A Meeting in Memory of Professor Andy King

  A picture of Andy


Professor Andy King, who died in January of this year, was an outstanding applied mathematician, with wide-ranging research interests. At this meeting, sixteen of his friends and collaborators will give invited talks on subjects relevant to Andy's work. Topics will include industrial mathematical modelling, reaction-diffusion equations, free surface flows and combustion. Confirmed speakers are:

Professor David Abrahams, University of Manchester
Professor John Billingham, University of Nottingham
Dr Stephen Decent, University of Birmingham
Professor Sam Falle, University of Leeds
Professor John King, University of Nottingham
Professor John Merkin, University of Leeds
Professor David Needham, University of Reading
Dr John Ockendon, University of Oxford
Professor David Parker, University of Edinburgh
Professor David Riley, University of Nottingham
Dr Ruben Schulkes, Norsk Hydro
Dr Nigel Scott, University of East Anglia
Dr Gary Sharpe, University of Leeds
Dr Yulii Shikhmurzaev, University of Birmingham
Professor Jean-Marc Vanden-Broeck, University of East Anglia
Professor Graham Wilks, University of Keele

The meeting will be held at the Department of Mathematics at the University of Reading, where Andy was a Special Professor. The meeting will start at 1.30pm on January 4th and end at 3.30pm on January 5th, 2006. All are welcome to attend, but a registration fee of £50 will be charged to cover overnight accommodation and the conference dinner. However, we have funds to cover the costs of eight research students. A special issue of the IMA Journal of Applied Mathematics dedicated to Andy, and containing contributions from most of the speakers, will appear after the meeting. The closing date for registration was 16th December 2005, and the final list of participants is given below.

This meeting has been made possible by funding from the London Mathematical Society and the University of Reading.

Accommodation and meals will be at St Patrick's Hall, Northcote Avenue, and the talks will be in the Engineering Gordon Lecture Theatre in the Engineering Building.
St Patrick's Hall and the Engineering Building are marked on this map (the Engineering Building is building number 25).

A draft programme is now available.


Final list of participants

Mr James Andrews, University of Birmingham
Professor Simon Chandler-Wilde, University of Reading
Professor John Chapman, Keele University
Miss Natalie Culverhouse, University of Birmingham
Mr Geoffrey Curtis, University of Birmingham
Professor Ulf Ehrenmark, London Metropolitan University
Professor David Evans, University of Bristol
Mr Robert Harter, University of Manchester
Dr Chris Howls, University of Southampton
Mr Zahir Hussain, University of Birmingham
Professor Oliver Jensen, University of Nottingham
Professor Andrew Lacey, Heriot Watt University
Dr John Leach, University of Reading
Dr Stephen Langdon, University of Reading
Mr Jeremy Marston, University of Birmingham
Mr Moss Mokgolele, Univesity fo Reading
Dr William Parnell, University of Manchester
Professor Howell Peregrine, University of Bristol
Mr Sylvain Reboux, University of Nottingham
Professor Norman Riley, University of East Anglia
Dr Mark Simmons, University of Birmingham
Dr Warren Smith, University of Birmingham
Mr James Sprittles, University of Birmingham
Dr Sharon Stephen, University of Birmingham
Dr Richard Tew, University of Nottingham
Dr Ian Thompson, University of Loughborough
Mr Jamal Uddin, University of Birmingham
Mr Paul Wakeley, University of Birmingham
Professor Adam Wheeler, University of Southampton
Mr Ian Williams, University of Bristol


Talks

David Abrahams and William Parnell: Wave propagation in periodic and random composite materials

This talk will examine models of wave propagation through materials containing large numbers of inclusions. Such problems arise in many areas; examples include carbon fibre materials in engineering structures  particle suspensions in fluids, porous media in geophysical applications, biological tissues such as bone. The inclusions often have material properties (eg density, shear modulus, refractive index) quite distinct from those of the host phase, may have complicated shape, and can be distributed randomly or in a periodic arrangement. The orientation of the bodies (eg elliptical shaped fibres) may also be distributed in a random fashion, or aligned in a fixed direction. We are interested in the long wavelength limit and seek the effective (homogenised) material properties of the composite structure.

