Mathematical Sciences: Diffusion in Fluid

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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. ***