Mathematical Methods for Chaotic Advection in Three-Dimensional Fluid Flows

三维流体流动中混沌平流的数学方法本文提出的研究目标是扩展二维、时变流动中现有的混沌平流和混合理论。特别是，我们希望用动力系统几何理论的方法来研究三维不可压缩的定常和非定常以及二维和三维可压缩的流动。我们还将研究反应和扩散对这些流动中粒子运动的影响，混沌平流的一般性质，以及在工程应用中非常相关的特殊模型：同心和偏心旋转圆柱体之间的流动。这些问题将从三维无散度矢量场几何理论的最新进展开始讨论。本文将通过理论分析和计算机模拟来解决保体图和流中的声域、共振和叶瓣动力学问题。将与三维流动中的混合实验联系起来。惯性和粘度对混沌混合的影响将通过结合动力系统输运理论的渐近大雷诺数分析来研究。这将允许讨论层流、不可压缩、粘性流的混合特性随雷诺数变化的变化。在此基础上，利用微扰多尺度方法结合遍历理论，通过数值模拟研究分子扩散对混沌平流的影响。我们将研究纯平流问题（可能有混沌混合）和完全平流扩散问题之间的一般关系。我们将通过前面提到的工具组合来研究反应-扩散-平流方程。我们将进行非线性稳定性分析，以寻找混沌平流不稳定性的影响。我们将研究的具体例子是同心旋转圆柱体之间的流动。可压缩流在混沌平流研究中几乎没有受到重视。我们建议通过对这个问题进行基础研究来纠正这种情况。简单的模型流将从圆柱形容器中的可压缩左旋流开始识别。将与不可压缩流的行为进行比较。上述研究在各种技术背景下都是有用的。最近人们对微纳米技术的兴趣激增，这带来了许多数学上的挑战。例如，大型混合器和燃烧室等流动装置的设计目的是通过在湍流状态下移动流动来实现良好的混合，从而通过流动中流体的快速随机运动来增强混合。这在微观设备中是不可能的。混合的过程需要更好地理解，以便设计微观的混合器、燃烧室等，这些都是微观处理设备和微型发动机的必要组成部分。这种装置中的混合过程通常是三维的。因此，微观混合装置的设计将受益于上述三维混合过程的基础研究。此外，对环境化学污染（NO_x）和声污染水平的新要求也对宏观器件的性能提出了挑战。这些装置设计的改进将基于对潜在混合过程的更好理解。上述研究将通过揭示三维流动中混合的基本原理，并使用这些概念来研究混合对燃烧和噪声产生的影响，为这种设计提供一些关键概念。

DMS 9803555Mathematical Methods for Chaotic Advection in Three-dimensional FluidFlowsPI: Igor MezicThe goal of the proposed research is to extend the existing theory ofchaotic advection and mixing in two-dimensional, time-dependent flows.In particular, we wish to study three-dimensional incompressible steadyand unsteady and two- and three-dimensional compressible flows usingthe methods of geometric theory of dynamical systems. We will alsostudy effects of reaction and diffusion on the motion of particles inthese flows, properties of chaotic advection in general, and particularmodels of great relevance in engineering applications: flows betweenconcentric and eccentric rotating cylinders. These problems will beaddressed starting from our recent developments in geometric theory ofthree-dimensional divergence-free vector fields. The issues ofcantori, resonances and lobe dynamics in volume-preserving maps andflows will be addressed through theoretical analysis and computersimulation. Connection with experiments on mixing in three-dimensionalflows will be made. Effects of inertia and viscosity on chaotic mixingwill be studied via asymptotic, large Reynolds number analysis inconjunction with the transport theory of dynamical systems. This willallow for discussion of the change of mixing properties of laminar,incompressible, viscous flows with the change of the Reynolds number.Further, we will study the effects of molecular diffusion on chaoticadvection based on the perturbative multiple-scales method combinedwith ergodic theory, and through numerical simulations. We willinvestigate a general relationship between the pure advection problem(possibly with chaotic mixing) and the full advection-diffusionproblem. We will study reaction-diffusion-advection equations throughthe combination of tools mentioned before. We shall pursue nonlinearstability analysis in search of effects of chaotic advection onstability. The particular examples we will be studying are flowsbetween concentric rotating cylinders. Compressible flows receivedvirtually no attention in chaotic advection studies. We propose toremedy this situation by pursuing a basic study of this problem.Simple model flows will be identified starting with a compressiblevortex flow in a cylindrical container. Comparison with the behavior ofincompressible flows will be pursued.The above study is useful in a variety of technological contexts. Thereis a recent surge of interest in micro- and nano-scale technology thatposes a number of mathematical challenges. For example, flows indevices such as large mixers and combustion chambers are designed tomix well by moving the flow in a turbulent regime and thus causingenhanced mixing by rapid random movements of fluid in the flow. This isnot possible in microscopic devices. The process of mixing needs to beunderstood much better in order to design microscopic mixers,combustion chambers etc. necessary as the building block ofmicroscopic processing devices and microengines. The mixing process insuch devices is typically three-dimensional. Thus, the design ofmicroscopic mixing devices will benefit from the fundamental study ofthree-dimensional mixing processes outlined above. In addition, theperformance of macroscopic devices is challenged by new requirementson the levels of environmental chemical pollution (NO_x) and soundpollution. The improvement in the design of these devices will be basedon a better understanding of the underlying mixing processes. The abovestudy will provide some of the crucial concepts for such design byunraveling the fundamentals of mixing in three-dimensional flows andusing these concepts to study the effect of mixing on combustion andnoise production.