Long-wave Dynamics of Single- and Two-layer Flows
Creators: name: Mavromoustaki, Aliki
Publisher: name: Imperial College London
Year: 2011

description: Thin-film flows are central to a number of industrial, biomedical and daily-life applications, which include coating flow technology, enhanced oil recovery, microfluidics, and surfactant replacement therapy. Though these systems have received a lot of attention in a variety of settings, the understanding of the dominant physics governing the flows is not completely thorough; this is especially true in cases where the free surface of the film or, in two-layer flows, the fluid-fluid interface is susceptible to instabilities leading to the break-up of the film and the formation of fingering patterns. The elucidation of the underlying mechanisms behind the onset of these instabilities is of utmost importance to several industrial processes. The work in this thesis focusses on modelling the dynamics of thin-film flows in the presence of complexities; the latter arise from the presence of surface-active chemicals and spatial confinement. The lubrication approximation, which is valid in the limit of small film aspect ratios, is used to simplify the governing equations; this facilitates the derivation of an evolution equation for the interfacial position. This methodology is employed extensively in the present thesis to examine co- and counter-current two-layer flows in a closed, rectangular channel and the dynamics of a thin film laden with surfactant, driven to climb up an inclined substrate. In the two-fluid case, the dynamics of the flow are described by a single, two-dimensional, fourth-order nonlinear partial differential equation. Analysis of the one-dimensional flow demonstrate the existence of travelling-wave solutions which take the form of Lax shocks, undercompressive shocks, and rarefaction waves. In unstably-stratified cases, a Rayleigh-Taylor mechanism spawns the formation of large-amplitude capillary waves. A wide range of parameters is studied, which include the density and viscosity ratios of the two fluids, the flow configuration (whether co- or counter-current), the heights of the films at the channel ends and the channel inclination. The stability of these structures to perturbations in the spanwise direction, is also examined through a linear stability analysis and transient, two-dimensional numerical simulations. These analyses demonstrate successfully that some of the structures observed in the one-dimensional flow are unstable to fingering phenomena. In the case of the climbing film, two configurations are examined, namely, constant flux and constant volume whereby the evolution equation for the interface is coupled to convective-diffusive equations for the concentration of surfactant, present in the form of monomers and micelles. The former are allowed to exist at the gas-liquid and liquid-solid interfaces, and in the bulk; the latter can only be present in the bulk. For the constant flux case, the flow is simulated by a continuously-fed uncontaminated fluid and surfactant at the flow origin allowed to spread on a solid substrate which has been prewetted by a thin, surfactant-free precursor layer. The constant volume configuration simulates the deposition of a finite drop, laden with surfactant, spreading on a thin, uncontaminated film. In the absence of spanwise disturbances, the one-dimensional solutions demonstrate how the climbing rate and the structural deformation of the film are influenced by gravity, and physico-chemical parameters such as surfactant concentration (whether above or below the critical micelle concentration), and rates of adsorption of monomers at the two interfaces. The stability of the flow is examined through linear theory and transient solutions of the full, nonlinear, two-dimensional system of equations revealing the growth of spanwise perturbations into full-length fingers. A brief introduction to the experimental design of an apparatus, aimed at validating channel flow results, is also described. The objective of the experiment was to investigate the physical features associated with the counter-current, pressure-driven flow of a gas-liquid system. Preliminary experimental results revealed that upon perturbing the flow, an initially uniform liquid film becomes unstable, resulting in the formation of fingers which elongated downstream as time progressed. Finally, we conclude with recommendations for future work, representing natural extensions to the theoretical work described in the present thesis.