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Introduction

## Publications

Publications (30)

We investigate self-similar sign-changing solutions to the thin-film equation, ht = −(| h | ⁿhxxx ) x , on the semi-infinite domain x ≥ 0 with zero-pressure-type boundary conditions h = hxx = 0 imposed at the origin. In particular, we identify classes of first- and second-kind compactly supported self-similar solutions (with a free-boundary x = s (...

The calculus of variations provides a well-defined methodology for formulating differential equation models from problems which seek solutions that optimise some property of interest. The derivation of Euler–Lagrange equations from variational principles will be given in detail. Applications to problems in mechanics will be discussed. The elementar...

Problems that can be written as singularly perturbed systems of first order differential equations can be solved using approaches that combine matched asymptotic expansions with phase plane analysis. The \(\varepsilon \rightarrow 0\) limit yields a separation of time-scales that reduces the overall system to different forms for the fast and slow dy...

Dimensional scaling is a process allowing for the specific physical units defining the original form of a problem to be factored out to leave a scaled mathematical problem. The solutions of the scaled problem will depend on a set of nondimensional parameters obtained from combinations of the original given quantities. Two scaling principles are int...

We describe how the construction of similarity solutions of partial differential equations extends naturally from concepts in dimensional analysis. In particular, we show how to obtain self-similar solutions through scaling invariances of linear and nonlinear PDE. We give examples illustrating how similarity solutions of PDEs can be obtained from s...

In many problems, not all details of the structure of the solution of a partial differential equation are of interest. In some cases, it is possible to obtain an essential understanding of the behaviour of the system without directly solving the full equation. This chapter outlines the method of moments which can provide information about the evolu...

For problems that can be described as small perturbations of linear oscillators, the usual regular perturbation expansions can be shown to have limited validity. Difficulties arise from resonant forcing terms interacting with the expansions to produce misleading secular growth in the solutions at moderate to large times. We describe two methods, ca...

Multi-dimensional partial differential equation problems on slender domains, having a small aspect ratio, can be analysed primarily in terms of solutions of simpler ODE problems for the variation of the solution in the slender direction. Using the aspect ratio as an asymptotic parameter, solutions having slow-variation (or “long-wave” dependence) i...

We introduce the standard terminology used in perturbation methods and asymptotic analysis. Asymptotic expansions will be employed to construct solutions to introductory problems in algebraic/transcendental equations and ordinary differential equations. In particular, we introduce the iteration and expansion methods for solving such equations and d...

We provide a brief introduction to applications of ordinary differential equation (ODE) rate equations in chemistry, biology and physics. In mechanics, Newton’s laws prescribe how to write the rate equations while in other fields, the ODEs are based on different principles. In the context of chemical reactions, the law of mass action yields systems...

Solutions of singularly perturbed ordinary differential equations exhibit non-uniform convergence as \(\varepsilon \rightarrow 0\). Regular and singular solutions (also called outer and inner solutions respectively) may be needed to satisfy all of the conditions imposed in a boundary value problem. The construction of matched asymptotic expansions...

When the property of interest in a system depends on secondary independent variables (such as spatial position, particle speed, particle size) then its evolution can be described by a partial differential equation (PDE). In the context of properties depending on space and time, such PDEs are called transport equations, and they are generally writte...

This chapter covers the modelling of two applied problems in fluid dynamics in brief case studies. The problems, on air bearing sliders and rivulet flows in wedges, build on a common core model called lubrication theory, which is derived in the beginning of the chapter. The case studies differ in many physical aspects (compressible versus incompres...

This book presents mathematical modelling and the integrated process of formulating sets of equations to describe real-world problems. It describes methods for obtaining solutions of challenging differential equations stemming from problems in areas such as chemical reactions, population dynamics, mechanical systems, and fluid mechanics. Chapters 1...

We consider the dynamics of a symmetrically heated thin incompressible
viscous fluid sheet. We take surface tension to be temperature dependent
and consequently the streamwise momentum equation includes the effects
of thermocapillarity, inertia, viscous stresses, and capillarity. Energy
transport to the surrounding environment is also included. We...

