Dispersive and diffusive-dispersive shock waves for nonconvex conservation laws

We consider two physically and mathematically distinct regularization mechanisms of scalar hyperbolic conservation laws. When the flux is convex, the combination of diffusion and dispersion are known to give rise to monotonic and oscillatory traveling waves that approximate shock waves. The zero-diffusion limits of these traveling waves are dynamically expanding dispersive shock waves (DSWs). A richer set of wave solutions can be found when the flux is non-convex. This review compares the structure of solutions of Riemann problems for a conservation law with non-convex, cubic flux regularized by two different mechanisms: 1) dispersion in the modified Korteweg--de Vries (mKdV) equation; and 2) a combination of diffusion and dispersion in the mKdV-Burgers equation. In the first case, the possible dynamics involve two qualitatively different types of DSWs, rarefaction waves (RWs) and kinks (monotonic fronts). In the second case, in addition to RWs, there are traveling wave solutions approximating both classical (Lax) and non-classical (undercompressive) shock waves. Despite the singular nature of the zero-diffusion limit and rather differing analytical approaches employed in the descriptions of dispersive and diffusive-dispersive regularization, the resulting comparison of the two cases reveals a number of striking parallels. In contrast to the case of convex flux, the mKdVB to mKdV mapping is not one-to-one. The mKdV kink solution is identified as an undercompressive DSW. Other prominent features, such as shock-rarefactions, also find their purely dispersive counterparts involving special contact DSWs, which exhibit features analogous to contact discontinuities. This review describes an important link between two major areas of applied mathematics, hyperbolic conservation laws and nonlinear dispersive waves.Comment: Revision from v2; 57 pages, 19 figure

Combustion of liquid sprays at high pressures

The combustion of pressure atomized fuel sprays in high pressure stagnant air was studied. Measurements were made of flame and spray boundaries at pressures in the range 0.1-9 MPa for methanol and n-pentane. At the higher test pressure levels, critical phenomena are important. The experiments are compared with theoretical predictions based on a locally homogeneous two-phase flow model. The theory correctly predicted the trends of the data, but underestimates flame and spray boundaries by 30-50 percent, indicating that slip is still important for the present experiments (Sauter mean diameters of 30 microns at atmospheric pressure under cold flow conditions). Since the sprays are shorter at high pressures, slip effects are still important even though the density ratio of the phases approach one another as the droplets heat up. The model indicates the presence of a region where condensed water is present within the spray and provides a convenient means of treating supercritical phenomena

Studies in Crystal Structure. Part I: alpha-Ethynylacetic Acid. Part II: n-Hexatriacontane

The first part of this thesis describes an investigation, by X-ray methods, of ethynylacetic acid (prop-2-yne-1-carboxylic acid). The acid was found to exist in two crystal forms and one of these - the alpha-form - was studied in some detail. The dimensions of the two crystallographically independent molecules in the unit cell were obtained from projections along two of the cell axes and more accurate values of the molecular parameters were found by a process of averaging. An attempt was then made to determine the nature of the acetylenic carbon-hydrogen bond at the end of the molecular chain, Part 2 describes an investigation of the monoclinic form of n-hexatriacontane. The signs of the reflecting planes in two of the lattice zones were obtained by means of sub-cell theory and the molecular structure was determined from projections down two of the cell axes. By assuming a regular repetition along the hydrocarbon chain, the molecular dimensions were determined with greater accuracy

