Collaborative Research: Topological Fluid Mechanics of Stirring

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StremlerDMS-0607606BoylandDMS-0604570 The broad objectives of this work are to develop the theoryof topological chaos and to advance the application of thistheory to fluid stirring. The main focus is on fluid systems forwhich a unique top-level periodic orbit of pseudo-Anosov type hasbeen "built in." This work is accomplished using an integratedprogram of theoretical, numerical, and experimental research. Inparticular, the investigators (1) advance the rigorousmathematical theory underlying topological chaos; (2) use thetheory to develop an understanding of the fundamental mechanismsof mixing in several canonical fluid flows of practical interest;and (3) leverage these developments to advance current laminarmixing technology. The mathematical/theoretical andnumerical/experimental components of this project are mutuallysupporting. The theory provides ideas and a basic framework forthe design and analysis of fluid mixing, while the fluidapplications provide evidence for or against the posedconjectures and suggest ideas for additional development of thetheory. The intellectual merit of this work includes itsextensions of the Thurston-Nielsen theory, its contributions tothe fundamental understanding of fluid stirring and the role oftopological methods in global predictions of chaos, and itsdetailed analysis of fluid systems of interest for practicalmixing enhancement. Laminar fluid flow systems are at the center of numerousmajor advances in medical, biological, chemical, and materialprocessing applications that are important for improving humanhealth, advancing scientific discovery, and maintaining nationalsecurity. Fluid mixing is known to play a significant role inthese applications, and mixing enhancement is most often achievedthrough efficient stirring. Further advances in laminar mixingenhancement are limited in part by the tools used to model,analyze, and predict efficient stirring in laminar flows. Atopological method based on a deep mathematical theory due toThurston and Nielsen has recently been applied to fluid stirringenhancement by the investigators and others with quite dramaticresults. The mathematical theory, when properly applied,provides a means to "design for chaos" predictively. The broaderimpacts of this work include promoting teaching and learning atthe undergraduate and graduate levels, seeking to broaden theparticipation of underrepresented groups in research, enhancinginteraction between the engineering and mathematics communities,and benefiting society by developing the techniques for mixingenhancement in laminar flow.