Dissertations / Theses on the topic 'Kinetic theory of gases'

A complete formulation of the recently developed. F_N method in kinetic theory is presented and the accuracy of this advanced semi-analytical-numerical technique is demonstrated by testing the method to several classical problems in rarefied gas dynamics. The method is based on the existing analysis for the vector transport equation arising from the decomposition of the linearized BGK equation. Using full-range orthogonality, a system of singular integral equations for the distribution functions at the boundaries is established. The unknown distribution functions are then approximated by a finite expansion in terms of a set of basis functions and the coefficients of the expansion are found by requiring the set of the reduced algebraic equations to be satisfied at certain collocation points. By studying the half-space heat transfer and weak evaporation problems and the problem of heat transfer between two parallel plates it is demonstrated that the F_N method is a viable solution technique yielding results of benchmark accuracy. Two different sets of basis functions are provided for half-space and finite media problems, respectively. In all cases, highly accurate numerical results are computed and compared to existing exact solutions. The obtained numerical results help in judging the accuracy to expect of the method and indicate that the F_N method may be applied with confidence to problems for which, more exact methods of analysis do not appear possible. Then, the cylindrical Poiseuille flow and thermal creep problems, which are not amenable to exact treatment, are solved. The F_N method is formulated and tested successfully for the first time in cylindrical geometry in kinetic theory. The complete solution of the two aforementioned problems is presented with the numerical results quoted as converged being of reference-quality good for benchmark accuracy.
Ph. D.
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In this thesis, we investigate the emergence of kinetic processes in finite quantum systems. We first generalize the Redfield theory to describe the dynamics of a small quantum system weakly interacting with an environment of finite heat capacity. We then study in detail the spin-GORM model, a model made of a two-level system interacting with a random matrix environment. By doing this, we verify our new theory and find a critical size of the environment over which kinetic processes occur. We finally study the emergence of a diffusive transport process, on a finite tight-binding subsystem interacting with a fast environment, when the size of subsystem exceeds a critical value.
Doctorat en sciences, Spécialisation chimie
info:eu-repo/semantics/nonPublished

En ce qui concerne les gaz polyatomiques, nous proposons deux hiérarchies distinctes formées d'équations de moments, qui permettent d'obtenir des lois de conservation de la densité de masse, de la quantité de mouvement et de l'énergie totale du gaz. Ces hiérarchies sont généralement coupées à un certain ordre. Une méthode qui fournit une solution appropriée au problème de fermeture est la méthode de la maximisation d'entropie. Nous formulons un problème variationnel et nous explorons en détail le cas physique de 14 moments. On étudie un mélange de gaz polyatomiques dans lequel la fonction de distribution de chaque espèce converge vers une Maxwellienne, chacune avec sa propre vitesse moyenne et température. Les lois pour la densité de masse, de quantité de mouvement et d'énergie peuvent être obtenues. En particulier, les coefficients phénoménologiques de la thermodynamique étendue peuvent être déterminés à partir des termes sources. On présente pour les mélanges de gaz monoatomiques l'asymptotique diffusive des équations de Boltzmann. Le développement de Hilbert de chaque fonction de distribution donne deux équations. La première équation permet d'affirmer que le mélange est proche de l'équilibre. La deuxième équation est une équation fonctionnelle linéaire en la variable de vitesse. Nous prouvons l'existence d'une solution de cette équation. D'une part, lorsque les masses moléculaires sont égales, les techniques introduites par Grad peuvent être utilisés. D'autre part, nous proposons une nouvelle approche qui est valable lorsque les masses moléculaires sont différentes
Considering polyatomic gases, we first propose two independent hierarchies of the moment equations, which allow to obtain conservation laws for mass density, momentum and total energy of a gas. Such hierarchies are usually truncated at some order. A method which provides an appropriate solution to the closure problem is the maximization of entropy method. We formulate a variational problem and explore in detail the physical case of 14 moments. We study mixtures of polyatomic gases in which the distribution function of each species converges towards a Maxwellian distribution function, each with its own bulk velocity and temperature. Balance laws for mass density, momentum and energy can be obtained. In particular, the phenomenological coefficients of extended thermodynamics can be determined from the source terms. Regarding mixtures of monatomic gases, we discuss the diffusion asymptotics of the Boltzmann equations. The Hilbert expansion yields two equations. The first equation allows to state that the mixture is close to equilibrium. The second equation is a linear functional equation in the velocity variable. We prove the existence of a solution to this equation. On the one hand, when molecular masses are equal, the techniques introduced by Grad can be used. On the other hand, we propose a new approach, which only holds when molecular masses are different

We construct two independent hierarchies of moment equations and we apply the maximum entropy principle for polyatomic gases. We formulate multivelocity and multitemperature model of Eulerian polyatomic gases starting from kinetic theory, that is compared in the neighborhood of global equilibrium state to the models based on extended thermodynamics. We analyze diffusion asymptotics of the Boltzmann equations for mixtures of monatomic gases.
Конструишу се две независне хијерархијеједначина момената и примењује се принципмаксимума ентропије за вишеатомске гасове.Формира се вишебрзински и вишетемпературнимодел Ојлерових вишеатомских гасова полазећиод кинетичке теорије и добијени модел сепореди у околини стања глобалне равнотеже самоделом проширене термодинамике. Анализирасе дифузиона асимптотика Болцмановихједначина за мешавине једноатомских гасова.
Konstruišu se dve nezavisne hijerarhijejednačina momenata i primenjuje se principmaksimuma entropije za višeatomske gasove.Formira se višebrzinski i višetemperaturnimodel Ojlerovih višeatomskih gasova polazećiod kinetičke teorije i dobijeni model seporedi u okolini stanja globalne ravnoteže samodelom proširene termodinamike. Analizirase difuziona asimptotika Bolcmanovihjednačina za mešavine jednoatomskih gasova.

