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.

Black holes in the early universe, in compact binaries, and as energy sources inside solar-type stars

This thesis consists of three separate studies of roles that black holes might play in our universe.

In the first part we formulate a statistical method for inferring the cosmological parameters of our universe from LIGO/VIRGO measurements of the gravitational waves produced by coalescing black-hole/neutron-star binaries. This method is based on the cosmological distance-redshift relation, with "luminosity distances" determined directly, and redshifts indirectly, from the gravitational waveforms. Using the current estimates of binary coalescence rates and projected "advanced" LIGO noise spectra, we conclude that by our method the Hubble constant should be measurable to within an error of a few percent. The errors for the mean density of the universe and the cosmological constant will depend strongly on the size of the universe, varying from about 10% for a "small" universe up to and beyond 100% for a "large" universe. We further study the effects of random gravitational lensing and find that it may strongly impair the determination of the cosmological constant.

In the second part of this thesis we disprove a conjecture that black holes cannot form in an early, inflationary era of our universe, because of a quantum-field-theory induced instability of the black-hole horizon. This instability was supposed to arise from the difference in temperatures of any black-hole horizon and the inflationary cosmological horizon; it was thought that this temperature difference would make every quantum state that is regular at the cosmological horizon be singular at the black-hole horizon. We disprove this conjecture by explicitly constructing a quantum vacuum state that is everywhere regular for a massless scalar field. We further show that this quantum state has all the nice thermal properties that one has come to expect of "good" vacuum states, both at the black-hole horizon and at the cosmological horizon.

In the third part of the thesis we study the evolution and implications of a hypothetical primordial black hole that might have found its way into the center of the Sun or any other solar-type star. As a foundation for our analysis, we generalize the mixing-length theory of convection to an optically thick, spherically symmetric accretion flow (and find in passing that the radial stretching of the inflowing fluid elements leads to a modification of the standard Schwarzschild criterion for convection). When the accretion is that of solar matter onto the primordial hole, the rotation of the Sun causes centrifugal hangup of the inflow near the hole, resulting in an "accretion torus" which produces an enhanced outflow of heat. We find, however, that the turbulent viscosity, which accompanies the convective transport of this heat, extracts angular momentum from the inflowing gas, thereby buffering the torus into a lower luminosity than one might have expected. As a result, the solar surface will not be influenced noticeably by the torus's luminosity until at most three days before the Sun is finally devoured by the black hole. As a simple consequence, accretion onto a black hole inside the Sun cannot be an answer to the solar neutrino puzzle.

Proteomic analysis of the Cdc48/Ubx network identifies a role for Ubx2 in the regulation of lipid biosynthesis

Cdc48/p97 is an essential, highly abundant hexameric member of the AAA (ATPase associated with various cellular activities) family. It has been linked to a variety of processes throughout the cell but it is best known for its role in the ubiquitin proteasome pathway. In this system it is believed that Cdc48 behaves as a segregase, transducing the chemical energy of ATP hydrolysis into mechanical force to separate ubiquitin-conjugated proteins from their tightly-bound partners.

Current models posit that Cdc48 is linked to its substrates through a variety of adaptor proteins, including a family of seven proteins (13 in humans) that contain a Cdc48-binding UBX domain. As such, due to the complexity of the network of adaptor proteins for which it serves as the hub, Cdc48/p97 has the potential to exert a profound influence on the ubiquitin proteasome pathway. However, the number of known substrates of Cdc48/p97 remains relatively small, and smaller still is the number of substrates that have been linked to a specific UBX domain protein. As such, the goal of this dissertation research has been to discover new substrates and better understand the functions of the Cdc48 network. With this objective in mind, we established a proteomic screen to assemble a catalog of candidate substrate/targets of the Ubx adaptor system.

Here we describe the implementation and optimization of a cutting-edge quantitative mass spectrometry method to measure relative changes in the Saccharomyces cerevisiae proteome. Utilizing this technology, and in order to better understand the breadth of function of Cdc48 and its adaptors, we then performed a global screen to identify accumulating ubiquitin conjugates in cdc48-3 and ubxΔ mutants. In this screen different ubx mutants exhibited reproducible patterns of conjugate accumulation that differed greatly from each other, pointing to various unexpected functional specializations of the individual Ubx proteins.

