Thesis (Ph. D.)--University of Hong Kong, 1999. / Includes bibliographical references (leaves 77-83).

The aim of this thesis is to derive and solve mathematical models for the flow of liquid in a foam. A primary concern is to investigate how so-called `Marangoni stresses' (i.e. surface tension gradients), generated for example by the presence of a surfactant, act to stabilise a foam. We aim to provide the key microscopic components for future foam modelling. We begin by describing in detail the influence of surface tension gradients on a general liquid flow, and various physical mechanisms which can give rise to such gradients. We apply the models thus devised to an experimental configuration designed to investigate Marangoni effects. Next we turn our attention to the flow in the thin liquid films (`lamellae') which make up a foam. Our methodology is to simplify the field equations (e.g. the Navier-Stokes equations for the liquid) and free surface conditions using systematic asymptotic methods. The models so derived explain the `stiffening' effect of surfactants at free surfaces, which extends considerably the lifetime of a foam. Finally, we look at the macroscopic behaviour of foam using an ad-hoc averaging of the thin film models.

The aim of this thesis is to seek a low-dimensional description of baroclinic instability in general, and of the Martian atmosphere in particular, where both forcing and spatial resonance are relevant to the dynamics of the system being analysed. The Proper Orthogonal Decomposition (POD) is used to determine a basis for the modal decomposition of climatic simulations of Mars, obtained by using two General Circulation Models (GCMs): (a) a simple GCM, which is an idealised model in which the meteorological primitive equations are solved on a sphere with simplified physical parameters and (b) the Martian GCM, a more realistic model in which a comprehensive range of the relevant Martian physical parameters and topography are represented. Results of these analyses are presented for a range of Martian seasons and climatic conditions. The effects of using different forms of energy norm in performing the analysis is considered, with the objective of providing analyses which represents the physically most significant components of the circulation, with optimal efficiency. Reduced low-dimensional models that replicated the full simple GCM streamfunction simulations are formulated by projecting the spherical quasi-geostrophic equations onto the PODs of the large-scale calculations. The resulting models are analysed by using a combination of solution continuation and numerical integration methods. A thorough analysis of the models reveals that a 6-D POD model is capable of reproducing the amplitude, frequency and behaviour of the leading oscillatory structures of the simple GCM, to within a 1% error. Such an excellent reproduction of the original system is shown to be due to (1) an accurate vertical formulation scheme, (2) the use of the correct norm, (3) a sufficiently high level of truncation and (4) the fact that the original system is a steady wave flow. The behaviour of the various regimes observed in the low-order models are comparable with observations from studies of large-scale waves and instabilities in planetary atmospheres, including a range of hydrodynamical experiments on baroclinic wave interactions of a stratified fluid in cylindrical containers.

Over the past 30 years the study of free boundary problems has stimulated much work. However, there exists a widely occurring, but little studied subclass of free boundary problems in which the free boundary has dimension two fewer than that of the underlying space rather than the more commonly studied case of one less. These problems are called `codimension-two' free boundary problems. In Chapter 1 the typical geometries required for such problems, the main mathematical techniques and the methodology used are discussed. Then, in Chapter 2, the techniques required to solve them are demonstrated using the particular example of the water entry problem. Further results for the water entry problem are then derived including an analysis of the relatively poorly understood water exit problem. In Chapter 3 a review is given of some classical contact and crack problems in solid mechanics. The inclusion of a cohesive zone in a dynamic type-III crack problem is considered. The Muskhelishvili potential method is presented and used to solve both a contact and crack problem. This enables the solution of a type-I crack problem relating to an ink delivery system to be found. In Chapter 4 a problem posed by car windscreen forming is addressed. A local solution near a corner is analysed to explain when and how point forces occur at the corners of the frame on which the simply supported windscreen rests. Then the full problem is solved numerically for different types of boundary condition. Chapters 5 and 6 deal with several sintering problems in viscous flow highlighting the value of the methodology introduced in Chapter 1. It will be shown how the Muskhelishvili potential method also carries over to Stokes flow problems. The difficulties of matching to an inner as opposed to an outer region are investigated. Last two interface problems between immiscible liquids are considered which show how the solution procedure is adapted when the field equation in the thin region is non-trivial. In the final chapter results are summarised, open problems listed and conclusions drawn.