The electron energy distribution function (EEDF) in a microwave discharge in hydrogen was calculated and measured in the pressure range 1 to 100 Torr. Other discharge parameters such as electron density, H atom density and gas temperature were also measured. More generally7 diagnostics suitable for discharges in H2 at moderate pressure were developed and basic data to aid these diagnostics was obtained. The EEDF was calculated using the two term approximation to the Boltzmann kinetic equation subject to the constraint of ionization balance. The influence of the excited states of H2 and H was investigated and it was concluded that collisions with vibrationally excited H2 had significant efiects on the EEDF but only at gas temperatures above 1500 K. Criteria for estimating the importance of superelastic collisions with excited species were also proposed. The effects of variations in the electron density, H atom density and gas temperature on the EEDF were also studied. The gas temperature was measured in two ways: from rotational structure in the G —B emission band of H2 and from the Doppler profile of ground state atomic hydrogen, measured by two photon laser induced fluorescence (LIF). The G — B rotational temperature was at least 300 K lower than the H atom temperature over the pressure range 1 to 40 Torr. The hydrogen atom density was measured using two photon LIF. A new calibration method based on decay of the fluorescence was devised for this measurement. The decay of atomic hydrogen density in the afterglow of a pulsed discharge was measured to deduce the H atom wall recombination probability 7 which was found to be about 5 X 10—3. Another estimate of ”y was obtained from measurements of Ha emission during pulsing of the discharge and was found to be about two times larger. The hydrogen atom density measurements were used to deduce quenching rates for argon by H2, which should enable wider application of argon actinometry for studying relative changes in H atom density. Three spectroscopic methods were used to test the electron kinetic model. The first was to measure the populations of the n = 3, 4 and 5 excited states of atomic hydrogen. Generally, agreement between the measured populations and the predictions of a corona model was satisfactory. Another method based on pulsing of the discharge to separate the contributions to Ha emission by dissociative excitation of H2 and direct excitation of H was also used to test the electron kinetic model. Good agreement between theory and the electron mean energy inferred from these measurements was obtained at pressures above 10 Torr. The third method involved measurement of the H2 dissociation frequency from the time evolution of Ha emission during pulsing of the discharge and the equilibrium hydrogen atom density. Comparison with the predicted dissociation frequency showed agreement to within a factor of two. The excited state population measurements were also used to obtain quenching rate coefficients for the n = 4 and 5 excited states of H by H2. Langmuir probe methods of measuring the EEDF were also investigated (Chapter 6). Higher than expected mean electron energies were obtained with both double and single Langmuir probes. Comparison with experiments in an argon discharge led to the conclusion that the short electron energy relaxation length in H2 meant that the sampled EEDF corresponded to a region localised about the probe and that this region was probably disturbed by the probe7 leading to anomalous measurements. In general, the results indicated that a positive column model of the discharge with spatially averaged quantities is an adequate model for calculation of the EEDF.

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)

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 L¹. 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.

Ce travail consiste à étudier des systèmes constitués par un grand nombre de grains, auxquels de l’énergie cinétique est fournie, et à étudier leurs similarités et leurs différences avec des fluides traditionnels. Je me concentre principalement sur la nature de non-équilibre de ces fluides granulaires, en montrant que, même si les méthodes de méchanique statistique y sont applicables, leurs propriétés sont très différentes de celles de systèmes à l’équilibre ou proches de l’équilibre :

• Les fluides granulaires présentent des phénomènes de transport qui n’ont pas d’équivalent dans des fluides moléculaires, tels qu’un couplage spécifique entre flux de chaleur et gradient de densité.

• Leur distribution de vitesse est en général différente de la distribution de Maxwell-Boltzmann, et présente une surpopulation pour les grandes vitesses.

• Dans le cas de mélanges, différentes espèces de grains sont en général caractérisées par des énergies cinétiques différentes, i.e. ces systèmes sont sujet à une non-equipartition de leur énergie.

• Ces fluides ont tendance à former des inhomogénéités spatiales spontanément. Cette propriété est illustrée en étudiant l’expérience du Demon de Maxwell appliquée aux systèmes granulaires.

Chacune de ces particularités est discutée en détail dans des chapitres distincts, où l’on applique différentes méthodes de méchanique statistique (équation de Boltzmann, transition de phase, mean field models…) et où l’on vérifie les prédictions théoriques par simulations numériques (MD, Monte Carlo…).

Doctorat en sciences, Spécialisation physique
info:eu-repo/semantics/nonPublished

Desenvolvemos urna teoria cinética para urna mistura de gases ionizados em presença de campos elétricos e magnéticos. As leis de Ohm, Fourier e Navier-Stokes são obtidas por dois métodos distintos que se baseiam na equação de Boltzmann. Verificamos que o emprego de teoremas de representação torna o método de Chapman-Enskog mais direto. Entretanto o método combinado mostrou-se extremamente simples, onde os coeficientes de transporte são determinados através da inversão de tensores de segunda e quarta ordens. Calculamos também a integral de colisão para as possíveis interações em gases ionizados tais como, entre partículas carregadas, partícula carregada e partícula neutra e entre partículas neutras. Como uma aplicação do método combinado, determinamos os coeficientes de condutividade elétrica, condutividade térmica, coeficiente termo-elétrico e o coeficiente de viscosidade cisalhante para um gás totalmente ionizado. Obtemos seus respectivos gráficos, considerando então um gás ionizado formado a partir do gás de hélio.
We develop a kinetic theory for ionized gases mixtures under the presence of electric and magnetic fields. The laws of Ohm, Fourier and Navier-Stokes are obtained by two different methods based on the Boltzmann equation. We verify that the use of representation theorems makes the Chapman-Enskog method more direct. However the combined method shows up as extremely simple where the transport coefficients are determined through inversion of second-order and fourth order tensors. We calculate also the collision integrals for possible interactions in ionized gases like: between charged particles, between charged particles and neutral particles and between neutral particles. As an application of the combined method, we determine the electrical and thermal conductivity coefficients, thermo-electric and shear viscosity coefficients for a completely ionized gas. We obtain their respective graphics considering an ionized gas of helium.

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.

学位授与大学：京都大学 ; 取得学位: 博士(工学) ; 学位授与年月日: 2007-09-25 ; 学位の種類: 新制・課程博士 ; 学位記番号: 工博第2860号 ; 請求記号: 新制/工/1420 ; 整理番号: 25545
Kyoto University (京都大学)
0048
新制・課程博士
博士(工学)
甲第13389号
工博第2860号
新制||工||1420(附属図書館)
25545
UT51-2007-Q790
京都大学大学院工学研究科航空宇宙工学専攻
(主査)教授 青木 一生, 教授 稲室 隆二, 教授 斧 髙一
学位規則第4条第1項該当

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The work in question discusses the usage of probabilistic arguments on the Kinetic Theory of gases by James Clerk Maxwell, more specifically when describing the speed of the gas particles. The reflection on the gas behavior and its constituents develops during the 19th century. In this context, it can be verified the use of probabilist arguments in formulation of this knowledge. This work aims to identify the factors that influenced the use of such probabilistic arguments in the development of the Kinetic Theory of gas by Maxwell, and in which sense his approach differentiates from the existing ones, since the probability was already considered in previous formulations of the same theory. In order to identify the subjects studied by Maxwell and the varieties of reflections to which he was exposed, we searched for information on the calendar of the Cambridge University in the period he frequented the institution, we also researched papers published which he might have had contact with and letters exchanged with family and friends
O trabalho em questão versa sobre a utilização de argumentos probabilísticos na Teoria Cinética dos Gases por James Clerk Maxwell, mais especificamente ao descrever a velocidade das partículas que compõem um gás. A reflexão sobre o comportamento dos gases e de seus constituintes ocorreu ativamente ao longo do século XIX. Nesse contexto, percebe-se a utilização de argumentos probabilísticos na construção desse conhecimento. Este trabalho tem como objetivo identificar fatores que influenciaram o uso de argumentos probabilísticos na construção da Teoria Cinética dos Gases por Maxwell e em que sua abordagem a diferencia das teorias já existentes, uma vez que a probabilidade já era considerada em formulações anteriores para a mesma teoria. A fim de identificar os assuntos estudados por Maxwell e a quais tipos de reflexão ele foi exposto, buscamos informações no calendário da Universidade de Cambridge nos anos em que ele a frequentou, em artigos publicados na época com que de alguma forma ele tenha tido contato e também em cartas trocadas com familiares e amigos

