Mathematical Sciences: Well-Posed Numerical Calculations of Free-Surface Flows

9308075贝克，研究人员通过研究一个相关解析问题中奇点的发展来研究流体流动中自由表面的运动。在许多自由表面流动中，粘度和表面张力的影响似乎很小，以至于可以忽略。然而，由于曲率奇点的形成，用数值方法(例如边界积分方法)来求解自由面运动的尝试很快就会遇到困难。只有当其中一种流体是有效的真空时，数值计算才表明自由面的运动是适定的。显然，表面张力和粘度的影响，无论系数多么小，在奇点发展的时间附近都变得非常重要。准确地说，这些正规化物质如何防止奇点的形成，以及随后的运动的性质是什么，是这一提议的主要推动力。为了实现这一目标，研究人员使用了一个简单的数学观点，它通过将边界积分解析地扩展到复弧长变量(或其他合适的参数化变量)来描述奇点的形成。在这个复杂的平面上，奇点产生了，这取决于初始条件的细节。然后，这些奇点在复平面中跟踪，并可能在有限时间内到达实轴。在这一点上，它们变得物理上相关。对表面张力影响的初步研究表明，奇点的速度变慢，在有限时间内不能到达实轴，但它们非常接近。这些奇点非常接近真实的轴，这困扰着标准的数值方法，因为捕捉极度扭曲的自由面需要高分辨率。取而代之的是，研究人员使用一种方法，显式地表示复平面上的任何奇点，以便自由表面的其余行为是解析的。特别是，这个解析部分可以非常精确地用傅立叶级数来表示。第一个任务是将这种方法用于一个更简单的经典自由面流动问题，即Hele-Shaw单元中的指进不稳定性。二是研究表面张力作用下两等密度不可压缩流体间界面的Kelvin-Helmholtz不稳定性问题。不相容的液体和气体之间的表面在自然界和技术中大量存在。一些例子包括化学反应单元中气泡的上升，落下的雨滴，喷雾射流，以及地下流动的石油和水。试图描述表面运动的数学模型通常对运动的细节表现出极大的敏感性。因此，除非格外小心，否则计算机模拟可能会有不准确之处。研究人员开发了一种概念上的新数学方法，可以可靠地跟踪严重变形的几何图形。特别是，该方法将能够跟踪Hele-Shaw Cell中石油渗透水的长期行为，Hele-Shaw Cell是一个在数学上类似的模型，用于描述油田二次采油过程中油和水的运动。一旦完全理解了这个重要的问题，这种方法将适用于研究晶体生长过程中上升的气泡和树枝状结构。***

9308075 Baker The investigator studies the motion of free surfaces in fluid flows by investigating the development of singularities in a related analytic problem. In many free-surface flows, the effects of viscosity and surface tension appear to be so small that they are neglected. However, attempts to solve the motion of the free surface by numerical techniques, boundary integral techniques for example, soon run into difficulties due to the formation of curvature singularities. Only when one of the fluids is effectively a vacuum do numerical calculations indicate that the motion of the free-surface is well-posed. Obviously, the effects of surface tension and viscosity, no matter how small the coefficients are, become very important close to the time of singularity development. Precisely how these regularizing agents prevent singularity formation, and what the nature of the subsequent motion is, are the main thrusts of this proposal. To achieve this objective, the investigator uses a simple mathematical view that describes singularity formation through the analytic extension of the boundary integrals into the complex arclength variable (or other suitable parametrization variable). In this complex plane, singularities are born, depending on the details of the initial conditions. These singularities then track through the complex plane, and may reach the real axis in finite time. At that point they become physically relevant. Preliminary studies of the influence of surface tension show that the singularities slow down and fail to reach the real axis in finite time, but they get extremely close. The close proximity of these singularities to the real axis plagues standard numerical methods, because of the high resolution needed to capture the extremely distorted free-surface. Instead, the investigator uses a method that represents any singularities in the complex plane explicitly, so that the rest of the behavior of the free-surface is analytic. In particular, this analytic part can be represented by Fourier series very accurately. The first task is to use this approach on a simpler classical free-surface flow problem, the fingering instability in a Hele-Shaw cell. The second is to explore the method on the Kelvin-Helmholtz instability of an interface between two incompressible fluids of equal density and in the presence of surface tension. Surfaces between immiscible liquids and gases occur abundantly in nature and technology. Some examples include the rise of bubbles in chemically reacting units, falling rain-drops, spray jets, and oil and water flowing in the ground. Mathematical models that attempt to describe the motion of surfaces typically exhibit great sensitivity to the details of the motion. Consequently, computer simulations can suffer from inaccuracies unless extraordinary care is taken. The investigator develops a conceptually new mathematical approach that will allow severely deformed geometries to be tracked reliably. In particular, the approach will be able to track the long term behavior of water penetrating oil in a Hele-Shaw Cell, a mathematically similar model for the motion of oil and water during secondary stage recovery from an oil-field. Once completely understood on this important problem, the approach will be adapted to the study of rising bubbles and dendritic formation during crystal growth. ***