Dave Hewett: Canonical solutions in wave scattering

By a "canonical solution" I have in mind a closed-form exact solution of the scalar wave equation in a simple geometry, for example the exterior of a circular cylinder, or the exterior of an infinite wedge. In this talk I hope to convince you that the study of such problems is (a) interesting; (b) important; and (c) a rich source of (difficult) open problems involving eigenfunction expansions, special functions, the asymptotic evaluation of integrals, and matched asymptotic expansions.

Surfactants are chemicals that adsorb onto the air-liquid interface and lower the surface tension there. Non-uniformities in surfactant concentration result in surface tension gradients leading to a surface shear stress, known as a Marangoni stress. This stress, if sufficiently large, can influence the flow at the interface.

Surfactants are ubiquitous in many aspects of technology and industry to control the wetting properties of liquids due to  their ability to modify surface tension. They are used in detergents, crop spraying, coating processes and oil recovery. Surfactants also occur naturally, for example in the mammalian lung. They reduce the surface tension within the liquid lining the airways, which assists in preventing the collapse of the smaller airways. In the lungs of premature infants, the quantity of surfactant produced is insufficient as the lungs are under- developed. This leads to a respiratory distress syndrome which is treated by Surfactant Replacement Therapy.

Motivated by this medical application, we theoretically investigate a model problem involving the spreading of a drop laden with an insoluble surfactant down an inclined and pre-wetted substrate.  Our focus is in understanding the mechanisms behind a “fingering” instability observed experimentally during the spreading process. High-resolution numerics reveal a multi-region asymptotic wave-like structure of the spreading droplet. Approximate solutions for each region is then derived using asymptotic analysis. In particular, a quasi-steady similarity solution is obtained for the leading edge of the droplet. A linear stability analysis of this region shows that the base state is linearly unstable to long-wavelength perturbations. The Marangoni effect is shown to be the dominant driving mechanism behind this instability at small wavenumbers. A small wavenumber stability criterion is derived and it's implication on the onset of the fingering instability will be discussed.

A Simple Generative Model of Collective Online Behavior (Mason Porter)

Human activities increasingly take place in online environments, providing novel opportunities for relating individual behaviors to population-level outcomes. In this paper, we introduce a simple generative model for the collective behavior of millions of social networking site users who are deciding between different software applications. Our model incorporates two distinct mechanisms: one is associated with recent decisions of users, and the other reflects the cumulative popularity of each application. Importantly, although various combinations of the two mechanisms yield long-time behav- ior that is consistent with data, the only models that reproduce the observed temporal dynamics are those that strongly emphasize the recent popularity of applications over their cumulative popularity.

This demonstrates --- even when using purely observational data with- out experimental design --- that temporal data-driven modeling can effectively distinguish between competing microscopic mechanisms, allowing us to uncover previously unidentified aspects of collective online behavior.

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Bubbles, Turing machines, and possible routes to Navier-Stokes blow-up (Robert van Gorder)

Navier-Stokes existence and regularity in three spatial dimensions for an incompressible fluid... is hard. Indeed, while the original equations date back to the 1840's, existence and regularity remains an open problem and is one of the six remaining Millennium Prize Problems in mathematics that were stated by the Clay Mathematics Institute in 2000. Despite the difficulty, a resolution to this problem may say little about real-world fluids, as many real fluid problems do not seem to blow-up, anyway.
In this talk, we shall briefly outline the mathematical problem, although our focus shall be on the negative direction; in particular, we focus on the possibility of blow-up solutions. We show that many existing blow-up solutions require infinite energy initially, which is unreasonable. Therefore, obtaining a blow-up solution that starts out with nice properties such as bounded energy on three dimensional Euclidean space is rather challenging. However, if we modify the problem, there are some results. We survey recent results on averaged Navier-Stokes equations and compressible Navier-Stokes equations, and this will take us anywhere from bubbles to fluid Turing machines. We discuss how such results might give insight into the loss of regularity in the incompressible case (or, insight into how hard it might be to loose regularity of solutions when starting with finite energy in the incompressible case), before philosophizing about whether mathematical blow-up solutions could ever be physically relevant.

Permanent deformations of crystalline materials are known to be carried out by a large
number of atomistic line defects, i.e. dislocations. For specimens on micron scales or above, it
is more computationally tractable to investigate macroscopic material properties based on the
evolution of underlying dislocation densities. However, classical models of dislocation
continua struggle to resolve short-range elastic interactions of dislocations, which are believed
responsible for the formation of various heterogeneous dislocation substructures in crystals. In
this talk, we start with discussion on formulating the collective behaviour of a row of
dislocation dipoles, which would be considered equivalent to a dislocation-free state in
classical continuum models. It is shown that the underlying discrete dislocation dynamics can
be asymptotically captured by a set of evolution equations for dislocation densities along with
a set of equilibrium equations for variables characterising the self-sustained dislocation
substructures residing on a shorter length scale, and the strength of the dislocation
substructures is associated with the solvability conditions of their governing equilibrium
equations. Under the same strategy, a (continuum) flow stress formula for multi-slip systems
is also derived, and the formula resolves more details from the underlying dynamics than the
ubiquitously adopted Taylor-type formulae.