Transport of amorphous and semi-crystalline, oriented, annealed and non-annealed PET films has been studied using pure and mixed gas/vapor feeds to understand the influence of flavor molecules on the efficacy of the barrier material. Methanol has been used as the flavor molecule simulant, and pure methanol vapor sorption studies show swelling and relaxation effects in the polymer. Multi-component transport of O2/methanol and O2/CO¬2/methanol mixtures, performed at different activities of methanol, shows that vapor induced plasticization leads to increases in O2 and CO2 permeability. Annealed, semi-crystalline PET is shown to be most resistant to plasticization effects. It has been shown that the non-annealed film is less stable despite similar crystallinity as the annealed film due to the presence of orientation related stress in the material. Presence of crystals also restricts the chain motion, and helps suppress the plasticization effects. The results have been compared with the predictions of the dual mode model for multi-component mixtures. Plasticization effects at the high activities have been analyzed within the framework of the free volume theory. It has been proposed that only the densified domains of a glassy polymer be considered when evaluating fractional free volume change due to swelling in the polymer-penetrant system. The free volume parameter- BA has been evaluated for O2 and CO2 in PET and is found to be different from that for other high permeability polymers.

O objetivo deste trabalho e a determinação dos coeficientes de viscosidade de cisalhamento e condutividade térmica de gases ideais quânticos. No cálculo dos coeficientes de transporte foram considerados dois aspectos: uma estatística quântica com seções transversais quânticas e uma estatística quântica com seções transversais quânticas. No primeiro caso, e utilizado um método alternativo para a determinação das aproximações sucessivas (ate a quinta ordem) para os coeficientes de transporte dos gases Helio 4, Helio 3, para-hidrogênio e orto-hidrogênio. No caso de estatística quântica desenvolvida uma teoria com base no método dos momentos de Grad e na equação de Uehling- Uhlenbeck, com a finalidade de determinar as express6es algébricas para os coeficientes de transporte. Esta teoria e baseada em 13 momentos, de densidade, velocidade, tensor pressão e fluxo de calor. Do conhecimento da função de distribuição em termos dos momentos, as equações constitutivas são determinadas e os coeficientes de transporte seguem de um método iterativo semelhante ao procedimento Maxwelliano.
The aim of this work is the determination of the coefficients of shear viscosity and thermal conductivity of quantum ideal gases. In the calculation of the transport coefficients two aspects have been taken into account: a classical statistical with quantum cross-sections and a quantum statistical with quantum cross-sections. In the first case, an alternative method is used for the determination of the successive approximations (up to the fifth order) to the transport coefficients of the gases helium 4, helium 3, para-hydrogen, ortho-hidrogen. In the case of quantum statistical a theory based on method of moments of Grad and on the Uehling-Uhlenbeck equation is developed in order to determine the algebraic expressions to the transport coefficients. This theory is based on 13 moments of density, velocity, pressure tensor and heat flux. From the knowledge of the distribution function the constitutive equations are determinate and the transport coefficients follow from an iterative method akin to the 11axwellian procedure.

As possíveis soluções da equação integro-diferencial de Boltzmann constituem uma importante ferramenta para o estudo de gases e plasmas. No entanto, suas soluções analíticas são difíceis de serem encontradas. Uma abordagem bastante utilizada na literatura para obter soluções aproximadas da equação de Boltzmann é através de hipóteses que simplificam a forma da integral de colisão. Nesta dissertação, discutimos dois modelos colisionais alternativos que generalizam o método originalmente proposto por Bhatnagar, Gross e Krook, usualmente referido na literatura como aproximação BGK. O primeiro deles é um modelo de relaxação de segunda ordem, no qual introduzimos um segundo tempo de relaxação, 2 , relacionado com efeitos não lineares. O segundo modelo é baseado em outra generalização do modelo BGK obtida através de uma lei de potência parametrizada por um índice . No limite 1 o modelo BGK padrão é recuperado. As duas aproximações são fisicamente interpretadas. Além disso, para ilustrar nossos resultados com algumas aplicações mais quantitativas, obtemos as expressões analíticas para diversos coeficientes de transporte, a saber: a condutividade térmica (), a viscosidade de cisalhamento () e a condutividade elétrica (). Em particular, no modelo de relaxação de segunda ordem, as correções dependem da razão 1/2, onde 1 é a escala de tempo característica do modelo BGK padrão e 2 a nova escala de tempo associada aos efeitos não lineares. Finalmente, como um resultado geral, mostramos também que todas as correções nos coeficientes de transporte dependem numa certa potência do chamado número de Knudsen.
The possible solutions of the integro-diferential Boltzmann equation constitute an important tool for studying gases and plasmas. |However, its analytical solutions are hardly derived. An approach often adopted in the literature for obtaining approximate solutions of the Boltzmann equation is to consider some simplifying hypothesis on the collisional term. In this Dissertation, we discuss two diferent alternative collisional models which generalize the method originally proposed by Bhatnagar, Gross e Krook, and usually referred to as BGK approximation. The first one is a second order relaxation model in which a second relaxation time, 2, related with the nonlinear efects, is introduced. The second one is based on a diferent generalization of the BGK model which is obtained through a power law parameterized by a index . In the limit 1, the BGK model is recovered. Both approximations are physically interpreted. Further, in order to illustrate our results with some more quantitative applications, we derive the analytical expressions for several transport coefficients, among them the thermal conductivity (), the shear viscosity () and the electric conductivity (). In particular, for the second order relaxation model, we find that the corrections depend on the ratio 1/2 where 1 is the characteristic time scale of the BGK model and 2 describe the nonlinear efects. Finally, as a general result, it is also shown that all the corrections on the transport coefficients depend on a given power of the so-called Knudsen number.

