Free Boundary Problems with Surface Tension

自由边界问题在过去的几十年里，自由边界问题由于其理论上的兴趣和在自然科学和工程科学中的大量应用而引起越来越多的关注。自由边界问题在材料科学、流体力学、流体力学、热力学、磁动力学、固体物理学、地质学、化学、生物和医学等许多领域都具有重要意义。对于工程师和数学家来说，恰当的数值和分析处理都是一个重大挑战。通常，自由边界问题由一个或多个偏微分方程组组成，这些偏微分方程组必须在先验未知的区域内求解，并且必须作为问题的一部分确定。自由边界问题通常比基本的微分方程式在规定的区域内更难解决，无论是在分析上还是在数值上。它们有一种内在的非线性结构，因为两个独立的解不能叠加。在这个项目中，PI建议系统地研究在存在表面张力的情况下具有移动接触线的各种自由边界问题。移动接触线在许多情况下都会发生，例如，在粘性流体对固体表面进行涂层、微芯片的旋转涂层、一种流体沿固体边界由另一种流体置换、液滴在固体表面上扩散、在自由表面与容器壁接触的容器中的流体(或流体-液体系统)的运动中。接触线运动的建模与表征是毛细管理论尚未解决的主要问题之一。

Free Boundary Problems with Surface Tension.Abstract of Proposed ResearchGerri Simonett Over the last decades, the subject of free boundary problems has attracted increasing attention because of its theoretical interest, and because of its numerous applications in the natural and engineering sciences. Free boundary problems are important in many fields such as material sciences, fluid mechanics, hydrodynamics, thermo-mechanics, magneto-dynamics, solid state physics, geology, chemistry, and the biological and medical sciences. The appropriate numerical and analytical treatment is a major challenge, both to the engineer and to the mathematician. Typically, a free boundary problem consists of one or more partial differential equations which have to be solved in a domain that is a priori unknown and that has to be determined as part of the problem. Free boundary problems are in general harder to solve, both analytically and numerically, than the underlying differential equations would be in a prescribed domain. They have an inherent nonlinear structure, as two separate solutions cannot be superposed. In this project, the PI proposes a systematic investigation of various free boundary problems with moving contact lines in the presence of surface tension. Moving contact lines occur in many situations, for example in processes such as the coating of solid surfaces by a viscous fluid, spin coating of micro chips, the displacement of one fluid by another fluid along a solid boundary, the spreading of drops on solid surfaces, the motion of a fluid (or a fluid-liquid system) in a container where the free surface is in contact with the wall of the container. The modeling and characterization of contact line motion is one of the major unsolved problems of capillary theory.