We prove exponential decay for a fractional power of the Green's function for some correlated Anderson models using the fractional moment method.

[pt] O processo de tratamento de tumores por embolização vascular é sensível ao conjunto de partículas poliméricas empregado, ditos agentes embólicos, cujos fatores como tamanho e morfologia influenciam no sucesso do procedimento e podem ocasionar complicações quando mal dimensionados. Partículas esféricas de poli(acetato de vinila-co-metacrilato de metila) apresentam a maioria das características desejadas após tratamento por hidrólise alcalina. Este material é relativamente novo, o que significa que há uma lacuna de conhecimento em relação ao estudo dos fenômenos que regem sua cinética. Dessa forma, o presente trabalho investigou a cinética de copolimerização responsável pela sua produção. No desenvolvimento matemático, o método dos momentos foi utilizado assumindo estado quase-estacionário para as espécies radicalares. Além disso, o modelo considera difusão das moléculas no meio para contabilização dos efeitos viscosos, comumente determinados empiricamente. Constatou-se que as características físicas dos monômeros, assim como os parâmetros cinéticos da homopolimerização, puderam ser utilizados na copolimerização. Entretanto, como relatado na literatura para outros sistemas, os efeitos viscosos se comportam de forma consideravelmente diferente na copolimerização, sendo necessário a reestimação de alguns parâmetros relativos aos mesmos. Assim, foi possível reproduzir de forma adequada perfis de conversão, massas molares médias e composição do copolímero. Concluiu-se que o modelo proposto é capaz de representar a cinética da copolimerização em suspensão do poli(acetato de vinila-co-metacrilato de metila), possibilitando um melhor controle das características do copolímero aplicado ao procedimento de embolização vascular. Até onde se tem conhecimento, este é o primeiro trabalho que investiga e implementa com sucesso a modelagem cinética desse sistema. / [en] The treatment of vascularized tumors through vascular embolization is sensible to the polymeric particles used during procedure. These embolic agents have attributes, like size and morphology, which play a significant role on the success of this technique and can promote complications when not well dimensioned. Among the many options available, spherical particles of poly(vinyl acetate-co-methyl methacrylate) present most desired characteristics after alkalyne hydrolysis treatment. Being relatively new, the literature lack studies related to the kinetics of production of this material. Therefore, this research investigated the copolymerization kinetics of poly(vinyl acetate-co-methyl methacrylate) production. In the mathematical development, the method of moments was used assuming quasi-steady state for the free radical species. Additionally, the model includes the viscous effects through the diffusion of the involved molecules, which is usually accounted empirically. It was possible to use the physical properties of the monomers as well as the homopolymerization kinetic parameters in the copolymerization. However, as reported in the literature, some parameters are sensible to the system and some viscous effects affect the copolymerization differently. Therefore, some parameters were reestimated. It was possible to predict the conversion, average molecular weights and composition. Consequently, the model was capable of representing the kinetics of the suspension copolymerization of poly(vinyl acetate-co-methyl methacrylate), meaning it could be used to improve the production of this polymer as an embolic agent for vascular embolization procedure. As far as known by the author, this is the first study to successfully perform the kinetic modeling of this specific system.

Liquid film flow is an important flow type in many applications of process engineering. For supporting experiments, theoretical and numerical investigations are required. The present state of the art is to model the liquid film flow with Navier--Stokes-based methods, whereas the lattice Boltzmann method is employed here. The final model has been developed within this treatise by means of a two-phase flow and a heat transfer model, and boundary and initial conditions. All these sub-models have been applied to simple test cases. It could be found that the two-phase model is capable of solving flow phenomena with a large density ratio which has been shown impressively in conjunction with wall boundary conditions. The heat transfer model was tested against spectral method results with a transient non-uniform flow field. It was possible to find optimal parameters for computation. The final model has been applied to steady-state film flow, and showed very good agreement to OpenFOAM simulations. Tests with transient film flow demonstrated that the model is also able to predict these flow phenomena. / Flüssigkeitsfilmströmungen kommen in vielen verfahrenstechnischen Prozessen zum Einsatz. Zur Unterstützung von Experimenten sind theoretische und numerische Untersuchungen nötig. Stand der Technik ist es, Navier--Stokes-basierte Modelle zu verwenden, wohingegen hier die Lattice-Boltzmann-Methode verwendet wird. Das finale Modell wurde unter Verwendung eines Zweiphasen- und eines Wärmeübertragungsmodell entwickelt und geeignete Rand- und Anfangsbedingungen formuliert. Alle Untermodelle wurden anhand einfacher Testfälle überprüft. Es konnte herausgefunden werden, dass das Zweiphasenmodell Strömungen großer Dichteunterschiede rechnen kann, was eindrucksvoll im Zusammenhang mit Wandrandbedingungen gezeigt wurde. Das Wärmeübertragungsmodell wurde gegen eine Spektrallösung anhand eines transienten und nichtuniformen Strömungsproblemes getestet. Stationäre Filmströmungen zeigten sehr gute Übereinstimmungen mit OpenFOAM-Lösungen und instationäre Berechungen bewiesen, dass das Model auch solche Strömungen abbilden kann.

