Swimming the Chaotic Seas: Invariant Manifolds, Tori, and the Transport of Swimmers in Fluid Flows

自行体的动力学和控制，从成群的自主水下机器人(AUV)和鱼群到游动的细菌，都是生物物理学和工程学非常感兴趣的问题。然而，虽然许多工作都集中在个人之间的相互作用所产生的集体行为上，但对这些个人在外部动态变化的环境中的动态了解较少。这项研究将解决一个基本问题，即孤立的自行式游泳者是如何在非稳定的流体中运输的。通过利用动力系统理论中的工具，例如那些用于分析流体流动中反应前沿传播的工具，该项目旨在更好地理解游泳者在自然界和实验室中的真实流动中移动的行为。这项工作将教育和培训下一代科学工作者，并将极大地影响加州大学默塞德分校的发展和教育使命，这所大学最近成立于加州最具经济挑战的地区之一。稳定和不稳定的流形是控制被动粒子在稳定的、与时间无关的流动中传输和混合的关键结构。这些歧管都将流体分割成涡流单元，并通过流体中的“旋转门”叶控制单元之间的传输。拉格朗日相干结构(LCS)在非定常、非周期流动中提供了一个类似的框架。这个项目的目标是使这些理论适用于既受流体平流又受自身动力推动的粒子。不变流形和被动LCS不再是最相关的结构。相反，明确依赖于游泳速度的游泳不变流形(游泳)与游泳边和不变环面一起形成了相空间中的关键结构。这项研究将：(I)调查这些结构在诱捕游泳者和产生弹道运动方面的作用，(Ii)探索游泳在混乱环境中限制和引导交通的作用，(Iii)开发一种严格的方法来解释涡流中游泳者的噪音，以及(Iv)通过与研究微流体流中细菌和藻类传输的实验者的合作，将这些想法与真实世界的生物和流动联系起来。该奖项反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。

The dynamics and control of self-propelled bodies, from swarms of autonomous underwater vehicles (AUVs) and fish schools to populations of swimming bacteria, are of great interest to biophysics and engineering. Yet, while much work has focused on the collective behavior arising from interactions between individuals, less is known about the dynamics of these individuals in external, dynamically changing environments. This research will tackle the fundamental question of how isolated, self-propelled swimmers are transported in unsteady fluid flows. By leveraging tools from dynamical systems theory, such as those used in analyzing the propagation of reaction fronts in fluid flows, this project aims to create better understanding of the behavior of swimmers moving in realistic flows in nature and the laboratory. This work will educate and train the next generation of the scientific work force and will greatly impact the growth and educational mission of UC Merced, a university that was recently established in one of California's most economically challenged areas.Stable and unstable manifolds are critical structures that control the transport and mixing of passive particles in steady, time-independent flows. These manifolds both partition the fluid into vortex cells and control the transport between cells via “turnstile” lobes in the fluid. Lagrangian coherent structures (LCSs) provide an analogous framework in unsteady, aperiodic flows. The objective of this project is to adapt these theories to particles that are both advected by the fluid and propelled under their own power. Invariant manifolds and passive LCS are no longer the most relevant structures. Rather, swimming invariant manifolds (SwIMs), which depend explicitly on the swimming speed, along with SwIM edges and invariant tori, form key structures in the phase space. This research will: (i) investigate the role of these structures in both trapping swimmers and generating ballistic motion, (ii) explore the role of SwIMs on restricting and guiding transport in chaotic environments, (iii) develop a rigorous approach to account for swimmer noise in vortex flows, and (iv) connect these ideas to real-world organisms and flows through collaborations with experimentalists that study the transport of bacteria and algae in microfluidic flows.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.