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

该奖项旨在继续支持 薄剪切层中流体行为的研究 大雷诺数的极限。 经典的治疗方法 这些现象把薄层看成是一种理想化的涡流 表，仅包括粘性耗散效应 （内部摩擦）以一种特别的方式， 掌握了粘性力的本质 相比之下 该奖项的研究围绕分析和 计算方法的近似解决方案的实际 在粘性消失极限下的粘性方程。 具体 研究的问题包括如何粘性效应的描述 影响涡面发展曲率的趋势 奇点 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极限。 例如，物体在空气中的运动速度远低于 可以在近似下研究声速， 空气是无粘性的。 当然，这是一个虚构的，因为空气 有一个小的，非零粘度，但忽略粘性 在计算一系列空气动力学参数时， 系数，因为粘度的小影响并不接近 与更大的影响，如压降一样重要。 然而，有许多流体动力学现象， 存在取决于形式上的小部队的存在 就像粘度一样，正是这种现象 研究对象在此建议。 这尤其 研究将研究粘性效应如何影响 流体薄层内部的漩涡。

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.