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.

# Past Industrial and Applied Mathematics Seminar

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 10^{5} 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.

Fluids and solids leave our bodies everyday. How do animals do it, from mice to elephants? In this talk, I will show how the shape of urinary and digestive organs enable them to function, regardless of the size of the animal. Such ideas may teach us how to more efficiently transport materials. I will show how the pee-pee pipe enables animals to urinate in constant time, how slippery mucus is critical for defecation, and how the motion of the gut is related to the density of its contents, and in turn to the gut’s natural frequency.

More info is in the BBC news here: http://www.bbc.com/news/science-environment-34278595

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.

While there have been recent advances for analyzing the complex deterministic

behavior of systems with discontinuous dynamics, there are many open questions about

understanding and predicting noise-driven and noise-sensitive phenomena in the

non-smooth context. Stochastic effects can often change the picture dramatically,

particularly if multiple time scales are present. We demonstrate novel approaches

for exploring and explaining surprising phenomena driven by the interplay of

nonlinearities, delays, randomness, in specific applications with piecewise smooth

dynamics - nonlinear models of balance, relay control, and impacting dynamics.

Effective techniques typically depend on the combination of mathematical techniques,

multiple scales techniques, and phenomenological intuition from seemingly unrelated

canonical models of biophysics, mechanics, and chemical dynamics. The appropriate

strategy is not always immediately obvious from the area of application or model

type. This gap may follow from the limited attention that stochastic models with

discontinuous dynamics have received in the past, or it may be the reason for this

limited attention. Combining the geometrical perspective with asymptotic approaches

in physical and phase space appears to be a critical part of developing effective

approaches.

Barbara Mahler: 15+5 min

Thomas Woolley: 15+5 min

Julian A. Garcia Grajales: 15+5 min

Respiratory illnesses, such as asthma and chronic obstructive pulmonary disease, account for one in five deaths worldwide and cost the UK over £6 billion a year. The main form of treatment is via inhaled drug delivery. Typically, however, a low fraction of the inhaled dose reaches the target areas in the lung. Predictive numerical capabilities have the potential for significant impact in the optimisation of pulmonary drug delivery. However, accurate and efficient prediction is challenging due to the complexity of the airway geometries and of the flow in the airways. In addition, geometric variation of the airways across subjects has a pronounced effect on the aerosol deposition. Therefore, an accurate model of respiratory deposition remains a challenge.

High-fidelity simulations of the flow field and prediction of the deposition patterns motivate the use of direct numerical simulations (DNS) in order to resolve the flow. Due to the high grid resolution requirements, it is desirable to adopt an efficient computational strategy. We employ a robust immersed boundary method developed for curvilinear coordinates, which allows the use of structured grids to model the complex patient-specific airways, and can accommodate the inter-subject geometric variations on the same grid. The proposed approach reduces the errors at the boundary and retains the stability guarantees of the original flow solver.

A Lagrangian particle tracking scheme is adopted to model the transport of aerosol particles. In order to characterise deposition, we propose the use of an instantaneous Stokes number based on the local properties of the flow field. The effective Stokes number is then defined as the time-average of the instantaneous value. This effective Stokes number thus encapsulates the flow history and geometric variability. Our results demonstrate that the effective Stokes number can deviate significantly from the reference value based solely on a characteristic flow velocity and length scale. In addition, the effective Stokes number shows a clear correlation with deposition efficiency.