As validation of our mass spectrometry findings, we then examined in detail the endoplasmic-reticulum bound transcription factor Spt23, which we identified as a putative Ubx2 substrate. In these studies ubx2Δ cells were deficient in processing of Spt23 to its active p90 form, and in localizing p90 to the nucleus. Additionally, consistent with reduced processing of Spt23, ubx2Δ cells demonstrated a defect in expression of their target gene OLE1, a fatty acid desaturase. Overall, this work demonstrates the power of proteomics as a tool to identify new targets of various pathways and reveals Ubx2 as a key regulator lipid membrane biosynthesis.

Robust near-threshold QDI circuit analysis and design

The two most important digital-system design goals today are to reduce power consumption and to increase reliability. Reductions in power consumption improve battery life in the mobile space and reductions in energy lower operating costs in the datacenter. Increased robustness and reliability shorten down time, improve yield, and are invaluable in the context of safety-critical systems. While optimizing towards these two goals is important at all design levels, optimizations at the circuit level have the furthest reaching effects; they apply to all digital systems. This dissertation presents a study of robust minimum-energy digital circuit design and analysis. It introduces new device models, metrics, and methods of calculation—all necessary first steps towards building better systems—and demonstrates how to apply these techniques. It analyzes a fabricated chip (a full-custom QDI microcontroller designed at Caltech and taped-out in 40-nm silicon) by calculating the minimum energy operating point and quantifying the chip’s robustness in the face of both timing and functional failures.

Three-dimensional nonlinear inelastic analysis of steel moment-frame buildings damaged by earthquake excitations

The Northridge earthquake of January 17, 1994, highlighted the two previously known problems of premature fracturing of connections and the damaging capabilities of near-source ground motion pulses. Large ground motions had not been experienced in a city with tall steel moment-frame buildings before. Some steel buildings exhibited fracture of welded connections or other types of structural degradation.

A sophisticated three-dimensional nonlinear inelastic program is developed that can accurately model many nonlinear properties commonly ignored or approximated in other programs. The program can assess and predict severely inelastic response of steel buildings due to strong ground motions, including collapse.

Three-dimensional fiber and segment discretization of elements is presented in this work. This element and its two-dimensional counterpart are capable of modeling various geometric and material nonlinearities such as moment amplification, spread of plasticity and connection fracture. In addition to introducing a three-dimensional element discretization, this work presents three-dimensional constraints that limit the number of equations required to solve various three-dimensional problems consisting of intersecting planar frames.

Two buildings damaged in the Northridge earthquake are investigated to verify the ability of the program to match the level of response and the extent and location of damage measured. The program is used to predict response of larger near-source ground motions using the properties determined from the matched response.

A third building is studied to assess three-dimensional effects on a realistic irregular building in the inelastic range of response considering earthquake directivity. Damage levels are observed to be significantly affected by directivity and torsional response.

Several strong recorded ground motions clearly exceed code-based levels. Properly designed buildings can have drifts exceeding code specified levels due to these ground motions. The strongest ground motions caused collapse if fracture was included in the model. Near-source ground displacement pulses can cause columns to yield prior to weaker-designed beams. Damage in tall buildings correlates better with peak-to-peak displacements than with peak-to-peak accelerations.

Dynamic response of tall buildings shows that higher mode response can cause more damage than first mode response. Leaking of energy between modes in conjunction with damage can cause torsional behavior that is not anticipated.

Various response parameters are used for all three buildings to determine what correlations can be made for inelastic building response. Damage levels can be dramatically different based on the inelastic model used. Damage does not correlate well with several common response parameters.

Realistic modeling of material properties and structural behavior is of great value for understanding the performance of tall buildings due to earthquake excitations.

Part I. Energy straggling of ^4He below 2.0 MeV in Al, Ni, Au and Pt. Part II. Studies of the Ti-W metallization system on Si

Part I.

