Spring 2018 Seminars and Abstracts Spring 2018 Seminars and Abstracts

Title: Algebraic methods for multidimensional Hamiltonian PDEs

Date: Monday 15th January, 2pm, (SCI 0.31)
Speaker: Dr Matteo Casati (Loughborough University)
Abstract: The class of nonlinear PDEs that have been proved to be integrable shares an interesting feature: even though the system may have 2 or 3 spatial dimensions, the Hamiltonian structure are always one-dimensional. A scalar two-dimensional system may, then, be traded for an infinite-component one-dimensional system or the additional space variables may be regarded as further times of an integrable hierarchy. I will present a novel algebraic framework for Hamiltonian PDEs that can be used for system of any dimension and review the aforementioned construction in a few explicit examples (Helmoltz's equation, KP equation, Vlasov's equation).

Title: Interfacial instabilities in confined geometries

Date: Monday 22nd January, 2pm, (SCI 0.31)
Speaker: Prof Anne Juel (University of Manchester)
Abstract: What links a baby's first breath to adhesive de-bonding, enhanced oil recovery, or even drop-on-demand devices? All these processes involve moving or expanding bubbles displacing fluid in a confined space, bounded by either rigid or elastic walls. In this talk, we show how spatial confinement may either induce or suppress interfacial instabilities and pattern formation in such flows.                                 

We demonstrate that a simple change in the bounding geometry of the containing vessel, e.g. a small height constriction within the cross-section of a rectangular channel, can radically alter the behaviour of a fluid-displacing air bubbles and fingers. A rich array of propagation modes, including symmetric, asymmetric and localised fingers, is uncovered when air displaces oil from axially uniform tubes that have local variations in flow resistance within their cross-sections. An unexpected and novel propagation mode exhibits spatial oscillations formed by periodic sideways motion of the interface at a fixed relative distance behind the moving finger-tip. We apply these findings to passively sort bubbles by size. We support our experimental findings with a complementary analysis based on a depth-averaged theory. The theoretical study reveals that the exchange of stability between different modes of bubble propagation relies on non-trivial interactions between capillary and viscous forces.

Viscous fingering in Hele-Shaw cells is an archetype for front propagation and pattern formation: when air is injected into the narrow, liquid-filled gap between parallel rigid plates, the propagating air-liquid interface is unstable to deformation with a maximum unstable wavenumber set by the ratio of viscous to surface tension forces. We show how the introduction of wall elasticity (via the replacement of the upper bounding plate by an elastic membrane) can weaken or even suppress the fingering instability by allowing changes in cell confinement through the formation of axial depth gradients from the deflection of the membrane.

Title: The Fokas method for linear evolution equations (Part 1)

Date: Monday 29th January, 2pm, (SCI 1.20)
Speaker: Dr David Smith
Abstract: The classical Fourier series / transform methods are effective in solving initial-boundary value problems for the second order heat and linear Schr\"{o}dinger equations. However these methods generally fail for equations of higher spatial order, except when very simple boundary conditions are assumed. In contrast, the Fokas method may be used to solve higher order problems with arbitrary boundary conditions. Over two seminars, we fully implement the method for the heat equation in $(1+1)$, and survey some other applications.
In the first seminar, we derive a formula, in terms of complex contour integrals, which must give the solution of the heat equation on a finite interval with inhomogeneous boundary conditions. We make use of the Fourier transform and the basic tools of complex analysis.

Title: The Fokas method for linear evolution equations (Part 2)

Date: Monday 5th February, 2pm, (SCI 1.20)
Speaker: Dr David Smith
Abstract: The classical Fourier series / transform methods are effective in solving initial-boundary value problems for the second order heat and linear Schr\"{o}dinger equations. However these methods generally fail for equations of higher spatial order, except when very simple boundary conditions are assumed. In contrast, the Fokas method may be used to solve higher order problems with arbitrary boundary conditions. Over two seminars, we fully implement the method for the heat equation in $(1+1)$, and survey some other applications.

