Reaction, Diffusion, and Fluid Flow

该项目专注于对流体流动中发生的反应过程模型的分析研究。这些模型包括非线性偏微分方程组，如反应扩散方程，它可以耦合到流体动力学的Navier-Stokes方程。这项拟议的研究旨在提高我们对流体运动对燃烧影响的理解，主要有两个目标。第一个是关于周期性介质中被动燃烧的当前工作的继续，以及适用于一般无序介质的新技术的发展。需要解决的问题包括周期性流动的几何性质对反应传播速度的影响以及猝灭现象。我们还将研究反应在无序介质中的传播，特别是这些特殊解的行波解的存在性和任意解的渐近逼近。第二个主要目标是研究通过浮力直接反馈流体运动的反应的主动燃烧。包含这种反馈的模型，特别是与高度湍流燃烧区域相关的模型，涉及耦合到流体动力学方程的反应-扩散方程，并且本质上非常复杂。我们将重点研究旅行锋面的存在和稳定性，反应传播速度的界限，以及重力诱导的混合。此外，我们打算将所开发的技术应用于相关液体悬浮液模型中的相变研究。该项目解决的问题涉及丰富而微妙的数学，但也具有跨学科的特点。在自然界、科学和工程中，内燃机燃烧、恒星中的核反应、森林火灾和大气中臭氧的产生等反应过程无处不在。底层液体或气体介质的运动通常通过加快反应或使其熄灭而发挥关键作用。拟议的研究旨在更好地从数学上了解由此产生的湍流的各种性质对反应过程的影响。它与天体物理、生物学、环境科学和化学工程等科学分支相关，并可能在现实生活现象中提供有用的定性见解。这位首席研究员还计划教授一门本科水平的课程，作为芝加哥大学本科生研究经验暑期项目的一部分，以及一门专门的研究生水平的反应扩散方程课程。

The project focuses on analytical study of models of reaction processes taking place in fluid flow. These models involve nonlinear partial differential equations such as reaction-diffusion equations, which may be coupled to the Navier-Stokes equations of fluid dynamics. The proposed research aims at improving our understanding of the effects of fluid motion on combustion and has two main goals. The first is a continuation of current work on passive combustion in periodic media as well as development of new techniques applicable to general disordered media. The questions to be addressed include the effect of geometric properties of periodic flows on speed-up of propagation of reaction as well as the phenomenon of quenching. We will also investigate propagation of reaction in disordered media, in particular, the existence of traveling front solutions and asymptotic approach of arbitrary solutions to these special ones. The second main goal is the study of active combustion with direct feedback of reaction on the fluid motion via the buoyancy force. Models incorporating such feedback, particularly relevant in highly turbulent combustion regimes, involve reaction-diffusion equations coupled to fluid dynamics equations and are inherently very complex. We will focus our efforts on the existence and stability of traveling fronts, bounds on the speed of propagation of reaction, and gravity-induced mixing. In addition, we intend to apply the developed techniques to the study of phase transitions in a related model of liquid suspensions. The problems addressed by the project involve rich and subtle mathematics but also have an interdisciplinary character. Reaction processes such as burning in internal combustion engines, nuclear reactions in stars, forest fires, and production of ozone in the atmosphere are ubiquitous in nature, science, and engineering. Motion of the underlying liquid or gaseous medium often plays a crucial role by either speeding up reaction or quenching it. The proposed research aims at a better mathematical understanding of the effects of various properties of the resulting turbulent flows on reactive processes. It is relevant to branches of science such as astrophysics, biology, environmental science, and chemical engineering, and may provide useful qualitative insights in real life phenomena. The principal investigator also plans to teach an undergraduate-level course as part of the Research Experience for Undergraduates summer program at the University of Chicago, as well as a specialized graduate-level course in reaction-diffusion equations.