Streamlines, chaotic

Reversible system on a cylinder) While studying chaotic streamlines inside a drop immersed in a steady Stokes flow, Stone et al. (1991) encountered the system... 

Stone, H. A Nadim, A., and Strogatz, S.H. (1991) Chaotic streamlines inside drops immersed in steady Stokes flows. J. Fluid Meeh. 232, 629. 

Laminar and Turbulent Flow, Reynolds Number These terms refer to two distinct types of flow. In laminar flow, there are smooth streamlines and the fuiid velocity components vary smoothly with position, and with time if the flow is unsteady. The flow described in reference to Fig. 6-1 is laminar. In turbulent flow, there are no smooth streamlines, and the velocity shows chaotic fluctuations in time and space. Velocities in turbulent flow may be reported as the sum of a time-averaged velocity and a velocity fluctuation from the average. For any given flow geometry, a dimensionless Reynolds number may be defined for a Newtonian fluid as Re = LU p/ I where L is a characteristic length. Below a critical value of Re the flow is laminar, while above the critical value a transition to turbulent flow occurs. The geometry-dependent critical Reynolds number is determined experimentally. 

Fig. 7.8 Poincare sections after 2000 cycles. Initially nine marker points were placed along the y axis and six along the x axis. The dimensionless amplitude was 0.5, as in Fig. 7.7. The parameter was the dimensionless period (a) 0.05 (h) 0.10 (c) 0.125 (d) 0.15 (e) 0.20 (f) 0.35 (g) 0.50 (h) 1.0 (i) 1.5. For the smallest values of the time period we see that the virtual marker points fall on smooth curves. The general shape of these curves would he the streamlines of two fixed continuously operating agitators. As the time period increases the virtual marker particles fall erratically and the regions indicate chaotic flow. With increasing time periods larger and larger areas become chaotic. [Reprinted by permission from H. Aref, Stirring Chaotic Advection, J. Fluid Meek, 143, 1-21 (1984).]...

Figure 4. A schematic of one period of the partitioned pipe mixer, along with the form of the streamlines in one cross section. Reprinted with permission from Chem. Eng. Sci., vol. 42, p. 2909, D. V. Khakhar, J. G. Franjione, and J. M. Ottino, A Case Study of Chaotic Mixing in Deterministic Rows The Partitioned Pipe Mixer, copyright 1987 [32], Pergamon Press PLC.

When the Lewis number is nonunity, the mass diffusivity can be greater than the thermal diffusivity. This discrepancy in diffusivities is important with respect to the reactant that limits the reaction. Ignoring the hydrodynamic instability, consider again the condition between a pair of streamlines entering a wrinkle in a laminar flame. This time, however, look more closely at the flame stmcture that these streamlines encompass, noting that the limiting reactant will diffuse into the flame zone faster than heat can diffuse from the flame zone into the unbumed mixture. Thus, the flame temperature rises, the flame speed increases, and the flame wrinkles bow further in the downstream direction. The result is a flame that looks very much like the flame depicted for the hydrodynamic instability in Fig. 45. The flame surface breaks up continuously into new cells in a chaotic... 

If a stress, however slight, is applied to a fluid, it will flow. A fluid may be a gas or a liquid, and we will primarily consider liquids. The flow may be laminar or turbulent. The latter is chaotic, implying that a volume element may at any moment move in any direction, though the average flow is in one direction. In laminar flow, the streamlines, i.e., the trajectories of small volume elements, exhibit a smooth and regular pattern. 

Thus, the time-dependence of the flow generates chaotic trajectories that will enhance the mixing of fluid within these regions. However, the KAM tori formed by the remaining quasiperiodic orbits separate the domain into a set of disconnected regions with no advec-tive transport between them. Therefore, when the time-dependence is weak the fluid is only mixed within narrow layers around the resonant streamlines of the original time-independent flow. The areas... 

Since the time-dependence of the velocity field is restricted to a finite region the complex chaotic orbits are also limited to this region. Advection in such flows is a chaotic scattering process (Ott and Tel, 1993 Ott, 1993) in which fluid elements approach the mixing zone along the inflow streamlines, they follow chaotic trajectories inside... 

The experiments also demonstrated presence of the mixing islands , where very little mixing took place. In consequence, the chaotic mixing could be schematically represented by the streamline diagram comprising the elliptic points located in the center of the blinking vortex and a hyperbolic... 

Figure 4, Schematics of (a) horizontal flow along the main flow channel, with a straight streamline showing the direction offlow (b) streamline folding affected by perturbation, and (c) stretching of a streamline after switching off a perturbation, which is favorable to chaotic mixing.

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