Apesar dos avanços na última metade de século na teoria de transporte em Física de Plasmas, muitos de seus aspectos ainda são pouco compreendidos. Grande parte dessa limitação se deve à carência de modelos de primeiros princípios minimamente capazes de reproduzir os resultados experimentais. De fato, sem o embasamento em hipóteses fundamentais, os modelos devem se restringir à descrição do comportamento observado nos diferentes regimes de transporte no plasma, sem necessariamente especificar por que ou quais são os mecanismos envolvidos; até mesmo a identificação dos elementos envolvidos no transporte, por exemplo, se partículas ou células convectivas, é prejudicada. Uma abordagem que vem ganhando destaque na comunidade de Física de Plasmas ao longo dos anos é a estatística não-extensiva. Em particular, o interesse na teoria de Tsallis está na sua capacidade de descrever sistemas distantes do equilíbrio termodinâmico, uma característica comum à maioria dos plasmas de laboratório e astrofísicos. De fato, nessas circunstâncias, é sabido que as funções de distribuição das partículas são distantes das distribuições Maxwellianas, com longas-caudas, especialmente para os elétrons. A capacidade da teoria de Tsallis em descrever fenômenos da Física de Plasmas é retratada nas suas diversas aplicações encontradas na literatura, por exemplo, o transporte anômalo, oscilações eletrostáticas, ventos solares, plasmas empoeirados, onde é sabido que as previsões dadas pela estatística de Maxwell-Boltzmann não são capazes de descrever corretamente os resultados experimentais. A proposta desta tese de doutoramento é utilizar a estatística não-extensiva para determinar o transporte colisional em plasmas intensamente magnetizados. O desenvolvimento completo do modelo de transporte no contexto não-extensivo é estabelecido rigorosamente: partindo da definição da entropia de Tsallis e da hipótese das interações fracas (a condição do transporte colisional), somos capazes de deduzir as equações de fluidos utilizando apenas métodos estatísticos genéricos, e sem hipóteses adicionais. Nesse percurso, apresentamos, sempre de maneira consistente com a estatística não-extensiva, a definição da temperatura; a dedução da equação cinética com o operador colisional para plasmas; a generalização do método utilizado por Braginskii para determinar as soluções aproximadas da equação cinética; e o cálculo dos coeficientes de transporte. Porém, também apresentamos a aplicação de nosso modelo no transporte de calor em ventos solares e no pulso frio em plasmas de laboratório.
Despite the advances in the last half century in the plasma transport theory, many aspects of such phenomena remain poorly understood. Most of this limitation is due to the lack o first principles models capable of reproducing experimental observations. In fact, without a fundamental hypothesis, the models are restricted to describing the behavior of the observed plasma transport in diferent regimes, without specifying why or which mechanisms take part in the process; even the determination of the elements involved in the transport, for instance, whether particles or convective cells, is impaired. One approach that has been attracting attention in Plasma Physics community over the years is the non-extensive statistics. In particular, the interest in the Tsallis\'s theory lies in its ability to describe systems far from thermodynamic equilibrium, a common feature in most laboratory and astrophysical plasmas. The capability of the non-extensive statistics in describing phenomena of Plasma Physics is portrayed in various applications, for example, the anomalous transport, electrostatic oscillations, solar winds, dusty plasmas, where it is know that the predictions given by Maxwell-Boltzmann statistics cannot describe the experimental results. Indeed, under such cases, it is well known that the particle distribution functions are quite distant from Maxwellian distributions, with long tails, especially for electrons. The purpose of this doctoral thesis is to use the non-extensive statistics in order to obtain a model for the collisional transport in strongly magnetized plasmas. The complete development of the model in the non-extensive context is strictly established; starting with the definition of the Tsallis entropy and the weak interactions hypothesis (the collisional transport condition), we are able to derive the fluid equations using only generic statistical methods, without additional hypotheses. For such task, we present, consistently with non-extensive statistics, the definition of temperature; the deduction of the kinetic equation with the collision operator for plasmas, which are also appropriated for the determination of the fluid equations; the generalization of the method used by Braginskii to approximate the solution of the kinetic equation for electrons; and the calculation of electron transport coeficients. Lastly, we present the application of our model in the heat transport in the solar winds and in the phenomena of the cold pulse in laboratory plasmas.

This thesis aimed at developing multi-scale models and computational methods for aerother-modynamics applications. The research on multi-scale models has focused on internal energy excitation and dissociation of molecular gases in atmospheric entry flows. The scope was two-fold: to gain insight into the dynamics of internal energy excitation and dissociation in the hydrodynamic regime and to develop reduced models for Computational Fluid Dynamics applications. The reduced models have been constructed by coarsening the resolution of a detailed rovibrational collisional model developed based on ab-initio data for the N2 (1Σ+g)-N (4Su) system provided by the Computational Quantum Chemistry Group at NASA Ames Research Center. Different mechanism reduction techniques have been proposed. Their appli-cation led to the formulation of conventional macroscopic multi-temperature models and vi-brational collisional models, and innovative energy bin models. The accuracy of the reduced models has been assessed by means of a systematic comparison with the predictions of the detailed rovibrational collisional model. Applications considered are inviscid flows behind normal shock waves, within converging-diverging nozzles and around axisymmetric bodies, and viscous flows along the stagnation-line of blunt bodies. The detailed rovibrational colli-sional model and the reduced models have been coupled to two flow solvers developed from scratch in FORTRAN 90 programming language (SHOCKING_F90 and SOLV-ER_FVMCC_F90). The results obtained have shown that the innovative energy bin models are able to reproduce the flow dynamics predicted by the detailed rovibrational collisional model with a noticeable benefit in terms of computing time. The energy bin models are also more accurate than the conventional multi-temperature and vibrational collisional models. The research on computational methods has focused on rarefied flows. The scope was to formu-late a deterministic numerical method for solving the Boltzmann equation in the case of multi-component gases with internal energy by accounting for both elastic and inelastic collisions. The numerical method, based on the weighted convolution structure of the Fourier trans-formed Boltzmann equation, is an extension of an existing spectral-Lagrangian method, valid for a mono-component gas without internal energy. During the development of the method, particular attention has been devoted to ensure the conservation of mass, momentum and en-ergy while evaluating the collision operators. Conservation is enforced through the solution of constrained optimization problems, formulated in a consistent manner with the collisional in-variants. The extended spectral-Lagrangian method has been implemented in a parallel com-putational tool (best; Boltzmann Equation Spectral Solver) written in C programming lan-guage. Applications considered are the time-evolution of an isochoric gaseous system initially set in a non-equilibrium state and the steady flow across a normal shock wave. The accuracy of the proposed numerical method has been assessed by comparing the moments extracted from the velocity distribution function with Direct Simulation Monte Carlo (DSMC) method predictions. In all the cases, an excellent agreement has been found. The computational results obtained for both space homogeneous and space inhomogeneous problems have also shown that the enforcement of conservation is mandatory for obtaining accurate numerical solutions.

The subject of the thesis is the derivation of non-local diffusion equations from kinetic models with heavy-tailed equilibrium in velocity. We are particularly interested in confining the kinetic equations and developing methods that allow us, from the confined kinetic models, to derive confined versions of non-local diffusion equations.

