Nonlinear dynamics of microscale interfacial flows and model nonlinear partial differential equations

这项海外旅行补助金(OTG)计划寻求资金，使国际和平研究所能够访问普林斯顿大学和东京理工大学进行以下两个子项目的研究：1.液滴运动与液滴动态变化(普林斯顿大学)。液滴运动在广泛的自然现象和技术应用中普遍存在。从各种表面，如植物叶子和我们房子的窗户与雨滴相互作用的方式，到快速增长的微流体和纳米流体领域。一个具有技术意义的问题是目前正在普林斯顿大学进行主要实验研究的聚合物电解质膜(PEM)燃料电池。该系统涉及复杂几何形状的多相流，特别是从气孔中涌出并生长到气体流动通道中的液滴。整个问题相当复杂，将考虑一个更简单的原型，即固体衬底上的液滴，液滴可以从小孔中泵入或泵出(因此，模拟PEM单元中液滴的生长过程)。特别令人感兴趣的是底物无序的影响，无论是化学的还是地形的，以及噪声的影响。事实上，在PEM燃料电池中，液滴不断地与无序的衬底接触，它们也受到系统中自然存在的波动的影响。尽管这个问题很简单，但正如我们在Support 2中讨论的那样，它具有相当复杂的动态。空间扩展系统(SES)中复杂时空行为的非线性预测分析(东京理工学院)。空间扩展系统是在大范围或无界区域中用偏微分方程(PDE)确定性或随机性描述的无限维动力系统，其典型特征是存在广泛的特征长度和时间尺度，这往往导致复杂的时空行为。这类系统的一个例子是广义Kuramoto-Sivashinsky(GKS)方程，它保留了任何涉及一维波动演化的非线性过程的基本元素。该方程在从等离子体和地球物理现象到液膜下落的各种物理和技术背景下都有报道。多年来，确定性GKS方程受到了相当大的关注。其中一个主要发现是，足够强的色散倾向于规则化KS方程的时空混沌，有利于空间周期细胞结构。噪声的GKS方程也出现在各种各样的物理和技术背景中，例如通过溅射的固体薄膜的演化。提出的OTG旨在探索噪声对GKS方程的影响，并建立条件，在该条件下可以区分GKS方程的混沌行为(对于小色散)和由噪声引起的随机效应。一个相关的问题是噪声SES中的同步问题。同步是许多应用和自然现象的核心，从电路到生物系统，例如生物的合作行为。在这里，我们将使用一个耦合的噪声GKS方程系统作为原型来研究噪声SES中的同步。

This Overseas Travel Grant (OTG) proposal seeks funding to enable visits by the PI to Princeton U. and Tokyo U. of Science to undertake research in the following two subprojects:1. Droplet motion with dynamic droplet variation (Princeton U.).Droplet motion is ubiquitous in a wide spectrum of natural phenomena and technological applications. From the way various surfaces such as plant leaves and windows of our houses interact with rain droplets, to the rapidly growing field of micro- and nanofluidics. A problem of technological significance is that of polymer electrolyte membrane (PEM) fuel cells currently being investigated primarily experimentally at Princeton U. This system involves multiphase flows in complex geometries and in particular droplets that emerge from pores and grow into gas flow channels. The full problem is quite involved and a simpler prototype, that of a droplet on a solid substrate with a small pore from which liquid can be pumped in or out (thus, emulating the growth process of droplets in the PEM cells), will be considered. Of particular interest are the effects of substrate disorder, either chemical or topographical and influence of noise. Indeed, in the PEM fuel cells the droplets are constantly in contact with disordered substrates and they are also subjected to fluctuations which are naturally present in the system. Despite its simplicity, this problem has a rather complex dynamics as we discuss in the Case of Support.2. Nonlinear forecasting analysis of complex spatiotemporal behavior in spatially extended systems (SES) (Tokyo U. of Science).SES are infinite-dimensional dynamical systems described through partial differential equations (PDEs) deterministic or stochastic in large or unbounded domains, and are typically characterized by the presence of a wide range of characteristic length and time scales which often leads to complex spatiotemporal behavior. An example of such systems is the generalized Kuramoto-Sivashinsky (gKS) equation, a prototype that retains the fundamental elements of any nonlinear process that involves wave evolution in one dimension. The equation has been reported in a wide variety of physical and technological contexts, from plasma and geophysical phenomena to falling liquid films.The deterministic gKS equation has received considerable attention over the years. One of the main findings is that sufficiently strong dispersion tends to regularise the spatiotemporal chaos of the KS equation in favor of spatially periodic cellular structures. The noisy gKS equation also appears in a wide variety of physical and technological contexts, e.g. evolution of solid films by sputtering. The proposed OTG seeks to explore the effects on noise on the gKS equation and to establish conditions under which it is possible to distinguish between the chaotic behavior of the gKS equation (for small dispersion) and the stochastic effects induced by noise. A related problem is that of synchronization in noisy SES. Synchronization is central to many applications and natural phenomena, from electric circuits to biological systems, e.g. the cooperative behavior of living beings. Here we shall examine synchronization in noisy SES using a system of coupled noisy gKS equations as a prototype.