Past Differential Equations and Applications Seminar

12 March 2009
16:30
Mark Kelmanson
Abstract
The classic coating-flow problem first studied experimentally by Moffat and asymptotically by Pukhnachov in 1977 is reconsidered in the framework of multiple-timescale asymptotics. Two-timescale approximations of the height of the thin film coating a rotating horizontal circular cylinder are obtained from an asymptotic analysis, in terms of small gravitational and capillary parameters, of Pukhnachov's nonlinear evolution for the film thickness. The transition, as capillary effects are reduced, from smooth to shock-like solutions is examined, and interesting large-time dynamics in this case are determined via a multiple-timescale analysis of a Kuramoto-Sivashinsky equation. A pseudo-three-timescale method is proposed and demonstrated to improve the accuracy of the smooth solutions, and an asymptotic analysis of a modified Pukhnachov's equation, one augmented by inertial terms, leads to an expression for the critical Reynolds number above which the steady states first analysed by Moffatt and Pukhnachov cannot be realised. As part of this analysis, some interesting implications of the effects of different scalings on inertial terms is discussed. All theoretical results are validated by either spectral or extrapolated numerics.
  • Differential Equations and Applications Seminar
5 March 2009
16:30
Jean-Marc Vanden-Broeck
Abstract
GIBSON BUILDING COMMON ROOM 2ND FLOOR (Coffee and Cakes in Gibson Meeting Room - opposite common room) The effects of electric fields on nonlinear free surface flows are investigated. Both inviscid and Stokes flows are considered. Fully nonlinear solutions are computed by boundary integral equation methods and weakly nonlinear solutions are obtained by using long wave asymptotics and lubrication theory. Effects of electric fields on the stability of the flows are discussed. In addition applications to coating flows are presented.
  • Differential Equations and Applications Seminar
26 February 2009
16:30
Oliver Jensen
Abstract
I will provide an overview of theoretical models aimed at understanding how self-excited oscillations arise when flow is driven through a finite-length flexible tube or channel. This problem is approached using a hierarchy of models, from one to three spatial dimensions, combining both computational and asymptotic techniques. I will explain how recent work is starting to shed light on the relationship between local and global instabilities, energy balances and the role of intrinsic hydrodynamic instabilities. This is collaborative work with Peter Stewart, Robert Whittaker, Jonathan Boyle, Matthias Heil and Sarah Waters.
  • Differential Equations and Applications Seminar
19 February 2009
16:30
Jan Bouwe van den Berg
Abstract
The harmonic map heat flow is a model for nematic liquid crystals and also has origins in geometry. We will introduce the model and discuss some of its mathematical properties. In particular, we will focus on the possibility that singularities may develop. The rate at which singularities develop is investigated in settings with certain symmetries. We use the method of matched asymptotic expansions and identify different scenarios for singularity formation. More specifically, we distinguish between singularities that develop in finite time and those that need infinite time to form. Finally, we discuss which results can be proven rigorously, as well as some open problems, and we address stability issues (ongoing work with JF Williams).
  • Differential Equations and Applications Seminar
12 February 2009
16:30
Jim Woodhouse
Abstract
Friction-driven vibration occurs in a number of contexts, from the violin string to brake squeal and machine tool vibration. A review of some key phenomena and approaches will be given, then the talk will focus on a particular aspect, the "twitchiness" of squeal and its relatives. It is notoriously difficult to get repeatable measurements of brake squeal, and this has been regarded as a problem for model testing and validation. But this twitchiness is better regarded as an essential feature of the phenomenon, to be addressed by any model with pretensions to predictive power. Recent work examining sensitivity of friction-excited vibration in a system with a single-point frictional contact will be described. This involves theoretical prediction of nominal instabilities and their sensitivity to parameter uncertainty, compared with the results of a large-scale experimental test in which several thousand squeal initiations were caught and analysed in a laboratory system. Mention will also be made of a new test rig, which attempts to fill a gap in knowledge of frictional material properties by measuring a parameter which occurs naturally in any linearised stability analysis, but which has never previously been measured.
  • Differential Equations and Applications Seminar
5 February 2009
16:30
Karima Khusnutdinova (Loughborough) CANCELLED - WILL NOW BE IN TRINITY TERM 2009
Abstract
Layered (or laminated) structures are increasingly used in modern industry (e.g., in microelectronics and aerospace engineering). Integrity of such structures is mainly determined by the quality of their interfaces: poor adhesion or delamination can lead to a catastrophic failure of the whole structure. Can nonlinear waves help us to detect such defects? We study the dynamics of a nonlinear longitudinal bulk strain wave in a split, layered elastic bar, made of nonlinearly hyperelastic Murnaghan material. We consider a symmetric two-layered bar and assume that there is perfect interface for x < 0 and splitting for x > 0, where the x-axis is directed along the bar. Using matched asymptotic multiple-scales expansions and the integrability theory of the KdV equation by the Inverse Scattering Transform, we examine scattering of solitary waves and show that the defect causes generation of more than one secondary solitary waves from a single incident soliton and, thus, can be used to detect the defect. The theory is supported by experimental results. Experiments have been performed in the Ioffe Institute in St. Petersburg (Russia), using holographic interferometry and laser induced generation of an incident compression solitary wave in two- and three-layered polymethylmethacrylate (PMMA) bars, bonded using ethyl cyanoacrylate-based (CA) adhesive.
  • Differential Equations and Applications Seminar
29 January 2009
16:30
Dave Smith
Abstract
Sperm cells have been an archetype for very low Reynolds number swimming since the pioneering work of Gray & Hancock in the 1950s. However, there are fundamental questions regarding the swimming and function of mammalian, and particularly human sperm, that are unanswered, and moreover scientific and technological developments mean that for the first time, answering these questions is now possible. I will present results of our interdisciplinary work on two topics: (1) the relatively famous problem of 'surface accumulation' of sperm, and (2) characterising the changes to the flagellar beat that occur in high viscosity liquids such as cervical mucus. The approach we use combines both mathematical modelling and high speed imaging experiments. I will then discuss areas we are currently developing: quantifying the energy transport requirements of sperm, and understanding chemotaxis - the remarkable ability of human sperm to 'smell' lily of the valley perfume, which may be important in fertilisation.
  • Differential Equations and Applications Seminar
22 January 2009
16:30
Eugene Benilov
Abstract
We consider an infinite plate being withdrawn from an infinite pool of viscous liquid. Assuming that the effects of inertia and surface tension are weak, Derjaguin (1943) conjectured that the 'load', i.e. the thickness of the liquid film clinging to the plate, is determined by a certain formula involving the liquid's density and viscosity, the plate's velocity and inclination angle, and the acceleration due to gravity. In the present work, Deryagin's formula is derived from the Stokes equations in the limit of small slope of the plate (without this assumption, the formula is invalid). It is shown that the problem has infinitely many steady solutions, all of which are stable - but only one of these corresponds to Derjaguin’s formula. This particular steady solution can only be singled out by matching it to a self-similar solution describing the non-steady part of the film between the pool and the film’s 'tip'. Even though the near-pool region where the steady state has been established expands with time, the upper, non-steady part of the film (with its thickness decreasing towards the tip) expands faster and, thus, occupies a larger portion of the plate. As a result, the mean thickness of the film is 1.5 times smaller than the load. The results obtained are extended to order-one inclinantion angles and the case where surface tension is present.
  • Differential Equations and Applications Seminar

Pages