Levitation of a cylinder by a thin viscous film

When a horizontal cylinder is placed on a vertically moving belt coated with a thin layer of viscous fluid, experiments reveal that, at a specific belt velocity, the cylinder can be levitated at a fixed height while rotating around its own axis at an a priori unknown rate. We develop and solve a model for this experiment, using a combination of asymptotic analysis and direct numerical simulation. We obtain a relationship between belt speed and cylinder rotation rate which we successfully compare with experimental results

Acid Polishing of Lead Crystal Glass

The industrial partner manufactures high quality lead crystal glassware. The cutting of decorative features in the glass damages the surface and the cuts are optically opaque; to restore transparency, the glass is polished in a solution of hydrofluoric (HF) and sulphuric acid (H2 SO4 .) The polishing process comprises three stages: 1. immersion in a polishing tank containing acid; 2. rinsing in a tank containing water; and 3. settlement of the solid reaction products in a settlement tank. The manufacturer hopes to optimise its polishing process to • minimise the health/environmental impact of the process; • maximise throughput; • maintain the sharpness of the cut edges while still polishing to an acceptable level of transparency. The study group was asked to focus on modelling three aspects of the process: • the chemical reactions involved in the etching at the glass-acid solution interface; • the removal of reaction products in the settlement tank; • flow within the polishing tank

A model for the break-up of a tuft of fibers

A simple model for the forces acting on a single fiber as it is withdrawn from a tangled fiber assembly is proposed. Particular emphasis is placed on understanding the dynamics of the reptating fiber with respect to the entanglement of fibers within the tuft. The resulting two-parameter model captures the qualitative features of experimental simulation. The model is extended to describe the break-up of a tuft. The results show good agreement with experiment and indicate where a fiber is most likely to fracture based on the density of fiber end-points

Resonant oscillations of gases and liquids in three dimensions

Although extensive work has been carried out on one-dimensional resonant oscillations of both liquids in a tank (where the free surface varies in only one spatial dimension) and gases in a resonator, little is known about two-dimensional solutions. This thesis aims to unite and extend the knowledge about one-dimensional solutions and also develop a theory for classifying two-dimensional motions and, as a consequence, understand the different types of responses that may be found in tanks and resonators of arbitrary geometry. To do this we focus on (i) the nonlinearity and (ii) the geometry (and, hence, the nature of the spectrum) and ignore dissipation to lowest order although it is, in general, important. However, we can easily include dissipative effects a posteriori and its initial absence makes it easier to analyse the new two-dimensional effects. For reasons which will become apparent, we will mainly consider cuboid-shaped geometries and perturb the sidewalls of such tanks and resonators, allowing for the gradual introduction of two-dimensional effects. The thesis is split into two parts, underlying the differences between the problems that arise when the spectrum of the relevant linear problem is commensurate or non-commensurate. After a general introduction in Chapter 1 and a discussion of the model and governing equations in Chapter 2, the first part, comprising Chapter 3, looks at oscillations in deep water where the response typically consists of a finite number of modes. The second part is more extensive, looking at shallow water sloshing and the analogies of this problem with acoustic oscillations, both of which have a spectrum containing an infinite set of commensurate frequencies and the solution is much more intricate. We develop the problem and its one-dimensional solutions in Chapter 4 and then extend these ideas to two-dimensions in Chapter 5. With all this in mind we then make some general remarks about oscillations in tanks and resonators of arbitrary geometry in Chapter 6.</p

Resonant oscillations of gases and liquids in three dimensions

Although extensive work has been carried out on one-dimensional resonant oscillations of both liquids in a tank (where the free surface varies in only one spatial dimension) and gases in a resonator, little is known about two-dimensional solutions. This thesis aims to unite and extend the knowledge about one-dimensional solutions and also develop a theory for classifying two-dimensional motions and, as a consequence, understand the different types of responses that may be found in tanks and resonators of arbitrary geometry. To do this we focus on (i) the nonlinearity and (ii) the geometry (and, hence, the nature of the spectrum) and ignore dissipation to lowest order although it is, in general, important. However, we can easily include dissipative effects a posteriori and its initial absence makes it easier to analyse the new two-dimensional effects. For reasons which will become apparent, we will mainly consider cuboid-shaped geometries and perturb the sidewalls of such tanks and resonators, allowing for the gradual introduction of two-dimensional effects. The thesis is split into two parts, underlying the differences between the problems that arise when the spectrum of the relevant linear problem is commensurate or non-commensurate. After a general introduction in Chapter 1 and a discussion of the model and governing equations in Chapter 2, the first part, comprising Chapter 3, looks at oscillations in deep water where the response typically consists of a finite number of modes. The second part is more extensive, looking at shallow water sloshing and the analogies of this problem with acoustic oscillations, both of which have a spectrum containing an infinite set of commensurate frequencies and the solution is much more intricate. We develop the problem and its one-dimensional solutions in Chapter 4 and then extend these ideas to two-dimensions in Chapter 5. With all this in mind we then make some general remarks about oscillations in tanks and resonators of arbitrary geometry in Chapter 6.</p

Waves and compressible flow

Includes bibliographical references and index

Waves and compressible flow

Now in its second edition, this book continues to give readers a broad mathematical basis for modelling and understanding the wide range of wave phenomena encountered in modern applications.  New and expanded material includes topics such as elastoplastic waves and waves in plasmas, as well as new exercises.  Comprehensive collections of models are used to illustrate the underpinning mathematical methodologies, which include the basic ideas of the relevant partial differential equations, characteristics, ray theory, asymptotic analysis, dispersion, shock waves, and weak solutions. Although the main focus is on compressible fluid flow, the authors show how intimately gasdynamic waves are related to wave phenomena in many other areas of physical science.   Special emphasis is placed on the development of physical intuition to supplement and reinforce analytical thinking. Each chapter includes a complete set of carefully prepared exercises, making this a suitable textbook for students in applied mathematics, engineering, and other physical sciences.    Reviews of the first edition: "This book … is an introduction to the theory of linear and nonlinear waves in fluids, including the theory of shock waves. … is extraordinarily accurate and free of misprints … . I enjoyed reading this book. … most attractive and enticing appearance, and I’m certain that many readers who browse through it will wish to buy a copy. The exercises … are excellent. … A beginner who worked through these exercises would not only enjoy himself or herself, but would rapidly acquire mastery of techniques used…in JFM and many other journals…" (C. J. Chapman, Journal of Fluid Mechanics, Vol. 521, 2004) "The book targets a readership of final year undergraduates and first year graduates in applied mathematics. In the reviewer’s opinion, it is very well designed to catch the student’s interest … while every chapter displays essential features in some important area of fluid dynamics. Additionally, students may practice by solving 91 exercises. This volume is mainly devoted to inviscid flows. … The book is very well written." (Denis Serre, Mathematical Reviews, 2004) 

Capillary and viscous perturbations to Helmholtz flows

Inspired by recent calculations by Thoraval et al. (Phys. Rev. Lett., vol. 108, 2012, p. 264506) relating to droplet impact, this paper presents an analysis of the perturbations to the free surface caused by small surface tension and viscosity in steady Helmholtz flows. In particular, we identify the regimes in which appreciable vorticity can be shed from the boundary layer to the bulk flow