We investigate the behavior of moving droplets and rivulets, driven by a combination of gravity and surface shear (wind). The problem is motivated by a desire to model the behavior of raindrops on aircraft wings. We begin with the Stokes equations and use the approximations of lubrication theory to derive the specific thin film equation relevant to our situation. This fourth-order partial differential equation describing the height of the fluid is then solved numerically from varying initial conditions, using a fully implicit discretization for time stepping, and a precursor film to avoid singularities at the drop contact line. Results describing general features of droplet deformation, limited parameter studies, and the applicability of our implementation to the long-term goal of modeling wings in rain are discussed.

A lubrication-flow model for a free film in a corner is presented. The model, written in the hyperbolic coordinate system ξ = x² – y², η = 2xy, applies to films that are thin in the η direction. The lubrication approximation yields two coupled evolution equations for the film thickness and the velocity field which, to lowest order, describes plug flow in the hyperbolic coordinates. A free film in a corner evolving under surface tension and gravity is investigated. The rate of thinning of a free film is compared to that of a film evolving over a solid substrate. Viscous shear and normal stresses are both captured in the model and are computed for the entire flow domain. It is shown that normal stress dominates over shear stress in the far field, while shear stress dominates close to the corner.

Materials can flow down a slope in a wide range of geophysical and industrial contexts, including lava flows on volcanoes and thin films on coated surfaces. The aim of my research is to provide quantitative insight into these forms of motion and their dependence on effects of the topography, the volume and the rheology of the flowing structure. Numerous different problems are investigated through mathematical models, which are developed analytically and confirmed by laboratory experiments. The initial advance of long lava flows is studied by considering the flow of viscous fluid released on sloping channels. A scaling analysis, in agreement with analog experiments and field data, offers a practical tool for predicting the advance of lava flows and conducting hazard analysis. A simple and powerful theory predicts the structure of flows resulting from any time-dependent release of fluid down a slope. Results obtained by the method of characteristics reveal how the speed of the advancing front depends importantly on the rate of fluid supplied at an earlier time. Viscous flows on surfaces with different shapes are described by similarity solutions to address problems motivated by engineering as well as geophysical applications. Pouring viscous fluid out of a container can be a frustratingly slow process depending on the shape and the degree of tipping of the container. The discharge rate of the fluid is analysed in simple cases, shedding light on how containers can be emptied most quickly in cosmetic and food industries. In a separate study motivated by coating industries, thin films are shown to evolve with uniform thickness as they drain near the top of a horizontal cylinder or sphere. The leading edge eventually splits into rivulets as predicted theoretically and confirmed by experiments. Debris flows can develop levees and trigger avalanches which are studied by considering dense granular flows down a rough inclined plane. Granular materials released down a slope can produce a flowing structure confined by levees or trigger avalanches at regular intervals, depending on the steady rate of supply. The experimental results are discussed using theoretical ideas of shallow granular flows. Finally, materials flowing in long and slender ducts are investigated theoretically to better understand the digestive and urinary systems in biology. The materials are pumped in an elastic tube by translating waves of muscular contraction and relaxation. The deformation of the tube is predicted by solving a free-boundary problem, a similar mathematical exercise to predicting the moving boundaries of materials spreading on slopes.

Thin liquid films driven by surface tension gradients are studied in diverse applications, including the spreading of a droplet and fluid flow in the lung. The nonlinear partial differential equations that govern thin films are difficult to solve analytically, and must be approached through numerical simulations. We describe the development of a numerical solver designed to solve a variety of thin film problems in two dimensions. Validation of the solver includes grid refinement studies and comparison to previous results for thin film problems. In addition, we apply the solver to a model of surfactant spreading and make comparisons with theoretical and experimental results.

We consider mathematical modeling of thin film flow between two rough surfaces which are in relative motion. For example such flows take place in different kinds of bearings and gears when a lubricant is used to reduce friction and wear between the surfaces. The mathematical foundations of lubrication theory is given by the Navier--Stokes equation, which describes the motion of viscous fluids. In thin domains several approximations are possible which lead to the so called Reynolds equation. This equation is crucial to describe the pressure in the lubricant film. When the pressure is found it is possible to predict vorous important physical quantities such as friction (stresses on the bounding surfaces), load carrying capacity and velocity field. In hydrodynamic lubrication the effect of surface roughness is not negligible, because in practical situations the amplitude of the surface roughness are of the same order as the film thickness. Moreover, a perfectly smooth surface does not exist in reality due to imperfections in the manufacturing process. Therefore, any realistic lubrication model should account for the effects of surface roughness. This implies that the mathematical modeling leads to partial differential equations with coefficients that will oscillate rapidly in space and time. A direct numerical computation is therefore very difficult, since an extremely dense mesh is needed to resolve the oscillations due to the surface roughness. A natural approach is to do some type of averaging. In this PhD thesis we use and develop modern homogenization theory to be able to handle the questions above. Especially, we use, develop and apply the method based on the multiple scale expansions and two-scale convergence. The thesis is based on five papers (A-E), with an appendix to paper A, and an extensive introduction, which puts these publications in a larger context. In Paper A the connection between the Stokes equation and the Reynolds equation is investigated. More precisely, the asymptotic behavior as both the film thickness <img src="http://www.diva-portal.org/cgi-bin/mimetex.cgi?%5Cepsilon" /> and wavelength <img src="http://www.diva-portal.org/cgi-bin/mimetex.cgi?%5Cmu" /> of the roughness tend to zero is analyzed and described. Three different limit equations are derived. Time-dependent equations of Reynolds type are obtained in all three cases (Stokes roughness, Reynolds roughness and high frequency roughness regime). In paper C we extend the work done in Paper A where we compare the roughness regimes by numeric computations for the stationary case. In paper B we present a mathematical model that takes into account cavitation, surfaces roughness and compressibility of the fluid. We compute the homogenized coefficients in the case of unidirectional roughness.In the paper D we derive a mathematical model of thin film flow between two close rough surfaces, which takes into account cavitation, surface roughness and pressure dependent density. Moreover, we use two-scale convergence to homogenize the model. Finally, in paper E we prove the existence of solutions to a frequently used mathematical model of thin film flow, which takes cavitation into account.