Composite materials have complex wave characteristics because of the multiple scattering interactions between the inclusions. For periodic composites, however, there are a number of techniques available to tackle the problem (see [1] for details) and we will discuss an asymptotic method developed recently [2] which yields an effective anisotropic wave equation of simple form. For random distributions there have been fewer approaches offered to solve the problem. The main method, still pursued actively today [3], is due to Foldy [4] and uses a closure assumption such as the quasi crystalline approximation to determine the effective wave characteristics. Whilst, offering the correct leading order (low frequency) effective wavenumber, this method usually yields an imaginary term at higher orders which indicates energy loss as the wave propagates. We discuss this energy loss for a very simple model problem, and compare the result with that found by an alterative averaging approach.

[1] Parnell, WJ & Abrahams, ID. A new integral equation approach to elastodynamic homogenisation. To be submitted, 2005.
[2] Parnell, WJ & Abrahams, ID. Dynamic homogenization in periodic fibre reinforced media; quasi-static Limit for SH Waves. Submitted to Wave Motion, 2005.
[3] Linton, CM & Martin, PA. Multiple scattering by random configurations of circular cylinders: second order corrections for the effective wavenumber. J. Acoust. Soc. Amer., 117:3413-3423, 2005.
[4] Foldy, LL. Multiple scattering theory of waves. Phys. Rev., 67:107-119, 1945.

John Billingham: Moving contact lines in slender fluid wedges

In this talk, I will consider the problem of the response to a change in contact angle of a slender, initially stationary wedge of inviscid fluid bounded by a free surface and a solid surface. The solution is of similarity form, with lengths scaling on t2/3. Andy published a paper in 1991 on the asymptotic solution of this problem for a small change in contact angle. I will show that the problem can be solved using Kuzmak's method, firstly when the contact angle is small, but much larger than the wedge angle, and secondly, when the contact angle is of O(1). In particular, this includes the case of a 90o contact angle, which is equivalent to the recoil of a slender wedge of inviscid fluid bounded by two free surfaces. I will also show that no simply-connected solution exists for contact angles greater than 90o when the wedge angle is sufficiently small.

Stephen Decent: Mathematical and experimental examination of breaking jets and threads

This talk will examine some of the work arising out of the theoretical-experimental links built up by Andy King during his time at Birmingham between the Schools of Mathematics and Chemical Engineering, and especially in breaking jets and threads. Industrially motivated flows, such as those arising in prilling, will be examined.

Sam Falle: The role of thermal instability in star formation

It has been known for many years that the material between the stars, the Interstellar Medium (ISM), can exist in two distinct states of thermal equilibrium in which heating by energetic particles balances radiative cooling.  These are: a cold phase with densities above ~ 5 atoms per cc and a temperature of ~100 K; a warm phase with density below ~ 0.5 atoms per cc and temperatures above ~6000K. At densities between these limits, the gas is unstable and is liable to form density inhomogeneities that can subsequently collapse to form stars.

In this talk I will discuss the interaction between this instability and magnetohydrodynamic waves and show that these can explain many of the observations of star forming regions. In particular, it seems that self-gravity may not play much of a role in the initial fragmentation of interstellar clouds, which is contrary to current received wisdom.

John King: Waiting-time behaviour for the thin-film equation

The small- and waiting-time properties of the thin-film equation will be discussed and contrasted with those of the porous-medium equation, for which a much narrower set of phemomena is possible.

A.V.Lukyanov, Y.D.Shikhmurzaev and A.C.King: Nonlocal dependence of dynamic contact angle on wetting speed in low-Reynolds-number curtain coating

The process of curtain coating by high-viscosity liquids is analyzed numerically in the framework of a theory that  describes it as a particular case of flow with disappearance/formation of interfaces. The results show that there is no unique dependence of the dynamic contact angle on the wetting speed: the contact angle is not a function of the wetting speed; it is a functional of the flow field/geometry.

J.H. Merkin: The stability of autocatalytic reaction fronts

Reaction fronts in autocatalytic systems can become unstable through diffusion-driven instabilities and through buoyancy-driven instabilities. The first case requires a difference between the diffusion coefficients of the autocatalyst and substrate. This can be achieved through an additional complexation reaction (starch). The second case arises through the differences in density of the reactant solution resulting from the reaction. These cases are described and some new results are presented.  

