William W. Roberts

Modeling and Computer Simulation of the Compressional Behavior of Fiber Assemblies Part I: Comparison to van Wyk's Theory

A model is presented that allows prediction of the properties of a fiber assembly under compression from the physical properties of its component fibers, taking into account both static and kinetic friction. Unlike previous models, this model is not based on the assumed behavior of idealized bending elements. Computer simulations are run for four cases with two different friction conditions in order to compare predictions of this model with experimental results and with van Wyks theory of the uniaxial compression of an initially random fiber assembly. These simulations are the first to show a reasonable ability to predict the undeter mined constant K in van Wyks equation. They also show a significandy greater number of fiber-fiber contacts being formed than theories based only on the diameter and arrangement of fibers have predicted. The predicted contacts have a wide range of contact forces, while only a small percentage of them do not slip. Further improvements to the predictive abilities of this model may be obtained by running larger simulations, modifying the boundary conditions, and incorporating a more realistic friction model.

Modeling and Computer Simulation of the Compressional Behavior of Fiber Assemblies Part II: Hysteresis, Crimp, and Orientation Effects

In this study, the model presented in Part I is used to investigate phenomena related to the compression of fiber assemblies that are not accounted for by van Wyks theory of the uniaxial compression of an initially random fiber assembly. In order to do this, the potential energy, the work done on the assembly, and a discrete orientation density function for the assembly are calculated. Realistic looking hysteresis plots are produced. and the model can predict the amount of frictional energy dissipated as a function of time. Irrecoverable compression does not increase for as many cycles as has been experimen tally reported, which may be a result of the neglect of viscoelastic effects. Crimp has a large effect on the compressional properties, in that more highly crimped fibers absorb more energy as they are compressed. They also absorb a higher proportion of their energy in the twisting mode, which previous investigators have neglected. A lower induced orientation effect than that resulting from Stearns theory [12] is predicted, but this discrepancy cannot currently be resolved. Future work may incorporate viscoelasticity into the model and predict the most probable initial volume fraction for an assembly with given fiber properties.

Theoretical aspects of galactic research

Abstract The luminosity of a spiral arm is believed to originate primarily from the very young, newly forming stars; and the spiral arm itself is believed to be a spiral wave which is capable of triggering the formation of the young stars selectively along the wave crest. Such a wave has been visualized from two different viewpoints: first, the density wave viewpoint in which gravitational forces are considered as dominant forces, with magnetic forces also playing a role but of secondary importance; and second, the hydromagnetic wave viewpoint in which magnetic fields are visualized as the dominant forces in the gas. At the present time, only the density wave viewpoint has been developed toward a coherent theory to provide a quantitative viewpoint from which to visualize spiral structure. In the density wave model, a galactic shock wave forms in the gaseous component of the galactic disk as a necessary consequence of the theory of waves for sufficiently large amplitudes and is the nonlinear counterpart of the small amplitude, linear density wave. The galactic shock wave is visualized as a possible triggering mechanism for the gravitational collapse of gas clouds, leading to star formation along a spiral arm. In the density wave model for galaxies which undergo sizeable differential rotation, a shock, a sharp H I gas peak, a narrow dust lane, and the strongest magnetic fields are predicted to lie in a narrow lane on the inner side of the bright optical arm of young stars and H II regions triggered into existence in the shock. Recent observational results from the high resolution Westerbork Synthesis Radio Telescope in the Netherlands indicate that one striking example which exhibits such features as these is the galaxy M 51. Other theoretical results have been determined with the density wave model, and a number of these have been confirmed through observational studies. For example, several spirals studied to date are found to have neutral hydrogen gas concentrated along the optical spiral arms and systematic motions which correspond to the systematic motions expected for density waves. For our own Galaxy the shock wave together with the differential rotation of the gas provides a natural explanation for the striking separation between the peaks of the abundance distributions of H II gas and H I gas. The distinction between narrow optical arms and broad optical arms in the density wave model is found to depend on whether the velocity component of basic rotation normal to a spiral arm is greater than or less than the acoustic speed of the gas, and this distinction may explain why some galaxies have narrow optical arms while other galaxies have broad optical arms. From a recent study of the density wave models for twenty-five external galaxies, it is found that those galaxies whose models predict the possibility for strong shock waves exhibit long, well-developed spiral arms, and those galaxies whose models predict weak shock waves exhibit less developed spiral structure. On the small scale, a physical picture for star formation is evolving based on the two-phase model for the interstellar medium. Galactic shocks are initiated in the hot intercloud phase, and the cold clouds are viewed simply as embedded bodies which expand or contract to adjust to changes of the ambient pressure of the intercloud phase. The pressure increase across the galactic shock occurring in the intercloud phase is in turn transmitted to the cold clouds, leading to star formation. To be sure, much further work needs to be done to suggest further physical results applicable to the physical phenomena and physical processes which occur in galaxies and to investigate those areas where unresolved questions and exciting problems still remain.

