Applied Mathematics Seminar
University of Leeds
Welcome to the Leeds Applied Mathematics Seminar
Autumn 2017
Monday 2 October 2017
Helen Wilson (University College London)
Dropping the ball: Using simulations to explain (or not) strange behaviour in viscous suspensions
In recent experiments, Frederic Blanc and his group in France dropped a heavy sphere through a concentrated suspension of smaller, neutrally buoyant particles. They found that the application of a lateral oscillatory shear flow caused the heavy ball to fall faster on average; and that for highly concentrated suspensions, at certain moments of the cycle of shear oscillation, the heavy ball moves upwards. We have used Stokesian Dynamics to model these experiments and other related scenarios. In this talk, I will discuss the key dimensionless parameters involved, and discuss how the motion of the heavy particle and the microstructure of the suspension depend on them. I will also describe a physical mechanism which describes some (but not all) of the observed behaviour.
Monday 16 October 2017
Victor Buchstaber (Steklov Mathematical Institute, All-Russian Scientific Research Institute for Physical and Radio-Technical Measurements)
In 1973 B.Josephson received Nobel Prize for discovering a new fundamental effect in superconductivity concerning a system of two superconductors separated by a very narrow dielectric (this system is called the Josephson junction): there could exist a supercurrent tunneling through this junction.
We will discuss the reduction of the overdamped Josephson junction to a family of first order non-linear ordinary differential equation that defines a family of dynamical systems on two-torus. Physical problems of the Josephson junction led to studying the rotation number of this dynamical system as a function of the parameters and to the problem on the geometric description of the phase-lock areas: the level sets of the rotation number function ρ with non-empty interiors.
In our case the phase-lock areas exist only for integer rotation numbers (quantization effect), and the complement to them is an open set. On their complement the rotation number function ρ is an analytic submersion that induces its fibration by analytic curves. It appears that the family of dynamical systems on torus under consideration is equivalent to a family of second order linear complex differential equations on the Riemann sphere with two irregular singularities, the well-known double confluent Heun equations. This family of linear equations depends on complex parameters λ, μ, n. Our dynamical systems on torus correspond to the equations with real parameters satisfying the inequality λ + μ 2 > 0. The monodromy of the Heun equations is expressed in terms of the rotation number.
This result has several applications. First of all, it gives the description of those values λ, μ, n and b for which the monodromy operator of the corresponding Heun equation has eigenvalue e 2πib . It also gives the description of those values λ, μ, n for which the monodromy is parabolic, i.e., has a multiple eigenvalue; they correspond exactly to the boundaries of the phase-lock areas. This implies the explicit description of the union of boundaries of the phase-lock areas as solutions of an explicit transcendental functional equation. For every θ ∈ / Z we get a description of the set {ρ ≡ ±θ(mod2Z)}.
The talk will be accessible for a wide audience and devoted to different connections between physics, dynamical systems on two-torus and applications of analytic theory of complex linear differential equations.
Tuesday 31 October 2017
Graham Donovan (University of Auckland)
Clustered ventilation defects in asthma
Clustered ventilation defects are a hallmark of asthma, typically seen via imaging studies during asthma attacks. The mechanisms underlying the formation of these clusters is of great interest in understanding asthma. Because the clusters vary from event to event, many researchers believe they occur due to dynamic, rather than structural, causes. This talk will cover recent progress in understanding the mathematics behind clustered ventilation defect formation, interactions with structural factors, and the implications for understanding asthma and its treatment.
Monday 13 November 2017
Sarah Dance (University of Reading)
End to end flood forecasting: a mathematical tour
This talk will provide a mathematician’s introduction to flood forecasting: from observations of clouds, to numerical weather prediction of rainfall, through river run-off to flood inundation forecasts. Data assimilation plays a key role at various stages in this process. Data assimilation is a powerful technique for combining forecasts from dynamic models with observations to give an improved forecast, taking account of uncertainty. The talk will include new ideas for uncertainty estimation, new results in numerical linear algebra and optimization and some complex fluid dynamics simulations.
Monday 11 December 2017
Colm-cille Caulfield (University of Cambridge)
Making a LIST and checking it twice: Length scales of Instabilities & Stratified Turbulence
Stratified shear flows, where the `background' velocity and density distribution vary over some characteristic length scales, are ubiquitous in the atmosphere and the ocean. At sufficiently high Reynolds number, such flows are commonly believed to play a key role in the transition to turbulence, and hence to be central to irreversible mixing of the density field. Parameterizations of such irreversible mixing within larger scale models of the ocean in particular is a major area of uncertainty, not least because there is a wide range of highly scattered and apparently inconsistent experimental and observational data. It is becoming increasingly appreciated that appropriately defined characteristic length scales of the flow are critically important to all stages of the flow's evolution, and that such data scatter is associated with differing length scales being important in different experiments and observations. Here, I review some of the recent progress using modern mathematical techniques in developing understanding of instability, transition, turbulence and mixing in stratified shear flows, focussing in particular on the crucial role of various length scales. I highlight certain non-intuitive aspects of the subtle interplay between the ostensibly stabilizing effect of stratification and destabilizing effect of velocity shear, especially when the density distribution has layers, i.e. relatively deep and well-mixed regions separated by relatively thin `interfaces' of substantially enhanced density gradient.