In this thesis, three fluid dynamical problems are studied. First in chapter 2 we investigate, via both theoretical and experimental methods, the swimming motion of a magnetotactic bacterium having the shape of a prolate spheroid in a viscous liquid under the influence of an imposed magnetic field. The emphasis of the study is placed on how the shape of the non-spherical magnetotactic bacterium, marked by the size of its eccentricity, affects the pattern of its swimming motion. It is revealed that the pattern/speed of a swimming spheroidal magnetotactic bacterium is highly sensitive not only to the direction of its magnetic moment but also to its shape. Secondly, an important unanswered mathematical question in the theory of rotating fluids has been the completeness of the inviscid eigenfunctions which are usually referred to as inertial waves or inertial modes. In chapter 3 we provide for the first time a mathematical proof for the completeness of the inertial modes in a rotating annular channel by establishing the completeness relation, or Parseval’s equality, for any piecewise continuous, differentiable velocity of an incompressible fluid. Thirdly, in chapter 4 we investigate, through both asymptotic analysis and direct numerical simulation, precessionally driven flow of a homogeneous fluid confined in a fluid-filled circular cylinder that rotates rapidly about its symmetry axis and precesses about a different axis that is fixed in space. A particular emphasis is placed on the spherical-like cylinder whose diameter is nearly the same as its length. An asymptotic analytical solution in closed form is derived in the mantle frame of reference for describing weakly precessing flow in the spherical-like cylinder at asymptotically small Ekman numbers. We also construct a three-dimensional finite element model, which is checked against the asymptotic solution, in attempting to elucidate the structure of the nonlinear flow.

Nous présentons une étude expérimentale de l'influence d'une rotation d'ensemble sur le déclin d'un écoulement turbulent dans une géométrie confinée. L'écoulement est généré en translatant rapidement une grille dans un récipient parallélépipédique, et nous mesurons les champs de vitesse dans un plan vertical (parallèle à l'axe de rotation) grâce à un dispositif de PIV embarqué. Nous montrons dans un premier temps qu'une partie significative de l'énergie est contenue dans un écoulement moyen reproductible, qui s'identifie à une superposition de modes d'inertie résonnants de la cuve. Le couplage possible entre cet écoulement et la turbulence suggère que la turbulence ainsi créée n'est pas en déclin libre. Nous montrons cependant qu'il est possible d'inhiber l'apparition de ces modes d'inertie en modifiant les caractéristiques géométriques de la grille. Cette nouvelle configuration permet alors de caractériser dans l'espace physique les transferts d'énergie pour une turbulence en déclin libre. L'énergie associée aux incréments de vitesse et la densité de flux d'énergie sont calculées à partir d'un grand nombre de réalisations indépendantes. Nous montrons que la rotation provoque une forte bidimensionalisation de la distribution d'énergie, et que celle-ci est contrôlée par une densité de flux d'énergie qui reste quasi-radiale, mais qui fait apparaître une dépendance angulaire marquée. Enfin, nous étudions également l'écoulement dans un cube, que nous soumettons à une libration longitudinale afin d'exciter les modes observés initialement avec le dispositif de turbulence de grille. En comparant les champs de vitesse expérimentaux aux prédictions numériques des modes inviscides, nous montrons que seul un certain nombre de modes, compatibles avec les symétries du forçage, peuvent être excités par libration. Nous caractérisons en particulier la résonance du mode de plus bas ordre compatible avec les symétries du forçage, et discutons du rôle de la viscosité. / We investigate the effect of a backgroung rotation on the decay of a turbulent flow in a confined geometry. Turbulence is generated by rapidly towing a grid in a parallelepipedic water tank. The velocity fields of a large number of independent decays are measured in a vertical plane parallel to the rotation axis using a corotating Particle Image Velocimetry system. We first show that a significant amount of the kinetic energy is stored in a reproducible flow composed of resonant inertial modes. The possible coupling between these modes and turbulence suggests that turbulence cannot be considered as freely decaying in this configuration. We demonstrate however that these inertial modes may be reduced by changing the geometrical features of the grid. Thanks to this new configuration, anisotropic energy transfers in freely decaying turbulence in a rotating frame can be characterized experimentally in the physical space. The anisotropic energy flux density is determined from large data sets of Particle Image Velocimetry measurements. We show that in the presence of a background rotation, the energy distribution reflects a trend towards a 2D flow. This anisotropy is proved to be essentially driven by a nearly radial, but orientation-dependent, energy flux density. We also analyze the flow in a cubic container submitted to a longitudinal libration in order to generate inertial modes that were originally observed when the "simple" grid was used. By comparing the measured flow fields to the expected inviscid inertial modes, we show that only a subset of inertial modes, matching the symmetries of the forcing, may be excited by the libration. We characterize in particular the resonance of the mode of lowest order compatible with the symmetries of the forcing, and discuss the role of the viscosity.

