Fluid mixing is a challenging problem in laminar flow systems. Even in microfluidic systems, diffusion is often negligible compared to advection in the flow. The idea of chaotic advection can be applied in these systems to enhance mixing efficiency. Topological chaos can also lead to efficient and rapid mixing. In this dissertation, an approach to enhance fluid mixing in laminar flows without internal rods is demonstrated by using the idea of topological chaos. Periodic motion of three stirrers in a two-dimensional flow can lead to chaotic transport of the surrounding fluid. For certain stirrer motions, the generation of chaos is guaranteed solely by the topology of that motion and continuity of the fluid. This approach is in contrast to standard techniques. Appropriate stirrer motions are determined using the Thurston-Nielsen classification theorem, which also predicts a lower bound on the complexity of the dynamics in the flow. Work in this area has focused largely on using physical rods as stirrers, but the theorem also applies when the ``stirrers'' are passive fluid particles. In this thesis, topological chaos is theoretically and numerically investigated in lid-driven cavities and rectangular channels without internal rods. When a two-dimensional incompressible Newtonian flow is in the Stokes flow regime, the stream function satisfies the two-dimensional biharmonic equation. When the flow occurs in a lid-driven cavity with solid side walls, this equation can be solved using a method that is similar to the traditional Fourier expansion but uses an asymptotic approximation for the sum of high order terms. When the flow occurs between two infinite plates with spatially periodic boundary conditions, an exact solution in a rectangle with finite width, which represents the flow in this infinitely-wide cavity, can be obtained by using the principle of superposition. A fully developed, three-dimensional flow in a rectangular channel can be decomposed into an unperturbed Poiseuille flow in the axial direction and a lid-driven cavity secondary flow in the cross section. This model can be applied to numerically simulate either pressure-driven flow in a rectangular channel with surface grooves or electro-osmotic flow in a rectangular channel with variations in surface potential. In this dissertation, the occurrence of topological chaos in unsteady two-dimensional flows as well as steady three-dimensional flows without internal rods is demonstrated. For appropriate choices of boundary velocity on the top and/or bottom walls, there exist three periodic points in the flows that produce a chaos-generating motion. In steady flow through a three-dimensional rectangular channel, the axial direction plays the role of time and the periodic points lie on streamtubes that "braid" the surrounding fluid as it moves through the duct. When appropriate motion is applied on the boundary of the wide cavity or channel, topological chaos can also be generated in the flow. The stretching rate of non-trivial material lines in all these flows agrees with the prediction of the lower bound of topological entropy provided by the Thurston-Nielsen theorem. Ghost rod structures are found and analyzed in the lid-driven cavity and rectangular channel flows with solid side walls. The results suggest that the no-slip boundary condition on the stationary internal surfaces is one of the reasons for poor mixing in steady laminar three-dimensional flows considered previously with solid braided internal rods. / Ph. D.

Since it is difficult or impossible to generate turbulent flow in a highly viscous fluid or a microfluidic system, efficient mixing becomes a challenge. However, it is possible in a laminar flow to generate chaotic particle trajectories (well-known as chaotic advection), that can lead to effective mixing. This dissertation studies mixing in flows with the limiting case of zero Reynolds numbers that are called Stokes flows and illustrates the practical use of different theories, namely the topological chaos theory, the set-oriented analysis and lobe dynamics in the analysis, design and optimization of different laminar-flow mixing systems. In a recent development, the topological chaos theory has been used to explain the chaos built in the flow only based on the topology of boundary motions. Without considering any details of the fluid dynamics, this novel method uses the Thurston-Nielsen (TN) classification theorem to predict and describe the stretching of material lines both qualitatively and quantitatively. The practical application of this theory toward design and optimization of a viscous-flow mixer and the important role of periodic orbits as "ghost rods" are studied. The relationship between stretching of material lines (chaos) and the homogenization of a scalar (mixing) in chaotic Stokes flows is examined in this work. This study helps determining the extent to which the stretching can represent real mixing. Using a set-oriented approach to describe the stirring in the flow, invariance or leakiness of the Almost Invariant Sets (AIS) playing the role of ghost rods is found to be in a direct relationship with the rate of homogenization of a scalar. The mixing caused by these AIS and the variations of their structure are explained from the point of view of geometric mechanics using transport through lobes. These lobes are made of segments of invariant manifolds of the periodic points that are generators of the ghost rods. A variety of the concentration-based measures, the important parameters of their calculation, and the implicit effect of diffusion are described. The studies, measures and methods of this dissertation help in the evaluation and understanding of chaotic mixing systems in nature and in industrial applications. They provide theoretical and numerical grounds for selection of the appropriate mixing protocol and design and optimization of mixing systems, examples of which can be seen throughout the dissertation. / Ph. D.

Fluid mixing is a challenging problem in laminar flow systems. Chaotic advection can play an important role in enhancing mixing in such flow. In this thesis, different approaches are used to enhance fluid mixing in two laminar flow systems. In the first system, chaos is generated in a flow between two closely spaced parallel circular plates by pulsed operation of fluid extraction and reinjection through singularities in the domain. A singularity through which fluid is injected (or extracted) is called a source (or a sink). In a bounded domain, one source and one sink with equal strength operate together as a source-sink pair to conserve the fluid volume. Fluid flow between two closely spaced parallel plates is modeled as Hele-Shaw flow with the depth averaged velocity proportional to the gradient of the pressure. So, with the depth-averaged velocity, the flow between the parallel plates can effectively be modeled as two-dimensional potential flow. This thesis discusses pulsed source-sink systems with two source-sink pairs operating alternately to generate zig-zag trajectories of fluid particles in the domain. For reinjection purpose, fluid extracted through a sink-type singularity can either be relocated to a source-type one, or the same sink-type singularity can be activated as a source to reinject it without relocation. Relocation of fluid can be accomplished using either "first out first in" or "last out first in" scheme. Both relocation methods add delay to the pulse time of the system. This thesis analyzes mixing in pulsed source-sink systems both with and without fluid relocation. It is shown that a pulsed source-sink system with "first out first in" scheme generates comparatively complex fluid flow than pulsed source-sink systems with "last out first in" scheme. It is also shown that a pulsed source-sink system without fluid relocation can generate complex fluid flow. In the second system, mixing and transport is analyzed in a two-dimensional Stokes flow system. Appropriate periodic motions of three rods or periodic points in a two-dimensional flow are determined using the Thurston-Nielsen Classification Theorem (TNCT), which also predicts a lower bound on the complexity generated in the fluid flow. This thesis extends the TNCT -based framework by demonstrating that, in a perturbed system with no lower order fixed points, almost invariant sets are natural objects on which to apply the TNCT. In addition, a method is presented to compute line stretching by tracking appropriate motion of finite size rods. This method accounts for the effect of the rod size in computing the complexity generated in the fluid flow. The last section verifies the existence of almost invariant sets in a two-dimensional flow at finite Reynolds number. The almost invariant set structures move with appropriate periodic motion validating the application of the TNCT to predict a lower bound on the complexity generated in the fluid flow. / Ph. D.