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

从自主水下航行器（auv）群和鱼群到游动细菌群，自推进体的动力学和控制是生物物理学和工程学的重要研究方向。然而，尽管许多工作都集中在个体之间相互作用产生的集体行为上，但对这些个体在外部动态变化的环境中的动态知之甚少。这项研究将解决一个基本问题，即孤立的、自我推进的游泳者是如何在非定常流体流动中被输送的。通过利用动力系统理论的工具，例如用于分析流体流动中反应锋传播的工具，该项目旨在更好地理解游泳者在自然界和实验室中实际流动中的行为。这项工作将教育和培训下一代的科学工作人员，并将极大地影响加州大学默塞德分校的发展和教育使命，这所大学最近在加州最具经济挑战的地区之一成立。稳定和不稳定流形是控制被动粒子在稳定、不依赖于时间的流动中的输运和混合的关键结构。这些流形既将流体划分为漩涡细胞，又通过流体中的“旋转门”叶控制细胞之间的运输。拉格朗日相干结构（LCSs）在非定常、非周期流动中提供了一个类似的框架。这个项目的目标是将这些理论应用于既被流体平流又被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，游动不变量流形（SwIMs）与游动边和不变环面一起，明确地依赖于游动速度，形成相空间中的关键结构。这项研究将：(i)调查这些结构在捕获游泳者和产生弹道运动中的作用，（ii）探索游泳者在混沌环境中限制和引导运输的作用，（iii）开发一种严格的方法来解释涡流中游泳者的噪音，（iv）通过与研究微流体中细菌和藻类运输的实验学家合作，将这些想法与现实世界的生物和流动联系起来。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值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.