Convection in a layer of fluid with a free surface due to a combination of thermocapillary stresses and buoyancy is a classic problem of fluid mechanics. It has attracted increasing attentions recently due to its relevance for two-phase cooling. Many of the modern thermal management technologies exploit the large latent heats associated with phase change at the interface of volatile liquids, allowing compact devices to handle very high heat fluxes. To enhance phase change, such cooling devices usually employ a sealed cavity from which almost all noncondensable gases, such as air, have been evacuated. Heating one end of the cavity, and cooling the other, establishes a horizontal temperature gradient that drives the flow of the coolant. Although such flows have been studied extensively at atmospheric conditions, our fundamental understanding of the heat and mass transport for volatile fluids at reduced pressures remains limited. A comprehensive and quantitative numerical model of two-phase buoyancy-thermocapillary convection of confined volatile fluids subject to a horizontal temperature gradient has been developed, implemented, and validated against experiments as a part of this thesis research. Unlike previous simplified models used in the field, this new model incorporates a complete description of the momentum, mass, and heat transport in both the liquid and the gas phase, as well as phase change across the entire liquid-gas interface. Numerical simulations were used to improve our fundamental understanding of the importance of various physical effects (buoyancy, thermocapillary stresses, wetting properties of the liquid, etc.) on confined two-phase flows. In particular, the effect of noncondensables (air) was investigated by varying their average concentration from that corresponding to ambient conditions to zero, in which case the gas phase becomes a pure vapor. It was found that the composition of the gas phase has a crucial impact on heat and mass transport as well as on the flow stability. A simplified theoretical description of the flow and its stability was developed and used to explain many features of the numerical solutions and experimental observations that were not well understood previously. In particular, an analytical solution for the base return flow in the liquid layer was extended to the gas phase, justifying the previous ad-hoc assumption of the linear interfacial temperature profile. Linear stability analysis of this two-layer solution was also performed. It was found that as the concentration of noncondensables decreases, the instability responsible for the emergence of a convective pattern is delayed, which is mainly due to the enhancement of phase change. Finally, a simplified transport model was developed for heat pipes with wicks or microchannels that gives a closed-form analytical prediction for the heat transfer coefficient and the optimal size of the pores of the wick (or the width of the microchannels).

Cette thèse est dédiée à l'étude mathématique et numérique d'une classe d'équations cinétiques collisionnelles, de type équation de Boltzmann. Nous avons porté un intérêt tout particulier à l'équation des milieux (ou gaz) granulaires, initialement introduite dans la littérature physique pour décrire le comportement hors équilibre de matériaux composés d'un grand nombre de grains, ou particules, non nécessairement microscopiques, et interagissant par des collisions dissipant l'énergie cinétique. Ces modèles se sont révélés avoir une structure mathématique très riche. Cette thèse se structure en trois partie pouvant être lues de manière indépendante, mais néanmoins en rapport avec des équations cinétiques collisionnelles en général, et l'équation des milieux granulaires en particulier. La première partie est dédiée à l'étude mathématique du comportement asymptotique de certaines équations cinétiques collisionnelles dans un cadre homogène en espace. Nous y montrons des résultats de type explosion et convergence vers la solution autosimilaire avec calcul explicite des taux, pour des opérateurs de type Boltzmann, grâce à l'utilisation (entre autre) d'une nouvelle méthode de changement de variables dépendant directement de la solution de l'équation considérée. En particulier, nous démontrons que pour un modèle de gaz granulaire - dit anormal - il est possible d'observer une explosion en temps fini. Dans la deuxième partie, orientée analyse numérique et calcul scientifique, nous nous intéressons développement et à l'étude de méthodes spectrales pour la résolution de problèmes multi-échelles, issus de la théorie des équations cinétiques collisionnelles. Les méthodes de changement de variables tiennent aussi une place importante dans cette partie, et permettent d'observer numériquement des phénomènes non triviaux qui apparaissent lors de l'étude de gaz granulaires, comme la création d'amas de matière ou la caractérisation précise du retour vers l'équilibre. La troisième et dernière partie est dédiée à l'étude spectrale de l'opérateur des milieux granulaires avec bain thermique, linéarisé au voisinage d'un équilibre homogène en espace, afin d'établir des résultats de type stabilité et convergence vers une limite hydrodynamique. Ce travail est en fait la généralisation d'un résultat célèbre dans la théorie de l'équation de Boltzmann, dû à R. Ellis et M. Pinsky, et établissant rigoureusement la première limite hydrodynamique vers les équations d'Euler compressibles linéaires puis Navier-Stokes de cette équation.