No description available.

Many known extra-solar giant planets lie close to their host stars. Around 60 have their semi-major axes smaller than 0.05 AU. In contrast to planets further out, the vast majority of these close-in planets have low eccentricity orbits. This suggests that their orbits have been circularized likely due to tidal dissipation inside the planets. These exoplanets share with our own Jupiter at least one trait in common: when they are subject to periodic tidal forcing, they behave like a lossy spring, with a tidal ``quality factor'', Q, of order 10^5. This parameter is the ratio between the energy in the tide and the energy dissipated per period. To explain this, a possible solution is resonantly forced internal oscillation. If the frequency of the tidal forcing happens to land on that of an internal eigenmode, this mode can be resonantly excited to a very large amplitude. The damping of such a mode inside the planet may explain the observed Q value. The only normal modes that fall in the frequency range of the tidal forcing (~ few days) are inertial modes, modes restored by the Coriolis force. We present a new numerical technique to solve for inertial modes in a convective, rotating sphere. This technique combines the use of an ellipsoidal coordinate system with a pseudo-spectral method to solve the partial differential equation that governs the inertial oscillations. We show that, this technique produces highly accurate solutions when the density profile is smooth. In particular, the lines of nodes are roughly parallel to the ellipsoidal coordinate axes. In particular, using these accurate solutions, we estimate the resultant tidal dissipation for giant planets, and find that turbulent dissipation of inertial modes in planets with smooth density profiles do not give rise to dissipation as strong as the one observed. We also study inertial modes in density profiles that exhibit discontinuities, as some recent models of Jupiter show. We found that, in this case, our method could not produce convergent solutions for the inertial modes. Additionally, we propose a way to observe inertial modes inside Saturn indirectly, by observing waves in its rings that may be excited by inertial modes inside Saturn.

Many known extra-solar giant planets lie close to their host stars. Around 60 have their semi-major axes smaller than 0.05 AU. In contrast to planets further out, the vast majority of these close-in planets have low eccentricity orbits. This suggests that their orbits have been circularized likely due to tidal dissipation inside the planets. These exoplanets share with our own Jupiter at least one trait in common: when they are subject to periodic tidal forcing, they behave like a lossy spring, with a tidal ``quality factor'', Q, of order 10^5. This parameter is the ratio between the energy in the tide and the energy dissipated per period. To explain this, a possible solution is resonantly forced internal oscillation. If the frequency of the tidal forcing happens to land on that of an internal eigenmode, this mode can be resonantly excited to a very large amplitude. The damping of such a mode inside the planet may explain the observed Q value. The only normal modes that fall in the frequency range of the tidal forcing (~ few days) are inertial modes, modes restored by the Coriolis force. We present a new numerical technique to solve for inertial modes in a convective, rotating sphere. This technique combines the use of an ellipsoidal coordinate system with a pseudo-spectral method to solve the partial differential equation that governs the inertial oscillations. We show that, this technique produces highly accurate solutions when the density profile is smooth. In particular, the lines of nodes are roughly parallel to the ellipsoidal coordinate axes. In particular, using these accurate solutions, we estimate the resultant tidal dissipation for giant planets, and find that turbulent dissipation of inertial modes in planets with smooth density profiles do not give rise to dissipation as strong as the one observed. We also study inertial modes in density profiles that exhibit discontinuities, as some recent models of Jupiter show. We found that, in this case, our method could not produce convergent solutions for the inertial modes. Additionally, we propose a way to observe inertial modes inside Saturn indirectly, by observing waves in its rings that may be excited by inertial modes inside Saturn.