In this thesis I investigate a number of problems in the nonlinear stability of density stratified plane Couette flow. I begin by describing the history of transient growth phenomena, and in particular the recent application of adjoint based optimisation to find nonlinear optimal perturbations and associated minimal seeds for turbulence, the smallest amplitude perturbations that are able to trigger transition to turbulence. I extend the work of Rabin et al. (2012) in unstratified plane Couette flow to find minimal seeds in both vertically and horizontally sheared stratified plane Couette flow. I find that the coherent states visited by such minimal seed trajectories are significantly altered by the stratification, and so proceed to investigate these states both with generalised Koopman mode analysis and by stratifying the self-sustaining process described by Waleffe (1997). I conclude with an introductory problem I considered that investigates the linear Taylor instability of layered stratified plane Couette flow, and show that the nonlinear evolution of the primary Taylor instability is not coupled to the form of the linearly unstable mode, in contrast to the Kelvin-Helmholtz instability, for example. I also include an appendix in which I describe joint work conducted with Professor Neil Balmforth of UBC during the 2015 WHOI Geophysical Fluid Dynamics summer programme, investigating stochastic homoclinic bifurcations.

Interfacial instabilities and wetting phenomena of phase separated colloid-polymer mixtures are addressed in this study. Colloidal particles offer certain advantages over molecular systems, due to their larger lengthscales and slower timescales. Moreover, the phenomena can be directly visualised using laser scanning confocal microscopy, and a perfect match with soft-lithography fabrication techniques can be exploited. In particular, we study the viscous fingering instability in three dimensions, focusing on the role of wetting conditions and of thermal fluctuations. Combined with results obtained by lattice Boltzmann simulations, we reveal that the cross-over of the meniscus in the direction across the channel thickness is controlled by the capillary and Peclet numbers, and viscosity contrast of the system. The curvature of the meniscus has a pronounced effect on the onset of the Saffman-Taylor instability, in which the formation of the viscous fingers is suppressed up to a certain threshold. Furthermore, we investigate a related contact line instability, which leads to entrainment and subsequent droplet pinch-off. A theoretical prediction for the onset of the instability is developed, which shows a good agreement with the experimental observations and yields a method to directly measure the slip length of the interface. The large thermal fluctuations of our interface play an important role in pinch-off events, leading to periodic emission of droplets of similar sizes. Finally, we study wetting phenomena at geometrically sculpted walls. We focus on the shape, the thickness and the radius of curvature of the adsorbed liquid film, and find good agreement with theory. Changing the curvature of the wedge from a flat surface to a capil- lary slit furthermore smoothly connects wetting behaviour and capillary condensation, again in qualitative agreement with theory. Non-equilibrium effects may interfere with the data and are difficult to rule out. We end with recommendations for future work.