Lubrication theory plays a fundamental role in all mechanical design as well as applications to biomechanics. All machinery are composed of moving parts which must be protected against wear and damage. Without effective lubrication, maintenance cycles will be shortened to impractical levels resulting in increased costs and decreased reliability. The focus of the work presented here is on the lubrication of rotating machinery found in advanced power systems and designs involving micro-turbines. One of the earliest studies of lubrication is due to Osborne Reynolds in 1886 who recorded what is now regarded as the canonical equation governing all lubrication problems; this equation and its extensions have become known as the Reynolds equation. In the past century, Reynolds equation has been extended to include three-dimensional effects, unsteadiness, turbulence, variable material properties, non-newtonian fluids, multi-phase flows, wall slip, and thermal effects. The bulk of these studies have focused on highly viscous liquids, e.g., oils. In recent years there has been increasing interest in power systems using new working fluids, micro-turbines and non-fossil fuel heat sources. In many cases, the design of these systems employs the use of gases rather than liquids. The advantage of gases over liquids include the reduction of weight, the reduction of adverse effects due to fouling, and compatibility with power system working fluids. Most treatments of gas lubrication are based on the ideal, i.e., low pressure, gas theory and straightforward retro-fitting of the theory of liquid lubrication. However, the 21st Century has seen interest in gas lubrication at high pressures. At pressures and temperatures corresponding to the dense and supercritical gas regime, there is a strong dependence on gas properties and even singular behavior of fundamental transport properties. Simple extrapolations of the intuition and analyses of the ideal gas or liquid phase theory are no longer possible. The goal of this dissertation is to establish the correct form of the Reynolds equation valid for both low and high pressure gases and to explore the dynamics predicted by this new form of the Reynolds equation. The dissertation addresses five problems involving our new Reynolds equation. In the first, we establish the form appropriate for the simple benchmark problem of two-dimensional journal bearings. It is found that the material response is completely determined by a single thermodynamic parameter referred to as the "effective bulk modulus". The validity of our new Reynolds equation has been established using solutions to the full Navier-Stokes-Fourier equations. We have also provided analytical estimates for the range of validity of this Reynolds equation and provided a systematic derivation of the energy equation valid whenever the Reynolds equation holds. The next three problems considered here derive local and global results of interest in high speed lubrication studies. The results are based on a perturbation analysis of our Reynolds and energy equation resulting in simplified formulas and the explicit dependence of pressure, temperature, friction losses, load capacity, and heat transfer on the thermodynamic state and material properties. Our last problem examines high pressure gas lubrication in thrust bearings. We again derive the appropriate form of the Reynolds and energy equations for these intrinsically three-dimensional flows. A finite difference scheme is employed to solve the resultant (elliptic) Reynolds equation for both moderate and high-speed flows. This Reynolds equation is then solved using perturbation methods for high-speed flows. It is found that the flow structure is comprised of five boundary layer regions in addition to the main ``core'' region. The flow in two of these boundary layer regions is governed by a nonlinear heat equation and the flow in three of these boundary layers is governed by nonlinear relaxation equations. Finite difference schemes are employed to obtain detailed solutions in the boundary layers. A composite solution is developed which provides a single solution describing the flow in all six regions to the same accuracy as the individual solutions in their respective regions of validity. Overall, the key contributions are the establishment of the appropriate forms of the Reynolds equation for dense and supercritical flows, analytical solutions for quantities of practical interest, demonstrations of the roles played by various thermodynamic functions, the first detailed discussions of the physics of lubrication in dense and supercritical flows, and the discovery of boundary layer structures in flows associated with thrust bearings.
Doctor of Philosophy
Lubrication theory plays a fundamental role in all mechanical design as well as applications to biomechanics. All machinery are composed of moving parts which must be protected against wear and damage. Without eective lubrication, maintenance cycles will be shortened to impractical levels resulting in increased costs and decreased reliability. The focus of the work presented here is on the lubrication of rotating machinery found in advanced power systems and designs involving micro-turbines. One of the earliest studies of lubrication is due to Osborne Reynolds in 1886 who recorded what is now regarded as the canonical equation governing all lubrication problems; this equation and its extensions have become known as the Reynolds equation. In the past century, Reynolds equation has been extended to include three-dimensional eects, unsteadiness, turbulence, variable material properties, non-newtonian uids, multi-phase ows, wall slip, and thermal eects. The bulk of these studies have focused on highly viscous liquids, e.g., oils. In recent years there has been increasing interest in power systems using new working uids, micro-turbines and non-fossil fuel heat sources. In many cases, the design of these systems employs the use of gases rather than liquids. The advantage of gases over liquids include the reduction of weight, the reduction of adverse eects due to fouling, and compatibility with power system working uids. Most treatments of gas lubrication are based on the ideal, i.e., low pressure, gas theory and straightforward retro-tting of the theory of liquid lubrication. However, the 21st Century has seen interest in gas lubrication at high pressures. At pressures and temperatures corresponding to the dense and supercritical gas regime, there is a strong dependence on gas properties and even singular behavior of fundamental transport properties. Simple extrapolations of the intuition and analyses of the ideal gas or liquid phase theory are no longer possible. The goal of this dissertation is to establish the correct form of the Reynolds equation valid for both low and high pressure gases and to explore the dynamics predicted by this new form of the Reynolds equation. The dissertation addresses ve problems involving our new Reynolds equation. In the rst, we establish the form appropriate for the simple benchmark problem of two-dimensional journal bearings. It is found that the material response is completely determined by a single thermodynamic parameter referred to as the eective bulk modulus. The validity of our new Reynolds equation has been established using solutions to the full Navier-Stokes-Fourier equations. We have also provided analytical estimates for the range of validity of this Reynolds equation and provided a systematic derivation of the energy equation valid whenever the Reynolds equation holds. The next three problems considered here derive local and global results of interest in high speed lubrication studies. The results are based on a perturbation analysis of our Reynolds and energy equation resulting in simplied formulas and the explicit dependence of pressure, temperature, friction losses, load capacity, and heat transfer on the thermodynamic state and material properties. Our last problem examines high pressure gas lubrication in thrust bearings. We again derive the appropriate form of the Reynolds and energy equations for these intrinsically threedimensional ows. A nite dierence scheme is employed to solve the resultant (elliptic) Reynolds equation for both moderate and high-speed ows. This Reynolds equation is then solved using perturbation methods for high-speed ows. It is found that the ow structure is comprised of ve boundary layer regions in addition to the main core region. The ow in two of these boundary layer regions is governed by a nonlinear heat equation and the ow in three of these boundary layers is governed by nonlinear relaxation equations. Finite dierence schemes are employed to obtain detailed solutions in the boundary layers. A composite solution is developed which provides a single solution describing the ow in all six regions to the same accuracy as the individual solutions in their respective regions of validity. Overall, the key contributions are the establishment of the appropriate forms of the Reynolds equation for dense and supercritical ows, analytical solutions for quantities of practical interest, demonstrations of the roles played by various thermodynamic functions, the rst detailed discussions of the physics of lubrication in dense and supercritical ows, and the discovery of boundary layer structures in ows associated with thrust bearings.

The hydrodynamic equations of change for a reacting gas mixture of monomers and dimers are studied. The gas is considered to be dilute and described by the kinetic theory of Lowry and Snider (J. Chem. Phys. 61, 2320 (1974)). From the kinetic equations for the density operators representing the monomer and dimer, the equations of change for one-molecule observables are obtained. Since the energy operator involves the intermolecular potential energy, it is necessary to derive the energy balance equation from the von Neumann equation, since this includes molecule-molecule correlations. As well, the kinetic theory formulated by Lowry and Snider is rewritten so that rearrangement collisions are emphasized. A collisional sum rule is derived involving the commutation properties of channel projectors and their respective potentials. A known property of the optical theorem is that it identifies the reactive loss terms as part of the non-reactive transition superoperators. The sum rule is applied to rewrite the non-reactive transition superoperators so as to display the reactive loss terms. This aids in establishing conservation laws for the physical observables of mass, linear momentum, angular momentum and energy. A form of the optical theorem in which kinetic energy off-diagonality is allowed for is also derived. Both the optical theorem and the sum rule are based on the strong orthogonality hypothesis, which plays a fundamental role in the Lowry-Snider theory. On localising the physical attributes at the centres of mass of the molecules, the contributions to the equations of change from collisional transfer (due to the forces and torques between the collision partners) and from the transfer of the physical attributes from the reactants to the products are identified. The transformation of dimer internal degrees of freedom into monomer translational degrees of freedom or vice versa when a dimer Is dissociated or formed is found to contribute to the equations of change by virtue of the differing locality of the collision partners. The decomposition of the kinetic energy operator into its components for radial and rotational motions allows the kinetic energy flux contributions associated with the pressure tensor and the molecular angular momentum flux to be identified.
Science, Faculty of
Chemistry, Department of
Graduate

A numerical study is done of a plasma in contact with a cold solid surface that is emitting a neutral gas. Two numerical models have been developed to describe the dominant phenomena of surface plasma structures. The first model entails a steady-state, kinetic treatment of the transport equations in one space dimension and one velocity dimension, to determine self-consistently the distribution functions of the interacting species and the electrostatic potential near the solid surface. The dominant phenomena in this region are the ionization of the neutral gas and the acceleration of the resulting ions by the electrostatic field in a pre-sheath region. Other effects involved are a Debye sheath structure between the solid surface and pre-sheath, and collisional trapping and untrapping of electrons in an electrostatic potential well that is predicted in the pre-sheath region. Results are presented from a nondimensional model with a monatomic returning neutral species and for diatomic molecular hydrogen returning from the surface. For each set of physical parameters chosen, a one parameter family of solutions is obtained. The second numerical model involves a steady-state treatment of the transport equations in a (x,v∥,v⊥) phase space for the interacting species. Included in this model are ionization of the refluxing monatomic neutrals, a self-consistently determined electrostatic potential and a nonlinear Fokker-Planck treatment of ion-ion Coulomb collisions. Both the region near the surface dominated by kinetic effects and the region away from the surface in which Coulomb collisional effects are significant are treated. Results are presented which identify the correct physical solution for the region near the surface from the permitted family found with the kinetic model. Additionally, results are shown which span a temperature range from the high temperature kinetic regime where Coulomb collisional effects are negligible, to the low temperature, highly collisional fluid regime. At low temperatures the collisional model agrees well with standard fluid techniques.

