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An attractive square-well is incorporated into the Enskog equation, in order to model the kinetic theory of a moderately dense gas with intermolecular potential. The existence of solutions to the Cauchy problem in <i>L</i>¹. global in time and for arbitrary initial data. is proved. A simple derivation of the square-well kinetic equation is given. Lewis's method is used~ which starts from the Liouville equation of statistical mechanics. Then various symmetries of the collisional integrals are established. An H-theorem for entropy, mass, and momentum conservation is obtained, as well as an energy estimate, and key gain-loss estimates. Approximate equations for the square-well kinetic equation are constructed that preserve symmetries of the collisional integral. Existence of nonnegative solutions of the approximate equations and weak compactness are obtained. The velocity averaging lemma of Golse is then a principal tool in demonstrating the convergence of the approximate solutions to a solution of the renormalized square well kinetic equation. The existence of weak solution of the initial value problem for the square well kinetic equation is thus proved. / Ph. D.

The metallic interconnect of a solid oxide fuel cell (SOFC) contains chromium in order to protect the metal from the corrosive environment in the fuel cell. Unfortunately, the chromium introduces chemical instability in the cathode as it migrates from the interconnect to the pores in the cathode. A model was developed previously in Asinari et al. [1] and Kasula et al [2] to model the flow of particles in a fuel cell electrode. To learn more about the migration of the chromium, the previous code is modified in this thesis work to include the effects of the chromium. The model uses Kinetic Theory to simulate the fuel cell at a mesoscopic scale. The discretized form of the Lattice Boltzmann equation is modified for enhanced performance and for use on a parallel processing system. With the new model, the migration of the chromium in the cathode and the performance degradation of the fuel cell are predicted. / Master of Science

A high-order macroscopic model for the accurate description of rarefied polyatomic gas flows is introduced based on a simplified kinetic equation. The different energy exchange processes are accounted for with a two term collision model. The order of magnitude method is applied to the primary moment equations to acquire the optimized moment definitions and the final scaled set of Grad's 36 moment equations for polyatomic gases. The proposed kinetic model, which is an extension of the S-model, predicts correct relaxation of higher moments and delivers the accurate Prandtl (Pr) number. Also, the model has a proven H-theorem. At the first order, a modification of the Navier-Stokes-Fourier (NSF) equations is obtained, which shows considerable extended range of validity in comparison to the classical NSF equations in modeling sound waves. At third order of accuracy, a set of 19 regularized PDEs (R19) is obtained. Furthermore, the terms associated with the internal degrees of freedom yield various intermediate orders of accuracy, a total of 13 different orders. Attenuation and speed of linear waves are studied as the first application of the many sets of equations. For frequencies were the internal degrees of freedom are effectively frozen, the equations reproduce the behavior of monatomic gases. Thereafter, boundary conditions for the proposed macroscopic model are introduced. The unsteady heat conduction of a gas at rest and steady Couette flow are studied numerically and analytically as examples of boundary value problems. The results for different gases are given and effects of Knudsen numbers, degrees of freedom, accommodation coefficients and temperature dependent properties are investigated. For some cases, the higher order effects are very dominant and the widely used first order set of the Navier Stokes Fourier equations fails to accurately capture the gas behavior and should be replaced by a higher order set of equations. / Graduate / 0346, 0791, 0548, 0759 / behnamr@uvic.ca

The objective of this thesis is to develop an easy-to-use and computationally economical numerical tool to investigate the flow field in the Pulsed Pressure Chemical Vapour Deposition (PP-CVD) reactor. The PP-CVD process is a novel thin film deposition technique with some advantages over traditional CVD methods. The numerical modelling of the PP-CVD flow field is carried out using the Quiet Direct Simulation (QDS) method, which is a flux-based kinetic-theory approach. Two approaches are considered for the flux reconstruction, which are the true directional manner and the directional splitting method. Both the true directional and the directional decoupled QDS codes are validated against various numerical methods which include EFM, direct simulation, Riemann solver and the Godunov method. Both two dimensional and axisymmetric test problems are considered. Simulations are conducted to investigate the PP-CVD reactor flow field at 1 Pa and 1 kPa reactor base pressures. A droplet flash evaporation model is presented to model the evaporation and transport of the liquid droplets injected. The solution of the droplet flash evaporation model is used as the inlet conditions for the QDS gas phase solver. The droplet model is found to be able to provide pressure rise in the reactor at the predicted rate. A series of parametric studies are conducted for the PP-CVD process. The numerical study confirms the hypothesis that the flow field uniformity is insensitive to the reactor geometry. However, a sufficient distance from the injection inlet is required to allow the injected precursor solution to diffuse uniformly before reaching the substrate. It is also recommended that placement of the substrate at the reactor’s centre axis should be avoided.