Hypersonic, stratified gas flows past an obstacle: direct simulation Monte Carlo calculations

Abstract Computational studies of the three-dimensional, hypersonic flow of rarefied, strongly stratified gas past an obstacle are carried out, the incident gas stratified in a direction transverse to the mean flow. An “ N -body” computational code based on Monte Carlo techniques is developed for these purposes. Our primary interest is centered on the three-dimensional effects induced in the gas flow by a solid obstacle comparable in size to the gas scale height and collisional mean free path. Of the different types and shapes of obstacles studied, we focus herein on a cylindrical obstacle, assumed to be a diffuse elastic scatterer. The cylindrical obstacle is a short uniform pipe whose upstream end is fully open, facing directly into the flow, and whose downstream end is covered by a flat circular endplate containing an “orifice” at its center. For different choices of “orifice” diameter, the obstacle serves as a useful model of an impact probe (closed orifice) or scoop (closed, partially open, or fully open orifice) in a rapidly rotating strongly stratified gas (as in a gas centrifuge). The computed results show that the obstacle (in all cases studied, spanning the range from completely closed orifice to fully open orifice) induces large systematic motions in the gas, with strong radially inward driven flow in the direction of the gradient of density stratification, and correspondingly large density perturbations in these regions. The radial inflow of gas is prominent not only in the neighborhood of the obstacle and downstream from it but also at considerable distances radially inward from it and at z -heights well above and below. The radially driven gas inflow is a striking three-dimensional effect induced when strongly stratified gas impinges upon an obstacle; it constitutes a major characteristic common to all the hypersonic, stratified flows studied despite differences in Mach number and gas scale height and regardless of whether the obstacle is a flat plate, a “long” solid rod, or a “short” cylindrical pipe. The resultant density distribution of the obstructed molecular gas exhibits a striking “asymmetry” in the (radial) direction of the gradient of density stratification but retains “symmetry” in the z -direction perpendicular to the stratification gradient. The “ r -asymmetry” is in striking contrast to the characteristic rotation-symmetry exhibited about the obstructing pipes central axis in corresponding cases of unstratified flows investigated. The bow shock that forms near the obstacle exhibits a characteristic thickness that broadens with mean free path in the direction of the gradient of density stratification and a characteristic shape that is substantially warped (in that direction) from the paraboloid-shaped bow shock that forms (with axis of revolution coincident with the pipes central axis) in corresponding cases of unstratified flows. The redirection of the gas flow at the shock away from the direction of the incident mean flow is particularly strong along that portion of the warped bow shock most closely aligned with the (radial) direction of the stratification gradient; the postshock density ridge in this direction is masked considerably by the strongly stratified density background.

Simulations of cloudy, gaseous galactic disks

Aspects of the simulation of cloudy, gaseous galactic disks are discussed. The motivation for conducting such studies is addressed, and the formulation of the fundamental physical problem in mathematical terms is considered. The computational solution study of the formulated physical problem is examined, and the results and interpretation of computational studies are discussed. 84 refs.

