Diffusion, Dissipation Enhancement, and Mixing

点击翻译按钮获取中文摘要

Diffusion and mixing are key physical phenomena that arise in applications ranging from micro-fluids to meteorology, to cosmology. Diffusion tends to spread out clusters and tries to disperse them, while mixing weaves the clusters and tends to intersperse layers of one cluster with another. This project seeks to quantify their interaction and determine how it impacts the dynamics of physical and chemical fluid systems. Improving the understanding of how they interact has both direct and indirect consequences in scientific areas, such as the study of chemotaxis, of phase separation of binary mixtures including alloys, and of the earth's magnetic field. The research will be carried out with active involvement from students and postdoctoral associates, who will be trained across a broad range of fields.This project will examine several issues concerning enhanced dissipation, diffusion, and mixing. In incompressible fluids, stirring induces mixing by filamentation and facilitates the formation of small scales. Diffusion, on the other hand, efficiently damps small scales and the balance between these two phenomena can lead to enhanced dissipation - the tendency of solutions to dissipate energy faster than normal. Four goals of the project are to construct simple and explicit family of flows that enhance dissipation by an arbitrarily large amount; to use dissipation enhancement to address physically meaningful questions concerning nonlinear partial differential equations; to obtain sharp estimates for the dissipation enhancement in terms of the mixing rates; and to investigate residual diffusivity, and the effect of dissipation enhancement on the long-time behavior of magnetic fields.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.