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

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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.