Stationary patterns

Figure 21. Examples of the stationary patterns observed in the CIMA reaction (a) Hexagonal structure (b) lamellar stripe structure (c) irregular (turbulent structure).

The original condition for the development of a Turing pattern, namely that the inhibitor diffuses faster than the activator, can be generalized allowing also for transport processes different from diffusion, in which case this necessary condition becomes A stationary pattern with a characteristic wavelength may develop if the... 

Fig. 71. Stationary patterns of different wavelength emerging in the camphor, periodate Au system for different compositions of the electrolyte [160], The images were obtained with a surface plasmon microscope.

Tordella [35] was probably the first to show HDPE and LDPE birefnngence patterns highly perturbed after the onset of flow instabilities. Discontinuities in the extinction bands and non stationary patterns were then observed, which was confirmed later by Vinogradov et al. [24] on polybutadienes or Oyanagi et al. [36] on HDPE and PS. Piau et al. [17] observed on polybutadiene slight pulsations at the die exit, correlated to the frequency of the sharkskin defect. 

Finally, with the reference electrode close to the working electrode, stationary patterns also formed under certain conditions. They consisted of an active area in the center of the disk surrounded by a passive ring. The... 

Fig. 1.17. Development of stationary patterns for medium switching rates - phase space. The system with small inhomogeneities (indicated by a cloud) is initially distributed around one of the fixed point. Once switching occurs the cloud starts to move to the second fixed point. If a second switching occurs just as the cloud is quickly passing from the left to the right it can be split into two. The second cloud then does not return to the fixed point of the deterministic system but moves to the other side where it stays. A spatial structure is formed.

Figure 8.1 sketches the basic mechanism underlying this type of stationary pattern formation [5] The concentration c of the activator (red line) is locally slightly perturbed from its steady-state value Css (s) This initiates the self-enhanced production of the activator that now slowly diffuses into neighboring regions (b). As a consequence, the inhibitor, that is, the double-layer potential grows (blue dashed line) that propagates more rapidly out of the activator spot (c). Since the inhibitor consumes the activator, the concentration of the latter will always be suppressed outside the spot (d). In this way, a stationary concentration profile may develop surrounded by a halo of increased inhibitor < dl-... 

The time series shows a. stationary pattern until a homogeneously oscillating flow rate around a constant level. However, an additive outlier occurs at time index 128. To analyse the time dependency pattern of the Naphtha time series, the effect of the outlier has to be removed. Therefore, the outlier s effect (i.e. the raise beyond the mean level) is estimated by modelling a simple linear regression model and the corresponding value of the time series is corrected by this estimate. For the corrected Naphtha time series, ACF and PACF are calculated (Figure 2.13a and Figure 2.13b). 

The Turing condition C4 = 0 for hyperbolic reaction-diffusion equations leads to exactly the same conditions as for the standard reaction-diffusion equation, namely (10.42) and (10.40). In other words, the Turing condition is independent of and Ty. If inertia in the transport is modeled by HRDEs, then the inertia has no effect whatsoever on the Turing instability to stationary patterns. 

Fig. 12.4 Stationary patterns in the CIMA reaction in a CFUR. (a) and (b) hexagons, (c) stripes, (d) mixed state. The bar beside each picture represents 1 mm the reactor is 25 mm in diameter. Reprinted with permission from Macmillan Publishers Ltd Nature [337]. Copyright 1991...

To determine which network topologies give rise to localized stationary patterns, we need to investigate the conditions on the topology that lead to localized eigen-... 

Fig. 13.17 Stationary pattern for the four graphs from Fig. 13.16 with inhibitory coupling and Lengyel-Epstein kinetics. The stationary value of the activator concentration is plotted vs the reactor number. (The inhibitor concentration shows quahtatively the same pattern.) Parameter values a = 10.0, Q = 0.1, a = 1.0, b = 5.0, and k = 23.56824264/ j8jo. Reprinted from [209]. Copyright 2006, with permission from Elsevier...

Castets et al. [31] have defined Turing structure as follows These are stationary patterns originating with Sole coupling of reaction and diffusion processes. They... 

In this reaction, ClOj is self-inhibitor and I is autocatalyst. Kepper et al. [40] investigated the pattern formation in CIMA reaction by using the above type of reactor. For the CIMA system, the composition of CSTR was kept constant whereas that of other CSTRs was varied stepwise. Different sequences of spatial and spatio-temporal patterns were observed. At large CIO2 flow rates, stationary patterns were observed. In the CIMA reaction in CR, the concentrations of Ij, MA and H were same in both the CSTRs but spatial gradient in CIOJ concentration may be imposed by applying different feed concentrations in the CSTR. Thus [ClOj] was allowed to vary with distance along the reactor but not with time. 

Transient Turing-like patterns in PA-MBO reaction (Polyacrylamide-methylene blue-sulphide-oxygen) system are obtained. Non-Turing stationary patterns in FIS reaction (Hexacyanoferrate (Il)-iodate-sulphite system) are obtained using an experimental technique similar to that used in CIMA reaction. These patterns develop through propagation of chemical fronts from the initial perturbation [54],... 

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