Este trabalho consiste do estudo das propriedades de transporte de sistemas dinâmicos caóticos por meio do uso de técnicas de operadores de projeção. Tais sistemas podem exibir difusão determinística e relaxação para o equilíbrio. Mostramos que esse comportamento difusivo pode ser visto como uma propriedade espectral do operador de Perron-Frobenius associado. Em particular, a ressonância dominante de Policott-Ruelle é calculada analiticamente para uma classe geral de mapas que preservam área. Sua dependência do número de onda determina os coeficientes de transporte normais. Calculamos uma fórmula geral exata para o coeficiente de difusão, obtida sem qualquer aproximação de alta estocasticidade, e um novo efeito emergiu: a evolução angular pode induzir modos rápidos ou lentos de difusão mesmo no regime de alta estocasticidade. Os aspectos não-Gaussianos do transporte caótico são também investigados para esses sistemas. O estudo é realizado por meio de uma relação entre a curtose, o coeficiente de difusão e o coeficiente de Burnett de quarta ordem, os quais são calculados analiticamente. Uma escala de tempo característica que delimita os regimes Gaussiano e Markoviano para a função densidade foi estabelecida. À parte os modos acelerados, cujas propriedades cinéticas são anômalas, todo os resultados estão em excelente acordo com as simulações numéricas / This work consists in the study of the transport properties of chaotic Hamiltonian systems by using projection operator techniques. Such systems can exhibit deterministic diffusion and display an approach to equilibrium. We show that this diffusive behavior can be viewd as a spectral property of the associated Perron-Frobenius operator. In particular, the leading Pollicott-Ruelle resonance is calculated analytically for a general class of two-dimensional area-preserving maps. Its wavenumber dependence determines the normal transport coefficients. We calculate a general exact formula for the diffusion coefficient, derived without any high stochasticity approximation and a new effect emerges: the angular evolution can induce fast or slow modes of diffusion even in the high stochasticity regime. The non-Gaussian aspects of the chaotic transport are also investigated for this systems. This study is done by means of a relationship between kurtosis and diffusion coefficient and fourth order Burnett coefficient, which are calculated analytically. A characteristic time scale which delimits the Markovian and Gaussian regimes for the density function was established. Despite the accelerator modes, whose kinetics properties are anomalous, all theoretical results are in excellent agreement with the numerical simulations

Esta dissertação descreve um experimento de Townsend pulsado cuja finalidade foi obter experimentalmente parâmetros de transporte de elétrons em gases, em particular em nitrogênio e isobutano, em uma faixa de campo elétrico reduzido entre 130 Td e 210 Td. Os parâmetros de transporte obtidos foram a velocidade média do centro de massa da nuvem eletrônica, a taxa de ionização e o coeficiente de ionização (primeiro coeficiente de Townsend). Essas grandezas foram determinadas a partir da análise do sinal eletrônico induzido pelos elétrons em deslocamento entre dois eletrodos de placas paralelas. Para efetuar essa análise, foi desenvolvido um modelo que relaciona a dinâmica dos elétrons no meio gasoso com o sinal induzido. Esse modelo permitiu a dedução de uma função explícita do tempo e de parâmetros relacionados com os parâmetros físicos, que foram estimados por meio do ajuste da função aos resultados experimentais pelo método dos mínimos quadrados. Além do método de análise, este trabalho descreve um código computacional de aquisição e controle, especialmente desenvolvido para integrar as diversas operações de controle instrumental e aquisição de dados. Foram realizadas várias séries de medições nas mesmas condições e verificou-se a repetitibilidade dos resultados. Os parâmetros de transporte em nitrogênio reproduziram os resultados da literatura e de cálculos teóricos dentro dos limites de erro dos resultados experimentais. No caso do isobutano, cujos parâmetros de transporte em elétrons são escassos na literatura, verificou-se que a estimativa do coeficiente de ionização é consistente com os resultados da literatura baseados em medições em um intervalo de campo elétrico reduzido mais elevado. / This thesis describes a pulsed Townsend experiment whose purpose was to obtain experimentally the transport parameters of electron in gases, particularly nitrogen and isobutane, within a range of reduced electric field between 130 Td and 210 Td. The transport parameters obtained were the average speed of center of mass of the electron cloud, the ionization rate and ionization coefficient (first Townsend coefficient). These quantities were obtained from the analysis of the electronic signal induced by electrons moving between two parallel plate electrodes. To perform this analysis, we developed a model that relates the dynamics of electrons in gaseous medium with the induced signal. This model allowed the deduction of an explicit function of time and parameters related to the physical parameters, which were estimated by fitting the function to the experimental results by the method of least squares. Besides the method of analysis, this paper describes a computer program especially designed to integrate the various processes of instrumental control and data acquisition. We performed several sets of measurements under the same conditions and verified the repeatability of results. The transport parameters in nitrogen reproduced the results of the literature and theoretical calculations within the error limits of the experimental results. In the case of isobutane, whose electron transport parameters have been poorly studied, it was found that the estimation of ionization coefficient agrees well with the literature results based on measurements in a range of higher reduced electric field.