David Needham: Travelling waves in the hyperbolic Fisher equation

This paper considers permanent form travelling wave (PTW) structures in a generalized hyperbolic Fisher equation. Particular attention is given to the existence of subsonic, sonic and supersonic PTW structures.

John Ockendon: Delta Shocks

This talk will discuss the use of delta shocks in hypercritical shallow water flow.

David Parker


David Riley, C.J.Noakes and J.R.King: A multiple scales approach to developing rational approximations for coating and rimming flows on rapidly rotating cylinders

New generations of aeroengines are operating at ever higher rotation rates. The associated lubrication problems are such that inertial effects cannot be neglected and classical lubrication theory fails. In this study, analysis of thin-film rimming and coating flows is extended to create a more general model that represents a new distinguished limit. Specifically, three-dimensional flows under the influence of gravity are considered in which the Reynolds number is large enough to necessitate consideration of both inertial and centrifugal effects, along with those of viscous, gravitational and surface-tension forces. Reduction of dimensionality is shown to be attainable within a systematic (multiple-timescale) asymptotic framework that encompasses all these effects, leading to a two-dimensional formulation that not only is more general than many of the existing ones but also represents a rational approximation.

Nigel Scott: The propagation of a temperature wave in generalized isotropic thermoelasticity

In one-dimensional isotropic thermoelasticity the longitudinal particle displacement and temperature increment are coupled together in the system of partial differential equations describing linear momentum balance and energy balance. This parabolic system of equations has one thing in common with the parabolic equation of temperature diffusion: an initial temperature disturbance of bounded support propagates to infinity in infinitesimal time. In fact, the same is true of an initial particle displacement. In the theory of generalized thermoelasticity Fourier's constitutive equation for heat conduction is replaced by one in which a small relaxation time is associated with the time derivative of the heat flux, thereby turning the parabolic system into a weakly hyperbolic one. The effect of this non-zero relaxation time is to give finite speeds of propagation at all frequencies for the particle displacement and temperature increment. This is known as the phenomenon of  second sound. Here, the nature of the connection between the weakly hyperbolic system and the parabolic system is explored for decreasing relaxation times

Gary Sharpe: Explosions with a three-step chain-branching chemistry model

The majority of theoretical work on gaseous explosions has used a simple one-step chemistry model. However, such a simple model does not properly describe the way in which heat is released in many real fuels. Here we examine a three-step chain-branching model in the context of constant volume explosions and shock-induced ignition, including the importance of the concept of 'chain-branching cross-over temperatures' which simpler models do not capture. However, it is shown that within particular parameter regimes, the three-step model can be effectively reduced to either the one-step model or to a simpler two-step model.

Ruben Schulkes: Stability of stratified gas-liquid flow

In this presentation we revisit the problem of linear stability of interfacial waves in a co-flowing gas-liquid system. The influence of profile coefficients on the neutral stability curve is investigated. It is shown that taking the profile coefficients slightly larger than unity has surprisingly large effects on the neutral stability curves. In particular at high velocities or at high pressures, the effect of the profile coefficients is significant.

Jean-Marc Vanden-Broeck: Nonlinear free surface flows over topography

In an interesting paper King and Bloor (Journal of Fluid Mechanics (1987) vol 182, pp193-208) developed an efficient boundary integral equation method to compute nonlinear free surface flows over a step. We will present further analytical work on the topic. In particular long wave asymptotics for values of the Froude number close to one will be described. In addition the numerical results of King and Bloor will be compared with  asymptotic results obtained in the limit as the Froude number approaches zero. In that limit the waves on the free surface are exponentially small.


Graham Wilks: Similarity solutions : A means to an end

Similarity solutions abound in fluid mechanics. In themselves they relate to relatively constrained configurations e.g. particular velocity fields, prescribed temperature distributions or designated forms of stretching boundary. However their role can be much wider if they are seen as local approximations about which perturbations can be developed. In particular many non-similar configurations can be characterised as a progression between limiting local similarity states, a fact which can be exploited both analytically and computationally. In this talk I shall demonstrate a variety of non-similar configurations, including jets, wakes and plumes which have proved susceptible to this approach.