The understanding of wave dynamics in interfacial microchannel flows is important for many technological applications in the micro-device industry. Here, a theoretical and numerical study is undertaken in order to understand the propagation of interfacial waves in a two-layer flow. The flow is considered to be driven by, in separate cases, the force of gravity and a pressure gradient. The results may provide steps towards more efficiently designed microfluidic products, and a better understanding of experimentally observed waves.

In this thesis we generalise the rigidity estimates of Friesecke et al. [2002] and Müller et al. [2014] to vector fields whose properties are constrained by both conditions on the support of their curl and the underlying discrete symmetries of the lattice Z2. These analytical estimates and other considerations are applied to a statistical model of a crystal containing defects based on work by Aumann [2015]. It is demonstrated in this thesis that we allow a finite density of defects. The main result is that regardless of crystal size, the ordering of the crystal, expressed via the L2-distance of a random vector field from the rotations, can be made arbitrarily small for sufficiently low temperature β-1.

We investigate the existence of phase transitions for a class of continuum multi-type particle systems. The interactions act on hyperedges between the particles, allowing us to define a class of models with geometry-dependent interactions. We establish the existence of stationary Gibbsian point processes for this class of models. A phase transition is defined with respect to the existence of multiple Gibbs measures, and we establish the existence of phase transitions in our models by proving that multiple Gibbs measures exist. Our approach involves introducing a random-cluster representation for continuum particle systems with geometry-dependent interactions. We then argue that percolation in the random-cluster model corresponds to the existence of a phase transition. The originality in this research is defining a random-cluster representation for continuum models with hyperedge interactions, and applying this representation in order to show the existence of a phase transition. We mainly focus on models where the interaction is defined in terms of the Delaunay hypergraph. We find that phase transitions exist for a class of models where the interaction between particles is via Delaunay edges or Delaunay triangles.

This work explores the nature of the normal modes of vibration for harmonic lattices with the inclusion of disorder in one-dimension (1D) and three-dimensions (3D). The model systems can be visualised as a `ball' and `spring' model in simple cubic configuration, and the disorder is applied to the magnitudes of the masses, or the force constants of the interatomic `springs' in the system. With the analogous nature between the electronic tight binding Hamiltonian for potential disordered electronic systems and the isotropic Born model for phonons in mass disordered lattices we analyse in detail a transformation between the normal modes of vibration throughout a mass disordered harmonic lattice and the electron wave function of the tight-binding Hamiltonian. The transformation is applied to density of states (DOS) calculations and is also particularly useful for determining the phase diagrams for the phonon localisation-delocalisation transition (LDT). The LDT phase boundary for the spring constant disordered system is obtained with good resolution and the mass disordered phase boundary is verified with high precision transfer-matrix method (TMM) results. High accuracy critical parameters are obtained for three transitions for each type of disorder by finite size scaling (FSS), and consequently the critical exponent that characterises the transition is found as = 1:550+0:020 -0:017 which indicates that the transition is of the same orthogonal universality class as the electronic Anderson transition. With multifractal analysis of the generalised inverse participation ratio (gIPR) for the critical transition frequency states at spring constant disorder width k = 10 and mass disorder width m = 1:2 we confirm that the singularity spectrum is the same within error as the electronic singularity spectrum at criticality and can be considered to be universal. We further investigate the nature of the modes throughout the spectrum of the disordered systems with vibrational eigenstate statistics. We find deviations of the vibrational displacement uctuations away from the Porter-Thomas distribution (PTD) and show that the deviations are within the vicinity of the so called `bosonpeak' (BP) indicating the possible significance of the BP.

