Cette thèse porte sur le traitement mathématique du problème de Hele-Shaw dans l’approximation de Stokes-Leibenson. À l’aide d’une transformation de type Helmholtz- Kirchhoff, nous explicitons une équation d’évolution du contour fluide valable pour tout type d’écoulement plan. Cette équation généralise des résultats précédents et permet alors d’établir un schéma numérique performant dit du quasi-contour, qui se réduit à un problème de Cauchy. Nous considérons ensuite l’étude du problème par transformations conformes menant à l’équation de Polubarinova-Galin. Dans le cas simple d’un contour représenté par un trinôme à coefficients réels, nous réussissons à expliciter la solution exacte du problème. Notons que les trajectoires des solutions exactes permettent de préciser la position de la frontière des domaines d’univalence décrits par les trinômes. Enfin, nous introduisons des paramètres de contrôle sous forme de coefficients d’un multipôle superposé à la source. Des conditions suffisantes de contrôlabilité sont établies et des résultats de contrôle optimal sont explicités pour les solutions binomiales et trinomiales. L’introduction de paramètres de contrôle permet de comprendre le lien qui relie les moments de Richardson à l’équation de Polubarinova-Galin. / This PhD thesis deals with the mathematical treatment of the Hele–Shaw problem in the Stokes–Leibenson approximation. By an Helmholtz–Kirchhoff transformation, we exhibited an evolutive equation of the fluid contour applicable to all type of planar fows. This equation generalizes previous results and also allows to state an efficient numerical scheme called quasi-contour’s, which is a simple Cauchy problem. We then consider the study of this problem using conformal transformations leading to the Polubarinova-Galin equation. In the simple case of a contour representing by a trinomial with real coefficients, we succeeded in exhibiting the exact solution of the problem. Notice that the trajectories of the exact solutions enable to precise the position of frontiers of univalent domains described by trinomials. Finally, we introduce control parameters under the form of coefficients of a multipole superposed to the source. Sufficient conditions of controllability are stated and results on optimal control established for the binomial and trinomial cases. Introduction of control parameters allows us to understand the link, which bound Richardson’s moments to the Polubarinova-Galin equation.

