A dynamical systems analysis of vortex pinch-off

Vortex rings constitute the main structure in the wakes of a wide class of swimming and flying animals, as well as in cardiac flows and in the jets generated by some moss and fungi. However, there is a physical limit, determined by an energy maximization principle called the Kelvin-Benjamin principle, to the size that axisymmetric vortex rings can achieve. The existence of this limit is known to lead to the separation of a growing vortex ring from the shear layer feeding it, a process known as `vortex pinch-off', and characterized by the dimensionless vortex formation number. The goal of this thesis is to improve our understanding of vortex pinch-off as it relates to biological propulsion, and to provide future researchers with tools to assist in identifying and predicting pinch-off in biological flows.

To this end, we introduce a method for identifying pinch-off in starting jets using the Lagrangian coherent structures in the flow, and apply this criterion to an experimentally generated starting jet. Since most naturally occurring vortex rings are not circular, we extend the definition of the vortex formation number to include non-axisymmetric vortex rings, and find that the formation number for moderately non-axisymmetric vortices is similar to that of circular vortex rings. This suggests that naturally occurring vortex rings may be modeled as axisymmetric vortex rings. Therefore, we consider the perturbation response of the Norbury family of axisymmetric vortex rings. This family is chosen to model vortex rings of increasing thickness and circulation, and their response to prolate shape perturbations is simulated using contour dynamics. Finally, the response of more realistic models for vortex rings, constructed from experimental data using nested contours, to perturbations which resemble those encountered by forming vortices more closely, is simulated using contour dynamics. In both families of models, a change in response analogous to pinch-off is found as members of the family with progressively thicker cores are considered. We posit that this analogy may be exploited to understand and predict pinch-off in complex biological flows, where current methods are not applicable in practice, and criteria based on the properties of vortex rings alone are necessary.

Effects of particle size ratio on single particle motion in colloidal dispersions

The motion of a single Brownian particle of arbitrary size through a dilute colloidal dispersion of neutrally buoyant bath spheres of another characteristic size in a Newtonian solvent is examined in two contexts. First, the particle in question, the probe particle, is subject to a constant applied external force drawing it through the suspension as a simple model for active and nonlinear microrheology. The strength of the applied external force, normalized by the restoring forces of Brownian motion, is the Péclet number, Pe. This dimensionless quantity describes how strongly the probe is upsetting the equilibrium distribution of the bath particles. The mean motion and fluctuations in the probe position are related to interpreted quantities of an effective viscosity of the suspension. These interpreted quantities are calculated to first order in the volume fraction of bath particles and are intimately tied to the spatial distribution, or microstructure, of bath particles relative to the probe. For weak Pe, the disturbance to the equilibrium microstructure is dipolar in nature, with accumulation and depletion regions on the front and rear faces of the probe, respectively. With increasing applied force, the accumulation region compresses to form a thin boundary layer whose thickness scales with the inverse of Pe. The depletion region lengthens to form a trailing wake. The magnitude of the microstructural disturbance is found to grow with increasing bath particle size -- small bath particles in the solvent resemble a continuum with effective microviscosity given by Einstein's viscosity correction for a dilute dispersion of spheres. Large bath particles readily advect toward the minimum approach distance possible between the probe and bath particle, and the probe and bath particle pair rotating as a doublet is the primary mechanism by which the probe particle is able to move past; this is a process that slows the motion of the probe by a factor of the size ratio. The intrinsic microviscosity is found to force thin at low Péclet number due to decreasing contributions from Brownian motion, and force thicken at high Péclet number due to the increasing influence of the configuration-averaged reduction in the probe's hydrodynamic self mobility. Nonmonotonicity at finite sizes is evident in the limiting high-Pe intrinsic microviscosity plateau as a function of bath-to-probe particle size ratio. The intrinsic microviscosity is found to grow with the size ratio for very small probes even at large-but-finite Péclet numbers. However, even a small repulsive interparticle potential, that excludes lubrication interactions, can reduce this intrinsic microviscosity back to an order one quantity. The results of this active microrheology study are compared to previous theoretical studies of falling-ball and towed-ball rheometry and sedimentation and diffusion in polydisperse suspensions, and the singular limit of full hydrodynamic interactions is noted.

Second, the probe particle in question is no longer subject to a constant applied external force. Rather, the particle is considered to be a catalytically-active motor, consuming the bath reactant particles on its reactive face while passively colliding with reactant particles on its inert face. By creating an asymmetric distribution of reactant about its surface, the motor is able to diffusiophoretically propel itself with some mean velocity. The effects of finite size of the solute are examined on the leading order diffusive microstructure of reactant about the motor. Brownian and interparticle contributions to the motor velocity are computed for several interparticle interaction potential lengths and finite reactant-to-motor particle size ratios, with the dimensionless motor velocity increasing with decreasing motor size. A discussion on Brownian rotation frames the context in which these results could be applicable, and future directions are proposed which properly incorporate reactant advection at high motor velocities.

