Modelling, computation and analysis of droplets guided by Faraday waves: a complex system with macroscopic quantum analogies.

当一个装满液体的容器被垂直摇动时，如果震动足够强烈，人们可以在容器表面观察到波浪。这些波产生于次谐波不稳定性：它们的频率是振动的一半，被称为法拉第波。在单独的实验中，液滴撞击静态液浴的高速薄膜表明，液滴在撞击时并不总是与液浴合并，但在合并之前可能会反弹几次。结合这两个实验事实，大约在10年前，人们发现毫米级的液滴在摇晃的容器中落在同一液体的表面时可以无限地弹跳。该现象发生在法拉第不稳定性的振动阈值以下。更令人惊讶的是，液滴还可以自发地沿着振动槽的表面“行走”。然后，这些行行者表现出许多以前被认为是量子力学独有的特征，如波粒二象性、量子化能态、单粒子衍射和隧穿行为。本提案的目的是探索该系统的流体力学方面。由于问题的复杂性，有许多未解决的挑战性问题。流体力学问题包括对非聚结跌落冲击的理解，以及反射步行者及其在壁面处的导波场的行为。我们还将试图理解一个纯粹的经典力学系统是如何表现出类似量子力学的行为的，并探索这种类比的局限性。提出的研究包括分析和数值方法的结合，以及与实验的比较。我们将与麻省理工学院最先进的流体动力学实验室合作，该实验室将提供实验数据和设计验证实验。我们计划研究的问题是流体力学、自由边界问题和动力系统理论中普遍感兴趣的问题。预计这些结果将有广泛的应用，特别是在了解液滴和颗粒与流体的影响方面。法拉第不稳定性也是连续产生大小一致的液滴的最可靠的方法。正因为如此，这项研究有几种可能的微流体应用，比如开发更好的吸入药物输送设备。

When a fluid filled container is shaken vertically, one may observe waves on the surface of that container if the shaking is sufficiently strong. These waves arise out of a subharmonic instability: they have half the frequency of the shaking, and are called Faraday waves.In separate experiments, high-speed films of droplet impacts on static fluid baths show that the droplet does not always coalesce with the bath on impact, but may bounce a few times before coalescing.Combining these two experimental facts, about 10 years ago it was discovered that millimetric liquid droplets can bounce indefinitely when dropped on the surface of a bath of the same liquid in a shaken container. The phenomenon occurs below the shaking threshold for the Faraday instability. More surprisingly, the droplets can also spontaneously "walk" along the surface of the vibrating bath. These walkers then exhibit many features previously thought to be exclusive to quantum mechanics such as wave-particle duality, quantised energy states, single particle diffraction and tunnelling behaviour.The aim of this proposal is to explore the fluid mechanical aspects of this system. There are many unanswered challenging questions due to the complexity of the problem. Fluid mechanics questions include the understanding of non coalescing drop impact, and the behaviour of reflecting walkers and their pilot wave field at walls. We will also seek to understand how a purely classical mechanics system exhibits quantum mechanical-like behaviour, and probe the limits of this analogy. The proposed research involves a combination of analytical and numerical approaches as well as comparisons with experiments. We will partner with an MIT state-of-the-art fluid dynamics laboratory which will provide both experimental data and design validation experiments. The problems we plan to study are of general interest in fluid mechanics and in the theory of free boundary problems and dynamical systems. It is expected that the results will have broad applications, in particular to the understanding of the impact of drops and particles with fluids.Faraday instabilities are also the most reliable way of generating consistently sized droplets continuously. Because of this there are several possible microfluidics applications for this research, such as developing better devices for delivering inhaled drugs.