Applications of Advection-Diffusion Based Methodologies to Fluid and Granular Flows.

摘要

The transport of material by a known velocity field is well-described by the advection-diffusion equation. In this dissertation, the advection-diffusion equation is investigated numerically to gain insights into mixing phenomena in both fluid and granular flows.;In fluid flows, mixing occurs through the interplay of chaotic advection and diffusion. By using a mapping method with operator splitting, the advection-diffusion equation is numerically solved and the time-evolution of the mixing in the system is determined. This method is applied to mixing in time-periodic sine flow, the rotated potential mixer, and three-dimensional ABC flow, and the physical mechanisms of mixing in these flows is investigated. The method is generalized to solve advection-reaction-diffusion systems as well, where a species is advected by a velocity field, diffuses, and undergoes a chemical reaction. Specifically, competitive autocatalytic reactions are investigated to gain insight into problems of chiral symmetry breaking and homochirality.;In granular flows, bidisperse mixtures of particles (two different particle sizes) have the tendency to segregate: small particles percolate downwards between shear generated voids and large particles percolate upwards. By considering discrete granular particles in an Eulerian framework, the advection-diffusion equation can be applied to study this phenomenon as well. This generalization of the advection-diffusion equation represents an interplay between advection due to mean flow, advection due to percolation driven segregation, and diffusion. Using a similar method to the mapping method with operator splitting introduced for mixing in fluid flows, the advection-diffusion equation, modified to include size segregation, is efficiently solved to determine segregation patterns in granular flows. The method is applied to bounded heaps and rotating tumblers, and, in both flows, quantitative agreement with experiments and simulations is obtained. Theoretical predictions of segregation patterns serve to elucidate the different mechanisms that cause mixing and segregation in both bounded heaps and rotating tumblers. The approach is generalized to describe multidisperse (more than two different particle sizes) and polydisperse (a continuum of different particle sizes) mixtures of particles, and results consistent with simulations are observed.