Three dimensional, stratified gas flows past an obstacle

Abstract An “ N -body” computer code is developed to study supersonic gas flows in a rarefied, strongly stratified medium. Results of numerical simulations both with and without an obstacle obstructing the flow are presented, focusing upon three dimensional effects. In particular, the systematic gas motions and density perturbations induced by a flat plate whose normal vector is parallel to the direction of umperturbed flow and perpendicular to the stratification gradient are discussed. Such a plate can be regarded as a useful approximation to a scoop in a gas centrifuge.

GLOBAL, LOCAL, AND INTERMEDIATE-SCALE STRUCTURES IN PROTOTYPE SPIRAL GALAXIES

The relationship between galacitc spiral structure and the matter in the underlying disk constitutes one of the central problems in galactic dynamics. In Bertin et al. (1989a, b), disk matter characterized by a low dispersive speed is shown to be capable of playing a key role in the generation of large-scale spiral structure. In Roberts et al. (1992), this self-gravitating, low-dispersion disk matter is shown to be capable of playing an essential role in the formation of structure on local and intermediate scales. Both in computed cases where large-scale spiral strucutre is present and in those where it is not, the same dominant physical processes and fundamental dynamical mechanisms are active on local scales. The new perception, in which large-scale and small-scale phenomena operate somewhat independently as evidenced in the computational studies, permits a range of flocculent, multi-armed, and grand design spiral types to be simulated. In particular, grand design galaxies with ragged appearances exhibiting spurs, arm branchings, and interarm bridges in addition to the major spiral amrs, similar to those often observed, can be generated.

On the distribution of pitch angles in external galactic spirals NGC 1232 and NGC 5457

A numerical method, originally developed to analyze the morphology of global and local structure in prototype galaxies, is modified for analyzing observed disk-shape galaxies. Two digitized spiral galaxies NGC 1232 and NGC 5457 with varying degrees of contrast between arm and interarm regions are analyzed. A synergism of partitioning methods and a geometric mean least-squares regression algorithm serves to isolate local arm segments, spurs, feathers, and secondary features and to measure their pitch angles and lengths. The global arms are actually highly disjointed, with arm segments frequently revealing pitch angles between 30 and 50 deg, certainly greater than those of the parent arms. Prominent spurs tend to exhibit a much greater pitch angle. The automated mathematical algorithm is shown to have negligible numerical biasing and could be applied to any number of spiral galaxies manifesting flocculent structure, either prototype or observed, and could possibly be used as a tool for classification of multiple-armed-type galaxies.

Fibrous assemblies: modeling/computer simulation of compressional behaviour

We develop a model to relate the mechanical properties of individual fibers and how they are arranged in a fibrous assembly to the bulk properties of the fibrous assembly. The model allows the prediction of the bulk properties of the fibrous assembly during compression from the physical properties of its component individual fibers, considering both static and kinetic friction at contacts between fibers. Computer simulations are run for several cases with specific friction conditions applied in order to compare predictions of this model with experimental results and with van Wyks theory of the uniaxial compression of an initially random fibrous assembly. These computer simulations demonstrate a reasonable ability to predict the undetermined constant K in van Wyks theory. The computer simulations also show a significantly greater number of fiber‐fiber contacts being formed than theories based only on the diameter and arrangement of fibers have predicted. The predicted contacts have a wide range of contact forces, while only a small percentage of them do not slip. The model may be used to investigate phenomena associated with the compression of fibrous assemblies, such as fiber crimp, hysteresis, and orientation effects.

Analysis of the distribution of pitch angles in model galactic disks - Numerical methods and algorithms

An automated mathematical method capable of successfully isolating the many different features in prototype and observed spiral galaxies and of accurately measuring the pitch angles and lengths of these individual features is developed. The method is applied to analyze the evolution of specific features in a prototype galaxy exhibiting flocculent spiral structure. The mathematical-computational method was separated into two components. Initially, the galaxy was partitioned into dense regions constituting features using two different methods. The results obtained using these two partitioning algorithms were very similar, from which it is inferred that no numerical biasing was evident and that capturing of the features was consistent. Standard least-squares methods underestimated the true slope of the cloud distribution and were incapable of approximating an orientation of 45 deg. The problems were overcome by introducing a superior fit least-squares method, developed with the intention of calculating true orientation rather than a regression line.