La capacité de modéliser, simuler et prédire le phénomène de reconnexion magnétique est un enjeu crucial pour de nombreuses applications (ITER, plasmas astrophysiques) et impacte la prédiction du « temps solaire » et des « orages magnétiques » pouvant perturber les satellites. L’enjeu scientifique fondamental est la description du transfert instationnaire d’énergie magnétique en énergie cinétique et thermique, encore hors d’atteinte des modèles magnéto-hydrodynamique (MHD) actuels. L’objectif premier de la thèse est le développement d’un modèle fluide cohérent de plasma magnétisé hors équilibrethermique et chimique avec une description détaillée des effets dissipatifs basée sur la théorie cinétique des gaz et une bonne structure mathématique. Le second repose sur le développement d’une stratégie numérique innovante, précise et robuste, dans un code de calcul massivement parallèle avec adaptation demaillage permettant de capturer tout le spectre d’échelle en jeu et la raideur numérique en résultant. L’ensemble des coefficients de transport, la thermodynamique et la chimie correspondante seront étudiés et comparés aux données préalablement utilisées dans le domaine. Puis on montrera que le modèle et sa simulation, issus d’un travail transdisciplinaire impliquant ingénierie, physique des plasmas, physique solaire, mathématique, et calcul scientifique et parallèle, est capable de reproduire correctement la physique du phénomène. La validation de l’approche à travers une série de cas test issus de l’application à la dynamique de l’atmosphère solaire en lien avec la NASA et le VKI permettra de disposer d’un outil, ouvert à la communauté, capable de lever plusieurs verrous scientifiques et technologiques
The ability to model, simulate and predict magnetic reconnection (MR) is a stumbling block in order to predict space weather and geomagnetic storms, which can lead to great perturbation of satellites. Some fundamental aspects of MR are not yet well understood. The scientific issue at stake is the proper description of the unsteady energy transfer from magnetic energy to kinetic and thermal energy, which is still out of reach for the standard Magneto-hydrodynamics (MHD) models. The first objective of the present project is to develop a coherent fluid model for magnetized plasmas out of thermal and chemical equilibrium with a detailed description of the dissipative effects based on kinetic theory of gases, which thus inherits a proper mathematical structure. The second goal is the development of a new numerical strategy, with high accuracy and robustness, based on a massively parallel code with adaptive mesh refinement able to cope with the full spectrum of scales of the model and related stiffness. The whole set of transport coefficients, thermodynamics relations and chemical rates in this magnetized two-temperature setting will be studied and compared to the one in the literature used in the field. Then, we will show that the model and related numerical strategy, obtained from this transdisciplinary work involving engineering, plasma physics, solar physics, mathematics, scientific computing and HPC, is able to properly reproduce the physics of MR. The validation of the approach through a series of test-cases relevant for the application to the dynamics of solar atmosphere in connection with VKI and NASA will provide a tool, open to the community, capable of resolving several critical scientific and technological issues

Un gaz à l'intérieur d’un microsystème ou d’un milieu poreux est dans un état hors équilibre, car le libre parcours moyen des molécules est comparable à la dimension caractéristique du milieu. Ce même état degaz, appelé raréfié, se retrouve en haute altitude ou dans un équipement de vide à basse pression. Ces gaz raréfiés suivent des types d’écoulements qui peuvent être décrits par des modèles cinétiques dérivés de l'équation de Boltzmann. Dans ce travail nous présentons les principaux modèles et leurs mises en oeuvre numériquepour la simulation des écoulements de gaz raréfiés. Parmi les modèles utilisés nous présentons les deux modèles complets de l'équation de Boltzmann, le modèle de Shakhov(S-model) pour un gaz monoatomique et le modèle de McCormack pour un mélange de gaz toujours monoatomiques. La méthode des vitesses discrètes est utilisée pour la discrétisation numérique dans l'espace des vitesses moléculaires et le schéma de type TVD est mis en œuvre dans l'espace physique. L’aspect original de ce travail se situe sur les régimes transitoires et, en particuliersur les comportements non-stationnaires des transferts de chaleur et de masse. Cependant, pour certaines configurations nous considérons uniquement les conditions stationnaires des écoulements et un schéma implicite est développé afin de réduire le coût de calcul. En utilisant ces approches numériques, nous présentons les résultats pour plusieurs types d’écoulements non-stationnaires, de gaz raréfiés monoatomiqueset de mélanges binaires de gaz monoatomiques
A gas inside the microsystems or the porous media is in its non-equilibrium state, due to the fact that the molecular mean free path is comparable to the characteristic dimension of the media. The same state of a gas, called rarefied, is found at high altitude or in the vacuum equipment working at low pressure. All these types of flow can be described by the kinetic models derived from the Boltzmann equation. This thesis presents the development of the numerical tools for the modeling and simulations of the rarefied gas flows. The two models of the full Boltzmann equation, the Shakhov model (S-model) for the single gas and the McCormack model for the gas mixture, are considered. The discrete velocity method is used to the numerical discretization in the molecular velocity space and the TVD-like scheme is implemented in the physical space. The main aspect of this work is centered around the transient properties of the gas flows and, especially, on the transient heat and mass transfer behaviors. However, for some configurations only steady-state solutions are considered and the implicit scheme is developed to reduce the computational cost. Using the proposed numerical approach several types of the transient rarefied single gas flows as well as the binary mixture of the monoatomic gases are studied

This thesis consists of four papers. the first is devoted to discrete velocity models, the second to hydrodynamic equation beyond Navier-Stokes level, the third to a multi-linear Maxwell model for economic or social dynamics and the fourth is devoted to a function related to the Riemann zeta-function. In Paper 1, we consider the general problem of construction and classification of normal, i.e. without spurious invariants, discrete velocity models (DVM) of the classical Boltzman equation. We explain in detail how this problem can be solved and present a complete classification of normal plane DVMs with relatively small number n of velocities (n≤10). Some results for models with larger number of velocities are also presented. In Paper 2, we discuss hydrodynamics at the Burnett level. Since the Burnett equations are ill-posed, we describe how to make a regularization of these. We derive the well-posed generalized Burnett equations (GBEs) and discuss briefly an optimal choice of free parameters and consider a specific version of these equations. Finally we prove linear stability for GBE and present some numerical result on the sound propagationbased on GBEs. In Paper 3, we study a Maxwell kinetic model of socio-economic behavior. The model can predict a time dependent distribution of wealth among the participants in economic games with an arbitrary, but sufficiently large, number of players. The model depends on three different positive parameters {γ,q,s} where s and q are fixed by market conditions and γ is a control parameter. In particular, we investigate the efficiency of control. Some exact solutions and numerical examples are presented. In Paper 4, we study a special function u(s,x), closely connected to the Riemann zeta-function ζ(s), where s is a complex number. We study in detail the properties of u(s,x) and in particular the location of its zeros s(x), for various x≥0. For x=0 the zeros s(0) coincide with non-trivial zeros of ζ(s). We perform a detailed numerical study of trajectories of various zeros s(x) of u(s,x).

