The majority of this work is concerned with the local-linear convective instability analysis of the incompressible boundary-layer flows over prolate spheroids and oblate spheroids rotating in otherwise still fluid. The laminar boundary layer and the perturbation equations have been formulated by introducing two distinct orthogonal coordinate systems. A cross-sectional eccentricity parameter e is introduced to identify each spheroid within its family. Both systems of equations reduce exactly to those already established for the rotating sphere boundary layer. The effects of viscosity and streamline-curvature are included in each analysis. We predict that for prolate spheroids at low to moderate latitudes, increasing eccentricity has a strong stabilizing effect. However, at high latitudes of ϴ ≥ 60, increasing eccentricity is seen to have a destabilizing effect. For oblate spheroids, increasing eccentricity has a stabilizing effect at all latitudes. Near the pole of both types of spheroids, the critical Reynolds numbers approach that for the rotating disk boundary layer. However, in prolate spheroid case near the pole for very large values of e, the critical Reynolds numbers exceed that for the rotating disk. We show that high curvature near the pole of prolate spheroids is responsible for the increase in critical Reynolds number with increasing eccentricity. For both types of spheroids at moderate eccentricity, we predict that the most amplified modes travel at approximately 76% of the surface speed at all latitudes. This is consistent with the existing studies of boundary-layer flows over the related rotating-disk, -sphere and -cone geometries. However, for large values of eccentricity, the traveling speed of the most amplified modes increases up to approximately 90% of the surface speed of oblate spheroids and up to 100% in the prolate spheroid case.

La thèse présente une étude numérique des instabilités de trajectoires de sphères, d'ellipsoïdes aplatis et de bulles en mouvement libre sous l'action de la gravité, de la poussée d'Archimède et des forces hydrodynamiques. Le chapitre sur les sphères reprend, complète et étend l'étude numérique de Jenny et al. (2004) en se concentrant sur la transition au chaos et sur les trajectoires chaotiques. Les résultats montrent la différence entre le scénario de transition au chaos de sphères de faible et de grand rapport de densité. Plusieurs grandeurs statistiques sont proposées afin de fournir une caractérisation quantitative des états chaotiques. Elle permettent de mettre en relation les états ordonnées et chaotiques et offrent une possibilité de comparaison objective de données aléatoires d'origine numérique ou expérimentale. L'étude, très extensive, du comportement d'ellipsoïdes aplatis établit le lien entre les disques et les sphères en faisant varier l'aplatissement des objets depuis infiniment plat jusqu'à presque sphérique. Les huit diagrammes d'état présentés permettent de comprendre l'effet de la forme des ellipsoïdes sur le scénario de transition. Le cas d'ellipsoïdes presque sphériques montre que de faibles imperfections de la forme peuvent avoir in impact significatif sur les trajectoires de sphères de très faible rapport de densité. Pour les bulles considérées dans la limite de rapport de densité et viscosité az/liquide nul, l'étude se concentre sur l'analyse de stabilité linéaire et aboutit à la courbe de stabilité marginale dans le plan des paramètres nombre de Bond – nombre de Galilée en tenant compte de la déformation des bulles au moment de la perte de leur axisymétrie. Plus deux décades de nombres de Bond, entre 0,1 et 20, sont couvertes. Les résultats montrent clairement l'effet de la déformation de la bulle sur le seuil de l'instabilité. / The thesis presents a numerical study of path instabilities for spheres, oblate spheroids and bubbles moving freely under the effect of the gravity, buoyancy and hydrodynamic forces. For spheres, the parametric study of Jenny et al. (2004) is revisited, improved end extended with a special focus on the chaotic states. The results reveal that the effect of density ratio responsible for different oblique oscillating states of low and high frequencies has a significant impact both on the onset of chaos and on the behavior of fully chaotic states. Several quantitative statistical quantities are proposed and shown to be relevant for establishing the relation between chaotic and ordered states and for an objective comparison of random data of numerical or experimental origin. The extensive study on freely moving spheroids establishes the link between disks and spheres by varying the aspect ratio of spheroids from infinitely flat to almost spherical. The state diagrams provided for eight different aspect ratios of spheroid show in detail how the transition scenario varies depending of the body shape. The investigation of almost spherical spheroids reveals the specificities of the dynamics of light imperfect spheres.For the deformable gas bubble in the limit of zero gas/liquid density and viscosity ratio, a marginal stability curve is given in the two-parameter plane of the Galileo and the Bond number indicating the critical Galileo numbers for the loss of stability of vertical trajectories. The numerical investigation covers more than two decades of Bond number going from 0.1 to 20. The results clearly show the crucial role of the surface deformation.