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

<p>We construct two independent hierarchies of&nbsp;moment equations and we apply the maximum&nbsp;entropy principle for polyatomic gases. We&nbsp;formulate multivelocity and multitemperature&nbsp;model of Eulerian polyatomic gases starting from&nbsp;kinetic theory, that is compared in the&nbsp;neighborhood of global equilibrium state to the&nbsp;models based on extended thermodynamics. We&nbsp;analyze diffusion asymptotics of the Boltzmann&nbsp;<br />equations for mixtures of monatomic gases.</p> / <p>Конструишу се две независне хијерархије<br />једначина момената и примењује се принцип<br />максимума ентропије за вишеатомске гасове.<br />Формира се вишебрзински и вишетемпературни<br />модел Ојлерових вишеатомских гасова полазећи<br />од кинетичке теорије и добијени модел се<br />пореди у околини стања глобалне равнотеже са<br />моделом проширене термодинамике. Анализира<br />се дифузиона асимптотика Болцманових<br />једначина за мешавине једноатомских гасова.</p> / <p>Konstruišu se dve nezavisne hijerarhije<br />jednačina momenata i primenjuje se princip<br />maksimuma entropije za višeatomske gasove.<br />Formira se višebrzinski i višetemperaturni<br />model Ojlerovih višeatomskih gasova polazeći<br />od kinetičke teorije i dobijeni model se<br />poredi u okolini stanja globalne ravnoteže sa<br />modelom proširene termodinamike. Analizira<br />se difuziona asimptotika Bolcmanovih<br />jednačina za mešavine jednoatomskih gasova.</p>

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.