for practical mixing enhancement.

Laminar fluid flow systems are at the center of numerous major advances in medical, biological, chemical, and material processing applications that are important for improving human health, advancing scientific discovery, and maintaining national security. Fluid mixing is known to play a significant role in these applications, and mixing enhancement is most often achieved through efficient stirring. Further advances in laminar mixing enhancement are limited in part by the tools used to model, analyze, and predict efficient stirring in laminar flows. A topological method based on a deep mathematical theory due to Thurston and Nielsen has recently been applied to fluid stirring enhancement by the investigators and others with quite dramatic results. The mathematical theory, when properly applied, provides a means to "design for chaos" predictively. The broader impacts of this work include promoting teaching and learning at the undergraduate and graduate levels, seeking to broaden the participation of underrepresented groups in research, enhancing interaction between the engineering and mathematics communities, and benefiting society by developing the techniques for mixing enhancement in laminar flow.

National Science Foundation (NSF)
Division of Mathematical Sciences (DMS)
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Michael H. Steuerwalt
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Vanderbilt University Medical Center
United States
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