We consider both a long-wave model and a first-order weighted-residual integral boundary layer (WIBL) model in the investigation of thin film flow down a topographical incline whilst under the effects of a normal electric field. The liquid is assumed to be a perfect dielectric, although is trivially extended to the case of a perfect conductor. The perfect dielectric case with no topography includes a simple modified electric Weber number which incorporates the relative electrical permittivity constant into itself. Linear stability analysis is carried out for both models, and critical Reynolds numbers which depend on the electric Weber number and the capillary number are produced. Regions of stability, convective instability and absolute instability are then determined for both models in terms of our electric Weber number and Reynolds number parameters in the case of no topography. Time-dependent simulations are produced to corroborate the aforementioned regions and investigate the effect of normal electric field strength in addition to sinusoidal and rectangular topographical amplitude on our system for various domain sizes. For the time-dependent simulations we find strong agreement with the linear stability analysis, and the results suggest that the inclusion of a normal electric field may have some stabilising properties in the long-wave model which are absent in the case of a flat wall, for which the electric field is always linearly destabilising. This stabilising effect is not observed for the same parameters in the WIBL model with a sinusoidal wall, although a similar effect is noticed in the WIBL model with a rectangular wall. We also investigate the simultaneous effect of domain size, wall amplitude and electric field strength on the critical Reynolds numbers for both models, and find that increasing the electric field strength can make large-amplitude sinusoidal topography stabilising rather than destabilising for the long-wave model. Continuation curves of steady solutions and bifurcation diagrams are also produced, and comparisons between the two models are made for various parameter values, which show excellent agreement with the literature. Subharmonic branches and time-periodic solutions are additionally included, similarly showing very good agreement with the literature.

Les écoulements rampants, recirculants et instationnaires d'un fluide viscoplastique autour d'un cylindre ont été étudiés.Numériquement, les morphologies des écoulements, la localisation des zones rigides, les champs de contraintes et pression autour du cylindre ainsi que le coefficient de traînée, ont été déterminés sur un large domaine des nombres de Reynolds et d'Oldroyd.Expérimentalement, les fluides étudiés sont des gels de polymère Carbopol®. Le comportement élastoviscoplastique de ces gels a été modélisé par une loi d'Herschel-Bulkley adaptée. Le montage expérimental conçu et réalisé a été validé par l'étude de l'écoulement d'un fluide newtonien autour d'un cylindre et la mise en place d'une procédure adaptée pour les fluides à seuil.On a pu constater l'influence des conditions d'interface avec l'apparition d'une morphologie de lâchers de tourbillons simultanés et symétriques.

The linear stability of high speed-viscous plane Couette and Couette-Poiseuille flows are investigated numerically. The conservation equations along with Sutherland&amp / #65533 / s viscosity law are studied using a second-order finite difference scheme. The basic velocity and temperature distributions are perturbed by a small-amplitude normalmode disturbance. The small-amplitude disturbance equations are solved numerically using a global method using QZ algorithm to find all the eigenvalues at finite Reynolds numbers, and the incompressible limit of these equations is investigated for Couette-Poiseuille flow. It is found that the instabilities occur, although the corresponding growth rates are often small. Two families of wave modes, Mode I (odd modes) and Mode II (even modes), were found to be unstable at finite Reynolds numbers, where Mode II is the dominant instability among the unstable modes for plane Couette flow. The most unstable mode for plane Couette &amp / #65533 / Poiseuille flow is Mode 0, which is not a member of the even modes. Both even and odd modes are acoustic modes created by acoustic reflections between a will and a relative sonic line. The necessary condition for the existence of such acoustic wave modes is that there is a region of locally supersonic mean flow relative to the phase speed of the instability wave. The effects of viscosity and compressibility are also investigated and shown to have a stabilizing role in all cases studied. Couette-Poiseuille flow stability is investigated in case of a choked channel flow, where the maximum velocity in the channel corresponds to sonic velocity. Neutral stability contours were obtained for this flow as a function if the wave number,Reynolds number and the upper wall Mach number. The critical Reynolds number is found as 5718.338 for an upper wall Mach number of 0.0001, corresponding to the fully Poiseuille case.

Les écoulements rampants, recirculants et instationnaires d’un fluide viscoplastique autour d’un cylindre ont été étudiés.Numériquement, les morphologies des écoulements, la localisation des zones rigides, les champs de contraintes et pression autour du cylindre ainsi que le coefficient de traînée, ont été déterminés sur un large domaine des nombres de Reynolds et d’Oldroyd.Expérimentalement, les fluides étudiés sont des gels de polymère Carbopol®. Le comportement élastoviscoplastique de ces gels a été modélisé par une loi d’Herschel-Bulkley adaptée. Le montage expérimental conçu et réalisé a été validé par l'étude de l'écoulement d'un fluide newtonien autour d'un cylindre et la mise en place d’une procédure adaptée pour les fluides à seuil.On a pu constater l'influence des conditions d’interface avec l’apparition d’une morphologie de lâchers de tourbillons simultanés et symétriques. / Creeping, recirculating and unsteady flows around a cylinder for yield stress fluid were studied.Numerically, morphologies of the flows, the location of the unyielded zones, the pressure and stress fields around the cylinder and the drag coefficient were determined over a wide range of Reynolds and Oldroyd numbers.Experimentally, fluids studied are polymer gels Carbopol®. The elastoviscoplastic behavior of these gels was modeled by a Herschel-Bulkley adapted law. The experimental setup was designed and validated by studying the flow of a newtonian fluid around a cylinder. An appropriate procedure for the viscoplastic fluid was implemented.We observe the influence of interface conditions with the appearance of a morphology showing simultaneous and symmetrical vortex shedding.