| Theoretical mixing studies have focused on the topology of structures that occur in partially chaotic laminar flows because these classes of structures are responsible for segregation, bulk transport, and the dispersion of scalars. Though some analogy has been made to three-dimensional systems, most of the previous work done in these areas has focused on the two-dimensional point of view from the mixing standpoint. The mathematics needed to study mixing in 3D is already available; however, it has yet to be applied in many situations. In this dissertation, three aspects of topology in three-dimensional systems are used to develop 3D tools to understand mixing: sets of nested 2-tori, 3D flow skeletons, and 3D injection deformation.; In this work, a simple mathematical model of a torus is exposed to symmetry breaking 3D perturbations. The bifurcation pathways and the hardiness of nested 2-tori are compared to both simulation and experiments in agitated stirred tanks with eccentric impeller placement. Moving from the regular regions to the chaotic domain, 3D behavior near non-elliptical critical points experiences a remarkable change from 2D. Where only three types of critical points are possible in 2D, as many as thirteen are possible in 3D. Additionally, the connections between these points also become more complex in 3D. The existence and connections of these critical points are uncovered through simulations in the stirred tank, and the implications for mixing are explored. Moving from the scaffold of the chaotic region to the bulk, the creation of structure from instantaneous and non-instantaneous injections is explored, leading to observed structures that range between tendrils and sheets. The sheet and tendril structures are described by a dynamical feature herein called the unstable-neutral angle, defined as the angle between the local neutral and unstable directions in a flow. This angle is used to analyze the topology generate in stirred tanks via simulations. Use of a simple convection diffusion model shows important impact on selectivity due to changing topology. Together the tools that have been introduced show the richness of 3D topology on laminar mixing that cannot always be analogized to 2D studies and theory. |
