Free Boundary Problems with Surface Tension

具有表面张力的自由边界问题。在过去的几十年里，自由边界问题由于其理论价值以及在自然科学和工程科学中的广泛应用而引起了越来越多的关注。自由边界问题在许多领域都很重要，如材料科学、流体力学、流体力学、热力学、磁动力学、固体物理学、地质学、化学以及生物和医学科学。对工程师和数学家来说，适当的数值和解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确定。一般来说，自由边界问题无论在解析上还是在数值上都比在规定的区域内求解基本微分方程要困难。它们具有固有的非线性结构，因为两个独立的解不能叠加。在这个项目中，PI提出了一个系统的研究各种自由边界问题与移动接触线在表面张力的存在。移动接触线发生在许多情况下，例如，粘性流体对固体表面的涂覆、微芯片的自旋涂覆、一种流体沿着固体边界被另一种流体所取代、液滴在固体表面上的扩散、容器中流体（或流-液系统）的运动，其中自由表面与容器壁接触。接触线运动的建模和表征是毛细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.