Submitted by Fabio Sobreira Campos da Costa (fabio.sobreira@ufpe.br) on 2016-06-29T13:52:59Z No. of bitstreams: 2 license_rdf: 1232 bytes, checksum: 66e71c371cc565284e70f40736c94386 (MD5) tese_final.pdf: 5635071 bytes, checksum: b300efb627e9ece412ad5936ab67e8e2 (MD5) / Made available in DSpace on 2016-06-29T13:52:59Z (GMT). No. of bitstreams: 2 license_rdf: 1232 bytes, checksum: 66e71c371cc565284e70f40736c94386 (MD5) tese_final.pdf: 5635071 bytes, checksum: b300efb627e9ece412ad5936ab67e8e2 (MD5) Previous issue date: 2013-08-26 / CNPq / No presente trabalho duas classes de problemas são abordadas. Primeiramente, são apresentados estudos computacionais sobre o modelo O(n) de spins na rede quadrada, e em seguida apresentamos novas soluções exatas para a dinâmica de bolhas na célula de Hele-Shaw. O estudo do modelo O(n) é feito utilizando sua representação em laços (cadeias fechadas), a qual é obtida a partir de uma expansão para altas temperaturas. Nesse representação, a função de partição do modelo possui uma expansão diagramática em que cada termo depende do número e comprimento total de laços e do número de (auto)interseções entre esses laços. Propriedades críticas do modelo de laços O(n) são obtidas através de conceitos oriundos da teoria de percolação. Para executar as simulações Monte Carlo, usamos o eficiente algoritmo WORM, o qual realiza atualizações locais através do movimento da extremidade de uma cadeia aberta denominada de verme e não sofre com o problema de "critical slowing down". Para implementar esse algoritmo de forma eficiente para o modelo O(n) na rede quadrada, fazemos uso de um nova estrutura de dados conhecida como listas satélites. Apresentamos estimativas para o ponto crítico do modelo para vários valores de n no intervalo de 0 < n ≤ 2. Usamos as estatísticas de laços e vermes para extrair, respectivamente, os expoentes críticos térmicos e magnéticos do modelo. No estudo de dinâmica de interfaces, apresentamos uma solução exata bastante geral para um arranjo periódico de bolhas movendo-se com velocidade constante ao longo de uma célula de Hele-Shaw. Usando a periodicidade da solução, o domínio relevante do problema pode ser reduzido a uma célula unitária que contém uma única bolha. Nenhuma imposição de simetria sobre forma da bolha é feita, de modo que a solução é capaz de produzir bolhas completamente assimétricas. Nossa solução é obtida por métodos de transformações conformes entre domínios duplamente conexos, onde utilizamos a transformação de Schwarz-Christoffel generalizada para essa classe de domínios. / In this thesis two classes of problems are discussed. First, we present computational studies of the O(n) spin model on the square lattice and determine its critical properties, whereas in the second part of the thesis we present new exact solutions for bubble dynamics in a Hele-Shaw cell. The O(n) model is investigated by using its loop representation which is obtained from a high-temperature expansion of the original model. In this representation, the partition function admits an diagrammatic expansion in which each term depends on the number and total length of loops (closed graphs) as well as on the number of intersections between these loops. Critical properties of the O(n) model are obtained by employing concepts from percolation theory. To perform Monte Carlo simulations of the model, we use the WORM algorithm, which is an efficient algorithm that performs local updates through the motion of one of the ends (called head) of an open chain (called worm) and hence does not suffer from “critical slowing down”. To implement this algorithm efficiently for the O(n) model on the square lattice, we make use of a new data structure known as a satellite list. We present estimates for the critical point of the model for various values of n in the range 0 < n ≤ 2. We use the statistics about the loops and the worm to extract the thermal and magnetic critical exponents of the model, respectively. In our study about interface dynamics, we present a rather general exact solution for a periodic array of bubbles moving with constant velocity in a Hele-Shaw cell. Using the periodicity of the solution, the relevant domain of the problem can be reduced to a unit cell containing a single bubble. No symmetry requirement is imposed on the bubble shape, so that the solution is capable of generating completely asymmetrical bubbles. Our solution is obtained by using conformal mappings between doubly-connected domains and employing the generalized Schwarz-Christoffel formula for this class of domains.

La dispersion de particules par une onde de souffle ou de choc induit la formation de structures régulières qui prennent la forme de jets de particules. Jusqu'à présent, les expériences n'ont été réalisées qu'en trois dimensions rendant difficile l'exploitation des données. Dans cette étude, une onde de souffle, générée à l'extrémité d'un tube à choc, débouche au centre d'un anneau de particules solides initialement confiné dans une cellule de Hele-Shaw. Pour la première fois, à partir d'une expérience de laboratoire, la formation de jets de particules est observée dans une configuration quasi bi-dimensionnelle et pour de faibles niveaux de pression. Grâce à un système de visualisation ultra-rapide, il a été mis en évidence que la sélection du nombre de jets de particules est un processus instationnaire. Nous avons observé que les jets de particules sont initialement formés à l'intérieur de l'anneau et sont ensuite expulsés à l'extérieur du front de particules en expansion. L'influence de nombreux paramètres, tels que la densité et le diamètre des particules, la surpression générée et la géométrie de l'anneau, ont été étudiées. La synthèse des résultats expérimentaux obtenus a permis d'établir certaines relations empiriques reliant le nombre de jets aux propriétés initiales. De plus, la formation de fines perturbations sur le front externe de la couche de particules a été observée. Ce phénomène est quant à lui indépendant des jets principaux et dépend seulement de la nature des particules. / The dispersion of particles by a blast or a shock wave induces the formation of coherent structures which take the form of particle jets. All the experiments conducted so far have been performed in three-dimensional geometry. In the present study, a blast wave, issuing from the discharge of a planar shock wave at the exit of a conventional shock tube, is generated in the center of a granular medium ring initially confined inside a Hele-Shaw cell. With the present experimental set-up, under impulsive acceleration, a solid particle jet formation is clearly obtained and observed in a quasi-two-dimensional configuration, for the first time. From fast flow visualizations, we highlighted that the selection of the number of jets is unsteady. We noticed, in all instances, that the jets are initially generated inside the particle ring and thereafter expelled outward. This point has not been observed in three-dimensional experiments. The influence of many parameters such as density and diameter of particles, the generated pressure and the geometry of the ring, has been studied. Empirical relationships were deduced from the experimental curves. Moreover, we observed in detail the formation of very thin perturbations created around the external surface of the dispersed particle layer. This phenomenon is independent of the main jet formation and solely depends on the nature of particles.

