Collaborative Research: Analysis and Computation of Dynamics of Elastic Structures in Stokes Flow

在自然界和工程中，弹性结构与周围流体相互作用的问题随处可见。这类流体结构相互作用(FSI)问题包括鸟类的飞行、鱼的游动以及血液在心脏和血管中的流动。一类特别重要的FSI问题是小空间尺度上的问题。细胞生物学过程的生物物理学就是一个非常重要的例子。在这样的FSI问题中，流体可以被视为非常粘性的流体(斯托克斯流体)。本项目将研究Stokes流体中的FSI问题。描述FSI问题的数学方程通常很难数学研究，但Stokes流体方程的相对简单使得对这类FSI问题进行详细的数学研究成为可能。研究计划分为两个部分。首先，我们研究了Stokes流体中膜的动力学。这样的模型经常被用来描述细胞膜的动力学。在第二章中，我们研究了斯托克斯流中细丝的动力学。这样的模型经常被用来描述微生物和精子的鞭毛的动态。对这类FSI问题进行详细的数学研究将有助于更好地理解FSI问题，也有助于开发有效的计算算法来模拟这类问题。该奖项还将为本科生和研究生参与研究提供支持。弹性结构与周围流体相互作用的流体结构相互作用(FSI)问题在科学和工程中广泛存在，许多作者通过计算方法对其进行了深入的研究。尽管它们很重要，但从分析的角度来看，控制偏微分方程组和这类问题的数值方法并没有得到很好的理解。在这项研究中，我们集中于一组正则的FSI问题，其中弹性结构与服从Stokes方程的流体相互作用。该项目由两部分组成。首先，我们将研究处于Stokes流中的一维共维弹性界面问题。在Mori和他的合作者关于一维弹性结构/二维Stokes流问题(Peskin问题)适定性工作的基础上，我们将把适定性理论扩展到更一般的相关问题，包括三维Stokes流中的二维弹性面问题。我们还将发展这类问题的边界积分方法的收敛理论，并开始研究将这些结果扩展到基于流体网格的方法的收敛分析。第二部分是细长体理论，它涉及三维Stokes流中细丝的动力学。我们最近成功地提供了细长身体近似的第一个数学证明，我们将利用它来进一步分析和开发细长身体计算的新计算方法。这一奖项反映了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。

Problems in which elastic structures interact with a surrounding fluid can be found throughout the natural world and in engineering. Such fluid structure interaction (FSI) problems include the flying of birds, the swimming of fish, and blood flow through the heart and blood vessels. A particularly important class of FSI problems are those at small spatial scales. The biophysics of cell biological processes is a very important example of this. In such FSI problems, the fluid can be treated as a very viscous fluid (Stokes fluid). This project will study FSI problems in a Stokes fluid. The mathematical equations describing FSI problems are generally difficult to study mathematically, but the relative simplicity of the equations of Stokes fluids makes possible a detailed mathematical study of this class of FSI problems. The research program is divided into two parts. In the first, we study the dynamics of membranes in a Stokes fluid. Such models are often used to describe the dynamics of the cell membrane. In the second, we study the dynamics of filaments in Stokes flow. Such models are often used to describe the dynamics of flagella of microorganisms and sperm. A detailed mathematical study of such FSI problems will lead to a better understanding of FSI problems in general and also to the development of efficient computational algorithms in the simulation of such problems. This award will also provide support for the involvement of undergraduate and graduate students in the research.Fluid structure interaction (FSI) problems in which an elastic structure interacts with the surrounding fluid abound in science and engineering and are studied intensively by computational methods by many authors. Despite their importance, the governing partial differential equations and the numerical methods for such problems are not well-understood from an analytical standpoint. In this research, we focus on a set of canonical FSI problems in which the elastic structures interact with a fluid obeying the Stokes equations. The project consists of two parts. In the first, we will study the problem of co-dimension one elastic interfaces immersed in Stokes flow. Building upon previous work by Mori and collaborators on the well-posedness of the 1D elastic structure/2D Stokes flow problem (Peskin problem), we shall extend the well-posedness theory to more general and related problems including the problem of a 2D elastic surface in 3D Stokes flow. We shall also develop a convergence theory for boundary integral methods for such problems and initiate a study to extend these results to the convergence analysis of fluid-grid based methods. The second part concerns slender body theory, which deals with the dynamics of thin filaments in 3D Stokes flow. We have recently succeeded in providing the first mathematical justification of slender body approximation, which we shall leverage to further our analysis and develop new computational methods for slender body computation.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.