Surface partial differential equations model several natural phenomena; for example in uid mechanics, cell biology and material science. The domain of the equations can often have complex and changing morphology. This implies analytic techniques are unavailable, hence numerical methods are required. The aim of this thesis is to design and analyse three methods for solving different problems with surface partial differential equations at their core. First, we define a new finite element method for numerically approximating solutions of partial differential equations in a bulk region coupled to surface partial differential equations posed on the boundary of this domain. The key idea is to take a polyhedral approximation of the bulk region consisting of a union of simplices, and to use piecewise polynomial boundary faces as an approximation of the surface and solve using isoparametric finite element spaces. We study this method in the context of a model elliptic problem. The main result in this chapter is an optimal order error estimate which is confirmed in numerical experiments. Second, we use the evolving surface finite element method to solve a Cahn- Hilliard equation on an evolving surface with prescribed velocity. We start by deriving the equation using a conservation law and appropriate transport formulae and provide the necessary functional analytic setting. The finite element method relies on evolving an initial triangulation by moving the nodes according to the prescribed velocity. We go on to show a rigorous well-posedness result for the continuous equations by showing convergence, along a subsequence, of the finite element scheme. We conclude the chapter by deriving error estimates and present various numerical examples. Finally, we stray from surface finite element method to consider new unfitted finite element methods for surface partial differential equations. The idea is to use a fixed bulk triangulation and approximate the surface using a discrete approximation of the distance function. We describe and analyse two methods using a sharp interface and narrow band approximation of the surface for a Poisson equation. Error estimates are described and numerical computations indicate very good convergence and stability properties.

No description available.

The thesis starts with a short introduction to smooth dynamical systems and Hamiltonian dynamical systems. The aim of the introductory chapter is to collect basic results and concepts used in the thesis to make it self–contained. The second chapter of the thesis deals with the interaction of two charges moving in R2 in a magnetic field B. This problem can be formulated as a Hamiltonian system with four degrees of freedom. Assuming that the magnetic field is uniform and the interaction potential has rotational symmetry we reduce this Hamiltonian system to one with two degrees of freedom; for certain values of the conserved quantities and choices of parameters, we obtain an integrable system. Furthermore, when the interaction potential is of Coulomb type, we prove that, for suitable regime of parameters, there are invariant subsets on which this system contains a suspension of a subshift of finite type. This implies non–integrability for this system with a Coulomb type interaction. Explicit knowledge of the reconstruction map and a dynamical analysis of the reduced Hamiltonian systems are the tools we use in order to give a description for the various types of dynamical behaviours in this system: from periodic to quasiperiodic and chaotic orbits, from bounded to unbounded motion. In the third chapter of the thesis we study the interaction of two charges moving in R3 in a magnetic field B. This problem can also be formulated as a Hamiltonian system, but one with six degrees of freedom. We keep the assumption that the magnetic field is uniform and the interaction potential has rotational symmetry and reduce this Hamiltonian system to one with three degrees of freedom; for certain values of the conserved quantities and choices of parameters, we obtain a system with two degrees of freedom. Furthermore, when the interaction potential is chosen to be Coulomb we prove the existence of an invariant submanifold where the system can be reduced by a further degree of freedom. The reductions simplify the analysis of some properties of this system: we use the reconstruction map to obtain a classification for the dynamics in terms of boundedness of the motion and the existence of collisions. Moreover, we study the scattering map associated with this system in the limit of widely separated trajectories. In this limit we prove that the norms of the gyroradii of the particles are conserved during an interaction and that the interaction of the two particles is responsible for a rotation of the guiding centres around a fixed centre in the case of two charges whose sum is not zero and a drift of the guiding centres in the case of two charges whose sum is zero.

This thesis focuses on the role played by fluctuations in both thermal optimisation techniques and diffusion limited aggregation. The key idea is that by tuning the level of input noise asymptotic results may be attained more rapidly. Stochastic optimisation problems are considered, where the function being optimised cannot be known exactly and may only be estimated. The noise in the estimates is used as the analogue of thermal fluctuations in simulated annealing. This analogy is made exact by use of an acceptance function, and stochastic annealing is seen to be the generalisation of simulated annealing to stochastic optimisation problems. The probabilistic travelling salesman problem (PTSP) is used as a test-bed for stochastic annealing, and significant new results are found. A good characterisation is found for the PTSP and scaling arguments are shown to be accurate for determining the expected length of the pruned and a priori tours, specifically E(Lpruned)=βpruned(p)√np. An oil field project, as a complex commercial problem, is considered and stochastic annealing is seen to make a large improvement in the expected return of the project. Noise reduction in diffusion limited aggregation (DLA) is known to be crucial to our understanding. A generalisation of noise reduction off-lattice is introduced, and noise reduction is shown to be a central parameter controlling the growth of DLA. In 2 dimensions, all quantities appear to be influenced by the slowest correction to scaling, with exponent ~ 0.3. In 3 dimensions, some quantities are not affected by the slowest correction to scaling, exponent ~ 0.2. The renormalisation of DLA is considered, and the noise reduction at the fixed point is measured. The noise, given as the relative variability in extremal cluster radius is found to be ε*2D ≃ 0.003 and ε*3D ≃ 0.006 in 2 and 3 dimensions, respectively.

