The focus of the research component of this project is analytical study of models of reactive processes, such as combustion, taking place in fluid flow. In recent years, there has been significant progress in the mathematical understanding of the effects of flow on combustion. Nevertheless, the inherent complexity and richness of the phenomena continues to present a challenge to our study of this problem. These models are given by nonlinear partial differential equations, in particular, by reaction-diffusion equations which may also be coupled to equations of fluid dynamics. One of the goals of the proposed research is a continuation of previous work on passive combustion in periodic, random, as well as general inhomogeneous media. The main questions of interest include the effects of flows on mixing and on speed-up of propagation of reaction. In addition, simplified models will be considered and experience gained from their study will be applied to the original problems. Another goal of the project is the study of propagation of reaction in a model of active combustion, with direct feedback of reaction on the fluid motion via the buoyancy force and gravity-induced mixing. In many situations this effect cannot be ignored, particularly in highly turbulent combustion regimes. Although this is a very difficult problem, limited progress has been achieved recently and it is proposed here to build upon these results. The developed techniques will also be applied to the study of other related models, including systems of reaction-diffusion equations and a model of liquid suspensions.
Reactive processes, such as burning in internal combustion engines, nuclear reactions in stars, and forest fires, are ubiquitous in nature. Their study is relevant to several branches of science and engineering such as astrophysics, biology, environmental science, and chemical engineering. The research proposed here focuses on the influence of the motion of the underlying medium, such as fluid flow, on the speed of spreading of reactive processes. It aims at improving our understanding of fundamental mathematical models describing such processes, which may provide useful qualitative insights into real life phenomena. An important part of this proposal is its educational component. The principal investigator (PI) plans to develop an undergraduate partial differential equations (PDE) course at the University of Wisconsin, mentor undergraduate mathematics majors in the study of advanced topics, and organize a Research Experience for Undergraduates program in analysis and PDEs. The PI will also teach specialized graduate courses and involve graduate students in working under his supervision.