L'instabilité de Rayleigh-Taylor (IRT) est notamment rencontrée lors des expériences de Fusion par Confinement Inertiel, et son développement est un obstacle à la réussite de ces expériences. L'objet de cette thèse est d'étudier la croissance de l'IRT pour différents régimes de compressibilité, au moyen de simulations numériques directes réalisées à l'aide d'un code pseudo-spectral multidomaine de type Chebyshev-Fourier-Fourier.La méthode du développement asymptotique permet d'établir des modèles à bas nombre de Mach pour lesquels la contribution acoustique est éliminée. L'implantation dans le code de simulation du modèle anélastique, qui met en jeu des fluides stratifiés et capture les effets thermiques, est améliorée. Le modèle de Boussinesq est ajouté au code. La précision de la méthode numérique est étudiée pour différents découpages en sous-domaines. Plusieurs éléments de validation sont présentés, dont la comparaison avec une expérience.La première simulation présentée, réalisée avec le modèle de Boussinesq, s'intéresse à la croissance auto-semblable de l'IRT. Les lois d'échelle de la vorticité et de la dissipation sont dégagées. La structure de la turbulence et du mélange entre les deux fluides est discutée. Certaines propriétés de la turbulence homogène et isotrope sont retrouvées, mais on note la persistance d'anisotropie aux petites échelles. Les premières simulations 3D de l'IRT avec le modèle anélastique sont présentées. L'influence des effets de compressibilité sur les premières phases de la croissance est étudiée. En outre, une couche de mélange anélastique en faible stratification est analysée et présente des effets de compressibilité non négligeables. / The Rayleigh-Taylor instability (RTI) is especially observed in inertial confinement fusion experiments, and its development prevents the success of these experiments. The purpose of this work is to study the growth of the RTI for different compressibility regimes by using a multidomain pseudospectral Chebyshev-Fourier-Fourier simulation code. The asymptotic expansion method allows to establish several low Mach number models which do not contains acoustics. The implantation of the anelastic model, which deals with stratified fluids and captures thermal effects, has been improved. Moreover, the Boussinesq model is added to the simulation code. The accuracy of the entire numerical method is studied, as a function of the subdomain separation, and several validation elements are shown, including a comparison with an experimental study. The first simulation to be analyzed is achieved with the Boussinesq model. We focus on the self-similarity of the RTI growth. The temporal scalings of vorticity and dissipation are displayed, and the structures of turbulence and mixing are discussed. Some properties of isotropic and homogeneous turbulence are observed, however some anisotropy remains at small scales. The first three-dimensional anelastic simulations are presented. The influence of compressibility effects on the first stages of the growth is studied. Finally, a developed anelastic mixing layer involving weakly stratified fluids is described and was found to display non-negligible compressibility effects.

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[pt] Uma abordagem experimental foi utilizada para o estudo da eficiência de varredura de fluidos viscoplásticos na célula de Hele-Shaw , que representa um meio poroso com porosidade constante. A análise baseia-se em um fluido empurrando outro onde, pela razão de viscosidades acima do punto de referência , manifesta-se a instabilidade de Saffman-Taylor, que desencadeia em uma interface instável com divisões sucessivas conhecidas como viscous fingers. A formação da instabilidade é considerada uma condição indesejável, e no caso da indústria petrolífera, torna-se de interesse na invasão da formação pelo fluido de perfuração, pasta de cimento, e no deslocamento de petróleo pesado em reservatórios entre outros. Apesar de que o fenômeno é vastamente estudado utilizando a célula de Hele-Shaw, a maioria dos trabalhos concentram-se no deslocamento de líquidos newtonianos por ar, deixando uma ampla gama de possibilidades de estudo, particularmente no que diz respeito aos fluidos não newtonianos. Utilizando equipamentos que permitem assegurar uma vazão constante de injeção, testou-se três concentrações diferentes de solução aquosa de carbopol em duas configurações diferentes onde o líquido viscoplástico é o deslocador e vice-versa. Fazendo uso do processamento digital de imagens, avaliou-se a forma da interface em função do desvio padrão, além da velocidade e da eficiência de varredura no deslocamento. Determinaram-se parâmetros adimensionais em função da geometria da bancada e dos fatores dinâmicos e reológicos dos fluidos onde se obtiveram duas zonas vem marcadas de varredura. Observou-se que a razão de viscosidades e o número de capilaridade modificada crítica são os indicadores que governam o deslocamento. / [en] An experimental approach has been used to study the sweep efficiency of viscoplastic fluid in a Hele-Shaw cell, which represents a porous medium with constant porosity. The analysis is based on a lower viscosity fluid pushing a more viscous through a geometry, which manifests the instability of Saffman-Taylor, that result in an unstable interface with successive divisions known as viscous fingers. The occurrence of instability is considered an undesirable condition, and in the petroleum oil industry, it becomes of interest in the reservoir oil invasion and heavy oil displacement in reservoirs. Despite being widely studied, most studies focus on the displacement of Newtonian liquids by air, leaving the field of study of non-Newtonian fluids without being evaluated where none of viscosities of the fluids involved can be considered negligible. Using equipment that ensures a constant flow rate of injection three different aqueous concentrations of carbopol were tested in two different configurations: carbopol as the displacing fluid and vice-versa. Using digital image processing, the shape interface was evaluated according to their standard deviation, average velocity and efficiency sweep during the displacement. Determined dimensionless parameters was evaluated in function of the experimental geometry, dynamic and rheological factors of the fluids in order to get benchmark that allow separating interfaces zones that tend to 100 per cent or plugs and the zone where the viscous fingers reveal themselves. It was found that the viscosity ratio and modified critical capillary govern the displacement.

