Ce travail de thèse est consacré à l'étude asymptotique d'écoulements de faible épaisseur et à la modélisation des conditions aux limites à imposer à l'interface fluide-solide dans différentes situations. Le chapitre 1 est consacré à l'etude asymptotique d'un écoulement fluide constitué d'une couche poreuse mince adjacente à un milieu fluide mince. On met en évidence l'existence d'un rapport critique entre la taille de la microstructure du milieu poreux et les deux épaisseurs, rapport pour lequel une équation de Reynolds modifiée est obtenue. De plus il est montré qu'on peut toujours pour une géométrie réelle se placer dans ce cas critique. Enfin, on présente des simulations numériques qui mettent en évidence les différences entre le modèle présenté ici et deux autres modèles utilisés en mécanique. Dans le chapitre 2, on s'intéresse à l'étude d'un écoulement de faible épaisseur quand une des surfaces est rugueuse. Ceci peut etre relié à l'étude du chapitre précédent en considérant un milieu poreux qui ne comporterait qu'une seule couche. On utilise la technique de la double échelle en homogénéisation pour obtenir rigoureusement les résultats de convergences. En outre, la convergence des contraintes normales et tangentielles sur les surfaces lisses et rugueuses est étudiée. Dans le chapitre 3, on étudie un écoulement d'un fluide non newtonien de type micropolaire avec de nouvelles conditions à l'interface fluide solide couplant la vitesse et la microrotation par l'introduction d'une viscosité de surface. On démontre l'existence et l'unicité de la solution et des estimations a priori qui conduisent, via l'étude asymptotique, à une équation de Reynolds micropolaire généralisée. Une étude numérique montre l'influence des conditions aux limites sur la charge et le coefficient de frottement. Les résultats sont comparés avec ceux d'autres modèles retenant une condition d'adhérence à la paroi.

At low Reynolds number flows, the effect of inertia becomes negligible and the fluid motion is dominated by the effect of viscous forces. Understanding of the behaviour of low Reynolds number flows underpins the prediction of the motion of microorganisms and particle sedimentation as well as the development of micro-robots that could potentially swim inside the human body to perform targeted drug/cell delivery and non-invasive microsurgery. The work in this thesis focuses on developing an understanding in the mathematical analysis of objects moving at low Reynolds numbers. A boundary element implementation of the Method of regularized Stokeslets (MRS) is applied to analyse the low Reynolds number flow field around an object of simple shape (sphere and cube). It also showed that the results obtained by a boundary element implementation for an unbounded cube, where singularities are presented in the corners of the cube, agrees with more complex solutions methods such as a GBEM and FEM.A methodology for analysing the effect of walls by locating collocation points on the surface of the walls and the object is presented. First at all, this methodology is validated with a boundary element implementation of the method of images for a sphere at different locations. Then, the method is extended when more than one wall is presented. This methodology is applied to predict the velocity filed of a cube moving in a tow tank at low Reynolds numbers for two different cases with a supporting rod similar to an experimental set-up, and without the supporting rod as in the CFD simulations based on the FVM. The results indicate a good match between CFD and the MRS, and an excellent approximation between the MRS and experimental data from PIV measurements. The drag, thrust and torque generated by helices moving at low Reynolds numbers in an unbounded medium is analysed by the resistive force theory, a slender body theory, and a boundary element method of the MRS. The results show that the resistive force theory predict accurately the drag, thrust and torque of moving helices when the resistive force coefficients are calculated from a slender body theory approximation by calculating independently the resistive force coefficients for translation and rotation, because it is observed that the resistive force coefficients depend also of the nature of motion. Moreover, the thrust generated by helices of different pitch angles is analysed calculated by a CFD numerical simulation based on the FVM and a boundary element implementation, an compared with experimental data. The results also show an excellent prediction between the boundary element implementation, the CFD results and the experimental data. Finally, a boundary element implementation of the MRS is applied to predict swimming of a biomimetic swimmer that mimics the motion of E.coli bacteria in an unbounded medium. The results are compared with the propulsive velocity and induced angular velocity measurement by recording the motion of the biomimetic swimmer in a square tank. It is observed that special care needs to be taken when the biomimetic swimmer is modelled inside the tank, as there is an apparent increment in the calculate thrust propulsion which does not represent a real situation of the biometic swimmer which propels by a power supply. However, this increment does not represent the condition of the biomimetic swimmer and a suggested methodology based on the solution from an unbounded case and when the swimmer is moving inside the tank is presented. In addition, the prediction of the free-swimming velocity for the biomimetic swimmer agrees with the results obtained by the MRS when the resistive force coefficients are calculated from a SBT implementation. The results obtained in this work have showed that a boundary element implementation of the MRS produces results comparable with more complex numerical implementations such as GBEM, FEM, FVM, and also an excellent agreement with results obtained from experimentation. Therefore, it is a suitable and easy to apply methodology to analyse the motion of swimmers at low Reynolds numbers, such as the biomimetic swimmer modelled in this work.