Neste trabalho, o fenômeno de evaporação em gases rarefeitos e analisado, para o caso de uma espécie de gás bem como de misturas binárias. Evaporação fraca e forte são consideradas para escoamentos de gases em canal e semi-espaco. Também e investigado o fenômeno conhecido como reverso de temperatura, típico de gases em estado de rarefação. O método ADO, uma versão analítica do método de ordenadas discretas, é utilizado para construção de soluções em forma fechada para os diversos problemas e quantidades de interesse, como perfis de temperatura e fluxos de calor. Para o caso de um gás, uma solução unificada e desenvolvida para problemas formulados a partir dos modelos cinéticos, derivados da equação de Boltzmann, BGK, S, Gross- Jackson e MRS. No caso de mistura binária de gases, a formulação matemática e baseada no modelo McCormack. Particularmente, quando a evaporação forte e abordada, e aspectos não lineares devem ser incluídos, a versão não linear do modelo BGK e utilizada. Neste caso, a solução ADO do modelo linear e utilizada em um processo chamado de pós-processamento para inclusão dos termos não lineares do problema e reavaliação das quantidades de interesse, evidenciando melhoria dos resultados obtidos pela formulação linear. Uma serie de resultados numéricos são listados e é observada, de forma geral, excelente exatidão e eficiência computacional.
In this work, evaporation phenomena in rarefied gas flow, for one gas case and binary mixtures, are analyzed. Weak and strong evaporation are considered in channel and half-space problems. The reverse of temperature problem, typical in rarefied gas dynamics, is also investigated. The ADO method, an analytical version of the discrete ordinates method, is used to develop closed form solutions, to several problems and quantities of interest, as temperature profiles and heat flows. For the one gas case, an unified solution is developed for the BGK, S, Gross-Jackson and MRS models, derived from the Boltzmann equation. For binary mixtures, the mathematical formulation is based on the McCormack model. Particularly, when strong evaporation is investigated, and nonlinear aspects have to be included, the nonlinear BGK model is used. In this case, the ADO solution, provided by the linear model, is considered in a post-processing procedure which takes into account the nonlinear terms to evaluate the quantities of interest, and improved results are obtained, in comparison with the linear version. A series of numerical results are listed and, in general, an excellent accuracy and good computational efficiency are observed.

Nous étudions certaines variantes de l'équation de Boltzmann, cette dernière décrivant via une approche classique les gaz raréfiés simples et monoatomiques à l'échelle mésoscopique. Dans un premier temps, nous proposons un cadre général de modélisation de Boltzmann des gaz polyatomiques, englobant une large classe de modèles pré-existants et permettant d'en construire de nouveaux. D'abord présenté pour un gaz simple, ce cadre est ensuite étendu aux mélanges gazeux, pour lesquels on autorise des réactions chimiques binaires. Dans un deuxième temps, nous nous intéressons à un type de gaz polyatomique singulier, au sein duquel les collisions sont résonantes. Nous prouvons une propriété de compacité pour l'opérateur linéarisé lié à ce modèle. Afin de rendre plus flexible le cadre résonant, nous proposons ensuite un formalisme de Boltzmann pour des collisions quasi-résonantes, étudions ses propriétés-clés et menons des expériences numériques pour étayer notre compréhension de celles-ci. Enfin, dans un troisième temps, nous nous tournons vers une équation de Boltzmann incluant le principe d'exclusion de Pauli, utile notamment pour la description de la distribution d'électrons dans les semi-conducteurs. Nous développons une méthode permettant de transférer certaines inégalités fonctionnelles, liées à l'entropie, connues dans le cas classique, vers ce cas quantique. Par suite, grâce à l'obtention de ces nouvelles inégalités, nous obtenons un taux explicite de relaxation à l'équilibre pour les solutions de l'équation de Boltzmann-Fermi-Dirac homogène pour les potentiels durs avec cut-off
We study some variants of the Boltzmann equation, the latter describing, via a classical approach, single and monatomic rarefied gases at the mesoscopic scale. First, we propose a general framework for Boltzmann modelling of polyatomic gases, encompassing a wide class of pre-existing models and allowing to build new ones. Primarily presented for a single gas, the framework is then extended to mixtures, within which we allow binary chemical reactions. Second, we focus on a singular type of polyatomic gas, the molecules of which undergo resonant collisions, and prove a compactness property of the linearized operator related to this model. In order to make the latter resonant framework more flexible, we then propose a Boltzmann formalism with quasi-resonant collisions, study its key properties and conduct numerical experiences to support our understanding of them. Third, we turn our attention towards a Boltzmann equation which includes Pauli's exclusion principle, notably used in the study of electron distributions in semi-conductors. We develop a method that allows to transfer some functional inequalities, related to entropy, which are known in the classical case, to this quantum case. As a consequence, we use these new inequalities to obtain an explicit rate of relaxation to equilibrium for solutions to the homogeneous Boltzmann-Fermi-Dirac equation with cut-off hard potentials

The Vlasov equation of kinetic theory is introduced and the Hamiltonian structure of its moments is presented. Then we focus on the geodesic evolution of the Vlasov moments [1.2]. As a first step, these moment equations generalize the Camassa-Holm equation [3] to its multi-component version [4]. Subsequently, adding electrostatic forces to the geodesic moment equations relates them to the Benney equations [5] and to the equations for beam dynamics in particle accelerators. Next, we develop a kinetic theory for self assembly in nano-particles. The Darcy law [6] is introduced as a general principle for aggregation dynamics in friction dominated systems (at different scales). Then, a kinetic equation is introduced [7,8] for the dissipative motion of isotropic nano-particles. The zeroth-moment dynamics of this equation recovers the classical Darcy law at the macroscopic level [7]. A kinetic-theory description for oriented nano-particles is also presented [9]. At the macroscopic level, the zeroth moments of this kinetic equation recover the magnetization dynamics of the Landau-Lifshitz-Gilbert equation [10]. The moment equations exhibit the spontaneous emergence of singular solutions (clumpons) that finally merge in one singularity. This behaviour represents aggregation and alignment of oriented nano-particles. Finally, the Smoluchowsky description is derived from the dissipative Vlasov equation for anisotropic interactions. Various levels of approximate Smoluchowsky descriptions are proposed as special cases of the general treatment. As a result, the macroscopic momentum emerges as an additional dynamical variable that in general cannot be neglected.

Kyoto University (京都大学)
0048
新制・課程博士
博士(人間・環境学)
甲第10956号
人博第243号
15||198(吉田南総合図書館)
新制||人||60(附属図書館)
UT51-2004-G803
京都大学大学院人間・環境学研究科人間・環境学専攻
(主査)教授 冨田 博之, 助教授 早川 尚男, 助教授 阪上 雅昭
学位規則第4条第1項該当