Diffusion-induced instability

Lengyel, I. I.R. Epstein. 1991. Diffusion-induced instability in chemically reacting systems Steady-state multiplicity, oscillation, and chaos. Chaos 1 69 76. 

We consider the stability of a general two-variable system, first in the absence of diffusion and then with diffusion terms. Because the Turing bifurcation is a diffusion-induced instability, we will first show that the system is stable... 

Again, the stability of the steady state is determined by the sign of X., and for a diffusion-induced instability of this state, we require X > 0. From the above analysis of the homogeneous system, we know this requires Tr(B) > 0 or det(B) < 0 however, we also know from Eq. [60] that Tr(B) < 0 for stability of the homogeneous system. We therefore conclude that a necessary requirement for the diffusion-induced instability is det(B) < 0. 

Point B is much farther from the Turing bifurcation, and we anticipate the pattern formation to be more complex. Following the same procedure as described above, we find the following range of wavenumbers over which the diffusion-induced instability occurs... 

The first unambiguous observation of a Turing instability in any experimental system did not occur until 1990. That year, the Bordeaux group found convincing evidence for Turing patterns in an in vitro system, the CIMA reaction (see Sect. 1.4.9). The gap of almost 40 yr between Turing s theoretical prediction of diffusion-induced instabilities and the experimental realization of stationary chemical pattern was caused by two main factors. 

The earliest studies of coupled chemical cells were theoretical investigations and focused on the diffusion-induced instability to nonuniform steady states [354, 443] in two coupled reactors. More recently, Epstein and Golubitsky investigated the Turing instability in linear and circular arrays of Brusselators [109]. Their approach makes use of underlying symmetry groups. 

Figure 12.20 Bifurcation diagram for two physically coupled Degn-Harrison oscillators. (Reprinted with permission from Lengyel, I. Epstein, I. R. 1991. Diffusion-Induced Instability in Chemically Reacting Systems Steady-State Multiplicity, Oscillation, and Chaos, Chaos /, 69-76. 1991 American Institute of Physics.)...

All chemical waves reflect the features of their fronts. We have described in this chapter how the most basic fronts are related and how they may give rise to differences in behavior - differences that are important in certain instances. The mixed-order description provides insights into the quadratic form, often employed as the prototype for propagating fronts, as well as the cubic form, which has features related to reaction-conduction flames. Diffusion-induced instabilities in 2-D and 3-D fronts offer interesting experimental and theoretical challenges for studies of dynamical complexity. 

Experimentally it has been shown that the threshold pressure at which combustion instability can be induced artificially in composite proplnts by pulsing is a function of the burning rate of the proplnt (in a motor size of 5-inch diameter and 40-inch length) (Ref 45). This relationship is shown in Figs 17 and 18 for both aluminized and non-aluminized composite proplnts. It was also found that potassium perchlorate, lithium perchlorate and AN proplnts were resistant to this induced instability. Since AP composites were the only proplnts, other than double-base, which were driven unstable, the rate controlling reactions and response function are those related to AP decompn and perhaps the diffusion flame between oxidizer and binder... 

The continuous microcellular process is also based on the concept of thermodynamic instability and a much shorter time is needed for saturation of polymer with gas [67]. As described by Park, when the polymer is melted in the extrusion barrel, a metered amount of gas is delivered to the polymer melt [67]. The injected gas diffuses into the polymer matrix at a much higher rate because of convective diffusion induced in the extmsion barrel at an elevated temperature [65]. As for a batch process, there are three specific steps in a continuous microcellular extrusion process formation of polymer/gas solution, cell nucleation, and shaping and cell growth [23,24,65-71]. A typical schematic of the overall continuous microcellular experimental equipment is illustrated in Figure 17.6. 

All these physically different systems can formally be cast in a common language. However since it is the recent experimental observations [19,20] of Turing patterns that has induced a renewed interest in diffusion driven instabilities, in the following, we mainly discuss the case of the Turing instability in reaction-diffusion systems. 

The Differential Diffusivity Induced or Turing Instability Turing demonstrated [4] that a reaction-diffusion system X = f(X) + DAX... 

While the occurrence of the Turing instability depends crucially on the ratio of diffusion coefficients 6 = Dinh/T>act, the flow-induced instability is determined merely by the magnitude v of the easily controllable relative flow velocity (or rather by the ratio v / /D when diffusion is included). Thus it is immaterial which of the two control species is immobilized as the present analysis and experiment show, fixing the inhibitor promotes the DIFICI, while this would prevent the Turing instability from occurring. 

The necessary condition for the differential transport induced instabilities - the DIFICI and Turing instability (TI) - is the existence of an unstable subsystem, an activator [2,3,4,7,8]. The physical cause of these instabilities is the following the differential transport of activator and inhibitor, be it through a differential flow (DIFICI) or through differential diffusivity (TI), spatially decouples the counteracting species, and thereby releases locally the natural tendency of the activator to grow. 

In the previous section we have taken care to keep well away from parameter values /i and k for which the uniform stationary state is unstable to Hopf bifurcations. Thus, instabilities have been induced solely by the inequality of the diffusivities. We now wish to look at a different problem and ask whether diffusion processes can have a stabilizing effect. We will be interested in conditions where the uniform state shows time-dependent periodic oscillations, i.e. for which /i and k lie inside the Hopf locus. We wish to see whether, as an alternative to uniform oscillations, the system can move on to a time-independent, stable, but spatially non-uniform, pattern. In fact the... 

Another explanation of the lithium gap in the Hyades could be found in terms of turbulent diffusion and nuclear destruction. Turbulence is definitely needed to explain the lithium abundance decrease in G stars. If this turbulence is due to the shear flow instability induced by meridional circulation (Baglin, Morel, Schatzman 1985, Zahn 1983), turbulence should also occur in F stars, which rotate more rapidly than G stars. Fig. 2 shows a comparison between the turbulent diffusion coefficient needed for lithium nuclear destruction and the one induced by turbulence. Li should indeed be destroyed in F stars This effect gives an alternative scenario to account for the Li gap in the Hyades. The fact that Li is normal in the hottest observed F stars could be due to their slow rotation. 

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