Orientador: Lucas Catão de Freitas Ferreira / Dissertação (mestrado) - Universidade Estadual de Campinas, Instituto de Matemática, Estatística e Computação Científica / Made available in DSpace on 2018-08-19T19:21:09Z (GMT). No. of bitstreams: 1 ValenciaGuevara_JulioCesar_M.pdf: 664823 bytes, checksum: 0c43bba776e592ed44fad0d1bc2f6998 (MD5) Previous issue date: 2012 / Resumo: Estudamos existência, unicidade, dependência contínua nos dados e comportamento assint ótico de soluções globais das equações de Navier-Stokes (com n >= 3), sob condições de pequenez no dado inicial e na força externa, em um espaço de distribuições (PMa) cuja construção é baseada na transformada de Fourier. Este espaço contém funções fortemente singulares e, em particular, funções homogêneas de um certo grau cuja correspondente solução (com tais dados) é auto-similar. Além disso, mostramos a existência de uma classe de soluções que são assintoticamente auto-similar. Estudamos também a existência de soluções estacionárias pequenas e analisamos a estabilidade assintótica delas. Finalmente, são dadas condições sob as quais a solução é uma função regular para t > 0 (mesmo com dado inicial singular) e satisfaz as equações de Navier-Stokes no sentido clássico para t > 0. Esta dissertação é baseada no artigo de M. Cannone and G. Karch, Journal of Diff. Equations 197 (2) (2004) / Abstract: We study existence, uniqueness, continuous dependence upon the data and asymptotic behavior of solutions for the Navier-Stokes equations (with n _ 3), under smallness conditions on the initial data and external force, in a space of distributions (PMa), whose construction is based on Fourier transform. This space contains strongly singular functions and, in particular, homogeneous functions with a certain degree whose corresponding solution (with such data) is self-similar. Moreover, the existence of a class of asymptotically self-similar solutions is proved. We also study the existence of small stationary solutions and their asymptotic stability. Finally, conditions are given for the obtained solution to be regular for t > 0 (even with singular initial data) and to satisfy the Navier-Stokes equations in the classical sense for t > 0. This master dissertation is based on the paper by M. Cannone and G. Karch, Journal of Diff. Equations 197 (2) (2004) / Mestrado / Matematica / Mestre em Matemática

The study of finite-sized, deformable droplets adverted by turbulent flows is an active area of research. It spans many streams of sciences and engineering, which include chemical engineering, fluid mechanics, statistical physics, nonlinear dynamics, and also biology. Advances in experimental techniques and high-performance computing have made it possible to investigate the properties of turbulent fluids laden with droplets. The main focus of this thesis is to study the statistical properties of the dynamics of such finite-size droplets in turbulent flows by using direct numerical simulations (DNSs). The most important feature of the model we use is that the droplets have a back-reaction on the advecting fluid: the turbulent fluid affects the droplets and they, in turn, affect the turbulence of the fluid. Our study uncovers (a) statistical properties that characterize the spatiotemporal evolution of droplets in turbulent flows, which are statistically homogeneous and isotropic, and (b) the modification of the statistical properties of this turbulence by the droplets. This thesis is divided into seven Chapters. Chapter 1 contains an introduction to the background material that is required for this thesis, especially the details about the equations we use; it also contains an outline of the problems we study in subsequent Chapters. Chapter 2 contains our study of “Droplets in Statistically Homogeneous Turbulence: From Many Droplets to a few Droplets”. Chapter 3 is devoted to our study of “Coalescence of Two Droplets”. Chapter 4 deals with “Binary-Fluid Turbulence: Signatures of Multifractal Droplet Dynamics and Dissipation Reduction”. Chapter 5 deals with “A BKM-type theorem and associated computations of solutions of the three-dimensional Cahn-Hilliard-Navier-Stokes equations”. Chapter 6 is devoted to our study of “Turbulence-induced Suppression of Phase Separation in Binary-Fluid Mixtures”. Chapter 7 is devoted to our study of “Antibubbles: Insights from the Cahn-Hilliard-Navier-Stokes Equations”.

