Differential Equations and Applications Seminar

Whispering gallery modes in optical resonators have received a lot of attention as a mechanism for constructing small, directional lasers. They are also potentially important as passive optical components in schemes for coupling and filtering signals in optical fibres, in sensing devices and in other applications. In this talk it is argued that the evanescent field outside resonators that are very slightly deformed from circular or spherical is surprising in a couple of respects. First, even very small deformations seem to be capable of leading to highly directional emission patterns. Second, even though the undelying ray families are very regular and hardly differ from the integrable circular or spherical limit inside the resonator, a calculation of the evanescent field outside it is not straightforward. This is because even very slight nonintegrability has a profound effect on the complexified ray families which guide the external wave to asymptopia. An approach to describing the emitted wave is described which is based on canonical perturbation theory applied to the ray families and extended to comeplx phase space.

Frank-Read sources are among the most important mechanisms of dislocation multiplication, and their operation signals the onset of yield in crystals. We show that the critical stress required to initiate dislocation production falls dramatically at high elastic anisotropy, irrespective of the mean shear modulus. We hence predict the yield stress of crystals to fall dramatically as their anisotropy increases. This behaviour is consistent with the severe plastic softening observed in alpha-iron and ferritic steels as the alpha − gamma martensitic phase transition is approached, a temperature regime of crucial importance for structural steels designed for future nuclear applications.

An overview of the experiments of Steinbergs group, Theory-and-models and comparison of the applicability of recent reduced models.

Traditional Faraday waves appear in a layer of liquid that is shaken vertically. These patterns can take the form of horizontal stripes, close-packed hexagons, or even squares or quasiperiodic patterns. Faraday waves are commonly observed as fine stripes on the surface of wine in a wineglass that is ringing like a bell when periodically forced.

Motivated by recent experiments on Faraday waves in Bose-Einstein condensates we investigate both analytically and numerically the dynamics of cigar-shaped Bose-condensed gases subject to periodic modulation of the strength of the transverse confinement's trap. We offer a fully analytical explanation of the observed parametric resonance yielding the pattern periodicity versus the driving frequency. These results, corroborated by numerical simulations, match extremely well with the experimental observations.

Platelet ice may be an important component of Antarctic land-fast sea ice. Typically, it is found at depth in first-year landfast sea ice cover, near ice shelves. To explain why platelet ice is not commonly observed at shallower depths, we consider a new mechanism. Our hypothesis is that platelet ice eventually appears due to the sudden deposition of frazil ice against the fast ice-ocean interface, providing randomly oriented nucleation sites for crystal growth.

Brine rejected in plumes from land-fast ice generates stirring sufficient to prevent frazil ice from attaching to the interface, forcing it to remain in suspension until ice growth rate and brine rejection slow to the point that frazil can stick. We calculate a brine plume velocity, and match this to frazil rise velocity.

We consider both laminar and turbulent environments. We find that brine plume velocities are generally powerful enough to prevent most frazil from sticking in the case of laminar flow, and that in the turbulent case there may be a critical ice thickness at which most frazil suddenly settles.

Transient or unstable behavior is often ignored in considering long time dynamics in the deterministic world. However, stochastic effects can change the picture dramatically, so that the transients can dominate the long range behavior. Coherence resonance is one relatively simple example of this transformation, and we consider others such as noise-driven synchronization in networks, recurrence of diseases, and stochastic stabilization in systems with delay. The challenge is to identify common features in these phenomena, leading to new approaches for other systems of this type. Some recurring themes include the influence of multiple time scales, cooperation of both discrete and continuous aspects in the dynamics, and the remnants of underlying bifurcation structure visible through the noise.

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.

In the context of the linear theory of elasticity with eigenstrains, the radiated fields, including inertia effects, and the energy-release rates (“driving forces”) of a spherically expanding and a plane inclusion with constant dilatational eigenstrains are calculated. The fields of a plane boundary with dilatational eigenstrain moving from rest in general motion are calculated by a limiting process from the spherical ones, as the radius tends to infinity, which yield the time-dependent tractions that need to be applied on the lateral boundaries for the global problem to be well-posed. The energy-release rate required to move the plane inclusion boundary (and to create a new volume of eigenstrain) in general motion is obtained here for a superposed loading of a homogeneous uniaxial tensile stress. This provides the relation of the applied stress to the boundary velocity through the energy-rate balance equation, yielding the “equation of motion” (or “kinetic relation”) of the plane boundary under external tensile axial loading. This energy-rate balance expression is the counterpart to the Peach-Koehler force on a dislocation plus the “self-force” of the moving dislocation.

An elastic sheet will buckle out of the plane when subjected to an in-plane compression. In the simplest systems the typical lengthscale of the buckled structure is that of the system itself but with additional physics (e.g. an elastic substrate) repeated buckles with a well-defined wavelength may be seen. We discuss two examples in which neither of these scenarios is realized: instead a small number of localized structures are observed with a size different to that of the system itself. The first example is a heavy sheet on a rigid floor - a ruck in a rug. We study the static properties of these rucks and also how they propagate when one end of the rug is moved quickly. The second example involves a thin film adhered to a much softer substrate. Here delamination blisters are formed with a well-defined size, which we characterize in terms of the material properties of the system. We then discuss the possible application of these model systems to real world problems ranging from the propagation of slip pulses in earthquakes to the manufacture of flexible electronic devices."

I will discuss the so-called Lasso method for signal recovery for high-dimensional data and show applications in computational biology, machine learning and image analysis.

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.

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.

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.

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).

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.

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.