In recent years, backscattering spectrometry has become an important tool for the analysis of thin films. An inherent limitation, though, is the loss of depth resolution due to energy straggling of the beam. To investigate this, energy straggling of ⁴He has been measured in thin films of Ni, Al, Au and Pt. Straggling is roughly proportional to square root of thickness, appears to have a slight energy dependence and generally decreases with decreasing atomic number of the adsorber. The results are compared with predictions of theory and with previous measurements. While Ni measurements are in fair agreement with Bohr's theory, Al measurements are 30% above and Au measurements are 40% below predicted values. The Au and Pt measurements give straggling values which are close to one another.

Part II.

MeV backscattering spectrometry and X-ray diffraction are used to investigate the behavior of sputter-deposited Ti-W mixed films on Si substrates. During vacuum anneals at temperatures near 700°C for several hours, the metallization layer reacts with the substrate. Backscattering analysis shows that the resulting compound layer is uniform in composition and contains Ti, Wand Si. The Ti:W ratio in the compound corresponds to that of the deposited metal film. X-ray analyses with Reed and Guinier cameras reveal the presence of the ternary Ti_xW_(1-x)Si₂ compound. Its composition is unaffected by oxygen contamination during annealing, but the reaction rate is affected. The rate measured on samples with about 15% oxygen contamination after annealing is linear, of the order of 0.5 Å per second at 725°C, and depends on the crystallographic orientation of the substrate and the dc bias during sputter-deposition of the Ti-W film.

Au layers of about 1000 Å thickness were deposited onto unreacted Ti-W films on Si. When annealed at 400°C these samples underwent a color change,and SEM micrographs of the samples showed that an intricate pattern of fissures which were typically 3µm wide had evolved. Analysis by electron microprobe revealed that Au had segregated preferentially into the fissures. This result suggests that Ti-W is not a barrier to Au-Si intermixing at 400°C.

Nuclear Energy Levels of ^31S and ^33Cl

The reaction ³²S(³He, α) ³¹S has been used to locate 42 levels in ³¹S. For 11 of the first 17 levels ℓ_n-values have been determined. The first 6 excited states of ³¹S have been studied by applying the particle-gamma correlation method of Litherland and Ferguson (their Method II) to the reaction ³²S(³He, αγ) ³¹S. The resulting spins and parities are: E_X, J^π = 1.25 MeV, 3/2⁺; 2.23 MeV, 5/2⁺; 3.08 MeV, 1/2⁺; 3.29 MeV, 5/2⁺, 3/2⁺; 3.35 MeV, 7/2, 3/2; 3.44 MeV, 3/2⁺. Mixing and branching ratios have also been determined. The ground state Q-value for the reaction ³²S(³He, α)³¹S has been measured to be 5.538 ± 0.006 MeV. Analysis of the spectra of the reaction ³²S(³He, α)³³Cl which were obtained as a by-product of the spectra of the reaction ³²S(³He, α) ³¹S located levels in ³³Cl at the following excitation energies: 0, 810 ± 9, (1978 ± 14), 2351 ± 9, 2686 ± 8, 2848 ± 9 (a known doublet), 2980 ± 9, and 4119 ± 10 keV. The 2.0 MeV level was only weakly populated, and to confirm its existence the reaction ³⁶Ar(p, α)³³Cl has been studied. In this reaction the 2.0 MeV level was strongly populated and the measured excitation energy was 1999 ± 20 keV. The experimental results for ³¹S and ³³Cl are compared with their analogs and with nuclear model predictions.

Information-theoretic Studies and Capacity Bounds: Group Network Codes and Energy Harvesting Communication Systems

