Mathematical Sciences: Diffusion in Fluid

[60019范江]本项目研究不可压缩流体流动中粒子弥散的大尺度行为。在大尺度上，粒子的扩散或非扩散取决于流体的流动。该项目的重点之一是扩散状态的尖锐标准。新的变分方法已被用来获得稳定流动的判据。将变分方法推广到非定常流场和弱可压缩流场。此外，在扩散状态下，流体速度对输运系数（有效扩散系数）的影响具有实际和理论意义，特别是在分子扩散系数很小的情况下。因此，流体速度通常会将有效扩散系数提高几个数量级。变分方法已成功地应用于二维定常流动中对流增强扩散的非平凡幂律。在分析三维随机流动中的湍流扩散和二维非定常流中的混沌扩散方面，有待进一步推广。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，现在已经建立的对称问题的数值方法可以应用于这个框架。在非扩散状态下，变分方法作为分析有限体积下物理量的有效工具，对于研究无限体积极限下的异常扩散具有潜在的实用价值。最后，了解扩散时间尺度和其他瞬态时间尺度是非常有趣的。提出了一种鞅方法，并将（对称过程的）收缩性估计推广到奇异非对称对流扩散过程，用于研究时间尺度问题。自从伟大的科学家爱因斯坦和维纳为研究布朗运动奠定了理论基础以来，人们一直在努力理解分子尺度上的布朗运动和普通尺度上的流体运动的综合作用。这一问题也具有巨大的实际意义，因为化学工程、地下水污染和大气中的全球变化中的大多数传质和传热过程都涉及这两种运动。事实证明，分子布朗运动的典型小尺寸，以及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，在不同尺度上传输过程表现不同的多尺度，以及尚未找到适当描述的异常扩散。本课题为对流扩散问题的研究提供了一个严谨的框架和有效的方法，并取得了许多重要成果。这个和其他有希望的研究成果将增加我们对影响我们生活的最基本运输过程之一的基本理解。＊＊＊

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. ***