The Morphology and Evolution of the Primate Brachial Plexus

Primate evolutionary history is inexorably linked to the evolution of a broad array of locomotor adaptations that have facilitated the clade’s invasion of new niches. Researchers studying the evolution of primates and of their individual locomotor adaptations have traditionally relied on bony morphology – a practical choice given the virtual non-existence of any other type of tissue in the fossil record. However, this focus downplays the potential importance of the many other structures involved in locomotion, such as muscle, cartilage, and neural tissue, which may each be influenced by separate selective forces because of their different roles in facilitating movement. This dissertation is an investigation into the evolution of primate anatomy with an emphasis on the peripheral nervous system, particularly that of the brachial plexus, its intraspecific patterning, and its interactions with muscle in relation to changes in locomotion across clades. As the primate nervous system directs voluntary motor movement to the limbs, thereby facilitating locomotion, its morphology may be expected to vary with primate locomotor proclivities and/or limb anatomy. This prediction has not been explicitly tested. The anatomy of the peripheral nervous system was studied using a comparative approach both within 29 genera of primates and among non-primate clades via extensive primary dissection and a broad literature search in order to better understand its evolution. Data on spinal nerve level contributions, axon combination and branching morphology, nerve distribution pattern, and neural relationships with other soft tissues are detailed with photographs and standardized descriptions for 79 specimens and 123 individual plexuses. 99 characters generated from observations made during dissection were then analyzed using a parsimony-based phylogenetics approach to evaluate the evolutionary patterns presented by the brachial plexus in primates. The phylogenies generated with the brachial plexus characters did not perfectly mirror commonly accepted primate phylogenies, suggesting that while there is some evolutionary signal contained in the plexus, its morphology may also be influenced by forelimb function. As robust hypotheses exist regarding extant primate phylogenetic relationships and evolutionary histories, character evolution was mapped onto existing molecular trees to better understand how the individual structures that comprise the brachial plexus may evolve independently or in concert at different taxonomic levels. The rate of brachial plexus evolution in clades and leaf taxa was then assessed, demonstrating a marked heterogeneity in the structure both within and among clades. Taxa that have undergone recent locomotor shifts since divergence from their most recent common ancestor, and particularly those who exhibit some amount of suspensory behaviors, exhibit the highest rates of evolution observed here. Notably, several ape genera exhibit brachial plexus evolutionary rates significantly higher than the primate mean, running counter to the notion that hominoids have undergone an evolutionary slowdown relative to other primates. As the true unit of homology in the peripheral nervous system is a subject of ongoing debate, several levels of discussion are necessary to understand the variation in primates and their place in the broader spectrum of tetrapod diversity. Macroanatomy, microanatomy, development, and comparative anatomy are explored in a broad context to evaluate the evolutionary trends of the primate peripheral nervous system and are discussed in detail

X-HALE: Designing the Atmospheric Surveillance Platforms of the Future

Imagine the benefits that battlefield commanders or intelligence analysts could derive from an airborne surveillance platform that would carry a 500-pound payload, operate above the range of small arms fire, remain on station for weeks or even years, cost much less than a satellite, and relocate around the globe to a new region of interest within a couple of weeks. Realizing this concept, known as a high-altitude, long-endurance HALE aircraft, is a 10-to-15-year goal of researchers at the Air Force Institute of Technology AFIT. In order to reach this goal, those researchers are following a developmental path similar to the one the Wright brothers used over a century ago by gathering new test data and building theoretical formulations for this aircraft. The brothers discovery that the existing aeronautical data of the day was inaccurate proved key to their success. Indeed, Wilbur Wright even wrote that having set out with absolute faith in the existing scientific data, we were driven to doubt one thing after another, until finally, after two years of experiment, we cast it all aside, and decided to rely entirely upon our own investigations

A new method to identify earthquake swarms applied to seismicity near the San Jacinto Fault, California

Understanding earthquake clustering in space and time is important but also challenging because of complexities in earthquake patterns and the large and diverse nature of earthquake catalogues. Swarms are of particular interest because they likely result from physical changes in the crust, such as slow slip or fluid flow. Both swarms and clusters resulting from aftershock sequences can span a wide range of spatial and temporal scales. Here we test and implement a new method to identify seismicity clusters of varying sizes and discriminate them from randomly occurring background seismicity. Our method searches for the closest neighbouring earthquakes in space and time and compares the number of neighbours to the background events in larger space/time windows. Applying our method to California's San Jacinto Fault Zone (SJFZ), we find a total of 89 swarm-like groups. These groups range in size from 0.14 to 7.23 km and last from 15 min to 22 d. The most striking spatial pattern is the larger fraction of swarms at the northern and southern ends of the SJFZ than its central segment, which may be related to more normal-faulting events at the two ends. In order to explore possible driving mechanisms, we study the spatial migration of events in swarms containing at least 20 events by fitting with both linear and diffusion migration models. Our results suggest that SJFZ swarms are better explained by fluid flow because their estimated linear migration velocities are far smaller than those of typical creep events while large values of best-fitting hydraulic diffusivity are found