Modeling reaction–diffusion pattern formation in the Couette flow reactor

Abstract

Slow passage through a supercritical Hopf bifurcation: Timedelayed response in the Belousov-Zhabotinsky reaction in a batch reactorWe report on a numerical and theoretical study of spatio-temporal pattern forming phenomena in a one-dimensional reaction-diffusion system with equal diffusion coefficients. When imposing a concentration gradient through the system, this model mimics the sustained stationary and periodically oscillating "front structures" observed in a recent experiment conducted in the Couette flow reactor. Conditions are also found under which oscillations of the nontrivial spatial patterns become chaotic. Singular perturbation techniques are used to study the existence and the linear stability of single-front and multi-front patterns. A nonlinear analysis of bifurcating patterns is carried out using a center manifold/normal form approach. The theoretical predictions of the normal form calculations are found in quantitative agreement with direct simulations of the Hopfbifurcation from steady to oscillating front patterns. The remarkable feature of these sustained spatio-temporal phenomena is the fact that they organize due to the interaction of the diffusion process with a chemical reaction which itself would proceed in a stationary manner if diffusion was negligible. This study clearly demonstrates that complex spatia-temporal patterns do not necessarily result from the coupling of oscillators or nonlinear transport.

Turing Patterns: From Myth to Reality

“…On the contrary, in polyacrylamide gels, at the same starch concentration, stationary Turing patterns are still observed. The pulsating fronts recall some of the wave dynamics observed in the Couette flow reactor [58,59,67]. The last observations in [65].…”

“…This is obvious for the orientation of the striped patterns and the arrangement of spots in rows parallel to the feed boundaries ( Figure 10). Moreover, since the wavelength is much smaller than the smallest dimension of the gel strip -in this case, the gel thickness corresponds to 5 to 6 A -the structure has a tridimensional The status of the multiple front patterns of region III is not solved: They could either correspond to gradient induced structures [67] for which no difference in diffusion coefficients is necessary or -considering the 3-D character of the observed patterns -to planar sheet structures predicted from the nonlinear analysis of developed Turing patterns in three-dimensional systems [19]. …”

Numerical Bifurcation Analysis for Reaction-Diffusion Equations

“…In particular, qualitative behavior of solutions has been studied extensively in various context, for examples, periodic solutions, traveling waves, spiral waves, long time behavior and bifurcations in the stationary problem Stationary bifurcations lead to time-independent patterns. Alexander j Auchmuty [1], Dun-canjEilbeck [96], Elezgaray j Arneodo [98], Fiedler [100], FieldsjGolubitsky 4 1. Spatial patterns of modes are resulted from properties of eigenfunctions of the Laplacian, Le., diffusion of the participating substances.…”

Dirichlet internal layers in an excitable reaction-diffusion system

“…Moreover, due to the development of open spatial reactors, a variety of experimental results exhibiting spatio-temporal patterns have been reported in chemistry; and reaction-diffusion models with the Dirichlet boundary conditions (reflecting the fact that the system is open) have been numerically, and theoretically to some extent, confirmed realistic and suitable to account for the phenomena observed (Elezgaray and Arneodo [1991]). Therefore, our problem (1.1),(1.2) is not only of mathematical interest, but also will be a stepping stone to understand differences between open systems (Dirichlet or third boundary conditions) and closed systems (Neumann boundary conditions or in infinite spatial domains).…”