Binding site size limitations of imidazole-pyrrole polyamides for recognition in the minor groove of DNA

The discovery that the three ring polyamide Im-Py-Py-Dp containing imidazole (Im) and pyrrole (Py) carboxamides binds the DNA sequence 5'-(A,T)G(A,T)C(A,T)-3' as an antiparallel dimer offers a new model for the design of ligands for specific recognition of sequences in the minor groove containing both G,C and A,T base pairs. In Chapter 2, experiments are described in which the sequential addition of five N- methylpyrrolecarboxamides to the imidazole-pyrrole polyamide Im-Py-Py-Dp affords a series of six homologous polyamides, Im-(Py)_2-7-Dp, that differ in the size of their binding site, apparent first order binding affinity, and sequence specificity. These results demonstrate that DNA sequences up to nine base pairs in length can be specifically recognized by imidazole-pyrrole polyamides containing three to seven rings by 2:1 polyamide-DNA complex formation in the minor groove. Recognition of a nine base pair site defines the new lower limit of the binding site size that can be recognized by polyamides containing exclusively imidazole and pyrrolecarboxamides. The results of this study should provide useful guidelines for the design of new polyamides that bind longer DNA sites with enhanced affinity and specificity.

In Chapter 3 the design and synthesis of the hairpin polyamide Im-Py-Im-Py-γ-Im- Py-Im-Py-Dp is described. Quantitative DNase I footprint titration experiments reveal that Im-Py-Im-Py-γ-Im-Py-Im-Py-Dp binds six base pair 5'-(A,T)GCGC(A,T)-3' sequences with 30-fold higher affinity than the unlinked polyamide Im-Py-Im-Py-Dp. The hairpin polyamide does not discriminate between A•T and T•A at the first and sixth positions of the binding site as three sites 5'-TGCGCT-3', 5'-TGCGCA-3', and 5 'AGCGCT- 3' are bound with similar affinity. However, Im-Py-Im-Py-γ-Im-Py-Im-PyDp is specific for and discriminates between G•C and C•G base pairs in the 5'-GCGC-3' core as evidenced by lower affinities for the mismatched sites 5'-AACGCA-3', 5'- TGCGTT-3', 5'-TGCGGT-3', and 5'-ACCGCT-3'.

In Chapter 4, experiments are described in which a kinetically stable hexa-aza Schiff base La³⁺ complex is covalently attached to a Tat(49-72) peptide which has been shown to bind the HIV-1 TAR RNA sequence. Although these metallo-peptides cleave TAR site-specifically in the hexanucleotide loop to afford products consistent with hydrolysis, a series of control experiments suggests that the observed cleavage is not caused by a sequence-specifically bound Tat(49-72)-La(L)³⁺ peptide.

Awakening a sleeping giant: The riddle of Latino political participation

For some time now, the Latino voice has been gradually gaining strength in American politics, particularly in such states as California, Florida, Illinois, New York, and Texas, where large numbers of Latino immigrants have settled and large numbers of electoral votes are at stake. Yet the issues public officials in these states espouse and the laws they enact often do not coincide with the interests and preferences of Latinos. The fact that Latinos in California and elsewhere have not been able to influence the political agenda in a way that is commensurate with their numbers may reflect their failure to participate fully in the political process by first registering to vote and then consistently turning out on election day to cast their ballots.

To understand Latino voting behavior, I first examine Latino political participation in California during the ten general elections of the 1980s and 1990s, seeking to understand what percentage of the eligible Latino population registers to vote, with what political party they register, how many registered Latinos to go the polls on election day, and what factors might increase their participation in politics. To ensure that my findings are not unique to California, I also consider Latino voter registration and turnout in Texas for the five general elections of the 1990s and compare these results with my California findings.

I offer a new approach to studying Latino political participation in which I rely on county-level aggregate data, rather than on individual survey data, and employ the ecological inference method of generalized bounds. I calculate and compare Latino and white voting-age populations, registration rates, turnout rates, and party affiliation rates for California's fifty-eight counties. Then, in a secondary grouped logit analysis, I consider the factors that influence these Latino and white registration, turnout, and party affiliation rates.

I find that California Latinos register and turn out at substantially lower rates than do whites and that these rates are more volatile than those of whites. I find that Latino registration is motivated predominantly by age and education, with older and more educated Latinos being more likely to register. Motor voter legislation, which was passed to ease and simplify the registration process, has not encouraged Latino registration . I find that turnout among California's Latino voters is influenced primarily by issues, income, educational attainment, and the size of the Spanish-speaking communities in which they reside. Although language skills may be an obstacle to political participation for an individual, the number of Spanish-speaking households in a community does not encourage or discourage registration but may encourage turnout, suggesting that cultural and linguistic assimilation may not be the entire answer.

With regard to party identification, I find that Democrats can expect a steady Latino political identification rate between 50 and 60 percent, while Republicans attract 20 to 30 percent of Latino registrants. I find that education and income are the dominant factors in determining Latino political party identification, which appears to be no more volatile than that of the larger electorate.

