Reaction-Diffusion Front Speeds in Chaotic and Stochastic Flows

混沌和随机流中的反应扩散前沿速度

• 批准号: 1211179
• 负责人: Jack Xin
• 金额: $ 41.97万
• 依托单位: University of California-Irvine
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2012
• 资助国家: 美国
• 起止时间: 2012-07-01 至 2017-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1211179&HistoricalAwards=false
• 关键词: Reaction; Diffusion; Front; Speeds; Chaotic

Reaction-diffusion front propagation in fluid flows appears in many scientific areas such as particle transport in convection, turbulent combustion and wild fire spread in winds. We aim to carry out mathematical analysis and computation on prototype equations to gain fundamental understanding of the highly nontrivial effects of flow on transport. Significant amount of asymptotic and numerical work in this direction has been accomplished in recent years when the flow lines (streamlines) are either well-structured (regular motion) or fully random (ergodic motion). The often encountered yet less studied case is when the streamlines consist of both regular and stochastic motions, while neither one takes up the entire phase space. An example is the Arnold-Beltrami-Childress (ABC) flow, a class of three dimensional incompressible mean zero periodic flow with chaotic flow lines. The research program is to study large time front speeds of reaction-diffusion-advection and Hamilton-Jacobi equations in various ABC flows with chaotic and stochastic streamlines in channel domains orin the entire space. A measure of the amount of disorder in the streamlines is given by the phase space volume occupied by the points on the Poincare sections. The project combines analytical and computational approaches to study the dependence of front speeds on the degree of chaos, nonlinearities, and correlation with effective diffusion, a related transport property. Front speed variational formulas and corrector equations are used to reduce the original nonlinear dynamical problems on unbounded spatial domains to a principal eigenvalue or Lyapunov exponent (growth rate) problem on a finite spatial domain. Related educational activities, postdoc mentoring, and data management are also planned.Carrying out the proposed work will advance our understanding of material transport in disordered flows arising in nature, and generate broad impact to the science and engineering of pollutant transport, forest fire spreading, internal combustion engine with power efficiency and low waste gas emission to name a few. The mathematical and numerical methods developed in the project are potentially estimation and consulting tools for resolving complex real-world problems.They may aid decision makers to act timely to minimize damage from fire hazards and pollutants, and help manufacturers improve energy efficiency in engine design for green environment. The results and data generated in the project will also benefit educators in curriculum development and course offerings, which in turn stimulates more US students to pursue higher degrees in science, technological, engineering and mathematical disciplines.

流体流动中的反应-扩散波前传播现象在对流中的颗粒输运、湍流燃烧和野火在风中的传播等许多科学领域都有着广泛的应用。我们的目标是进行数学分析和计算的原型方程，以获得基本的理解的高度非平凡的影响，流动的运输。近年来，当流线（流线）是结构良好的（规则运动）或完全随机的（遍历运动）时，在这个方向上已经完成了大量的渐近和数值工作。经常遇到但较少研究的情况是，当流线由规则和随机运动组成，而没有一个占据整个相空间。一个例子是Arnold-Beltrami-奇尔德里斯（ABC）流，一类具有混沌流线的三维不可压缩平均零周期流。本文的研究计划是在通道域或整个空间中研究具有混沌和随机流线的各种ABC流中反应扩散平流和Hamilton-Jacobi方程的大时间波前速度。流线中的无序量的度量由庞加莱截面上的点所占据的相空间体积给出。该项目结合了分析和计算方法，以研究前端速度对混沌程度的依赖性，非线性，以及与有效扩散的相关性，一个相关的传输属性。 利用前速变分公式和校正方程，将无界空间域上的非线性动力学问题转化为有限空间域上的主特征值或李雅普诺夫指数（增长率）问题.本项目的开展将进一步加深我们对自然界中无序流动中物质输运的理解，并对污染物输运、森林火灾蔓延、高能效内燃机和低废气排放等科学与工程产生广泛的影响。该项目开发的数学和数值方法是解决复杂现实问题的潜在评估和咨询工具，可以帮助决策者及时采取行动，最大限度地减少火灾和污染物造成的损失，并帮助制造商提高发动机设计的能源效率，以实现绿色环境。该项目产生的结果和数据也将使教育工作者在课程开发和课程设置方面受益，从而刺激更多的美国学生攻读科学，技术，工程和数学学科的更高学位。