Le schéma Marker-And-Cell (MAC) est un schéma de discrétisation des équations aux dérivées partielles sur maillages cartésiens, très connu en mécanique des fluides. Nous nous intéressons ici à son analyse mathématique dans le cadre des écoulements incompressibles sur des maillages cartésiens non-uniformes en dimension 2 ou 3. Dans un premier temps nous discrétisons les équations de Navier-Stokes pour un écoulement incompressible stationnaire; nous établissons des estimations a priori sur les suites de vitesses et pressions approchées qui permettent d’une part d'établir l’existence d’une solution au schéma, et d’obtenir la compacité de ces suites lorsque le pas d’espace tend vers 0. Nous montrons alors la convergence de ces suites (à une sous-suite près) vers une solution faible du problème continu, ce qui nécessite une analyse fine du terme de convection non linéaire. Nous nous intéressons ensuite aux équations de Navier-Stokes en régime instationnaire avec une discrétisation en temps implicite. Nous démontrons que le schéma préserve les propriétés de stabilité du problème continu et obtenons ainsi l’existence d’une solution au schéma. Puis, grâce à des techniques de compacité et en passant à la limite dans le schéma, nous démontrons qu’une suite de vitesses approchées converge. Si l’on se restreint au problème de Stokes, et en supposant de plus que la condition initiale de la vitesse est dans H 1 , nous obtenons une estimation sur la pression qui permet de montrer la convergence forte des pressions approchées. Enfin nous étendons l’analyse aux écoulements incompressibles à masse volumique variable. On montre la convergence du schéma. / The Marker-And-Cell (MAC) scheme is a discretization scheme for partial derivative equations on Cartesian meshes, which is very well known in fluid mechanics. Here we are concerned with its mathematical analysis in the case of incompressible flows on two or three dimensional non-uniform Cartesian grids. We first discretize the steady-state incompressible Navier-Stokes equations. We show somea priori estimates that allow to show the existence of a solution to the scheme and some compactness and consistency results. By a passage to the limit on the scheme, we show that the approximate solutions obtained with the MAC scheme converge (up to a subsequence) to a weak solution of the Navier-Stokes equations, thanks to a careful analysis of the nonlinear convection term. Then, we analyze the convergence of the unsteady-case Navier-Stokes equations. The algorithm is implicit in time. We first show that the scheme preserves the stability properties of the continuous problem, which yields, the existence of a solution. Then, invoking compactness arguments and passing to the limit in the scheme, we prove that any sequence of solutions (obtained with a sequence of discretizations the space and time step of which tend to zero) converges up to the extraction of a subsequence to a weak solution of the continuous problem. If we restrict ourselves to the Stokes equations and assume that the initial velocity belongs to H 1, then we obtain estimates on the pressure and prove the convergence of the sequences of approximate pressures. Finally, we extend the analysis of the scheme to incompressible variable density flows. we show the convergence of the scheme.

