Mathematical Sciences: Physical Regularizations of the Motion of a Vortex Sheet

该奖项为在大雷诺数范围内对薄剪切层中流体行为的研究提供持续支持。对这些现象的经典处理将薄层视为理想化的涡旋层，并且只以一种特殊的方式包括粘性(内耗)的耗散效应，而不涉及粘性力的真实性质。相比之下，该奖项的研究集中在分析和计算方法上，以逼近实际粘性方程在粘度为零的极限下的解。具体的研究问题包括描述粘性效应如何影响涡片发展曲率奇异性的趋势。自然界中的许多流体动力学现象都是在零粘度或零表面张力的理想化极限下工作的。例如，物体在空气中以远低于音速的速度运动，可以在空气无粘性的近似下进行研究。当然，这是虚构的，因为空气确实有一个小的、非零的粘性，但在计算一系列空气动力学系数时，忽略粘性影响是合理的，因为粘性的小影响远不如压降等更大的影响重要。然而，有许多流体动力学现象的存在依赖于形式上像粘性这样的小力的存在，而这正是本提议中研究的对象。特别是，这项研究将研究粘性效应如何影响薄层流体内部的涡旋卷起。

This award provides for the continued support of research on the behavior of fluids in thin shear layers in the limit of large Reynolds numbers. The classical treatment of these phenomena regards the thin layer as an idealized vortex sheet and only includes the dissipative effects of viscosity (internal friction) in an ad hoc fashion that never comes to grips with the true nature of viscous forces. In contrast the research in this award centers around analytical and computational methods for approximating solutions of the actual viscous equations in the limit of vanishing viscosity. Specific problems for study include the description of how viscous effects influence the tendency for vortex sheets to develop curvature singularities. Many fluid dynamical phenomena in nature are operative in the idealized limit of zero viscosity or zero surface tension. For example, the motion of bodies in air at speeds well below the speed of sound can be studied under the approximation that the air is inviscid. Of course, this is a fiction since air does have a small, nonzero viscosity, but the neglect of viscous effects is justified when calculating a host of aerodynamical coefficients, since the small effect of viscosity is not nearly as important as a larger effect, such as a pressure drop. However there are many fluid dynamical phenomena whose very existence depends upon the presence of a formally small force like viscosity, and it is just such phenomena that are the objects of study in this proposal. In particular, this research will study how viscous effects influence the roll-up of vortices inside of thin layers of fluids.