There are many branches of engineering science that require solution of fluid flow problems. Some of these examples are aerodynamics of aircraft and vehicles, hydrodynamics of ships, electrical and electronic engineering and many others. Some of these flows may involve complex geometrical shapes which are usually modelled using the unstructured mesh discretisation techniques. There are well established methods that are used in such simulations. The aim of this project is to investigate the staggered positioning of variables on an unstructured based context and hence compare it to well known methods such as the cell-centred approach. A two dimensional unstructured staggered mesh discretisation method for the solution of fluid flow and heat transfer problems has been developed. This method stores and solves the vector variables at the cell faces and other scalar variables are stored at the cell centres. The very well known pressure based scheme SIMPLE is employed for pressure and velocity coupling. Three different approaches on unstructured staggered meshes are proposed. The first method solves for normal velocity component and interpolates the tangential velocity component, the second method solves for normal and tangential velocity components whereas the third method also solves for normal and tangential velocity components but uses a different upwind scheme for convection. The discretisation on unstructured staggered mesh methods is validated for a variety of fluid flow and heat transfer problems and comparisons are made between unstructured staggered mesh methods, the cell-centred approach and benchmark solutions. The first and third unstructured staggered mesh methods are shown to perform well and give comparable results to benchmark solutions. The third unstructured staggered mesh method does not always work.

The research work reported herein addresses the problem of mathematical modelling of fluid flow and combustion in internal combustion engines. In particular, the investigation of three topics that constitute prime sources of uncertainty, in current numerical models, namely turbulence modelling, inaccuracies in the solution procedure specific to moving grids, and combustion modelling. Two and three-dimensional computations of the in-cylinder turbulent flow in a diesel engine are described first, with emphasis on the modifications made to the standard k- model of turbulence to account for rapid compression/expansion, and on the k-W model also used in the computations. It is concluded that the standard k- model may lead to poor predictions when used for internal combustion engine simulations, and that the modified model leads to more reasonable length-scale distributions, improving significantly the overall agreement of velocity predictions with experiment. It is also demonstrated that the k-W model provides better turbulence predictions than the unmodified k- model for the cases considered. The moving boundary within a reciprocating engine poses the problem that as it moves toward the cylinder head it compresses the computational grid cells, creating large aspect ratios that can adversely affect the numerical accuracy and convergence. A conservative scheme has therefore been devised that allows for the removal or addition of grid cells during the simulation, so as to maintain reasonable aspect ratios. It is concluded that with the proposed scheme convergence is obtained within fewer iterations, computational cost is therefore reduced, and that the results are generally in better agreement with experimental data. The third part of this study investigates and compares the performance of the two most commonly used combustion models (the eddy-break-up and the Arrhenius models) and proposes a new formulation of a flame-front model. Calculations have been performed for a one-dimensional test case and for a representative spark-ignition engine in order to determine the grid and time step requirements for numerical accuracy, the sensitivity of results to empirical input and the physical realism of the predictions by comparison with experimental data. It has been found for the cases considered that neither the eddy-break-up nor the Arrhenius models are appropriate for predicting engine combustion. The Arrhenius model does not represent well the combustion process for the cases considered. The eddy-break-up model is not capable of predicting the observed flame front, and the empirical constants in the model require extensive tuning to obtain predictions that match experiments. The flame-front model however, in spite of many simplifications, produces much more realistic flame-front propagation and the empirical input of the model, i.e. the flame speed, can in principle be obtained by other means other than ad-hoc tuning. It is concluded that the flame-front model requires refinement, but for the cases considered, it provides the basis of a very promising combustion model for predicting premixed combustion in engines.