We derive a general reduced model for the flow of a slender thread of viscous fluid on a grooved substrate. Specific choices of the substrate topography allow further analytic progress to be made, and we subsequently focus on a convection–diffusion equation governing the evolution of viscous liquid in a wedge geometry. The model equation that arise...

Research article writing has received a great deal of attention from ESP researchers. Analyses of the general structure of Introduction-Method-Results-Discussion (IMRD) articles, as well as detailed analyses of individual sections, including introductions, results, and discussions sections, have dominated the ESP literature, especially following Sw...

We consider the dynamics of a thin symmetric fluid sheet subject to an initial temperature profile, where inertia, viscous stresses, disjoining pressures, capillarity, and thermocapillarity are important. We apply a long-wave analysis in the limit where deviations from the mean sheet velocity are small, but thermocapillary stresses and heat transfe...

As discussed above, singular perturbation theory tackles difficult problems by investigating various reduced problems and then assembling the results together in an appropriate form. These reductions could, for example, simply be to a lower order polynomial in an algebraic problem, or could be more significant, such as in the reduction of a PDE to...

RESUMEN RESUMEN
We<sup> </sup> investigate self-similar solutions of the dipole problem for the one-dimensional<sup> </sup> thin film equation on the half -line { x \ge 0}. We<sup> </sup>study compactly supported solutions of the linear moving boundary problem<sup> </sup>and show how they relate to solutions of the nonlinear <sup> </sup>problem....

We consider the evolution of a thin viscous fluid sheet subject to thermocapillary effects. Using a lubrication approximation we find, for symmetric interfacial deflections, coupled evolution equations for the interfacial profile, the streamwise component of the fluid velocity and the temperature variation along the surface. Initial temperature pro...

We consider the problem of a thin film driven by a thermal gradient with an opposing gravitational force. Under appro-priate conditions, an advancing film front develops a leading undercompressive shock followed by a trailing compressive shock. Here, we investigate the nonlinear dynamics of these shock structures that describe a surprisingly stable...

We investigate self-similar solutions of the thin film equation in the case of zero contact angle boundary conditions on a finite domain. We prove existence and uniqueness of such a solution and determine the asymptotic behaviour as the exponent in the equation approaches the critical value at which zero contact angle boundary conditions become unt...

Alternating Direction Implicit (ADI) schemes are constructed for the solution of two-dimensional higher-order linear and nonlinear diffusion equations, particularly including the fourth-order thin film equation for surface tension driven fluid flows. First and second-order accurate schemes are derived via approximate factorization of the evolution...

This paper is based on a series of four lectures, by the first author, on thin films and moving contact lines. Section 1 presents an overview of the moving contact line problem and introduces the lubrication approximation. Section 2 summarizes results for positivity preserving schemes. Section 3 discusses the problem of films driven by thermal grad...

RESUMEN RESUMEN
We investigate similarity solutions of the " thin film" equation . In particular we look at solutions on the half - line $ x\ge0 $ with compact support and zero contact angle boundary conditions in x=3D0. Such " dipole " solutions feature an anomalous exponent and are therefore called similarity solutions of the second kind . Usin...

A variety of mass preserving moving boundary problems for the thin film equation, u t = Gamma(u n u xxx ) x , are derived (by formal asymptotics) from a number of regularisations, the case in which the substrate is covered by a very thin pre-wetting film being discussed in most detail. Some of the properties of the solutions selected in this fashio...

We study the porous medium equation with sign changes and examine the way sign changes disappear. We give a formal classification of selfsimilar and non-selfsimilar scenarios for their disappearance, for N ? 1 restricting attention to the radial case. The results we present on the classification of similarity solutions are rigorous except where ind...

We investigate the extinction behaviour of a fourth order degenerate diffusion equation in a bounded domain, the model representing the flow of a viscous fluid over edges at which zero contact angle conditions hold. The extinction time may be finite or infinite and we distinguish between the two cases by identification of appropriate similarity sol...