The process of fluid injection into granular media is relevant to a wide number of applications such as enhanced oil recovery, grouting, and the construction of permeable reactive barriers. The response of the subsurface is dependent on multiple factors such as in-situ stresses, fluid properties, flow rate, and formation type. Based on these conditions a variety of response mechanisms can be initiated ranging from simple porous infiltration to hydraulic fracturing. Currently, the mechanics of fluid injection into competent rock are well understood and can be sufficiently modeled using linear elastic fracture mechanics. Because the grains in rock formations are individually cemented together, they exhibit cohesion and are able to support tensile stresses. The linear elastic method assumes tensile failure due to stress concentrations at the fracture tip. A fracture propagates when the stress intensity factor exceeds the material toughness (Detournay, 1988) However, understanding fluid injection in cohesionless granular media presents a much larger obstacle. Currently, no theoretical models have been developed to deal with granular media displacements due to fluid injection. Difficulty arises from the complexity of fluid rheology and composition used in engineering processes, the strong coupling between fluid flow and mechanical deformation, the non-linear response of subsurface media, and the multi-scale nature of the problem. The structure of this thesis is intended to first give the reader a basic background of some of the fundamental concepts for non-Newtonian fluid flow in granular media. Fluid properties as well as some interaction mechanisms are described in relation to the injection process. Next, the results from an experimental series of injection tests are presented with a discussion of the failure/flow processes taking place. We developed a novel technique which allows us to visualize the injection process by use of a transparent Hele-Shaw cell. Specifically, we will be using polyacrylamide solutions at a variety of concentrations to study non-Newtonian effects on the response within the Hele-Shaw cell. By performing tests at a range of solution concentrations and injection rates we are to be able to identify a transition from an infiltration dominated flow regime to a fracturing dominated regime.

We study the regularity of weak solutions for the Stefan and Hele- Shaw problems with Gibbs-Thomson law under special conditions. The main result says that whenever the free boundary is Lipschitz in space and time it becomes (instantaneously) C[superscript 2,alpha] in space and its mean curvature is Hölder continuous. Additionally, a similar model related to the Signorini problem is introduced, in this case it is shown that for large times weak solutions converge to a stationary configuration. / text

The dynamics of A + B → C reaction fronts is studied under modulated gravitational acceleration by means of a combination of parabolic flight experiments and numerical simulations. During modulated gravity the front position undergoes periodic modulation with an accelerated front propagation under hyper-gravity together with a slowing down under low gravity. The underlying reason for this is an amplification and a decay, respectively, of the buoyancy-driven double vortex associated with the front propagation under standard gravitational acceleration, as explained by reaction–diffusion–convection simulations of convection around an A + B → C front. Deeper insights into the correlation between grey-value changes in the experimental shadowgraph images and characteristic changes in the concentration profiles are obtained by a numerical simulation of the imaging process. / Dieser Beitrag ist mit Zustimmung des Rechteinhabers aufgrund einer (DFG-geförderten) Allianz- bzw. Nationallizenz frei zugänglich.

