Front propagation in cellular flows for fast reaction and small diffusivity

Area of Science:

• Chemical kinetics
• Fluid dynamics
• Mathematical modeling

Background:

• Fisher-Kolmogorov-Petrovsky-Piskunov (FKPP) models describe phenomena like population dynamics and chemical reactions.
• Understanding the influence of external forces, such as fluid flow, is crucial for accurately predicting front propagation.
• Previous studies often simplified flow conditions or focused on different parameter regimes.

Purpose of the Study:

• To investigate how fluid flows affect chemical front speeds in FKPP models.
• To develop an asymptotic theory for front propagation in cellular flows.
• To derive efficient methods for calculating front speeds under specific limiting conditions.

Main Methods:

• Development of an asymptotic theory for front speed in cellular flow.
• Analysis in the limit of small molecular diffusivity and large Péclet (Pe) and Damköhler (Da) numbers.
• Utilizing instanton theory to identify paths minimizing a specific functional for front speed calculation.

Main Results:

• The front speed is determined by a periodic path (instanton) that minimizes a functional.
• Efficient procedures for calculating front speed were established.
• Closed-form expressions for front speed were derived for two distinct parameter regimes: (logPe)(-1) ≪ Da ≪ Pe and Da ≫ Pe.

Conclusions:

• The study provides a robust theoretical framework for understanding advection-dominated chemical front propagation.
• Theoretical predictions align well with numerical solutions and simulations.
• The findings offer insights into optimizing reaction-diffusion processes in flowing systems.