Volcanic activity, and the resultant deposits and structures at the Earth's surface, are the outcome of the inner workings of underground magmatic plumbing systems. These systems, essentially, consist of magma reservoirs which supply magma to the surface through volcanic conduits feeding volcanic eruptions. The mechanics and structure of plumbing systems remain largely unknown due to the obvious challenges involved in inferring volcanic processes occurring underground from observations at the surface. Nevertheless, volcanologists are beginning to gain a deeper understanding of the workings and architecture of magmatic plumbing systems from geophysical observations on active volcanoes, as well as from geological studies of the erosional remnants of ancient volcanic systems. In this work, I explore the relationship between the structure and mechanics of shallow plumbing systems and the volcanic eruptions these systems produce. I attempt to contribute to the understanding of this complex relationship by linking geological and geophysical observations of an eroded basaltic subvolcanic system, and the eruptive and tectonic activity of an active volcano, with mathematical models of magma ascent and stress transfer. The remarkable exposures of the Carmel outcrop intrusions, near the San Rafael swell, southeast Utah, U. S. A., allow detailed geological and geophysical observations of the roots of volcanic conduits that emerge from a subhorizontal magma feeder reservoir. These observations reveal a new mechanism for magma ascent and eruption triggering through gravitational instabilities created from an underlying feeding sill, and shed light on the mechanics of sill emplacement. Geophysical and geological observations of the 1999 and xii 1992 eruptions of the Cerro Negro volcano, Nicaragua, are used to explore the coupling between changes in the stress field and the triggering of volcanic eruptions, and magma ascent through the shallow crust. Modeling results of stress transfer and conduit flow highlight the importance of the surrounding stress field and geometry of the volcanic conduits that comprise shallow plumbing systems.

[pt] A superposição de um escoamento circular de Couette e um fluxo com gradiente de pressão axial, através de um espaço anular ocorre em muitas aplicações práticas, tais como: reatores químicos catalíticos, filtros, extratores líquido- líquido, mancais e o fluxo de retorno de lamas de perfuração entre a coluna de perfuração rotatória e a formação rochosa na perfuração de poços produtores de petróleo e gás. As linhas de corrente curvadas do fluxo circular de Couette podem causar uma instabilidade centrífuga que produz vórtices toroidais, conhecidos como vórtices de Taylor. A presença destes vórtices muda as características hidrodinâmicas e a transferência de calor no processo. Em conseqüência, é muito importante ser capaz de prever o aparecimento da instabilidade. A maioria das análises numéricas e experimentais disponíveis na literatura são para fluidos Newtonianos e viscoelásticos (soluções polimericas). Neste trabalho, o efeito das propriedades viscoplásticas de suspensões de altas concentrações neste tipo de escoamento e nas condições críticas para o aparecimento de vórtices são determinadas teoricamente através da solução das equações de conservação. As equações diferenciais foram integradas pelo método de elementos finitos-Galerkin e o sistema de equações algébricas não lineares resultante foi resolvido pelo método de Newton. / [en] The superposition of a circular Couette flow and a pressure- driven axial flow in an annulus occurs in many practical applications, such as catalytic chemical reactors, filtration devices, liquid-liquid extractors, journal bearings, and the return flow of drilling mud between the rotating drill string and the stationary wall in oil and gas well drilling. The curved streamlines of the circular Couette flow can cause a centrifugal instability leading to toroidal vortices, well known as Taylor vortices. The presence of these vortices changes the hydrodynamic and heat transfer characteristics of the process. Therefore, it is very important to be able to predict the onset of instability. Most of the available theoretical and experimental analyses are for Newtonian and viscoelastic (polymeric solutions) liquids. In this work, the effect of the viscoplastic properties of high concentration suspensions on the onset of the Taylor vortices are determined theoretically by solving the conservation equations and searching the critical conditions. The differential equations were solved by the Galerkin / finite element method and the resulting set of non-linear algebraic equations, by Newtons method.

