Reactive Processes, Mixing, and Fluid Dynamics

University Of California-San Diego, La Jolla CA

Investigators

Abstract

Reactive processes such as forest fires, nuclear reactions in stars, or burning in internal combustion engines are ubiquitous in nature, science, and engineering. Mixing due to motion of a liquid or gaseous medium in which reactive processes occur is frequently an important component in their dynamics, and is also relevant to other processes, such as reliable manufacturing of glasses and alloys. This motion may be subject to fluid turbulence, the effects of which are of paramount importance in many areas of physics and engineering. The central aim of this project is a better understanding of the long term behavior of these physical processes through the analytical study of their mathematical models, which are expressed in the form of partial differential equations. The main questions addressed will be the question of dependence of the speed of spreading of reactive processes on the properties and structure of combustive media in which they occur; the question of how an underlying mixing process can enhance this speed and which types of mixing are most efficient at achieving this; and the question of spontaneous development of turbulence and unexpected singular behaviors in the motion of fluids. The goal is to obtain mathematically rigorous results which can also shed further light on the dynamical behavior of the actual physical processes being modeled. This research project studies mathematical models of several important physical processes, which include reactive processes, fluid dynamics, and mixing. The models are given by linear and nonlinear partial differential equations, in particular, by reaction-diffusion equations, transport equations, and equations of fluid dynamics. The main interest is in the long term dynamics of their solutions as well as in the formation of singularities. The goal of the reaction-diffusion portion of the project is the understanding and description of long term dynamics of reactive processes spreading through inhomogeneous media in one and several dimensions, including existence of traveling fronts, asymptotic convergence of general solutions to them, and homogenization of solutions in random media. The goal of the mixing portion of the project is the study of mixing efficiency of flows and the search for those which are best at mixing substances advected by them. The goal of the fluid dynamics portion of the project is the study of turbulence, particularly creation of small scales and finite time singularity formation in models of fluid and atmospheric motion in two dimensions. Another goal of the research is the study of active combustion, where all three of these processes come together due to a direct feedback of the reaction on fluid motion via the buoyancy force. Models incorporating such feedback involve reaction-diffusion equations coupled to fluid dynamics equations, and the focus will be on existence and stability of traveling fronts and on gravity-induced mixing. To address these questions, the research will make use of techniques recently developed by the investigator and collaborators, as well as the development of new methods capable of further advancing understanding of the dynamical behavior of reactive processes.