Free Boundary Problems with Surface Tension

自由边界问题与表面张力。摘要建议researchGerri Simonett在过去的几十年里，自由边界问题的主题已经引起了越来越多的关注，因为它的理论兴趣，并因为它在自然科学和工程科学的众多应用。自由边界问题在材料科学、流体力学、流体动力学、热力学、磁动力学、固体物理学、地质学、化学、生物学和医学等许多领域都有重要的应用。适当的数值和分析处理是一个重大的挑战，无论是工程师和数学家。通常，自由边界问题由一个或多个偏微分方程组成，这些偏微分方程必须在先验未知的域中求解，并且必须作为问题的一部分来确定。自由边界问题一般来说，无论是解析的还是数值的，都比在规定区域内的基本微分方程更难求解。它们具有固有的非线性结构，因为两个单独的解不能叠加。 在这个项目中，PI提出了一个系统的调查各种自由边界问题与移动接触线在表面张力的存在。移动接触线出现在许多情况下，例如在诸如通过粘性流体涂覆固体表面、微芯片的旋涂、一种流体被另一种流体沿固体边界沿着置换、液滴在固体表面上的扩散、流体（或流体-液体系统）在容器中的运动（其中自由表面与容器壁接触）的过程中。接触线运动的建模和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.