Les différents dimensionnements et expériences de Fusion par Confinement Inertiel (FCI) en attaque directe comme indirecte montrent qu'une des principales limites à l'atteinte de l'ignition est l'instabilité de Rayleigh-Taylor (IRT) qui cause la rupture de la coquille de la cible en vol et potentiellement le mélange du combustible chaud du coeur avec celui, froid, de la coquille. La connaissance, la compréhension et la maîtrise des conditions initiales de ce mécanisme sont donc d'un grand intérêt. Nous présentons ainsi une étude expérimentale et théorique des conditions initiales de l'IRT ablative en attaque directe au travers de deux campagnes expérimentales réalisées sur le laser OMEGA (LLE, Rochester). La première campagne concerne l'étude de l'instabilité de Richtmyer-Meshkov (IRM) ablative imprimée par laser ; cette instabilité commence à se développer au début de l'irradiation laser et fixe l'ensemencement de l'IRT. Nous avons mis en place une configuration expérimentale qui a permis de mesurer l'évolution temporelle de l'IRM ablative imprimée par laser pour la première fois. Nous présentons ensuite une interprétation des résultats de cette expérience par des simulations hydrodynamiques réalisées avec le code CHIC, ainsi que par un modèle théorique de l'IRM ablative imprimée par laser. Nous montrons que le moyen le plus direct de contrôler cette instabilité est de réduire l'amplitude des défauts d'intensité laser. Ceci peut être accompli en utilisant des cibles couvertes par une couche de mousse de basse densité. Ainsi, lors de la deuxième campagne, nous avons étudié pour la première fois l'effet de mousses sous-denses sur la croissance de l'IRT ablative. Au cours de ces expériences, des feuilles de plastique recouvertes d'une couche de mousse ont été irradiées par un faisceau laser portant une perturbation d'intensité destinée à imprimer des modulations sur la cible. Différentes données expérimentales sont présentes : rétrodiffusion de l'énergie laser, dynamique de la cible obtenue par mesure de côté d'auto-émission et radiographies de face faisant apparaître l'effet des mousses sur les modulations de densité surfacique des cibles. Ces données ont ensuite été interprétées à l'aide de simulations CHIC et du code d'interaction laser-plasma PARAX. Nous montrons qu'une des mousses réduit l'amplitude des modulations de l'intensité laser d'un facteur 2. Par conséquent, cette thèse a donné lieu au développement de configurations expérimentales et d'un ensemble d'outils de dépouillement numériques pour l'étude approfondie des instabilités hydrodynamiques en FCI. / Numerous designs and experiments in the domain of Inertial Confinement Fusion (ICF) show that, in both direct and indirect drive approaches, one of the main limitations to reach the ignition is the Rayleigh-Taylor instability (RTI). It may lead to shell disruption and performance degradation of spherically imploding targets. Thus, the understanding and the control of the initial conditions of the RTI is of crucial importance for the ICF program. In this thesis, we present an experimental and theoretical study of the initial conditions of the ablative RTI in direct drive, by means of two experimental campaigns performed on the OMEGA laser facility (LLE, Rochester). The first campaign consisted in studying the laser-imprinted ablative Richtmyer-Meshkov instability (RMI) which starts at the beginning of the interaction and seeds the ablative RTI.We set up an experimental configuration that allowed to measure for the first time the temporal evolution of the laser-imprinted ablative RMI. The experimental results have been interpreted by a theoretical model and numerical simulations performed with the hydrodynamic code CHIC. We show that the best way to control the ablative RMI is to reduce the laser intensity inhomogeneities. This can be achieved with targets covered by a layer of a low density foam. Thus, in