Thesis (Ph. D.)--University of Wisconsin--Madison, 1958. / Typescript. Vita. eContent provider-neutral record in process. Description based on print version record. Includes bibliographical references (leaves 133-134).

Thesis (Ph.D.)--Brown University, 2005. / Vita. Thesis advisor: Yan Guo. Includes bibliographical references (leaves 196-200). Also available online.

Thesis (Ph. D.)--University of Chicago, 1916. / "Private Edition, Distributed by the University of Chicago Libraries, Chicago, Illinois." "Reprinted from the Physical Review, N.S., Vol. X, No. 4, October 1917." Includes bibliographical references. Also available on the Internet.

This study entails an investigation of a novel moment closure, originally constructed for rarefied-gas prediction, to the modelling of inert, dilute, disperse, particle flows. Such flows are important in many engineering situations. As one example, in internal-combustion engines, fuel is often injected as a spray of tiny droplets and, during combustion, a cloud of tiny soot particles can be formed. These particle phases are often difficult to model, especially when particles display a range of velocities at each location in space. Lagrangian methods are often too costly and many Eulerian field-based methods suffer from model deficiencies and mathematical artifacts. Often, Eulerian formulations assume that all particles at a location and time have the same velocity. This assumption leads to nonphysical results, including an inability to predict particle paths crossing and a limited number of boundary conditions that can be applied. The typical multi-phase situation of many particles is, in many ways, similar to that of a gas compressed of a huge number of atoms or molecules. It is therefore expected that powerful techniques from the kinetic theory of gases could be applied. This work explores the advantages of using a modern fourteen-moment model, originally derived for rarefied gases, to predict multi-phase flows. Details regarding the derivation, the mathematical structure, and physical behaviour of the resulting model are explained. Finally, a numerical implementation is presented and results for several flow problems that are designed to demonstrate the fundamental behaviour of the models are presented. Comparisons are made with other classical models.

No description available.

We consider a model of N particles interacting through a Kac-style collision process, with m particles among them interacting, in addition, with a thermostat. When m = N, we show exponential approach to the equilibrium canonical distribution in terms of the L2 norm, in relative entropy, and in the Gabetta-Toscani-Wennberg (GTW) metric, at a rate independent of N. When m < N , the exponential rate of approach to equilibrium in L2 is shown to behave as m/N for N large, while the relative entropy and the GTW distance from equilibrium exhibit (at least) an "eventually exponential” decay, with a rate scaling as m/N^2 for large N. As an allied project, we obtain a rigorous microscopic description of the thermostat used, based on a model of a tagged particle colliding with an infinite gas in equilibrium at the thermostat temperature. These results are based on joint work with Federico Bonetto, Michael Loss and Hagop Tossounian.

