Chaotic structure

When a fluid flows rapidly, its flow pattern typically exhibits a subtle mixture of order and chaos, and it is this structured chaotic motion that is referred to as turbulence [27]. Turbulence can be defined as a property of an incompressible fluid flow at a very high Reynolds number given by... 

In Figure 1.15 the dependence of the density of the macromolecular entanglements clnster network on is shown. As might be expected, chaos intensification X increase) reduces the value, i.e., the local ordering degree in the polymer amorphous state structure [68]. More precise interpretation of the polymer structure chaotic character within the frameworks of multifractal formalism will be given below. 

It is one of the wonders of the history of physics that a rigorous theory of the behaviour of a chaotic assembly of molecules - a gas - preceded by several decades the experimental uncovering of the structure of regular, crystalline solids. Attempts to create a kinetic theory of gases go all the way back to the Swiss mathematician, Daniel Bernouilli, in 1738, followed by John Herapath in 1820 and John James Waterston in 1845. But it fell to the great James Clerk Maxwell in the 1860s to take... 

FIG. 8 Compact seaweed originating from the simulation of an isotropic phase by a phase-field model [120]. A doublon structure is just about to emerge from the chaotic background. 

Class c2 Simple stable states or periodic and separated structures emerge. Class c3 Chaotic nonperiodic patterns are generated. 

Note that while a system s static complexity certainly influences its dynamical complexity, the two measures are clearly not equivalent. A system may be structurally rather simple (i.e. have a low static complexity), but have a complex dynamical behavior. (Think of the chaotic behavior of Feigenbaum s logistic equation, for example). 

Catalytic reactors can roughly be classified as random and structured reactors. In random reactors, catalyst particles are located in a chaotic way in the reaction zone, no matter how carefully they are packed. It is not surprising that this results in a nonuniform fiow over the cross-section of the reaction zone, leading to a nonuniform access of reactants to the outer catalyst surface and, as a consequence, undesired concentration and temperature profiles. Not surprisingly, this leads, in general, to lower yield and selectivity. In structured reactors, the catalyst is of a well-defined spatial structure, which can be designed in more detail. The hydrodynamics can be simplified to essentially laminar, well-behaved uniform fiow, enabling full access of reactants to the catalytic surface at a low pressure drop. 

Small angle X-ray-scattering studies and light-scattering studies of lens extracts show that the transparency of the lens is the result of the short-range spatial order of lens proteins (Delaye and Tardieu, 1983). The molecular structure of the individual crystallins is well ordered even though the overall pattern in the lens may appear chaotic. 

Dye structures of passive tracers placed in time-periodic chaotic flows evolve in an iterative fashion an entire structure is mapped into a new structure with persistent large-scale features, but finer and finer scale features are revealed at each period of the flow. After a few periods, strategically placed blobs of passive tracer reveal patterns that serve as templates for subsequent stretching and folding. Repeated action by the flow generates a lamellar structure consisting of stretched and folded striations, with thicknesses s(r), characterized by a probability density function, f(s,t), whose... 

Illustration Role of dispersed phases onflow structure. Dispersed fluid phases that have properties different from those of the continuous phase can disrupt the structure of chaotic and regular regions obtained for single... 

We focus on aggregation in model, regular and chaotic, flows. Two aggregation scenarios are considered In (i) the clusters retain a compact geometry—forming disks and spheres—whereas in (ii) fractal structures are formed. The primary focus of (i) is kinetics and self-similarity of size distributions, while the main focus of (ii) is the fractal structure of the clusters and its dependence with the flow. 

Here we consider aggregation in a physically realizable chaotic flow, the journal bearing flow or the vortex mixing flow described earlier. The computations mimic fast coagulation particles seeded in the flow are convected passively and aggregate upon contact. In this example the clusters retain a spherical structure and the capture radius is independent of the cluster size. 

Illustration Aggregation of fractal structures in chaotic flows. In a... 

Liu, Y. H., and Zumbrunnen, D. A., Toughness enhancement in polymer blends due to the in-situ formation by chaotic mixing of fine-scale extended structures, J. Mater. Sci. 34, 1921-1931 (1999). 

Zhang, D. F., and Zumbrunnen, D. A., Influences of fluidic interfaces on the formation of fine scale structures by chaotic mixing. J. Fluids Eng. 118,40-47 (1996b). 

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