Long Time Dynamics in Combustion, Mixing, and Fluids Models

University Of California-San Diego, La Jolla CA

Investigators

Abstract

The aim of this project is a better understanding of mathematical models of several important real world phenomena, including propagation of reactive processes, turbulent motion of fluids, and mixing of substances by such motion. Reactive processes (e.g., forest fires) frequently occur in a patchwork of different environments (trees, brush, grass), through which the process propagates at different rates. The theory of homogenization aims to better understand how the dynamics of the process over large scales (long distances and times) depends on smaller scale variations in the environment. The main goal of this part of the project is to demonstrate that thanks to averaging over large regions, the long term behavior of the process can be predicted with a large degree of confidence without the need for overly detailed information about the composition of the environment. Mixing via fluid motion is crucial in the production of materials such as alloys, as well as in enhancement of processes such as chemical reactions. Quantifying mixing efficiency of flows and identifying those that stand out in this respect is therefore of great importance. The PI recently constructed universal mixers, flows that are particularly efficient in mixing. These have a relatively simple structure but may be too irregular for practical applications. The main goal of this part of the project is to identify more regular universal mixers, as well as to study how such fast mixing affects diffusive processes (e.g., heat transport). Even without considering its effects on mixing of substances, our understanding of the motion of fluids is still far from satisfactory. The study of onset of turbulence and the associated creation of small scale structures in fluids is important in mathematics as well as in physics and engineering, and while it has seen great progress in recent years, many fundamental questions remain unresolved. In this part of the project, the PI proposes to study how fast the development of turbulent structures in fluids can be and how severe this turbulence can become, both in the bulk of the fluid and in the vicinity of regular as well as irregular boundaries (walls of the fluid container). Physical processes such as combustion, mixing, and fluid turbulence, whose study motivates this project, are modeled by linear and nonlinear partial differential equations, including reaction-diffusion equations, transport equations, drift-diffusion equations, and equations of fluid dynamics. The primary focus of this project, which consists of three parts, is the study of long term dynamics of the solutions of these equations as well as their possible formation of singularities in finite time. The aim of the first part of the project is the understanding of large scale behavior of reactive processes spreading through heterogeneous media, including obtaining a satisfactory homogenization theory for these models in multi-dimensional random media. The aim of the second part of the project is the study of mixing efficiency of flows and the search for sufficiently regular universal mixers, flows that are highly efficient in mixing of substances advected by them regardless of the initial configuration of the latter. Enhancement of diffusion in drift-diffusion equations via flow-induced mixing will also be studied. The aim of the third part of the project is the study of turbulence in two-dimensional Euler and related equations of fluid dynamics, including rapid growth of gradients of solutions in the bulk of the fluid and possible formation of finite time singularities in more singular models. The question of well-posedness of Euler equations in planar domains with irregular boundaries will also be addressed. 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.