In the second seminar, we show that the function defined by our claimed solution representation from the first seminar indeed satisfies the original initial-boundary value problem. We also indicate the extensions necessary to apply the Fokas method to a number of other classes of problems.

Title: Modelling dynamics of sea ice and ocean in the marginal ice zone. Effects of ice rheology and waves and applications for marine industries

Date: Thursday 8th February, 2pm, (SCI 1.20)
Speaker: Dr Yevgeny Aksenov (National Oceanographic Centre)
Abstract: Exposure of large, previously ice-covered areas of the Arctic Ocean to the wind and surface ocean waves results in the Arctic pack ice cover becoming more fragmented and mobile, with large regions of ice cover evolving into the Marginal Ice Zone (MIZ). The need for better climate predictions, along with increase in the Arctic offshore operations and shipping due to improving accessibility, necessitates climate and forecasting models that can simulate fragmented sea ice with greater fidelity. We use sea ice-ocean general circulation model NEMO (stands for Nucleus for European Modelling of the Ocean) coupled with the ocean wave model WAM output from model of the European Centre for Medium-Range Weather Forecasts (ECMWF) and examine several key mechanisms through which the waves affect dynamics of the ocean and sea ice. The wave-ice interactions include ice fragmentation due to break–up by waves in the vicinity of the ice edge, wave attenuation due to multiple scattering and non-elastic losses in the ice, wave advection and evolution of ice fragmentation (floe sizes). The project develops analysis and forecasting technologies to provide key information for the maritime operations and marine information services. The project brings together physical oceanography and the mathematics of fluid structure interaction and address the likely extreme loads on a selection of structures. The study was a part of the EU FP7 Project ‘Ships and waves reaching Polar Regions (SWARP)’ under the European Union's Seventh Framework Programme (FP7/2007-2013), grant agreement N°607476 and was funded from the NERC UK Innovation Grant 'Safer Operations at Sea - Supported by Operational Simulations (SOS-SOS)'.

Title: Wave induced Vibrations of Ice Shelves

Date: Monday 26th February, 2pm, (Queens 0.13)
Speaker: Dr Mike Meylan
Abstract: I will present methods to model the impact of very long ocean surface waves on ice shelves, primarily waves in the tsunami--infragravity regime. These wave induced vibrations have recently been measured in the Ross Ice Shelf in Antarctica. I will show how methods developed to predict the hydroelastic motion of ships can be employed. The analysis is extended from the frequency domain to the time domain, and the resonant behaviour of the system is studied.
I will also discuss various interesting mathematical features of this problem and connections with Lax-Philips scattering and the theory of non-self-adjoint dissipative operators.

Title: The biggest bangs since the big one: How to detect Gravitational Waves

Date: Monday 16th April, 2pm, (ARTS 2.02) Joint Pure and Applied Seminar
Speaker: Mr Iain Dorrington (Cardiff University)
Abstract: In February 2016, LIGO (the Laser Interferometer Gravitational-wave Observatory) announced the first ever direct detection of gravitational-waves. The source of the gravitational-waves was two black holes, each approximately 30 solar masses, crashing together at about 60% of the speed of light. At it’s peak, this system was emitting more energy per second than every light source in the universe combined. Despite the enormous amount of energy released, gravitational-waves have very weak effects. Detecting this signal involved measuring distances to a precision of less than 0.002fm, less than 1% the diameter of a proton. This is not just a huge technological challenge, but a data analysis problem too: How can we be sure we really have detected gravitational-waves? In this seminar I will give an overview of gravitational-wave astronomy. I will cover the basics of gravitational-wave theory, the instruments used to make the detections, and my own work into the data analysis techniques we use.

