Mathematical Investigations of Nonlinear Free Boundary Problems in Stokes Flow

DMS-9803167 Stokes流中非线性自由边界问题的数学研究研究者：Michael Brenner和Darren Crowdy项目概述该提案概述了二维和三维慢粘性（Stokes）流体区域的一类自由边界问题的数学研究。该建议的动机是最近的结果值得注意的数学结构的一个基本问题，在研究二维斯托克斯流。 Crowdy和Tanveer设计了一种新的全局理论方法来解决二维Stokes流问题，从而确定了与简单连通流体区域的一大类精确解相关的无穷多个守恒量，并发现了双连通流体区域的第一个已知精确解。 这些有意义的数学结果将在各个方向得到发展。 主要的兴趣是推广到三维Stokes流的重要和高度非平凡的问题。 Nie和Tanveer的初步数值研究表明，轴对称（3D）Stokes泡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Stokes流的结果将被推广到各个方向。 连通度大于2的区域中的精确解是一个很有意义的问题。 此外，还将研究其他物理因素，包括静电（电场）和热毛细效应。所要研究的数学问题具有很大的物理意义和应用。这些解为粘性烧结的物理过程提供了有用的模型。烧结是一个广义的术语，指的是颗粒集合体的固结，其中质量传输是由表面张力驱动的，并且存在关于该主题的大量文献，涵盖了从材料科学到环境工程的许多学科。 了解这种烧结过程是非常重要的，例如，在光纤和光电技术。 研究电场作用下粘性流体小液滴的边界演化对于理解喷墨打印技术是至关重要的。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通过边界上的表面张力保持在一起的非常粘稠的流体（带电）团块。该项目将提供对这些重要物理模型下的数学的理解。

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.