Spiral Cores

Figure 4.17 The conversion of soot (a) into carbon onions (b) may also be affected by the electron beam under an electron microscope (black and white arrows indicate onions with concentric or with spiral core, respectively, ACS 2002).

Fig. 9.4(b) shows the spiral tip trajectory obtained experimentally under this feedback control. After a short transient the spiral core center drifts in parallel to the line detector. The asymptotic drift trajectory reminds the resonance attractor observed under one-channel control, because a small variation of the initial location of the spiral wave does not change the final distance between the detector and the drift line. To construct the drift velocity field for this control algorithm an Archimedean spiral approximation is used again. Assume the detector line is given as a = 0 and an Archimedean spiral described by Eq. (9.5) is located at a site x,y) with a > 0. A pure geometrical consideration shows that the spiral front touches the detector each time ti satisfying the following equation ... 

Fig.21 shows an example of a spiral-wave solution of eq.(159) obtained numerically. The left figure is a snap-shot corresponding to a particular moment of time. The spiral is rotating counterclockwise with constant frequency. One can see that far from the spiral core the wavenumber is constant and so is the amplitude A that is related to the wavenumber by eq.(166). In the core center A = 0. 

A great deal is known about the behavior of spiral waves in excitable media from both the mathematical and the experimental points of view. One feature of particular interest is the center or core from which the spiral waves emanate. The eikonal equation (6.31) allows us to obtain a rough estimate of the size of a spiral core. Chemical waves typically travel at speeds of millimeters per minute, or perhaps (4-10) xl0 cms. Diffusion constants of monomeric species in aqueous solution tend to be around 2 x 10 cm s. If we consider an expanding circle (which has a negative curvature) on the perimeter of the core, and plug these values for c and D into eq. (6.31), we find that A = 0 when... 

Pre-heated mixtures of methanol and water are injected under high pressure into the central core of the heat exchanger fitted with catalyst deposited on a spiral core. This drives reaction (4) ... 

Spiral cores (Figure 2.43) made from plastic or aluminum have been used for a long time for cooling the cores. The temperature control medium is fed via one or two threads to the core and then removed again. 

FIGURE 2.43 Spiral core, single or double-threaded made from aluminum or plastic material... 

FIGURE 2.90 Single-threaded spiral core (source MASCO Normalien GmbH)... 

The quantitative analysis of spirals yields, on the one hand, a detailed picture of the concentration distribution inside and outside the spiral core. This information can be used to clarify details in the mechanistic steps of the reaction coupled with diffusion. On the other hand, a large number of topological scenarios in the core region have been explored by now that are based on algorithms for the extraction of the essential dynamic features. These are commonly represented by the trajectory of the spiral tip. Unfortunately, in the experimenal evaluation the precise location of the tip is still a rather ill defined quantity and each experimentalist has his own procedure to trace the tip trajectories. In numerical simulation a clearer definition can be provided in terms of the concentration levels of the two variables. 

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