La présente thèse traite de deux questions de stabilité en mécanique des fluides. Les deux premiers résultats de la thèse sont consacrés au problème de la limite non-visqueuse pour les équations de Navier-Stokes. Il s'agit de déterminer si une famille de solutions de Navier-Stokes dans un demi-espace avec une condition de Navier au bord converge vers une solution du modèle non visqueux, l'équation d'Euler, lorsque les paramètres de viscosité tendent vers zéro. Dans un premier temps, on considère le modèle incompressible 2D. Nous obtenons la convergence dans L2 des solutions faibles de Navier-Stokes vers une solution forte d'Euler, et une instabilité dans L∞ en temps très court pour certaines données initiales qui sont des solutions stationnaires de l'équation d'Euler. Ces résultats ne sont pas contradictoires, et on construit un exemple de donnée initiale permettant de voir se réaliser les deux phénomènes simultanément dans le cadre périodique. Dans un second temps, on s'intéresse au modèle compressible isentropique (température constante) en 3D. On démontre l'existence de solutions dans des espaces de Sobolev conormaux sur un temps qui ne dépend pas de la viscosité lorsque celle-ci devient très petite, et on obtient la convergence forte de ces solutions vers une solution de l'équation d'Euler sur ce temps uniforme par des arguments de compacité. Le troisième résultat de cette thèse traite d'un problème de stabilité d'ondes solitaires. Précisément, on considère un fluide isentropique et non visqueux avec capillarité interne, régi par le modèle d'Euler-Korteweg, et on montre l'instabilité transverse non-linéaire de solitons, c'est-à-dire que des perturbations 2D initialement petites d'une solution sous forme d'onde progressive 1D peuvent s'éloigner de manière importante de celle-ci. / This thesis deals with a couple of stability problems in fluid mechanics. In the first two parts, we work on the inviscid limit problem for Navier-Stokes equations. We look to show whether or not a sequence of solutions to Navier-Stokes in a half-space with a Navier slip condition on the boundary converges towards a solution of the inviscid model, the Euler equation, when the viscosity parameters vanish. First, we consider the 2D incompressible model. We obtain convergence in L2 of weak solutions of Navier-Stokes towards a strong solution of Euler, as well as the instability in L∞ in a very short time of some initial data chosen as stationary solutions to the Euler equation. These results are not contradictory, and we construct initial data that allows both phenomena to occur simultaneously in the periodic setting. Second, we look at the 3D isentropic (constant temperature) compressible equations. We show that solutions exist in conormal Sobolev spaces for a time that does not depend on the viscosity when this is small, and we get strong convergence towards a solution of the Euler equation on this uniform time of existence by compactness arguments. In the third part of the thesis, we work on a solitary wave stability problem. To be precise, we consider an isentropic, compressible, inviscid fluid with internal capillarity, governed by the Euler-Korteweg equations, and we show the transverse nonlinear instability of solitons, that is that initially small 2D perturbations of a 1D travelling wave solution can end up far from it.

Orientador: João Frederico da Costa Azevedo Meyer / Tese (doutorado) - Universidade Estadual de Campinas, Instituto de Matemática, Estatística e Computação Científica / Made available in DSpace on 2018-08-18T17:19:29Z (GMT). No. of bitstreams: 1 Krindges_Andre_D.pdf: 12331441 bytes, checksum: ca3fa7d1c704c02f04ba59043413e0f7 (MD5) Previous issue date: 2011 / Resumo: Um dos problemas enfrentados pelo grupo de Ecologia Matemática do IMECC da UNICAMP é o de trabalhar com difusão de uma pluma poluente com 3 variáveis espaciais, além da temporal. Esta tese não só aborda esta questão, propondo, inclusive um algoritmo computacional para esta situação, mas fá-lo resolvendo aproximadamente a Equação de Navier-Stokes num domínio irregular. A primeira parte consiste na formulação do modelo matemático para o estudo de um sistema que inclui o campo de velocidades e o comportamento evolutivo de um material poluente. Na segunda parte, é feita a formulação variacional, são constituídas aproximações via o método de Galerkin para Elementos Finitos no espaço e Crank-Nicolson no tempo para a equação de difusão-advecção, e o método da projeção para a equação de Navier-Stokes. Em seguida, faz-se a descrição do algoritmo, indicando dificuldades operacionais do ponto de vista de computação científica e apontando soluções. O domínio utilizado para o estudo de caso é o da represa do rio Manso, que, discretizada em três dimensões, foi tratado com o software livre GMSH. Finalmente, um código numérico em plataforma MATLAB foi executado e resultados são apresentados no texto. O programa e diversas considerações técnicas essenciais fazem parte dos anexos / Abstract: One of the challenges faced by the Mathematical Ecology group at the Mathematics Institute at Campinas State University is that of working with the diffusion of a pollutant plume in three spatial variables, besides time. This work not only addresses this issue by proposing an approximation strategy as well as a computer algorithm for this situation, but it also includes a three-dimensional numerical approximation for the Navier-Stokes equation in an irregular domain. The first part consists in formulating the mathematical model for the study of a system that includes the velocity field and the evolutionary behavior of a polluting material. The second part begins with the variational formulation of the Navier-Stiokes system, and approximations are undertaken via the Galerkin method for finite elements in space and Crank-Nicolson in time for both the advection-diffusion equation and the method of projection for the Navier-Stokes equations. The algorithm is described, indicating operational difficulties in terms of scientific computing as well as the way in which these aforementioned difficulties are solved. The domain used for the case study is the Manso River reservoir, which, discretized in three dimensions, was treated with the free software GMSH. Finally, a numeric code in MATLAB environment was completed and results are presented in the text. The program and various essential technical considerations are part of the annexes / Doutorado / Matematica Aplicada / Doutor em Matemática Aplicada

