Mixing chaotic

The efficiency of mixing is significantly higher in chaotic flows due to the complex trajectories of fluid elements brought out by simple time-periodic or space-periodic velocity fields. The time series of fluid element locations undergoing chaotic advection is stochastic, and the Fourier spectra reveal multiple frequencies [61, 62]. The fluid-fluid interfaces periodically stretch and fold in chaotic flows. 

The periodic stretching and folding create new interfaces, which are essential for the rapid progress of chemical reactions [62]. 

A chaotic flow produces either transverse homocHnic or transverse heterocHnic intersections, and/or is able to stretch and fold material in such a way that it produces what is called a horseshoe map, and/or has positive Liapunov exponents. These definitions are not equivalent to each other, and their interrelations have been discussed by Doherty and Ottino [63]. The time-periodic perturbation of homoclinic and heteroclinic orbits can create chaotic flows. In bounded fluid flows, which are encountered in mixing tanks, the homoclinic and heteroclinic orbits are separate streamlines in an unperturbed system. These streamhnes prevent fluid flux from one region of the domain to the other, thereby severely limiting mixing. These separate streamlines generate stable and unstable manifolds upon perturbation, which in turn dictate the mass and energy transports in the system [64-66]. 

Chaotic flow can also be distinguished by the positive value of the Liapunov exponent k), which indicates that the nearby fluid element trajectories diverge exponentially with time because of stretching and folding. Note that X is close to zero in nonchaotic laminar flows in this case, the neighboring fluid element trajectories stay close to each other. 

The position of the species particle develops depending on the local velocity field, which can be obtained by solving (neglecting molecular diffusion) the following equation  

TABLE 6.3 Interfacial Tension Values for Typical Polymer Pairs 

Aref (1984) indicated that the equations which describe the particle trajectories in a two-dimensional flow have a Hamiltonian structure, that is. 

A flow can be termed chaotic if it satisfies any of the following criteria  

FIGURE 6.26 Elliptic and hyperbolic points. A blinking vortex system with vortex centers at the elliptic points can produce this streamline pattern. 

The Liapunov exponent, a, is related to the long time behavior of the lineal stretch, and it is equal to  

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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). 

Schepens, F. A. O., Chaotic mixing in the extended periodic cavity flow. Masters Thesis, Eindhoven University of Technology, 1996. 

Sundaraj, U., Dori, Y., and Macosko, C. W., Sheet formation in immiscible polymer blends model experiments on an initial blend morphology. Polymer 36,1957-1968 (1995). Swanson, P. D., and Ottino, J. M., A comparative computational and experimental study of chaotic mixing of viscous fluids, J. Fluid Mech. 213, 227-249 (1990). 

Zhang, D. F., and Zumbrunnen, D. A., Chaotic mixing of two similar fluids in the presence of a third dissimilar fluid. AIChE J. 42, 3301-3309 (1996a). 

Figure 8.4 Photographs of the deformation of dye with the chaotic mixing program a) photograph of the trajectory and stretching of the dye for the first several periods of flow, and b) photograph of a downstream midsection of the device after several periods of flow...

Figure 8.5 Photograph of the deformation of the dye with chaotic mixing near the discharge of the free helix extruder...

Figure 8.6 Average stretching (5,) for operational modes of a conventional extrusion and the chaotic mixing. The j/-axis is logarithmic. The top wall moving curve corresponds to a conventional single-screw extruder with a rotating screw. The results are shown for an /r/FF aspect ratio of one and a period of 4 s...

Figure 8.8 Numerical simulation of the stretching of four randomly placed compact matrices of particles into a square channel for the free helix geometry operated using the chaotic mixing program a) end view of the particles at 100 s of rotation, and b) three-dimensional view of the particles after 100 s of rotation...

Campbell, G. A., Bomma, S., St. John, S., and Chempath, S., Chaotic Mixing in a Free Helix Single Screw Device, SPEANTEC Tech. Papers, 48, 1441 (2002)... 

Kinematic simulations were undertaken to describe the folding of material lines and identify parameter settings which give chaotic mixing [48],... 

M 70a] [P 61] It is known that stirring and chaotic mixing can be achieved by an unsteady potential flow [154], This can also be utilized for helical flows in curved... 

Passive Chaotic Mixing by Posing Grooves to Viscous Flows Most Relevant Citations... 

This passive chaotic mixing concept is achieved with only minor modification of the flow channel, i.e. placement of only small obstacles, so that a low pressure loss for a given mixing task is expected [58],... 

C., An optimized split-and-recombine micro mixer with uniform chaotic mixing, Lab Chip 2004, 4, 65-69. 

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