Im Fokus dieser Diplomarbeit steht ein korreliertes Anderson Modell. Unser Modell beschreibt kurzreichweitige Einzelplatzpotentiale, wobei negative Korrelationen zugelassen werden. Für dieses korrelierte Modell wird mittels der fraktionalen Momentenmethode im Falle genügend großer Unordnung exponentieller Abfall der Greenschen Funktion bewiesen. Anschließend wird daraus für den nicht korrelierten Spezialfall Anderson Lokalisierung bewiesen. / This thesis (diploma) is devoted to a correlated Anderson model. Our model describes short range single site potentials, whereby negative correlations become certified. For this correlated model exponential decay of the Greens' function is proven in the case sufficient large disorder according to the fractional moment method. Subsequently, we prove Anderson localization for the not correlated special case.

Liquid film flow is an important flow type in many applications of process engineering. For supporting experiments, theoretical and numerical investigations are required. The present state of the art is to model the liquid film flow with Navier--Stokes-based methods, whereas the lattice Boltzmann method is employed here. The final model has been developed within this treatise by means of a two-phase flow and a heat transfer model, and boundary and initial conditions. All these sub-models have been applied to simple test cases. It could be found that the two-phase model is capable of solving flow phenomena with a large density ratio which has been shown impressively in conjunction with wall boundary conditions. The heat transfer model was tested against spectral method results with a transient non-uniform flow field. It was possible to find optimal parameters for computation. The final model has been applied to steady-state film flow, and showed very good agreement to OpenFOAM simulations. Tests with transient film flow demonstrated that the model is also able to predict these flow phenomena. / Flüssigkeitsfilmströmungen kommen in vielen verfahrenstechnischen Prozessen zum Einsatz. Zur Unterstützung von Experimenten sind theoretische und numerische Untersuchungen nötig. Stand der Technik ist es, Navier--Stokes-basierte Modelle zu verwenden, wohingegen hier die Lattice-Boltzmann-Methode verwendet wird. Das finale Modell wurde unter Verwendung eines Zweiphasen- und eines Wärmeübertragungsmodell entwickelt und geeignete Rand- und Anfangsbedingungen formuliert. Alle Untermodelle wurden anhand einfacher Testfälle überprüft. Es konnte herausgefunden werden, dass das Zweiphasenmodell Strömungen großer Dichteunterschiede rechnen kann, was eindrucksvoll im Zusammenhang mit Wandrandbedingungen gezeigt wurde. Das Wärmeübertragungsmodell wurde gegen eine Spektrallösung anhand eines transienten und nichtuniformen Strömungsproblemes getestet. Stationäre Filmströmungen zeigten sehr gute Übereinstimmungen mit OpenFOAM-Lösungen und instationäre Berechungen bewiesen, dass das Model auch solche Strömungen abbilden kann.

The fast-paced growth in microelectromechanical systems (MEMS), microfluidic fabrication, porous media applications, biomedical assemblies, space propulsion, and vacuum technology demands accurate and practical transport equations for rarefied gas flows. It is well-known that in rarefied situations, due to strong deviations from the continuum regime, traditional fluid models such as Navier-Stokes-Fourier (NSF) fail. The shortcoming of continuum models is rooted in nonequilibrium behavior of gas particles in miniaturized and/or low-pressure devices, where the Knudsen number (Kn) is sufficiently large. Since kinetic solutions are computationally very expensive, there has been a great desire to develop macroscopic transport equations for dilute gas flows, and as a result, several sets of extended equations are proposed for gas flow in nonequilibrium states. However, applications of many of these extended equations are limited due to their instabilities and/or the absence of suitable boundary conditions. In this work, we concentrate on regularized 13-moment (R13) equations, which are a set of macroscopic transport equations for flows in the transition regime, i.e., Kn≤1. The R13 system provides a stable set of equations in Super-Burnett order, with a great potential to be a powerful CFD tool for rarefied flow simulations at moderate Knudsen numbers. The goal of this research is to implement the R13 equations for problems of practical interest in arbitrary geometries. This is done by transformation of the R13 equations and boundary conditions into general curvilinear coordinate systems. Next steps include adaptation of the transformed equations in order to solve some of the popular test cases, i.e., shear-driven, force-driven, and temperature-driven flows in both planar and curved flow passages. It is shown that inexpensive analytical solutions of the R13 equations for the considered problems are comparable to expensive numerical solutions of the Boltzmann equation. The new results present a wide range of linear and nonlinear rarefaction effects which alter the classical flow patterns both in the bulk and near boundary regions. Among these, multiple Knudsen boundary layers (mechanocaloric heat flows) and their influence on mass and energy transfer must be highlighted. Furthermore, the phenomenon of temperature dip and Knudsen paradox in Poiseuille flow; Onsager's reciprocity relation, two-way flow pattern, and thermomolecular pressure difference in simultaneous Poiseuille and transpiration flows are described theoretically. Through comparisons it is shown that for Knudsen numbers up to 0.5 the compact R13 solutions exhibit a good agreement with expensive solutions of the Boltzmann equation.