Orientador: Donato Giorgio Torrieri / Dissertação (mestrado) - Universidade Estadual de Campinas, Instituto de Física Gleb Wataghin / Made available in DSpace on 2018-08-31T06:57:06Z (GMT). No. of bitstreams: 1 Coelho_DavidMontenegro_M.pdf: 2238497 bytes, checksum: 78f745f9d544a31d97a2d7a80b7dc505 (MD5) Previous issue date: 2016 / Resumo: O interesse científico cresceu após confirmado por testes experimentais o comportamento do Plasma de Quark-Glúon como um fluido quase perfeito no LHC e RHIC. O objetivo desse trabalho é fornecer as bases teóricas da Effective Field Theory (EFT) na abordagem da Hidrodinâmica, pois vários recursos não-triviais na dinâmica relativística dos fluidos são claramente explicados por esse formalismo. Problemas teóricos na EFT sugerem a inclusão de uma nova formulação do Princípio de Hamilton compatível com o princípio da causalidade, através do Closed-Time-Path. Após resolvido esse problema, alcançamos o requisito necessário para derivar a hidrodinâmica dissipativa em altas ordens por meio da ação. Assim, conseguimos caracterizar a Lagrangeana de Navier-Stokes ao introduzir a quebra de simetria na preservação do difeomorfismo pelo volume por meio do termo $B^{-1}_{IJ}$. No entanto, uma análise pelo método de Ostrogradski levou à supressão dessa equação, através da inclusão da Lagrangeana de Israel-Stewart na expansão que é justificada por meios de argumentos de estabilidade e causalidade. Por fim, propomos uma variável $X_{IJ}$ na Lagrangeana de Israel-Stewart, simétrica, anisotrópica e dependente das condições iniciais que juntamente com os já estabelecidos graus de liberdade de campo, formam a base para a derivação bottom-up em altas ordens da EFT e propicia medidas para estudar turbulência e instabilidade no vácuo e outras situações que chegam da relação entre graus de liberdade macroscópico e microscópico / Abstract: Scientific interest grew after the behavior of the quark-gluon Plasma as a nearly perfect fluid in the LHC and RHIC. The objective of this dissertation is offer support to use the Effective Field Theory (EFT) approach to study hydrodynamics because many non-trivial features in relativistic fluid dynamics are clearly explained by this Lagrangian formalism. Theoretical problems in EFT considering by including a new formulation of the Hamiltonian principle that is compatible with the principle of causality for non-conservative field through the Closed-Time-Path formalism. After solving this problem, we reached requirement to derive the dissipative hydrodynamics in higher orders of action. We were able to characterize Navier-Stokes' Lagrangian by introducing the symmetry breaking of preserving diffeomorphism through the volume with the term $B^{-1}_{IJ} $ to the Lagrangian of Navier-Stokes. An analyse of Ostrogradski's method led to the removal of equation by including the Israel-Stewart term in the Lagrangian expansion that provides an extra justification by means of symmetry and causality arguments. Finally, we propose a variable $ X_ {IJ} $, Israel-Stewart's Lagrangian, symmetric, anisotropic and dependent on initial conditions together with an established degree of freedom of the field, which form the basis for the derivation of higher orders of the bottom up and promote steps to the study of turbulence by instability in the vacuum, and other situations arising from the relationship between macroscopic and microscopic degrees of freedom / Mestrado / Física / Mestre em Física / 147435/2014-5 / CNPQ

