Nonlinear water waves driven by the motion of a solid body

长期以来，工程师、物理学家和应用数学家一直对水波感兴趣。部分浸没固体的运动产生的波浪在海洋和海军工程中具有明显的意义，在波浪发电中也具有重要的意义。这方面的大部分理论研究都是基于运动方程的数值解。然而，在自由表面与固体相遇的流动中，起始流动通常会形成一个通用的内部渐近区域，在该区域内，流动会在小范围内演化。这一区域可以控制对物体的初始载荷的计算，并对后续流动有显著影响。最近，李约瑟、比林厄姆和他的同事们一直在研究由刚性移动的墙驱动的自由表面流的最初发展。特别地，对于恒加速度斜板的情况，我们证明了存在一个临界板角度，在这个临界角处，内区的自由面上形成一个120度角--一个非周期的类似于斯托克斯最高周期波的角。对于更大的倾角，没有解决方案。当考虑表面张力并且接触角使得自由表面最初是水平的时，解存在有限时间，直到在自由表面上形成的非线性毛细管波的振幅变得足够大，足以将气泡掐掉。对于一般的接触角，在相关参数空间中存在丰富的结构，我们在本项目中试图通过分析、数值和实验相结合的工作来阐明这一点。我们还将把我们的分析扩展到平板平稳地从流体层中抽出，以及平板被脉冲移动的情况。这个项目是对流体力学中一个容易陈述但很难解决的基本问题的研究。该项目的所有方面都涉及重大的技术挑战--渐近的、数值的和实验的。此外，我们最近的结果使我们相信，数学解的定性结构和相应的流动既复杂又有趣，将使我们能够深入了解移动固体所产生的水波及其随后的传播。

Water waves have long been of interest to engineers, physicists and applied mathematicians. The generation of waves by the motion of partially immersed solid bodies is of obvious relevance in marine and naval engineering, as well as being important in wave power generation. Most theoretical studies in this area have been based on numerical solutions of the equations of motion. However, in flows where a free surface meets a solid body, the starting flow usually develops a generic inner asymptotic region where the flow evolves on a small scale. This region can dominate the calculation of the initial load on the body, and have a significant effect on the subsequent flow. Recently, Needham, Billingham and co-workers have been studying the initial development of free surface flow driven by a rigid, moving wall. In particular, for the case of an inclined plate with constant acceleration, we have shown that there is a critical plate angle at which a 120 degree corner is formed on the free surface within the inner region -- an aperiodic analogue of Stokes' highest periodic wave. For larger inclination angles, no solution is possible. When surface tension is taken into account and the contact angle is such that the free surface is initially horizontal, the solution exists for a finite time until the amplitude of the nonlinear capillary wave that forms on the free surface becomes large enough for a bubble to be pinched off. For general contact angles, there is a rich structure in the relevant parameter space, which we seek to elucidate in this project through a combination of analytical, numerical and experimental work. We will also extend our analyses to the case of a plate smoothly withdrawn from a fluid layer, and an impulsively-moved plate. This project is an investigation into a fundamental problem in fluid mechanics that is easy to state, but difficult to tackle. There are significant technical challenges involved in all aspects of the project -- asymptotic, numerical and experimental. Moreover, our recent results lead us to believe that the qualitative structure of the mathematical solutions, and the corresponding flows, is both intricate and interesting, and will allow us significant insight into the generation of water waves by moving solid bodies, and their subsequent propagation.