Propagation of target waves in the presence of obstacles

Abstract

The propagation of target waves in the presence of walls and windows is considered. It is shown that in a finite system, for sufficiently small passages no target waves are triggered. Propagation through a large opening can inhibit the onset of waves from smaller windows.PACS 03.40.Kf Concentric waves of chemical activity have been observed in reacting systems as well as in other media. The theoretical aspect of the problem has also drawn much attention from researchers in diverse fields. 1,2 In this paper we report numerical simulations of target waves in a reaction-diffusion medium where a partition is introduced in the system. Two openings of various size insure the communication between the two compartments. In such an environment, target waves show some unexpected behavior which is the object of this Letter.The reaction is assumed to proceed in a twodimensional square vessel in the presence of diffusion. The compartment walls are subject to "no-flux" boundary conditions. The general equation describing this system is -^=F(X)+Z>V 2 X,(1) dt where X(r,/) is a vector representing the concentration variables and D is the diffusion matrix taken here to be a multiple of the identity matrix.It is assumed that the reaction kinetics is such that for a range of parameter values a supercritical Hopf bifurcation leading to bulk oscillations takes place. Near the instability point, such a system may be reduced to a Ginzburg-Landau-type equation which characterizes the behavior of the slowly varying complex amplitude A(p,r) defined aswhere Xo describes the concentration values of a steady solution of Eq. (1) which has become unstable via the Hopf bifurcation and £ is the critical mode, i.e., the eigenvector related to the eigenvalue icoo of the Jacobian matrix of F computed at Xo. The small parameter e<£ 1 is a measure of the distance from the bifurcation point and the complex field A(p,r) evolves on the slow time scale r = et and on the large length scale p^is/D) 1/2 r. In the postcritical regime Eq. (1) gives 3 ^--nA-(g, + ig,)\A?A+\ 2 A.The real constants g r , g/, and JJ. can be related to the parameters of the model. In the case of a supercritical bifurcation, we have ,u and g r > 0. With the help of the amplitude equation (3), one shows that to the lowest order in e, the bulk frequency of the homogeneous oscillation of concentrations X is r = ft>o~" £/*£//#>• When the nonlinear dispersion g t is normal, i.e., when the period of the oscillations increases with the amplitude, then g, > 0.We first study the events in the compartment I when only one window is considered [ Figs. 1(a) and 1(b)]. In the system described by Eq. (2), target waves may be generated around centers by introducing in the system local inhomogeneities that we shall call pacemaker centers. First, traveling waves are generated under the action of a small local fluctuation in the concentration variable. The waves generated by such a mechanism have a tendency to vanish after some time. 4 Target waves also appear if the parameters of the pacemaker r...

Laser-Induced Target Patterns in the Oscillatory CO Oxidation on Pt(110)

“…Since the laser beam, in first approximation, produces a Gaussian temperature profile, it is assumed that the frequency parameter ω in the CGLE can be replaced by the space-dependent frequency parameter ω( x ) in the following way: where Δω is the frequency shift in the center (given by x 0 ) of the heterogeneity, and σ characterizes the width of the frequency perturbation. Many aspects of target pattern formation in the CGLE with a spatially nonuniform, fixed frequency distribution have been studied previously. − …”

“…Many aspects of target pattern formation in the CGLE with a spatially nonuniform, fixed frequency distribution have been studied previously. [15][16][17][18] It is well-known that a local frequency increase in an oscillatory reaction-diffusion system leads to an expanding target pattern that entrains the rest of the system only if the dispersion of the waves in the medium is positive, i.e., if the wave frequency increases for increasing wavenumbers. Since in the CGLE the frequency of one-dimensional plane waves is given by where Ω 0 ) ω + R is the frequency of uniform oscillations and k is the wavenumber of the waves emitted by the pacemaker, positive dispersion means that β -R > 0.…”

“…Therefore, it is conjectured that many pacemakers in the BZ reaction are formed by small local impurities such as dust particles or gas bubbles, which appear in a random way. The properties of such heterogeneous pacemakers and their target wave patterns have been analyzed in a number of studies, mostly in the context of the BZ reaction − or the complex Ginzburg-Landau equation, − a general model for oscillatory media. However, no systematic study of target patterns and pacemakers in the CO oxidation reaction has been reported until now.…”

Belousov-Zhabotinsky autonomic hydrogel composites: Regulating waves via asymmetry

“…Wave propagation, including the initiation site, wave type (target, traveling, and spiral), and direction, is critical for embedded logic and chemical computing. Initial studies on the interaction of chemical waves with diffusion obstacles, such as walls and windows, demonstrated the potential for geometrically regulated excitable and oscillatory media ( 17 ). For example, the chemical waves of the BZ reaction have been used to determine the optimal path through a maze ( 18 ) and to create chemical logic gates through oxidation within capillary tubes ( 19 , 20 ).…”

Excitability Modulation of Oscillating Media in 3D-Printed Structures

“…More recently, Steinbock and coworkers studied BZ wave propagation inside glass capillaries, reporting specific wave transmission phenomena, including backfiring pulses, in addition to linear pulse propagation [15,21]. A closely related experimental approach is to exploit wave propagation properties through BZ compartments that are separated by a solid wall with a small opening for BZ pulse transmission [26].…”