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# Barriers and Transport in Unsteady Flows: A Melnikov Approach

### by Sanjeeva Balasuriya

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2017 / xiv + 264 pages / Softcover / 978-1-611974-57-7 / List Price $84.00 / SIAM Member Price$58.80 / Order Code: MM21

Keywords: Unsteady flows; coherent structures; stable and unstable manifolds; chaotic flux; fluid mixing; nonautonomous flows.

Fluids that mix at geophysical or microscales tend to form well-mixed areas and regions of coherent blobs. The Antarctic circumpolar vortex, which mostly retains its structure while moving unsteadily in the atmosphere, is an example of a coherent structure. How do such structures exchange fluid with their surroundings? What is the impact on global mixing? What is the "boundary" of the structure, and how does it move? Can these questions be answered from time-varying observational data?

This book addresses these issues from the perspective of the differential equations that must be obeyed by fluid particles. In these terms, identification of the boundaries of coherent structures (i.e., "flow barriers"), quantification of transport across them, control of the locations of these barriers, and optimization of transport across them are developed using a rigorous mathematical framework. The concepts are illustrated with an array of theoretical and applied examples that arise from oceanography and microfluidics.

Barriers and Transport in Unsteady Flows: A Melnikov Approach provides

• an extensive introduction and bibliography, specifically elucidating the difficulties arising when flows are unsteady and highlighting relevance in geophysics and microfluidics;
• careful and rigorous development of the mathematical theory of unsteady flow barriers within the context of nonautonomous stable and unstable manifolds, richly complemented with examples; and
• chapters on exciting new research in the control of flow barriers and the optimization of transport across them.

Audience
The core audience is researchers and students interested in fluid mixing and so-called Lagrangian coherent structures, i.e., moving structures within fluids that have a dominant influence on global mixing. Some background in differential equations or dynamical systems is necessary for an in-depth understanding of the theoretical parts of Chapters 2 and 3. Researchers in oceanography, atmospheric science, engineering fluid mechanics, and microfluidics will also find it an excellent reference, particularly Chapter 1.

Sanjeeva Balasuriya is an Australian Research Council Future Fellow at the School of Mathematical Sciences, University of Adelaide. He has held positions at the University of Peradeniya (Sri Lanka), Oberlin College (USA), Connecticut College (USA), and the University of Sydney (Australia). His work in ordinary differential equations is inspired by many applied areas, and he has published in the Journal of Fluid Mechanics; Journal of Theoretical Biology; Journal of Micromechanics and Microengineering; Combustion Theory and Modeling; and Physical Review Letters, among other journals. He was the advisor of a University of Adelaide team that won the INFORMS Prize at the 2015 Mathematical Contest in Modeling and was awarded the 2008 J.H. Michell Medal for outstanding early career researcher in applied mathematics by Australian and New Zealand Industrial and Applied Mathematics (ANZIAM).

ISBN: 9781611974577

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