Este trabalho consiste do estudo das propriedades de transporte de sistemas dinâmicos caóticos por meio do uso de técnicas de operadores de projeção. Tais sistemas podem exibir difusão determinística e relaxação para o equilíbrio. Mostramos que esse comportamento difusivo pode ser visto como uma propriedade espectral do operador de Perron-Frobenius associado. Em particular, a ressonância dominante de Policott-Ruelle é calculada analiticamente para uma classe geral de mapas que preservam área. Sua dependência do número de onda determina os coeficientes de transporte normais. Calculamos uma fórmula geral exata para o coeficiente de difusão, obtida sem qualquer aproximação de alta estocasticidade, e um novo efeito emergiu: a evolução angular pode induzir modos rápidos ou lentos de difusão mesmo no regime de alta estocasticidade. Os aspectos não-Gaussianos do transporte caótico são também investigados para esses sistemas. O estudo é realizado por meio de uma relação entre a curtose, o coeficiente de difusão e o coeficiente de Burnett de quarta ordem, os quais são calculados analiticamente. Uma escala de tempo característica que delimita os regimes Gaussiano e Markoviano para a função densidade foi estabelecida. À parte os modos acelerados, cujas propriedades cinéticas são anômalas, todo os resultados estão em excelente acordo com as simulações numéricas / This work consists in the study of the transport properties of chaotic Hamiltonian systems by using projection operator techniques. Such systems can exhibit deterministic diffusion and display an approach to equilibrium. We show that this diffusive behavior can be viewd as a spectral property of the associated Perron-Frobenius operator. In particular, the leading Pollicott-Ruelle resonance is calculated analytically for a general class of two-dimensional area-preserving maps. Its wavenumber dependence determines the normal transport coefficients. We calculate a general exact formula for the diffusion coefficient, derived without any high stochasticity approximation and a new effect emerges: the angular evolution can induce fast or slow modes of diffusion even in the high stochasticity regime. The non-Gaussian aspects of the chaotic transport are also investigated for this systems. This study is done by means of a relationship between kurtosis and diffusion coefficient and fourth order Burnett coefficient, which are calculated analytically. A characteristic time scale which delimits the Markovian and Gaussian regimes for the density function was established. Despite the accelerator modes, whose kinetics properties are anomalous, all theoretical results are in excellent agreement with the numerical simulations

