Reaction-Diffusion Front Speeds in Chaotic and Stochastic Flows

University Of California-Irvine, Irvine CA

Investigators

Abstract

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.