Chaotic Flows

As already mentioned, the motion of a chaotic flow is sensitive to initial conditions [H] points which initially he close together on the attractor follow paths that separate exponentially fast. This behaviour is shown in figure C3.6.3 for the WR chaotic attractor at /c 2=0.072. The instantaneous rate of separation depends on the position on the attractor. However, a chaotic orbit visits any region of the attractor in a recurrent way so that an infinite time average of this exponential separation taken along any trajectory in the attractor is an invariant quantity that characterizes the attractor. If y(t) is a trajectory for the rate law fc3.6.2] then we can linearize the motion in the neighbourhood of y to get... 

For regular motion, T> t) grows only linearly with time, so that the exponents are all zero. On the other hand, because chaotic flows are characterized by exponential divergences of initial nearby trajectories, a characteristic signature of such flows is the existence of at least one positive Lyapunov exponent. 

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... 

Filaments in chaotic flows experience complex time-varying stretching histories. Computational studies indicate that within chaotic regions, the distribution of stretches, A, becomes self-similar, achieving a scaling limit. 

Muzzio, Swanson, and Ottino (1991a) demonstrated that the distribution of stretching values in a globally chaotic flow approaches a log-normal distribution at large n A log-log graph of the computed distribution approaches a parabolic shape (Fig. 8a) as required for a log-normal distribution. Furthermore, as n increases, an increasing portion of the curves in the figure (Fig. 8b) overlap when the distribution is rescaled as... 

Illustration Drop size distributions produced by chaotic flows. Affinely deformed drops generate long filaments with a stretching distribution based on the log-normal distribution. The amount of stretching (A) determines the radius of the filament locally as... 

Fig. 23. (a) Distribution of drop sizes for mother droplets and satellite droplets (solid lines) produced during the breakup of a filament (average size = 2 x 10 5 m) in a chaotic flow. The total distribution is also shown (dashed line). A log-normal distribution of stretching with a mean stretch of 10 4 was used, (b) The cumulative distribution of mother droplets and satellite droplets (solid line) approaches a log-normal distribution (dashed line). 

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. 

Illustration Aggregation in chaotic flows with constant capture radius. 

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 area-conserving clusters in two dimensional chaotic flows. Particles, converted passively in a two-dimensional chaotic flow, aggregate on contact to form clusters. The capture radius of the clusters increases with the size of the cluster. Since these simulations are in two dimensions, the area of the aggregating clusters is conserved. 

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

Note that only one system, the one corresponding to constant capture radius clusters in chaotic flows, behaves as expected via mean field predictions. In general, the average cluster size grows fastest in the well-mixed system. However, in some cases the average cluster size in the regular flow grows faster than in the poorly mixed system. 

Fractal dimension depends on mixing in chaotic flows good mixing (no islands) gives lower fractal dimensions. 

Franjione. J. G and Ottino, J. M., Feasibility of numerical tracking of material lines and surfaces in chaotic flows. Phys. Fluids 30, 3641-3643 (1987). 

Hansen, S., Aggregation and fragmentation in chaotic flows of viscous fluids. Ph.D. Thesis, Northwestern University (1997). 

Tjahjadi, M., and Ottino, J. M., Stretching and breakup of droplets in chaotic flows. J. Fluid Mech. 232,191-219 (1991). 

We find experimentally that when ite>O(103) the flow is no longer laminar, i.e. the flow becomes unstable and vortices are formed. These are first seen at the boundaries between the fluid and solid surfaces. These chaotic flows should be avoided in our laboratory equipment in the course of viscometric characterisation. The conclusion that we draw is that we should make measurements at low Reynolds numbers in order to ensure that only the viscous dissipation is making a significant contribution to our measurements. 

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