Mathematical Investigations of Nonlinear Free Boundary Problems in Stokes Flow

研究人员：Michael Brenner和Darren CrowdyPROJECT摘要：本文概述了一类慢粘性（Stokes）流体区域的二维和三维自由边界问题的数学研究。这一提议的动机是最近在研究二维斯托克斯流的一个基本问题的基础上的一个了不起的数学结构的结果。Crowdy和Tanveer设计了一种新的全局理论方法来解决二维Stokesflow问题，从而确定了与单连通流体区域的大量精确解相关的无穷守恒量，并发现了双连通流体区域的第一已知精确解。这些重要的数学结果将向各个方向发展。主要的兴趣是对三维斯托克斯流的推广这一重要而又非常重要的问题。Nie和Tanveer的初步数值研究表明，轴对称（3D） Stokesbubble表现出与其二维类似物非常相似的定性行为（例如，近尖点的形成和拓扑变化）。二维斯托克斯流的结果将在各个方向上推广。在连通性大于2的区域的精确解非常有趣。此外，还将研究其他物理因素，包括静电（电场）和热毛细效应。所研究的数学问题具有很大的物理意义和应用价值。这些解可作为粘性烧结物理过程的有用模型。烧结是一个广义的术语，指的是由表面张力驱动的颗粒组合的固结，从材料科学到环境工程的许多学科都有关于这个主题的大量文献。了解这种烧结过程非常重要，例如在光纤和光电技术中。研究电场作用下粘性流体小块的边界演化对于理解喷墨打印技术至关重要。该项目还具有潜在的生物技术相关性——细胞结构和膜通常被建模为（带电的）非常粘稠的流体团，通过边界上的表面张力保持在一起。这个项目将提供对这些重要物理模型背后的数学的理解。

DMS-9803167Mathematical Investigation of Nonlinear Free Boundary Problems inStokes FlowInvestigators: Michael Brenner and Darren CrowdyPROJECT SUMMARYThis proposal outlines a mathematical investigation of a class of freeboundary problems for slow viscous (Stokes) fluid regions in both twoand three dimensions. The proposal is motivated by recent results on aremarkable mathematical structure underlying a fundamental problem inthe study of two-dimensional Stokes flow. Crowdy and Tanveer havedevised a new global theoretical approach to the problem of 2D Stokesflow which resulted in the identification of an infinity of conservedquantities associated with a large class of exact solutions for asimply-connected fluid region, and the discovery of the first-knownexact solutions for doubly-connected fluid regions. These significantmathematical results will be developed in various directions. Ofprimary interest is the important and highly non-trivial question ofgeneralization to three-dimensional Stokes flow. Preliminary numericalstudies by Nie and Tanveer have shown that axisymmetric (3D) Stokesbubble exhibits very similar qualitative behavior to itstwo-dimensional analogue (e.g. formation of near-cusps and topologicalchanges). The results on two-dimensional Stokes flow will begeneralized in various directions. Exact solutions in regions ofconnectivity greater than two is of great interest. Moreover,additional physical factors will be studied including electrostatic(electric fields) and thermo-capillary effects.The mathematical problems to be studied are of great physical interestand application. The solutions serve as useful models in the physicalprocess of viscous sintering. Sintering is a term broadly referring tothe consolidation of an assemblage of particles in which the masstransport is driven by surface tension and there exists a hugeliterature on the subject spanning many disciplines from materialsscience to environmental engineering. Understanding such sinteringprocesses is very important, for example, in fiber-optic andopto-electronic technologies. The study of the boundary evolution ofsmall blobs of viscous fluid in the presence of an electric field iscrucial for the understanding of ink-jet printing technologies. Theproject also has potential biotechnological relevance -- cellsstructures and membranes are often modelled as (charged) blobs of veryviscous fluid kept together by surface tension forces on the boundary.This project will provide an understanding of the mathematicsunderlying these important physical models.