High-Frame-Rate Oil Film Interferometry Jonathan Charles White This thesis presents the design and implementation of a high-frame-rate oil film interferometry technique (HOFI) used to directly measure skin friction in time dependent flows. Experiments were performed to determine the ability of a high-speed camera to capture oil film interferometry images. HOFI was found to be able to capture these interferometry images at frequencies up to 105 Hz. Steady laminar and turbulent flows were tested. Transient flows tested consisted of a wind tunnel ramping up in velocity and a laminar boundary layer which was intermittently tripped to turbulence by puffing air out of a pressure tap. Flow speeds ranged from 0 to 108 ft/sec and 10 and 50 cSt Dow Corning 200 dimethylpolysiloxane silicone oil was used. The skin friction was determined from the rate of change of the height of the oil film using lubrication theory. The height of the oil film was determined from the high speed camera interferogram images using a MATLAB script which determined fringe spacing by fitting a four-parameter sine wave to the intensity levels in each image. The MATLAB script was able to determine the height of the oil film for thousands of interferogram images in only a few minutes with sub-pixel error in fringe spacing. The skin friction was calculated using the oil film height history allowing for the direct measurement of skin friction in time dependent flows.

Il est bien connu que les particules inertielles dans un écoulement périodique ont tendance à se focaliser sur des trajectoires privilégiées. Le but de ce travail de thèse est d'étudier l'influence de cette focalisation sur le transport et la sédimentation de particules dans une fracture plane à rugosité périodique. Tout d'abord, un écoulement monophasique dans une fracture est analysé asymptotiquement dans le cas de faible rugosité. Les résultats classiques de la théorie de la lubrification inertielle sont généralisés au cas de fractures avec des parois asymétriques. Les corrections non linéaires à la loi de Darcy sont calculées explicitement en fonction des facteurs géométriques de la fracture. Le transport de particules dans une fracture horizontal est étudié asymptotiquement dans le cas de particules de faible inertie. Les particules se focalisent sur une trajectoire attractrice, si le débit d'écoulement est assez fort par rapport à la gravité. Un diagramme complet de focalisation a été obtenu, qui prédit l'existence de l'attracteur en fonction du nombre de Froude et des facteurs géométriques de la fracture. Les paramètres quantitatifs du transport ont été calculés également. L'influence de la force de portance sur la migration de particules a été étudiée également. Dans un canal vertical, la portance (provoquée par la gravité) modifie le nombre d'attracteurs et leurs positions. En absence de gravité, la portance peut provoquer une dynamique chaotique des particules. En outre, le captage des particules par une paire de tourbillons a été étudié. Le diagramme d'accumulation obtenu démontre que toute paire de tourbillons peut être un piège à particules / It is well-known that inertial particles tend to focus on preferential trajectories in periodic flows. The goal of this thesis was to study the joint effect of particle focusing and sedimentation on their transport through a model 2D fracture with a periodic corrugation. First, single-phase flow though the fracture has been considered: the classical results of the inertial lubrication theory are revisited in order to include asymmetric fracture geometries. Cubic corrections to Darcy's law have been found analytically and expressed in terms of two geometric factors, describing channel geometry. For weakly-inertial particles in a horizontal channel it has been shown that, when inertia is strong enough to balance out the gravity forces, particles focus to some attracting trajectory inside the channel. The full trapping diagram is obtained, that predicts the existence of such attracting trajectory regime depending on the Froude number and on geometric factors. Numerical simulations confirm the asymptotic results for particles with small response times. The influence of the lift force on particle migration has also been studied. In a vertical channel the lift is induced by gravity and leads to complex trapping diagrams. In the absence of gravity the lift is caused by inertial lead/lag of particles and can lead to chaotic particle dynamics. Finally, for dust particles in a vortex pair it has been shown that particles can be trapped into one or two equilibrium points in a reference frame rotating with the vortices. A full trapping diagram has been obtained, showing that any pair of vortices can trap particles, independently of their strength ratio and the direction of rotation