Next, when I consider registration and turnout in Texas, I find that Latino registration rates are nearly equal to those of whites but that Texas Latino turnout rates are volatile and substantially lower than those of whites.

Low turnout rates among Latinos and the volatility of these rates may explain why Latinos in California and Texas have had little influence on the political agenda even though their numbers are large and increasing. Simply put, the voices of Latinos are little heard in the halls of government because they do not turn out consistently to cast their votes on election day.

While these findings suggest that there may not be any short-term or quick fixes to Latino participation, they also suggest that Latinos should be encouraged to participate more fully in the political process and that additional education may be one means of achieving this goal. Candidates should speak more directly to the issues that concern Latinos. Political parties should view Latinos as crossover voters rather than as potential converts. In other words, if Latinos were "a sleeping giant," they may now be a still-drowsy leviathan waiting to be wooed by either party's persuasive political messages and relevant issues.

Essays on Correlated Equilibrium and Voter Turnout

This thesis consists of three essays in the areas of political economy and game theory, unified by their focus on the effects of pre-play communication on equilibrium outcomes.

Communication is fundamental to elections. Chapter 2 extends canonical voter turnout models, where citizens, divided into two competing parties, choose between costly voting and abstaining, to include any form of communication, and characterizes the resulting set of Aumann's correlated equilibria. In contrast to previous research, high-turnout equilibria exist in large electorates and uncertain environments. This difference arises because communication can coordinate behavior in such a way that citizens find it incentive compatible to follow their correlated signals to vote more. The equilibria have expected turnout of at least twice the size of the minority for a wide range of positive voting costs.

In Chapter 3 I introduce a new equilibrium concept, called subcorrelated equilibrium, which fills the gap between Nash and correlated equilibrium, extending the latter to multiple mediators. Subcommunication equilibrium similarly extends communication equilibrium for incomplete information games. I explore the properties of these solutions and establish an equivalence between a subset of subcommunication equilibria and Myerson's quasi-principals' equilibria. I characterize an upper bound on expected turnout supported by subcorrelated equilibrium in the turnout game.

Chapter 4, co-authored with Thomas Palfrey, reports a new study of the effect of communication on voter turnout using a laboratory experiment. Before voting occurs, subjects may engage in various kinds of pre-play communication through computers. We study three communication treatments: No Communication, a control; Public Communication, where voters exchange public messages with all other voters, and Party Communication, where messages are exchanged only within one's own party. Our results point to a strong interaction effect between the form of communication and the voting cost. With a low voting cost, party communication increases turnout, while public communication decreases turnout. The data are consistent with correlated equilibrium play. With a high voting cost, public communication increases turnout. With communication, we find essentially no support for the standard Nash equilibrium turnout predictions.

I. Stokes flow past a thin screen. II. Viscous flows past porous bodies of finite size

Part I

The slow, viscous flow past a thin screen is analyzed based on Stokes equations. The problem is reduced to an associated electric potential problem as introduced by Roscoe. Alternatively, the problem is formulated in terms of a Stokeslet distribution, which turns out to be equivalent to the first approach.

Special interest is directed towards the solution of the Stokes flow past a circular annulus. A "Stokeslet" formulation is used in this analysis. The problem is finally reduced to solving a Fredholm integral equation of the second kind. Numerical data for the drag coefficient and the mean velocity through the hole of the annulus are obtained.

Stokes flow past a circular screen with numerous holes is also attempted by assuming a set of approximate boundary conditions. An "electric potential" formulation is used, and the problem is also reduced to solving a Fredholm integral equation of the second kind. Drag coefficient and mean velocity through the screen are computed.

Part II

The purpose of this investigation is to formulate correctly a set of boundary conditions to be prescribed at the interface between a viscous flow region and a porous medium so that the problem of a viscous flow past a porous body can be solved.

General macroscopic equations of motion for flow through porous media are first derived by averaging Stokes equations over a volume element of the medium. These equations, including viscous stresses for the description, are more general than Darcy's law. They reduce to Darcy's law when the Darcy number becomes extremely small.

The interface boundary conditions of the first kind are then formulated with respect to the general macroscopic equations applied within the porous region. An application of such equations and boundary conditions to a Poiseuille shear flow problem demonstrates that there usually exists a thin interface layer immediately inside the porous medium in which the tangential velocity varies exponentially and Darcy's law does not apply.

With Darcy's law assumed within the porous region, interface boundary conditions of the second kind are established which relate the flow variables across the interface layer. The primary feature is a jump condition on the tangential velocity, which is found to be directly proportional to the normal gradient of the tangential velocity immediately outside the porous medium. This is in agreement with the experimental results of Beavers, et al.

The derived boundary conditions are applied in the solutions of two other problems: (1) Viscous flow between a rotating solid cylinder and a stationary porous cylinder, and (2) Stokes flow past a porous sphere.