Quenching of combustion by shear flows

Abstract

We consider a model describing premixed combustion in the presence of fluid flow: a reaction-diffusion equation with passive advection and ignition-type nonlinearity. What kinds of velocity profiles are capable of quenching (suppressing) any given flame, provided the velocity's amplitude is adequately large? Even for shear flows, the solution turns out to be surprisingly subtle. In this article we provide a sharp characterization of quenching for shear flows; the flow can quench any initial data if and only if the velocity profile does not have an interval larger than a certain critical size where it is identically constant. The efficiency of quenching depends strongly on the geometry and scaling of the flow. We discuss the cases of slowly and quickly varying flows, proving rigorously scaling laws that have been observed earlier in numerical experiments. The results require new estimates on the behavior of the solutions to the advection-enhanced diffusion equation (also known as passive scalar in physical literature), a classical model describing a wealth of phenomena in nature. The technique involves probabilistic and partial-differential-equation (PDE) estimates, in particular, applications of Malliavin calculus and the central limit theorem for martingales