Microorganisms and drops are ubiquitous in nature: while drops can be found in sneezes, ink-jet printers, oceans etc, microorganisms are present in our stomach, intestine, soil, oceans etc. In most situations they are present in complex conditions: drop spreading on a rigid or soft substrate, drop covered with impurities that act as surfactants, marine microbe approaching a surfactant laden drop in density stratified oceanic waters in the event of an oil spill etc. In this thesis, we extract the physics underlying the influence of two such complicated effects (surfactant redistribution and density-stratification) on the motion of drops and swimming microorganisms when they are in isolation or in the vicinity of each other. This thesis is relevant in understanding the bioremediation of oil spill by marine microbes.<div><br></div><div>We divide this thesis into two themes. In the first theme, we analyze the motion of motile microorganisms near a surfactant-laden interface in homogeneous fluids. We begin by calculating the translational and angular velocities of a swimming microorganism outside a surfactant-laden drop by assuming the surfactant is insoluble, incompressible, and non-diffusing, as such system is relevant in the context of bioremediation of oil spill. We then study the motion of swimming microorganism lying inside a surfactant-laden drop by assuming the surfactant is insoluble, compressible, and has large surface diffusivity. This system is ideal for exploring the nonlinearities associated with the surfactant transport phenomena and is relevant in the context of targeted drug delivery systems wherein one uses synthetic swimmers to transport the drops containing drug. We then analyze the motion of a swimming organism in a liquid film covered with surfactant without making any assumptions about the surfactant and this system is relevant in the case of free-standing films containing swimming organisms as well as in the initial stages of the biofilm formation. In the second theme, we consider a density-stratified background fluid without any surfactants. In this theme, we examine separately a towed drop and a swimming microorganism, and find the drag acting on the drop, drop deformation, and the drift volume induced by the drop as well as the motility of the swimming microorganism.</div>

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.