Nonlinearity, chaos, and self-similarity: Effects of mixing, aggregation, reaction and breakup on structure formation

• Main Author: Muzzio, Fernando Javier
• Language: ENG
• Published: ScholarWorks@UMass Amherst 1991
• Subjects: Chemical engineering|Physics|Statistics
• Online Access: https://scholarworks.umass.edu/dissertations/AAI9132890

This dissertation focuses on four problems: stretching and stirring in chaotic flows, aggregation in chaotic and regular flows, diffusion and reaction in lamellar structures, and breakup in chaotic flows. Stretching and stirring in chaotic flows is simulated for O(10$\sp5$) material points and is modelled as the product of multipliers, defined as the ratios between stretchings accumulated by the points during successive periods. As expected for chaotic flows, the mean stretching increases exponentially with time. The probability density function of multipliers converges--in just two periods or so--to a time-invariant distribution, producing distributions of stretching that become self-similar in about ten periods. Aggregation is simulated in a simple shear flow and in a chaotic flow. The motion of the particles is due solely to the flow; particles aggregate each time they come closer than a distance d. For simple shear flow, segregation is the main element in the dynamics. For extensive enough aggregation, the cluster mass distribution becomes independent of d and exhibits scaling behavior. Chaotic flows continuously destroy segregation and generate "well-mixed" conditions, and under these circumstances a mean field Smoluchowski equation predicts the cluster size distribution, which is self-similar. Simulations of lamellar structures undergoing diffusion and reaction exhibit self-similar striation thickness distributions, allowing us to develop fractal kinetic models that predict the evolution of the concentration of reactants. For long times diffusion becomes the dominant process, the kinetic parameters become asymptotically irrelevant, and the average concentration of reactants C decays with a power law t$\sp{-1/4}$ for a wide range of reaction orders and reactions rate constants. Experiments on breakup of immiscible fluids in chaotic flows produce self-similar drop size distributions which belong to one of two different self-similar families: for low viscosity ratios, mean drop sizes decrease with increasing viscosity and/or shear rate; for higher viscosity ratios, drop size distributions become independent of fluid and/or flow parameters. Each family exhibits a different shape, presumably due to changes in the breakup mechanism. The presence of self-similar distributions in all these systems suggests that approaches based on self-similar concepts might have wide applicability in many other problems as well.