Collaborative Research: Topological Fluid Mechanics of Stirring

DMS-0607606Boyland DMS-0604570这项工作的主要目标是发展拓扑混沌理论，并推动该理论在流体搅拌中的应用。主要的焦点是流体系统，其中一个独特的顶层周期轨道的伪阿诺索夫类型已经“内置”。这项工作是通过一个理论、数值和实验研究相结合的程序来完成的。特别是，研究人员(1)提出了拓扑混沌背后的严格数学理论；(2)使用该理论来发展对几种具有实际意义的正则流体流动中混合的基本机制的理解；以及(3)利用这些发展来推进当前的分层混合技术。该项目的数学/理论和数值/实验部分是相互支持的。该理论为流体混合的设计和分析提供了思路和基本框架，而流体应用提供了支持或反对假设的证据，并为该理论的进一步发展提供了建议。这项工作的学术价值包括它对瑟斯顿-尼尔森理论的扩展，它对流体搅拌的基本理解的贡献，以及拓扑方法在全局混沌预测中的作用，以及它对实际混合强化感兴趣的流体系统的详细分析。层流流体系统是医学、生物、化学和材料加工应用领域众多重大进展的核心，这些应用对改善人类健康、推进科学发现和维护国家安全具有重要意义。众所周知，流体混合在这些应用中扮演着重要的角色，而强化混合通常是通过有效的搅拌来实现的。层流混合强化方面的进一步进展在一定程度上受到用于模拟、分析和预测层流中有效搅拌的工具的限制。一种基于瑟斯顿和尼尔森的深刻数学理论的拓扑学方法最近被研究人员和其他人应用于流体搅拌强化，取得了相当引人注目的结果。当恰当地应用数学理论时，它提供了一种预测性地“为混沌设计”的方法。这项工作的广泛影响包括促进本科生和研究生的教与学，寻求扩大未被充分代表的群体对研究的参与，加强工程界和数学界之间的互动，并通过开发层流混合强化技术来造福社会。

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.