Mathematical Investigations of Nonlinear Free Boundary Problems in Stokes Flow

DMS-9803167斯托克斯流动中非线性自由边界问题的数学研究这项建议概述了一类二维和三维慢粘(Stokes)流体区域自由边界问题的数学研究。这一建议的提出是基于最近关于可标记的数学结构的结果，该结构是二维斯托克斯流动研究中的一个基本问题。Crowdy和Tanveer提出了解决二维Stokes流问题的一种新的整体理论方法，其结果是确定了与简单连通流体区域的一大类精确解相关的无穷大守恒量，并发现了双连通流体区域的第一个已知的精确解。这些重要的数学结果将在不同的方向得到发展。主要的兴趣是将其推广到三维Stokes流的重要且非平凡的问题。Nie和Tanveer的初步数值研究表明，轴对称(3D)Stokes气泡表现出与其二维类似物非常相似的定性行为(例如，近尖的形成和拓扑变化)。关于二维Stokes流的结果将在不同方向得到推广。连通度大于2的区域的精确解是人们非常感兴趣的。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价值。这些解在粘性烧结的物理过程中是有用的模型。烧结是一个术语，广义上指的是一组颗粒的固结，其中的运动是由表面张力驱动的，在这个问题上存在着一个巨大的文献，涉及从材料科学到环境工程的许多学科。例如，在光纤和光电子技术中，了解这种烧结过程是非常重要的。研究电场作用下小颗粒粘性流体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模拟为(带电的)粘性液体团块，通过边界上的表面张力保持在一起。该项目将提供对这些重要物理模型背后的数学基础的理解。

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.