Submitted by Fabio Sobreira Campos da Costa (fabio.sobreira@ufpe.br) on 2015-05-08T14:05:16Z No. of bitstreams: 2 license_rdf: 1232 bytes, checksum: 66e71c371cc565284e70f40736c94386 (MD5) 1 Dissertação - João Vitor Nogueir.pdf: 7740367 bytes, checksum: e311056ecafb8741fc3538682907dcc1 (MD5) / Made available in DSpace on 2015-05-08T14:05:16Z (GMT). No. of bitstreams: 2 license_rdf: 1232 bytes, checksum: 66e71c371cc565284e70f40736c94386 (MD5) 1 Dissertação - João Vitor Nogueir.pdf: 7740367 bytes, checksum: e311056ecafb8741fc3538682907dcc1 (MD5) Previous issue date: 2014-01-30 / A instabilidade de Saffman-Taylor se dá na interface entre dois fluidos viscosos no interior de uma célula de Hele-Shaw (CHS). A CHS é um aparato experimental que consiste em duas placas, usualmente planas e paralelas, separadas por uma distância muito pequena. Sabe-se que quando um fluido viscoso desloca outro mais viscoso, em uma CHS, a interface entre eles se torna instável e estruturas chamadas de “dedos viscosos” surgem. Em função da geometria da CHS e da força motriz do fluxo, tais dedos podem bifurcar, afinar, competir em tamanho e interferirem uns com os outros, formando as mais diversas estruturas morfológicas. Estudos com fluidos newtonianos em CHS vêm sendo feitos desde o final do século XIX. No entanto, apenas recentemente, a cerca de trinta anos, estudos com fluidos não newtonianos em CHS vêm sendo conduzidos, e em sua maioria estudos analíticos lineares ou não lineares puramente numéricos. Este trabalho tenta preencher essa lacuna com um estudo analítico não linear de fluidos não newtonianos em CHS. Estudamos a CHS de geometria radial com levantamento da placa superior e injeção como forças motrizes do fluxo. Nestes casos estudamos o fluido yield stress, um tipo especial de fluido não newtoniano que se comporta como um “semissólido”. Estudamos também o fluido power-law que introduz, na viscosidade, uma dependência do fluxo. Em cada caso fomos capazes de estabelecer conexões entre as nossas previsões teóricas e os experimentos e simulações encontrados na literatura. Tais previsões são acerca da estabilidade e morfologia da interface como: bifurcação de dedos, a estrutura conhecida como side branching e a competição entre dedos.

The dynamics of A + B → C reaction fronts is studied under modulated gravitational acceleration by means of a combination of parabolic flight experiments and numerical simulations. During modulated gravity the front position undergoes periodic modulation with an accelerated front propagation under hyper-gravity together with a slowing down under low gravity. The underlying reason for this is an amplification and a decay, respectively, of the buoyancy-driven double vortex associated with the front propagation under standard gravitational acceleration, as explained by reaction–diffusion–convection simulations of convection around an A + B → C front. Deeper insights into the correlation between grey-value changes in the experimental shadowgraph images and characteristic changes in the concentration profiles are obtained by a numerical simulation of the imaging process. / Dieser Beitrag ist mit Zustimmung des Rechteinhabers aufgrund einer (DFG-geförderten) Allianz- bzw. Nationallizenz frei zugänglich.