Le contact Segment-Piston-Chemise (SPC) est essentiel pour la réduction des émissions de CO2 et de polluants dans les moteurs automobile. Pour optimiser ces deux contraintes antagonistes, il est nécessaire d’étudier l’influence de la microgéométrie afin de trouver de nouvelles architectures de ce contact. L’étude expérimentale étant très coûteuse, il est impératif de pouvoir modéliser les performances du contact SPC. Les méthodes multigrilles permettent de réduire considérablement les temps de calcul. Pour le contact SPC, les méthodes multigrilles classiques perdent leur efficacité en présence de texture. Un nouvel algorithme, basé sur des idées développées par Alcouffe et al. a été conçu qui permet l’étude du contact hydrodynamique en présence de texturation importante. Il a été validé par comparaison avec un modèle analytique 1D, ses performances ont été étudiées et une première étude de l’influence des paramètres sur la portance pour une surface striée a été faite. L’algorithme a ensuite été modifié pour imposer la conservation des flux. Un processus de relaxation plus proche de la physique du phénomène est utilisé. Le code obtenu n’atteint pas l’efficacité multigrille optimale. Cependant sa convergence est suffisamment rapide pour étudier l’influence de pores pour le segment plat et montrer l’importance d’une texturation partielle. Pour finir, le terme transitoire de l’équation de Reynolds est pris en compte. Pour un segment parabolique et une chemise striée, l’influence des paramètres des stries (profondeur, largeur, distance et angle des stries) sur la variation moyenne de la hauteur de film par rapport à une surface lisse est analysée. / The piston ring cylinder liner (PRCL) contact is essential in the CO2 and pollutant emission reduction in internal combustion engine. These two constraints being antagonistic, a compromise has to be found and it is necessary to study the microgeometry influence in order to find new PRCL contact architectures. As experimental study can be very expensive, modeling PRCL contact performance becomes crucial. Multigrid methods allow a huge reduction of the computational time. Unfortunately, in presence of texture, classical multigrid efficiency decays significantly and are not usable. A new algorithm based on an approach developed by Alcouffe et al. was designed that allows the study of the hydrodynamic lubrication of this contact even in the case of a highly varying geometry due to the texture. The obtained code was validated by comparison with a 1D analytical model, its performance was evaluated and a first study of an analytic cross hatched geometry parameter influence on the load carrying capacity was completed. The code was modified to include flow conservation. A relaxation process based closer to the physics phenomenon is used. The code efficiency is not the one that is expected from multigrid technics. However it convergence is sufficiently fast to study the dimple influence for the oil control ring and show the importance of partial texturing. Finally the transient term of the Reynolds equation is accounted for. In the case of a parabolic ring and for a cross hatched liner, the influence of the groove parameters ( depth, width, distance and groove angle) on the average minimum film thickness with respect to the smooth case is analyzed.