Front propagation in cellular flows for fast reaction and small diffusivity

Abstract

We investigate the influence of fluid flows on the propagation of chemical fronts arising in Fisher-Kolmogorov-Petrovsky-Piskunov (FKPP) type models. We develop an asymptotic theory for the front speed in a cellular flow in the limit of small molecular diffusivity and fast reaction, i.e., large Péclet (Pe) and Damköhler (Da) numbers. The front speed is expressed in terms of a periodic path--an instanton--that minimizes a certain functional. This leads to an efficient procedure to calculate the front speed, and to closed-form expressions for (logPe)(-1) ≪ Da ≪ Pe and for Da ≫ Pe. Our theoretical predictions are compared with (i) numerical solutions of an eigenvalue problem and (ii) simulations of the advection-diffusion-reaction equation.

Chemical Front Propagation in Periodic Flows: FKPP versus G

“…with the first two terms previously derived in [44]. In a similar manner, the minimising trajectory associated with the variational principle (3.12) for (G) is at leading order a straight line.…”

“…We find an approximation to c FK by approximating G (c, c 0 ) in (3.5) for c 1. We previously found [44] that the minimising periodic trajectory in (3.5) may be divided into two regions that we now describe. In region I, X(t) 1 and therefore we may seek a regular expansion in powers of c of the form (4.1)…”

“…6] and [20]) to establish an expression for c FK in terms of a single trajectory that minimises an action functional. This was subsequently exploited in [44] to obtain explicit results for cellular flows by carrying out a minimisation over periodic trajectories. For (G), Fermat's principle in a moving medium determines c G .…”

Barriers to front propagation in laminar, three-dimensional fluid flows

“…In many of these processes, the front propagation is strongly affected by fluid flows in the system. This generalized advectionreaction-diffusion (ARD) problem [1,2] has applications in a wide variety of systems, including microfluidic chemical and biological devices [3,4]; cellular-and embryonicscale biological processes [5]; oceanic-scale algal blooms [6,7]; the ignition stages of a supernova explosion [8]; and the propagation of a disease in a mobile society [9]. Previous experiments [10][11][12][13] have identified dynamicallydefined, one-way barriers that block reaction fronts propagating in a wide range of two-dimensional (2D), laminar flows.…”

“…The extension from 2D to 3D is accompanied by several topological questions: (1) Are there generalized BIMs that also act as barriers that impede the motion of reaction fronts for 3D flows? (2) What is the topology of these barriers -if they exist -for a 3D flow? (3) Are the barriers one-way, similar to their 2D counterparts?…”