Pinning of reaction fronts by burning invariant manifolds in extended flows

Abstract

We present experiments on the behavior of reaction fronts in extended, vortexdominated flows in the presence of an imposed wind. We use the ferroin-catalyzed, excitable Belousov-Zhabotinsky chemical reaction, which produces pulse-like reaction fronts. Two time-independent flows are studied: an ordered (square) array of vortices and a spatially disordered flow. The flows are generated with a magnetohydrodynamic forcing technique, with a pattern of magnets underneath the fluid cell. The magnets are mounted on a translation stage which moves with a constant speed V d under the fluid, resulting in motion of the vortices within the flow. In a reference frame moving with magnets, the flow is equivalent to one with stationary vortices and a uniform wind with speed W = V d. For a wide range of wind speeds, reaction fronts pin to the vortices (in a co-moving reference frame), propagating neither forward against the wind nor being blown backward. We analyze this pinning phenomenon and the resulting front shapes using a burning invariant manifold (BIM) formalism. The BIMs are one-way barriers to reaction fronts in the advection-reaction-diffusion process. Pinning occurs when several BIMs overlap to form a complete barrier that spans the width of the system. In that case, the shape of the front is determined by the shape of the BIMs. For the ordered array flow, we predict the locations of the BIMs numerically using a simplified model of the velocity field for the ordered vortex array and compare the BIM shapes to the pinned reaction fronts. We also explore transient behavior of the fronts (before reaching their steady state) to highlight the one-way nature of the BIMs.

Generalized Lagrangian coherent structures

“…Coherence of co-evolving variables may be different, and LCSs are not guaranteed to reveal it. The need for methods that can capture coherence of something other than fluid trajectories have urged researchers to extend the concept of LCSs to some chemical and bio-physical applications by using a modified velocity field in (1) [32,33,34,35,36,111,112,113,114]. In the next section we present a unified notation of generalized LCSs that naturally incorporate these prior developments and are readily applicable to other co-evolving variables not considered in prior literature.…”

“…We refer to these techniques as 'diagnostic methods' for determining LCSs, but our primary discussion will be essentially independent of which method is used. The key point is whatever the definitions of LCSs, their formulation is typically (with a few exceptions, such as, for example, the burning manifolds of [32,33,34,35,36]) based solely on the Lagrangian trajectories associated with (1), and the goal is to identify fluid parcel groups that behave similarly, and are thus 'coherent.' This has been a rich area of study using both experimental [37,38,39,40] and observational [41,42,43,44,45,46,47,48,49,50,51,52,53] data.…”

Frozen fronts selection in flow against self-sustained chemical waves

“…These fronts are analogous to flames in combustion [4] and autocatalytic reactions are a kind of "cold combustion model" especially in the thin flame limit. In contrast to flame propagation in combustion [4], where it has been analyzed thoroughly theoretically and experimentally, the effect of fluid flow (laminar or turbulent) on reaction fronts has not been explored in detail until recently [5][6][7][8][9][10][11][12][13][14][15][16][17][18]. In the presence of an hydrodynamic flow, it has already been observed and understood that such fronts while propagating at a new constant velocity, adapt their shape in order to achieve a balance between reaction diffusion and flow advection.…”

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

“…To quantitatively analyze the behavior of reaction fronts in a 3D flow, we extend the previous 2D BIM theory [10][11][12][13][14][15][16][17], in which we directly model the motion of an infinitesimal element of the reaction front. In 2D, a front element is parameterized by two spatial coordinates and a single orientation angle.…”

“…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. These barriers have been explained theoretically [10,[14][15][16][17] as burning invariant manifolds (BIMs) that are generalizations of passive invariant manifolds [18][19][20][21][22] that impede passive mixing in a flow.…”