Network information theory and channels with memory are two important but difficult frontiers of information theory. In this two-parted dissertation, we study these two areas, each comprising one part. For the first area we study the so-called entropy vectors via finite group theory, and the network codes constructed from finite groups. In particular, we identify the smallest finite group that violates the Ingleton inequality, an inequality respected by all linear network codes, but not satisfied by all entropy vectors. Based on the analysis of this group we generalize it to several families of Ingleton-violating groups, which may be used to design good network codes. Regarding that aspect, we study the network codes constructed with finite groups, and especially show that linear network codes are embedded in the group network codes constructed with these Ingleton-violating families. Furthermore, such codes are strictly more powerful than linear network codes, as they are able to violate the Ingleton inequality while linear network codes cannot. For the second area, we study the impact of memory to the channel capacity through a novel communication system: the energy harvesting channel. Different from traditional communication systems, the transmitter of an energy harvesting channel is powered by an exogenous energy harvesting device and a finite-sized battery. As a consequence, each time the system can only transmit a symbol whose energy consumption is no more than the energy currently available. This new type of power supply introduces an unprecedented input constraint for the channel, which is random, instantaneous, and has memory. Furthermore, naturally, the energy harvesting process is observed causally at the transmitter, but no such information is provided to the receiver. Both of these features pose great challenges for the analysis of the channel capacity. In this work we use techniques from channels with side information, and finite state channels, to obtain lower and upper bounds of the energy harvesting channel. In particular, we study the stationarity and ergodicity conditions of a surrogate channel to compute and optimize the achievable rates for the original channel. In addition, for practical code design of the system we study the pairwise error probabilities of the input sequences.

A fully-nonlocal energy-based formulation and high-performance realization of the quasicontinuum method

The quasicontinuum (QC) method was introduced to coarse-grain crystalline atomic ensembles in order to bridge the scales from individual atoms to the micro- and mesoscales. Though many QC formulations have been proposed with varying characteristics and capabilities, a crucial cornerstone of all QC techniques is the concept of summation rules, which attempt to efficiently approximate the total Hamiltonian of a crystalline atomic ensemble by a weighted sum over a small subset of atoms. In this work we propose a novel, fully-nonlocal, energy-based formulation of the QC method with support for legacy and new summation rules through a general energy-sampling scheme. Our formulation does not conceptually differentiate between atomistic and coarse-grained regions and thus allows for seamless bridging without domain-coupling interfaces. Within this structure, we introduce a new class of summation rules which leverage the affine kinematics of this QC formulation to most accurately integrate thermodynamic quantities of interest. By comparing this new class of summation rules to commonly-employed rules through analysis of energy and spurious force errors, we find that the new rules produce no residual or spurious force artifacts in the large-element limit under arbitrary affine deformation, while allowing us to seamlessly bridge to full atomistics. We verify that the new summation rules exhibit significantly smaller force artifacts and energy approximation errors than all comparable previous summation rules through a comprehensive suite of examples with spatially non-uniform QC discretizations in two and three dimensions. Due to the unique structure of these summation rules, we also use the new formulation to study scenarios with large regions of free surface, a class of problems previously out of reach of the QC method. Lastly, we present the key components of a high-performance, distributed-memory realization of the new method, including a novel algorithm for supporting unparalleled levels of deformation. Overall, this new formulation and implementation allows us to efficiently perform simulations containing an unprecedented number of degrees of freedom with low approximation error.

Energy inequalities and error estimates for axisymmetric torsion of thin elastic shells of revolution

The problem motivating this investigation is that of pure axisymmetric torsion of an elastic shell of revolution. The analysis is carried out within the framework of the three-dimensional linear theory of elastic equilibrium for homogeneous, isotropic solids. The objective is the rigorous estimation of errors involved in the use of approximations based on thin shell theory.

The underlying boundary value problem is one of Neumann type for a second order elliptic operator. A systematic procedure for constructing pointwise estimates for the solution and its first derivatives is given for a general class of second-order elliptic boundary-value problems which includes the torsion problem as a special case.

The method used here rests on the construction of “energy inequalities” and on the subsequent deduction of pointwise estimates from the energy inequalities. This method removes certain drawbacks characteristic of pointwise estimates derived in some investigations of related areas.

Special interest is directed towards thin shells of constant thickness. The method enables us to estimate the error involved in a stress analysis in which the exact solution is replaced by an approximate one, and thus provides us with a means of assessing the quality of approximate solutions for axisymmetric torsion of thin shells.

Finally, the results of the present study are applied to the stress analysis of a circular cylindrical shell, and the quality of stress estimates derived here and those from a previous related publication are discussed.