My main purpose in this talk is try and convey a sense of my enthusiasm for mathematical modelling generally and how I've come to use it in a range of transport applications. For concreteness, I am going to talk in particular about work I have been doing on EPSRC grant EP/K000438/1 (PI: Jillian Anable, Aberdeen) where we are using the UK government's Department for Transport MOT data to estimate mileage totals and study how they are broken down across the population in various different ways. Embedded inside this practical problem is a whole set of miniature mathematical puzzles and challenges which are quite particular to the problem area itself, and one wider question which is rather deeper and more general: whether it is possible (and how) to convert usage data that is low-resolution in time but high-resolution in individuals to knowledge that is high-resolution in time but only expressed at a population level.

Surface plasmons are collective electron-density oscillations at a metal-dielectric interface. In particular, highly localised surface-plasmon modes of nanometallic structures with narrow nonmetallic gaps, which enable a tuneable resonance frequency and a giant near-field enhancement, are at the heart of numerous nanophotonics applications. In this work, we elucidate the singular near-contact asymptotics of the plasmonic eigenvalue problem governing the resonant frequencies and modes of such structures. In the classical regime, valid for gap widths > 1nm, we find a generic scaling describing the redshift of the resonance frequency as the gap width is reduced, and in several prototypical dimer configurations derive explicit expressions for the plasmonic eigenvalues and eigenmodes using matched asymptotic expansions; we also derive expressions describing the resonant excitation of such modes by light based on a weak-dissipation limit. In the subnanometric ``nonlocal’’ regime, we show intuitively and by systematic analysis of the hydrodynamic Drude model that nonlocality manifests itself as a potential discontinuity, and in the near-contact limit equivalently as a widening of the gap. We thereby find the near-contact asymptotics as a renormalisation of the local asymptotics, and in particular a lower bound on plasmon frequency, scaling with the 1/4 power of the Fermi wavelength. Joint work with Vincenzo Giannini, Richard V. Craster and Stefan A. Maier. 

Life is different because it is inherited. All life comes from a blueprint (genes) that can only make proteins. Proteins are studied by more than one hundred thousand scientists and physicians every day because they are so important in health and disease. The function of proteins is on the macroscopic scale, but atomic details control that function, as is shown in a multitude of experiments. The structure of proteins is so important that governments spend billions studying them. Structures are known in exquisite detail determined by crystallographic measurement of more than 10⁵ different proteins. But the forces that govern the movement and function of proteins are not visible in the structure. Mathematics is needed to compute both function and forces so comparison with experiment can be made. Experiments report numbers, typically sets of numbers in the form of graphs. Verbal models, however beautifully written in the biological tradition, do not provide numerical outputs, and so it is difficult to tell which verbal model better fits data.

The mathematics of molecular biology must be multiscale because atomic details control macroscopic function. The device approach of the engineering and English physiological tradition provides the dimensional reduction needed to solve the multiscale problem. Mathematical analysis of hundreds of experiments (reported in some fifty papers) has been successful in showing how some properties of an important class of proteins—ion channels— work. Ion channels are natural nanovalves as important to animals as Field Effect Transistors (FETs) are to computers. I will present the Fermi Poisson approach started by Jinn Liang Liu. The Fermi distribution is used to describe the saturation of space produced by crowded spherical ions. The Poisson equation (and continuity of current) is used to describe long range electrodynamics. Short range correlations are approximated by the Santangelo equation. A fully consistent mathematical description reproduces macroscopic properties of bulk solutions of sodium and calcium chloride solutions. It also describes several different channels (with quite different atomic detailed structures) quite well in a wide range of conditions using a handful of parameters never changed. It is not clear why the model works as well it does, nor is it clear how well the model will work on other channels, transporters or proteins.

The aim of this talk is to describe effective media for wave propagation through periodic, or nearly periodic, composites. Homogenisation methods are well-known and developed for quasi-static and low frequency regimes. The aim here is to move to situations of more practical interest where the frequencies are high, in some sense, and to compare the results of the theory with large scale simulations.

Honey poured from a sufficient height onto toast undergoes the well-known `liquid rope coiling’ instability.

We have studied this instability using a combination of laboratory experiments, theory, and numerics, with the aim of determining phase diagrams and scaling laws for the different coiling modes. Finite-amplitude coiling has four distinct modes - viscous, gravitational, inertio-gravitational, and inertial - depending on how the viscous forces that resist deformation of the rope are balanced. The inertio-gravitational mode is particularly interesting as it involves resonance between the coiling portion of the rope and its long trailing `tail’. Further experiments using less viscous fluids reveal that the rope can exhibit five different morphologies, of which steady coiling is only one. We determine the detailed phase diagram of these morphologies, which includes a novel `liquid supercoiling’

state in which the coiled cylinder formed by the primary coiling instability undergoes in turn its own complex buckling instability.  We show that the onset of these different patterns is determined by a non-penetrability condition which takes different forms in the viscous, gravitational and inertial limits. To close, we will briefly evoke two additional related phenomena: spiral waves of bubbles generated by coiling, and the `fluid mechanical sewing machine’ in which the fluid falls onto a moving belt.