Integrated theoretical and experimental analysis is carried out in this work to investigate the fundamental failure mechanisms and flow patterns involved in the process of fluid injection into dense granular media. The experimental work is conducted with aqueous glycerin solutions, utilizing a novel setup based on a Hele-Shaw cell filled with dense dry sand. The two dimensional nature of the setup allows direct visualization and imaging analysis of the real-time fluid and grain kinematics. The experimental results reveal that the fluid flow patterns show a transition from simple radial flow to a ramified morphology while the granular media behaviors change from that of rigid porous media to localized failure that lead to development of fluid channels. Based on the failure/flow patterns, four distinct failure/flow regimes can be identified, namely, (i) a simple radial flow regime, (ii) an infiltration-dominated regime, (iii) a grain displacement-dominated regime, and (iv) a viscous fingering-dominated regime. These distinct failure/flow regimes emerge as a result of competition among various energy dissipation mechanisms, namely, viscous dissipation through infiltration, dissipation due to grain displacements, and viscous dissipation through flow in thin channels and can be classified based on the characteristic times associated with fluid injection, hydromechanical coupling and viscoelastoplasticity. The injection process is also analyzed numerically using the discrete element method (DEM) coupled with two fluid flow scheme, a fixed coarse grid scheme based on computational fluid dynamics (CFD) and a pore network modeling scheme. The numerical results from the two complementary methods reproduce phenomena consistent with the experimental observations and justify the concept of associating the displacement regimes with the partition among energy dissipation mechanisms. The research in this work, though fundamental in nature, will have direct impacts on many engineering problems in civil, environmental and petroleum engineering such as ground improvement, environmental remediation and reservoir stimulation.

Submitted by Daniella Sodre (daniella.sodre@ufpe.br) on 2015-04-08T12:57:46Z No. of bitstreams: 2 TESE Eduardo Olímpio Dias.pdf: 4555696 bytes, checksum: a1e7972a45dbba945f075adf6ee33606 (MD5) license_rdf: 1232 bytes, checksum: 66e71c371cc565284e70f40736c94386 (MD5) / Made available in DSpace on 2015-04-08T12:57:46Z (GMT). No. of bitstreams: 2 TESE Eduardo Olímpio Dias.pdf: 4555696 bytes, checksum: a1e7972a45dbba945f075adf6ee33606 (MD5) license_rdf: 1232 bytes, checksum: 66e71c371cc565284e70f40736c94386 (MD5) Previous issue date: 2014-02-28 / CNPq / A formação de padrões na natureza tem sido uma fonte de grande interesse acadêmico e tecnológico. Nesta tese, estudamos a dinâmica de interfaces instáveis de três sistemas fundamentais: a instabilidade hidrodinâmica de Sa man-Taylor (os chamados dedos viscosos ), a instabilidade de Mullins-Sekerka (refere-se ao crescimento de cristais) e a propaga ção de ondas de ionização através de descargas elétricas. Dentre estes sistemas, o problema Sa man-Taylor é o mais abordado. Esta instabilidade surge na interface que separa dois uidos viscosos con nados entre duas placas paralela, num dispositivo conhecido como célula de Hele-Shaw, ou imersos em um meio poroso. Ao longo da tese, estudamos meios de selecionar o número de onda dominante dos padrões na interface uido- uido, os efeitos da inércia do uido na instabilidade e, principalmente, métodos de controle e minimização das perturbações na interface. Um dos meios de controle das instabilidade é através de uma suave modi cação geométrica no aparato no qual o uido está con nado. Nesse contexto, fomos capazes tanto de favorecer quanto de inibir e, eventualmente, suprimir os dedos viscosos. Um outro protocolo de estabilização, que consiste no principal método de controle desta tese, é através do cálculo variacional. O objetivo desse procedimento é encontrar uma forma funcional para a taxa de injeção de uido que minimize as instabilidades. Comprovamos experimentalmente e por simulações numéricas a e cácia desse processo. Tal método também é utilizado para minimizar instabilidades em crescimento de cristais e em processos de descarga elétrica. Além da investigação de interfaces instáveis, nós estudamos a força e a energia de adesão de uidos complexos con nados entre placas paralelas. Neste caso, nosso método variacional visa obter a equação de levantamento ideal da placa superior a m de minimizar a energia de adesão. Finalmente, efeitos de inércia na força de levantamento são estudados.