Cílem této diplomové práce je vytvořit 3D model horních dýchacích cest podle originálního modelu segmentovaného z CT dat, aplikovat různé podmínky na průtok vzduchu v modelu, a poté hodnotit změnu charakteristik rychlosti a tlaku. Model horních dýchacích cest byl vytvořen v prostředí softwaru ANSYS, který využívá výpočetní dynamiku tekutin, a byly použity Navier-Stokesovy rovnice pro modelování průtoku vzduchu v modelu. Nejprve byl vytvořen jednoduchý 2D model za účelem seznámení se s prostředím ANSYS. Dále byl zkonstruován 3D model horních dýchacích cest a byly modelovány charakteristiky rychlosti a tlaku za různých podmínek. Tyto podmínky zahrnují různé umístění a množství míst pro odběr vzorků v modelu a výběr různých kombinací vstupů. Nakonec byly prezentovány a hodnoceny výsledky spolu s ilustracemi modelů modelovaných za různých podmínek. 3D model lze považovat ze kompromis mezi výpočetní náročností a složitostí modelu a lze jej použít jako základ pro další výzkum.

This thesis focuses on fluid simulation, particularly on coupling between particle based simulation and grid based simulation and thus modeling evaporation. Mentioned coupling is based on the article Evaporation and Condensation of SPH-based Fluids of authors Hendrik Hochstetter a Andreas Kolb. The goal of this thesis is not purely implementing ideas of the mentioned article, but also study of different methods used for fluid simulation.

Im Rahmen des Modellierungsseminars wurde die Kopplung einer Strömung mit der Ausbreitung einer reaktiven Substanz im Strömungsgebiet untersucht. Die Strömung wurde dabei durch die inkompressiblen Navier-Stokes-Gleichungen beschrieben. Zusätzlich wurde ein mathematisches Modell für die Ausbreitung der Substanz durch eine Diffusions-Konvektions-Gleichung bestimmt. Beide wurden durch die FEM- Sofware NAVIER berechnet und simuliert.:0 Notation 3 1 Aufgabenstellung 4 1.1 Projekt 4 1.2 Navier-Stokes-Strömung gekoppelt mit (passivem) Transport einiger (reaktiver) Substanzen 4 2 Einleitung 6 2.1 Strömungslehre 6 2.2 Regelungstheorie 7 3 Mathematische Modellierung 8 3.1 Strömungsmodellierung 8 3.2 Modell zur Ausbreitung des gelösten Stoffes 8 3.3 Gekoppeltes Modell und Randbedingungen 12 3.4 Grenzschichten 13 3.5 Feedback-Steuerung 15 4 Numerische Simulation 19 4.1 Umsetzung des Modells in NAVIER 19 4.1.1 Mathematische Grundlagen für NAVIER 20 4.1.2 Problemimplementation 21 4.2 Beispielsimulation 24 4.2.1 Gebietstriangulation 24 4.2.2 Konfigurationsdaten sowie Anfangs- und Randbedingungen 25 4.2.3 Ergebnisauswertung 28 4.3 Zusatzfunktionen für die Regelung 34 4.3.1 outflow-Funktion 34 4.3.2 Systemmatrizen 34 4.3.3 Feedback-Funktion 34 5 Zusammenfassung/Ausblick 35 5.1 Zusammenfassung 35 5.2 Ausblick 36 6 Schlusswort 37 7 Quellenverzeichnis 38 8 Verzeichnisse 39

The aim of this work is to provide new results of global existence for dif- ferent evolution equations of fluid mechanics. We are in general interested in finding weak solutions without restrictions on the size of initial data. The proofs of existence are based on several different approaches including en- ergy methods, convergence analysis of finite numerical methods and convex integration. All these techniques significantly exploit results of mathematical analysis and other branches of mathematics. 1