Plasmas seeded with solid particulates of nanometer to micron sizes (complex plasma systems) are a ubiquitous feature of intergalactic, interstellar and planetary environments but also of plasma processing applications or even fusion devices. Their novel aspects compared with ideal multi-component plasmas stem from (i) the large number of elementary charges residing on the grain surface, (ii) the variability of the charge over mass ratio of the dust component, (iii) the inherent openness and dissipative nature of such systems.   Their statistical description presents a major challenge; On one hand by treating dust grains as point particles new phase space variables must be introduced augmenting the classical Hamiltonian phase space, while the microphysics of interaction between the plasma and the grains will introduce additional coupling between the kinetic equations of each species, apart from the usual fine-grained electromagnetic field coupling. On the other hand complex plasma systems do not always exist in a gaseous state but can also condensate, i.e. form liquid, solid or crystalline states.   In this thesis we study gaseous partially ionized complex plasma systems from the perspective of the Klimontovich technique of second quantization in phase space. Initially, in regimes typical of dust dynamics. Starting from the Klimontovich equations for the exact phase space densities, theory deliverables such as the permittivity, the spectral densities of fluctuations and the collision integrals are implemented either for concrete predictions related to low frequency electrostatic waves or for diagnostic purposes related to the enhancement of the ion density and electrostatic potential fluctuation spectra due to the presence of dust grains. Particular emphasis is put to the comparison of the self-consistent kinetic model with multi-component kinetic models (treating dust as an additional massive charged species) as well as to the importance of the nature of the plasma particle source. Finally, a new kinetic model of complex plasmas (for both constant and fluctuating sources) is formulated. It is valid in regimes typical of ion dynamics, where plasma discreteness can no longer be neglected, and, in contrast to earlier models, does not require relatively large dust densities to be valid. / QC 20120316

In this thesis the nonlinear effects and heating are studied in inductively coupled plasma (ICP) in a regime of anomalous skin effect (nonlocal regime). In this regime the thermal motion of plasma electrons plays an important role, significantly influencing the processes associated with the penetration of electromagnetic field into plasma, such as the ponderomotive effect and heating of plasma by the field. We have developed a linear kinetic theory that describes the electron dynamics in ICP taking into account the electron thermal motion and collisions of electrons. This theory yields relatively simple expressions for the electron current in plasma, the ponderomotive force, and plasma heating. It describes correctly the thermal reduction of ponderomotive force in the nonlocal regime, which has been previously observed experimentally. It also describes the collisionless heating of plasma due to resonant interaction between the electromagnetic wave and plasma electrons. There is a good overall agreement of the results of our theory with the experimental data on ponderomotive force and plasma heating. Using our theory, we predicted a new effect of reduction of plasma heating compared to the purely collisional value, occurring at low frequencies. This effect has not been previously reported. The nonlinear effects of the electromagnetic field on the electron distribution function and on plasma heating, that are not accounted for in the linear kinetic theory, have been studied using a quasilinear kinetic theory, also developed in this thesis. Within the quasilinear approximation we have formulated the system of equations describing the slow response of plasma electrons to the fast oscillating electromagnetic field. As an example, these equations have been solved in the simplest case of cold plasma with collisions, and the nonlinear perturbation of the electron distribution function and its effect on the plasma heating have been found. It has been shown that the nonlinear modification of plasma heating occurs mainly due to the nonlinear effect of the magnetic component of the electromagnetic field. It has also been shown that at high frequencies the nonlinear effects vanish, and the heating is well described by the linear theory. To verify the predicted new effect of plasma heating reduction at low frequencies, as well as to investigate the nonlinear effect of the magnetic field on plasma heating for arbitrary amplitudes of electromagnetic field in plasma, we have developed a 1d3v Particle-In-Cell (PIC) numerical simulation code with collisions. The collisions were implemented into the PIC code using two different techniques: the direct Monte-Carlo technique for the electron-atom collisions, and the stochastic technique based on the Langevin equation for the electron-electron collisions. The series of numerical simulations by this code confirmed the results of our linear theory, particularly the effect of heating reduction at low frequencies that we predicted theoretically. Also, the nonlinear effects of electromagnetic field on plasma heating were studied using the PIC code in the cases of weak and strong electromagnetic fields. It has been shown that in the case of weak electromagnetic fields (corresponding to weak nonlinearity) the nonlinear effects lead to some enhancement of heating (compared to the linear theory) at low frequencies, followed by a small reduction of heating at higher frequencies. This observed nonlinear perturbation of heating in warm plasma with collisions is similar to that predicted by the quasilinear theory for the case of cold plasma with collisions. In the case of strong electromagnetic fields (corresponding to strong nonlinearity) the nonlinear effects lead to a further reduction of heating (compared to the linear theory) at low frequencies, as shown by the simulation, thus adding to the effect of reduction of heating predicted by the linear theory. The nonlinear effects are shown to vanish at high frequencies, as expected.