Bulk Burning Rate in Passive - Reactive Diffusion

Abstract

We consider a passive scalar that is advected by a prescribed mean zero divergence-free velocity field, diffuses, and reacts according to a KPP-type nonlinear reaction. We introduce a quantity, the bulk burning rate, that makes both mathematical and physical sense in general situations and extends the often ill-defined notion of front speed. We establish rigorous lower bounds for the bulk burning rate that are linear in the amplitude of the advecting velocity for a large class of flows. These "percolating" flows are characterized by the presence of tubes of streamlines connecting distant regions of burned and unburned material and generalize shear flows. The bound contains geometric information on the velocity streamlines and degenerates when these oscillate on scales that are finer than the width of the laminar burning region. We give also examples of very different kind of flows, cellular flows with closed streamlines, and rigorously prove that these can produce only sub-linear enhancement of the bulk burning rate.

32 pages, 4 figures

Cited by in corpus (23)

• Enhancement of the Traveling Front Speeds in Reaction-Diffusion Equations with Advection
• Biomixing by chemotaxis and efficiency of biological reactions: the critical reaction case
• Front speed enhancement in cellular flows
• A Bound on Mixing Efficiency for the Advection-Diffusion Equation
• Flame Enhancement and Quenching in Fluid Flows
• Thin front propagation in steady and unsteady cellular flows
• Superfast front propagation in reactive systems with anomalous diffusion
• Chaotic mixing induced transitions in reaction-diffusion systems
• Model Flames in the Boussinesq Limit: The Effects of Feedback
• Reactive Rayleigh-Taylor Turbulence
• A Variational Principle Based Study of KPP Minimal Front Speeds in Random Shears
• Propagating Fronts in Fluids with Solutal Feedback
• Pulsating Front Speed-up and Quenching of Reaction by Fast Advection
• An upper bound for the bulk burning rate for systems
• Stirring Speeds Up Chemical Reaction
• A Numerical Study of Turbulent Flame Speeds of Curvature and Strain G-equations in Cellular Flows
• Mixing and reaction efficiency in closed domains
• The speed of traveling waves in a FKPP-Burgers system
• A PDE hierarchy for directed polymers in random environments
• Front speed in reactive compressible stirred media
• Reaction fronts in persistent random walks with demographic stochasticity
• Long-time behavior for a nonlocal model from directed polymers
• Thin front propagation in random shear flows