Mathematical Sciences: Diffusion in Fluid

数学科学：流体中的扩散

• 批准号: 9600119
• 负责人: Albert Fannjiang
• 金额: $ 5.11万
• 依托单位: University of California-Davis
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 1996
• 资助国家: 美国
• 起止时间: 1996-08-01 至 2000-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9600119&HistoricalAwards=false
• 关键词: Mathematical; Sciences; Diffusion; Fluid

960019 Fannjiang This project concerns the large scale behavior of particle dispersion in incompressible fluid flows. On large scales, particles evolve diffusively or nondiffusively depending on the fluid flows. One of the focuses of this project is the sharp criteria for the diffusive regime. Novel variational methods have been employed to obtain such criteria for steady flows. The extension of the variational methods to unsteady flows and weakly compressible flows are proposed. Furthermore, within the diffusive regime, the influence of fluid velocities on the transport coefficients, the effective diffusivity, is of both practical and theoretical importance, particularly when the molecular diffusivity is small. As such, the fluid velocities typically enhance the effective diffusivity by orders of magnitude. The variational methods have been successfully applied to obtain nontrivial power laws for convection enhanced diffusion in two-dimensional steady flows. It remains to be extended to analyze turbulent diffusion in three dimensional random flows and chaotic diffusion in two dimensional unsteady flows. The variational methods also provide naturally a framework for numerical computations of the transport coefficients to which the well established numerical methods for symmetric problems can now be applied. In the non-diffusive regime, being an effective tool for analyzing physical quantities for finite volume, the variational methods are potentially useful in studying anomalous diffusion in the infinite volume limit. Finally, it is of enormous interest to understand the diffusion time scale and other transient time scales. The martingale method and the extension of the contractivity estimates (for symmetric processes) to the singularly non-symmetric convection-diffusion processes are proposed to study the time scales problem. %%% Since the great scientists A. Einstein and N. Wiener laid the theoretical foundation for the study of the Brownian motion, great effort has been made to understand the combined effect of the Brownian motion on the molecular scale and the fluid motion on the ordinary scale. Such problem is also of enormous practical importance since most of the mass and heat transfer processes in chemical engineering, ground water contamination and global change in the atmosphere involve both kinds of motion. As it turns out, the typical smallness in size of molecular Brownian motion, and chaos and randomness in convective motion give rise to numerous interesting phenomena on large scales such as convection enhanced diffusion where the size of the Brownian motion is greatly enhanced, multiple scales where the transport processes behave differently on different scales, and anomalous diffusion for which an appropriate description is yet to be found. This project proposes a rigorous framework and effective methods for studying the convection- diffusion problem, which has yielded many important results. This and other hopeful research outcomes will add to our fundamental understanding of one of the most basic transport processes affecting our lives. ***

960019范江本项目涉及不可压缩流体流动中颗粒弥散的大尺度行为。在大尺度上，颗粒的扩散或非扩散演化取决于流体流动。这个项目的重点之一是对扩散制度的严格标准。新的变分方法被用来获得定常流动的这种判据。将变分方法推广到非定常流动和弱可压缩流动。此外，在扩散区内，流体速度对输运系数--有效扩散系数的影响具有重要的理论和实际意义，特别是当分子扩散系数较小时。因此，流体速度通常会使有效扩散系数增加数量级。用变分方法成功地得到了二维定常流动中对流增强扩散的非平凡幂定律。对于三维随机流动中的湍流扩散和二维非定常流动中的混沌扩散的分析还有待推广。变分方法也自然地为输运系数的数值计算提供了一个框架，现在可以将成熟的对称问题的数值方法应用于该框架。在非扩散区，变分方法作为分析有限体积物理量的有效工具，在研究无限大体积极限中的反常扩散方面具有潜在的应用价值。最后，了解扩散时间尺度和其他瞬时时间尺度是非常有意义的。为了研究时间尺度问题，提出了一种研究时间尺度问题的方法，并将对称过程的可逆性估计推广到奇异非对称对流扩散过程。自从伟大的科学家A·爱因斯坦和N·维纳为布朗运动的研究奠定了理论基础以来，人们一直致力于理解分子尺度上的布朗运动和普通尺度上的流体运动的组合效应。这一问题也具有巨大的实际意义，因为化学工程、地下水污染和全球大气变化中的大多数质量和热传递过程都涉及这两种运动。结果表明，典型的分子布朗运动的小尺度和对流运动的混沌和随机性在大尺度上产生了许多有趣的现象，如对流增强扩散，布朗运动的大小大大增强，多个尺度上输运过程在不同尺度上表现不同，以及尚未找到合适描述的反常扩散。本项目为对流扩散问题的研究提供了一个严密的框架和有效的方法，并取得了许多重要成果。这一和其他充满希望的研究成果将增加我们对影响我们生活的最基本的运输过程之一的基本理解。***