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

弹性结构与周围流体相互作用的问题可以在整个自然界和工程中找到。这种流体结构相互作用（FSI）问题包括鸟的飞行、鱼的游泳以及通过心脏和血管的血液流动。 一类特别重要的FSI问题是那些在小空间尺度。细胞生物过程的生物物理学就是一个非常重要的例子。在这样的流固耦合问题中，流体可以被视为非常粘滞的流体（斯托克斯流体）。本计画将研究Stokes流体中的流固耦合问题。描述流固耦合问题的数学方程通常很难进行数学研究，但斯托克斯流体方程的相对简单性使得对这类流固耦合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。在第一部分中，我们研究了Stokes流体中膜的动力学。这种模型通常用于描述细胞膜的动力学。第二部分研究了Stokes流中细丝的动力学。这类模型常用于描述微生物鞭毛和精子的动力学。一个详细的数学研究，这样的FSI问题将导致更好地理解FSI问题一般，也发展有效的计算算法，在模拟这样的问题。该奖项还将为本科生和研究生参与研究提供支持。流体结构相互作用（FSI）问题，其中弹性结构与周围流体相互作用，在科学和工程中大量存在，许多作者通过计算方法进行了深入研究。尽管它们的重要性，这些问题的偏微分方程和数值方法的治理没有很好地理解从分析的角度来看。在这项研究中，我们专注于一组典型的流固耦合问题，其中的弹性结构与流体的相互作用服从斯托克斯方程。该项目包括两个部分。首先，我们将研究Stokes流中的余维一维弹性界面问题。基于Mori和他的合作者在1D弹性结构/2D Stokes流问题（Peskin问题）的适定性上的先前工作，我们将适定性理论扩展到更一般和相关的问题，包括3D Stokes流中的2D弹性表面的问题。我们还将发展一个收敛理论的边界积分方法，这样的问题，并开始研究这些结果的收敛性分析的流体网格为基础的方法。第二部分是细长体理论，研究三维Stokes流中细丝的动力学。我们最近成功地提供了细长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体计算的新计算方法。该奖项反映了NSF的法定使命，并通过使用基金会的智力价值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.