Bulk burning rate in passive-reactive diffusion

Peter Constantin, Alexander Kiselev, Adam Oberman, Leonid Ryzhik

Research output: Contribution to journalArticle

88 Scopus citations

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.

Original languageEnglish (US)
Pages (from-to)53-91
Number of pages39
JournalArchive for Rational Mechanics and Analysis
Volume154
Issue number1
DOIs
StatePublished - Jan 1 2000
Externally publishedYes

All Science Journal Classification (ASJC) codes

  • Analysis
  • Mathematics (miscellaneous)
  • Mechanical Engineering

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