the second campaign, we studied for the first time the effect of underdense foams on the growth of the ablative RTI. A layer of low density foam was placed in front of a plastic foil, and the perturbation was imprinted by an intensity modulated laser beam. Experimental data are presented : backscattered laser energy, target dynamic obtained by side-on selfemission measurement, and face-on radiographs showing the effect of the foams on the target areal density modulations. These data were interpreted using the CHIC code and the laser-plasma interaction code PARAX. We show that the foams noticeably reduce the amplitude of the laser intensity inhomogeneities and the level of the subsequent imprinted ablation front modulations. In conclusion, this thesis allowed us to develop an experimental platform and a suite of numerical tools for future, more detailed studies of hydrodynamic instabilities for ICFapplications.

L’approximation dite « anélastique » permet de filtrer les ondes acoustiques grâce à un développement asymptotique deséquations de Navier-Stokes, réduisant ainsi le pas en temps moyen, lors de la simulation numérique du développement d’instabilités hydrodynamiques. Ainsi, les équations anélastiques sont établies pour un mélange de deux fluides pour l’instabilité de Rayleigh-Taylor. La stabilité linéaire de l’écoulement est étudiée pour la première fois pour des fluides parfaits, par la méthode des modes normaux, dans le cadre de l’approximation anélastique. Le problème de Stokes issu des équations de Navier-Stokes sans les termes non linéaires (une partie de la poussée d’Archiméde est prise en compte) est défini ; l’éllipticité est démontrée, l’étude des modes propres et l’invariance liée à la pression sont détaillés. La méthode d’Uzawa est étendue à l’anélastique en mettant en évidence le découplage des vitesses en 3D, le cas particulier k = 0 et les modes parasites de pression. Le passage au multidomaine a permis d’établir les conditions de raccord (raccord Co de la pression sans condition aux limites physiques). Les algorithmes et l’implantation dans le code AMENOPHIS sont validés par les comparaisons de l’opérateur d’Uzawa développé en Fortran et à l’aide de Mathematica. De plus des résultats numériques ont été comparés à une expérience avec des fluides incompressibles. Finalement, une étude des solutions numériques obtenues avec les options anélastique et compressible a été menée. L’étude de l’influence de la stratification initiale des deux fluides sur le développement de l’instabilité de Rayleigh-Taylor est amorcée. / The « anelastic » approximation allows us to filter the acoustic waves thanks to an asymptotic development of the Navier-Stokes equations, so increasing the averaged time step, during the numerical simulation of hydrodynamic instabilitiesdevelopment. So, the anelastic equations for a two fluid mixture in case of Rayleigh-Taylor instability are established.The linear stability of Rayleigh-Taylor flow is studied, for the first time, for perfect fluids in the anelastic approximation.We define the Stokes problem resulting from Navier-Stokes equations without the non linear terms (a part of the buoyancyis considered) ; the ellipticity is demonstrated, the eigenmodes and the invariance related to the pressure are detailed.The Uzawa’s method is extended to the anelastic approximation and shows the decoupling speeds in 3D, the particular casek = 0 and the spurius modes of pressure. Passing to multidomain allowed to establish the transmission conditions.The algorithms and the implementation in the existing program are validated by comparing the Uzawa’s operator inFortran and Mathematica langages, to an experiment with incompressible fluids and results from anelastic and compressiblenumerical simulations. The study of the influence of the initial stratification of both fluids on the development of the Rayleigh-Taylor instability is initiated.