This dissertation considers the genesis and dynamics of interfacial instability in vertical laminar gas-liquid flows, using as a model the two-dimensional channel flow of a thin falling film sheared by counter-current gas. The methodology is linear stability theory by means of Orr-Sommerfeld analysis together with direct numerical simulation of the two-phase flow in the case of nonlinear disturbances. The influence of two main flow parameters on the interfacial dynamics, namely the film thickness and pressure drop applied to drive the gas stream, is investigated. To make contact with existing studies in the literature, the effect of various density and viscosity contrasts as well as surface tension is also examined. Energy budget analyses based on the Orr-Sommerfeld theory reveal various coexisting unstable modes (interfacial, shear, internal) in the case of high density contrasts, which results in mode coalescence and mode competition, but only one dynamically relevant unstable interfacial mode for low and intermediate density contrast. Furthermore, high viscosity contrast and increases in surface tension lead to some amount of mode competition for thin film. A study of absolute and convective instability for low density contrast shows that the system is absolutely unstable for all but two narrow regions of the investigated parameter space. These regions are extended at intermediate density contrast and exhibit only small changes with increased viscosity contrast or surface tension. Direct numerical simulations of the system with low density contrast show that linear theory holds up remarkably well upon the onset of large-amplitude waves as well as the existence of weakly nonlinear waves. For high density contrasts corresponding more closely to an air-water-type system, linear stability theory is also successful at determining the most-dominant features in the interfacial wave dynamics at early-to-intermediate times. Nevertheless, the short waves selected by the linear theory undergo secondary instability and the wave train is no longer regular but rather exhibits chaotic motion. Furthermore, linear stability theory also predicts when the direction of travel of the waves changes - from downwards to upwards. The practical implications of this change in terms of loading and flooding is discussed. The change in direction of the wave propagation is represented graphically for each investigated system in terms of a flow map based on the liquid and gas flow rates and the prediction carries over to the nonlinear regime with only a small deviation. Besides the semi-analytical and numerical analyses, experiments with an practically relevant setup and flow system have been carried out to benchmark and validate the models developed in this work.

This thesis is devoted to the study of some homogenization problems with applications in lubrication theory. It consists of an introduction, five research papers (I–V) and a complementary appendix.Homogenization is a mathematical theory for studying differential equations with rapidly oscillating coefficients. Many important problems in physics with one or several microscopic scales give rise to this kind of equations, whence the need for methods that enable an efficient treatment of such problems. To this end several mathematical techniques have been devised. The main homogenization method used in this thesis is called multiscale convergence. It is a notion of weak convergence in  Lp spaces which is designed to take oscillations into account. In paper II we extend some previously obtained results in multiscale convergence that enable us to homogenize a nonlinear problem with a finite number of microscopic scales. The main idea in the proof is closely related to a decomposition of vector fields due to Hermann Weyl. The Weyl decomposition is further explored in paper III.Lubrication theory is devoted to the study of fluid flows in thin domains. More generally, tribology is the science of bodies in relative motion interacting through a mechanical contact. An important aspect of tribology is to explain the principles of friction, lubrication and wear. The mathematical foundations of lubrication theory are given by the Navier–Stokes equation which describes the motion of a viscous fluid. In thin domains several simplifications are possible, as shown in the introduction of this thesis. The resulting equation is named after Osborne Reynolds and is much simpler to analyze than the Navier--Stokes equation.The Reynolds equation is widely used by engineers today. For extremely thin films, it is well-known that the surface micro-topography is an important factor in hydrodynamic performance. Hence it is important to understand the influence of surface roughness with small characteristic wavelengths upon the solution of the Reynolds equation. Since the 1980s such problems have been increasingly studied by homogenization theory. The idea is to replace the original equation with a homogenized equation where the roughness effects are “averaged out”. One problem consists of finding an algorithm for computing the solution of the homogenized equation. Another problem consists of showing, on introducing the appropriate mathematical definitions, that the homogenized equation is the correct method of averaging. Papers I, II, IV and V investigate the effects of surface roughness by homogenization techniques in various situations of hydrodynamic lubrication. To compare the homogenized solution with the solution of the deterministic Reynolds equation, some numerical examples are also included. / Godkänd; 2011; 20110408 (johfab); DISPUTATION Ämnesområde: Matematik/Mathematics Opponent: Professor Guy Bayada, Institut National des Sciences Appliquées de Lyon (INSA-LYON), Lyon, France, Ordförande: Professor Lars-Erik Persson, Institutionen för teknikvetenskap och matematik, Luleå tekniska universitet Tid: Tisdag den 7 juni 2011, kl 10.00 Plats: D2214/15, Luleå tekniska universitet