Collaborative Research: Topological Fluid Mechanics of Stirring

本工作的主要目标是发展拓扑混沌理论，并促进该理论在流体搅拌中的应用。主要关注的是流体系统，其中一个独特的伪阿诺索夫型顶层周期轨道已经“内置”。这项工作是通过理论、数值和实验研究的综合程序来完成的。特别是，研究人员(1)提出了拓扑混沌的严格数学理论；(2)运用理论发展对几种典型流体混合的基本机制的理解；(3)利用这些发展来推进当前的层流混合技术。这个项目的数学/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想和基本框架，而流体应用为所提出的猜想提供了证据，并为该理论的进一步发展提出了想法。这项工作的智力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和拓扑方法在混沌全局预测中的作用的贡献，以及它对实际混合增强感兴趣的流体系统的详细分析。层流系统是医学、生物、化学和材料加工应用领域众多重大进展的核心，对改善人类健康、推进科学发现和维护国家安全至关重要。众所周知，流体混合在这些应用中起着重要作用，而混合增强通常是通过有效的搅拌来实现的。层流混合增强的进一步发展部分受到用于模拟、分析和预测层流中有效搅拌的工具的限制。基于thurston和Nielsen的深层数学理论的拓扑方法最近被研究者和其他人应用于流体搅拌增强，并取得了相当显著的成果。数学理论在适当的应用下，提供了一种预测“混沌设计”的方法。这项工作的广泛影响包括促进本科和研究生水平的教学和学习，寻求扩大代表性不足的群体在研究中的参与，加强工程和数学社区之间的互动，并通过开发层流混合增强技术来造福社会。

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.