This work is about convergence to equilibrium problems for equations coming from kinetic theory. The bulk of the work is about Hypocoercivity. Hypocoercivity is the phenomenon when a semigroup shows exponentially relaxation towards equilibrium without the corresponding coercivity (dissipativity) inequality on the Dirichlet form in the natural space, i.e. a lack of contractivity. In this work we look at showing hypocoercivity in weak measure distances, and using probabilistic techniques. First we review the history of convergence to equilibrium for kinetic equations, particularly for spatially inhomogeneous kinetic theory (Boltzmann and Fokker-Planck equations) which motivates hypocoercivity. We also review the existing work on showing hypocoercivity using probabilistic techniques. We then present three different ways of showing hypocoercivity using stochastic tools. First we study the kinetic Fokker-Planck equation on the torus. We give two different coupling strategies to show convergence in Wasserstein distance, $W_2$. The first relies on explicitly solving the stochastic differential equation. In the second we couple the driving Brownian motions of two solutions with different initial data, in a well chosen way, to show convergence. Next we look at a classical tool to show convergence to equilibrium for Markov processes, Harris's theorem. We use this to show quantitative convergence to equilibrium for three Markov jump processes coming from kinetic theory: the linear relaxation/BGK equation, the linear Boltzmann equation, and a jump process which is similar to the kinetic Fokker-Planck equation. We show convergence to equilibrium for these equations in total variation or weighted total variation norms. Lastly, we revisit a version of Harris's theorem in Wasserstein distance due to Hairer and Mattingly and use this to show quantitative hypocoercivity for the kinetic Fokker-Planck equation with a confining potential via Malliavin calculus. We also look at showing hypocoercivity in relative entropy. In his seminal work work on hypocoercivity Villani obtained results on hypocoercivity in relative entropy for the kinetic Fokker-Planck equation. We review this and subsequent work on hypocoercivity in relative entropy which is restricted to diffusions. We show entropic hypocoercivity for the linear relaxation Boltzmann equation on the torus which is a non-local collision equation. Here we can work around issues arising from the fact that the equation is not in the H\"{o}rmander sum of squares form used by Villani, by carefully modulating the entropy with hydrodynamical quantities. We also briefly review the work of others to show a similar result for a close to quadratic confining potential and then show hypocoercivity for the linear Boltzmann equation with close to quadratic confining potential using similar techniques. We also look at convergence to equilibrium for Kac's model coupled to a non-equilibrium thermostat. Here the equation is directly coercive rather than hypocoercive. We show existence and uniqueness of a steady state for this model. We then show that the solution will converge exponentially fast towards this steady state both in the GTW metric (a weak measure distance based on Fourier transforms) and in $W_2$. We study how these metrics behave with the dimension of the state space in order to get rates of convergence for the first marginal which are uniform in the number of particles.

The influence of periodic vibrations on the granular flow of materials is of great interests to scientists and engineers due to both theoretical and practical reasons. In this paper, the stability of a vertically vibrated granular layer is examined by linear stability analysis. This includes two major steps, firstly, the base state at various values of mass holdup (Mt) and energy input (Qt) is calculated and secondly, small perturbations are introduced to verify the stability of the base state by solving the resultant eigenvalue problem derived from the linearized governing equations and corresponding boundary conditions. Results from the base state solution show that, for a given pair of Mt and Qt, solid fraction tends to increase at first along the layer height and then decrease after a certain vertical position while granular temperature decreases rapidly from the bottom plate to the top surface. This may be due to the existence of inelastic collisions between particles that dissipate the energy input from the bottom. It is also found that more energy input results in a lower solid fraction and a higher granular temperature. The stability diagram is constructed by checking the stability property at different points in the Mt-Qt plane. For a fixed Mt, the base state is stable at low energy inputs, and becomes unstable if Qt is larger than a critical value Qtc1. A higher value of Mt corresponds to a larger Qtc1. There also exists a critical mass holdup (Mtc), for Mt larger than Mtc, the patterns corresponding to the instabilities are standing waves (stationary mode); otherwise the flat layer appears (layer mode). Moreover, the stationary mode turns into the layer mode when Qt is increased beyond a critical value Qtc2. These findings agree with the experimental observations of other researchers (Hsiau and Pan, 1998). The effects of restitution coefficients (ep, ew) and material properties (dp, ρp) on the stability diagram are also investigated. Together with Mt and Qt these variables can be classified into two groups, i.e. the stabilizing factors (Mt, dp, ρp) and the destabilizing factors (Qt, ep, ew). The stability of the system is enhanced with increasing stabilizing factors and decreasing destabilizing factors. / Singapore-MIT Alliance (SMA)

Thesis (Ph.D.)--City University of Hong Kong, 2009. / "Submitted to Department of Mathematics in partial fulfillment of the requirements for the degree of Doctor of Philosophy." Includes bibliographical references (leaves [90]-94)

Thesis (M. Phil.)--Hong Kong University of Science and Technology, 2003. / Includes bibliographical references (leaves 108-109). Also available in electronic version. Access restricted to campus users.