Title: Particle-segregation within, and rheology of, dense granular free-surface flows

Date: Monday 23rd April, 2pm, (JSC 2.03)
Speaker: Prof. Nico Gray (University of Manchester)
Abstract: Geophysical granular flows, such as landslides, pyroclastic flows and snow avalanches, consist of particles with varying surface roughnesses or shapes that have a tendency to segregate during flow due to size differences. Such segregation leads to the formation of regions with different frictional properties, which in turn can feed back on the bulk flow. This talk introduces a well-posed depth-averaged model for these segregation mobility feedback effects. The full segregation equation for dense granular flows is integrated through the avalanche thickness by assuming inversely graded layers with large particles above fines, and a Bagnold shear profile. The resulting large particle transport equation is then coupled to depth-averaged equations for conservation of mass and momentum, with the feedback arising through a basal friction law that is composition dependent, implying greater friction where there are more large particles. The new system of equations includes viscous terms in the momentum balance, which are derived from the mu(I)-rheology for dense granular flows and represent a singular perturbation to previous models. Linear stability calculations of the steady uniform base state demonstrate the significance of these higher-order terms, which ensure that, unlike the inviscid equations, the growth rates remain bounded everywhere. The new system is therefore mathematically well posed. Two-dimensional simulations of bidisperse material propagating down an inclined plane show the development of an unstable large-rich flow front, which subsequently breaks into a series of finger-like structures, each bounded by coarse-grained lateral levees. The key properties of the fingers are independent of the grid resolution and are controlled by the physical viscosity. This process of segregation-induced finger formation is observed in laboratory experiments, and numerical computations are in qualitative agreement. The theory is also shown to be able model mono-disperse flows with erosion and deposition when solved in conjunction with a non-monotonic effective basal friction law.

Title: Gravity Driven Flow of Two Fluids of Equal Depth

Date: Monday 30th April, 2pm, (QUEENS 01.11)
Speaker: Prof. Oguz Yilmaz (Izmir Institue of Technology)
Abstract: Small time behaviour of gravity driven free surface flow of two fluids of equal depth is studied. Initially the fluids of different densities are at rest and separated with a vertical plate. The plate disappears suddenly and gravity driven flow of the fluids starts. The flow in early stage is described by the potential theory. Attention is paid to the motion of the interface and the free surfaces and the singular behaviours of the velocity field at the bottom point, where the interface meet the rigid bottom, and the top point, where the interface meets the free surfaces. The flow velocity is log  singular at the bottom point where a leading order inner asymptotic solution is found near that point. The shapes of the free surfaces and the interface in the leading order show a jump discontinuity of the free surface near the top point where the free surfaces and the interface meet. Inner region formulations are derived near the top point.

Title: Modelling the Formation of Microtubule Rings

Date: Monday 14th May, 2pm, (QUEENS 2.21)
Speaker: Dr Simon Pearce (University of Manchester)

Abstract: Microtubules (MTs) are one of the main components of cells, and are essential for many biological functions. As the stiffest cytoskeletal polymer, they are generally seen to be very straight over cellular lengthscales. However, in areas of neurodegeneration highly curved MTs are seen with radius of curvature of a micron. Similarly curved MT rings are also sometimes seen in gliding assays, where MTs are moved over a surface by the motor protein kinesin, amongst other MTs translocating as rigid rods.
Recent evidence suggests that some microtubule-associated proteins such as kinesin are able to sense and alter MT curvature, and so we model MTs as inextensible rods with a preferred curvature, which is controlled by the differential binding of the kinesin. We find that there exist parameter regimes wherein metastable rings can form, and hence offer this differential binding as an explanation for these highly curved MTs seen both in vitro and in vivo.
For certain parameter regimes, this model predicts that both straight and curved MTs can exist simultaneously as stable steady-states, as has been seen experimentally. Additionally, unsteady solutions are found, where a wave of differential binding propagates down the MT as it glides across the surface, which can lead to chaotic motion via a period doubling bifurcation.
I will also briefly mention the use of the compound matrix method to calculate the Evans function for solving eigenvalue boundary-value problems, and present a Mathematica package for calculating this.

Autumn 2017 Seminars and Abstracts Autumn 2017 Seminars and Abstracts

Title: A stable and dual consistent finite difference method

Date: Monday 2nd October, 2pm, (SCI 0.31)
Speaker: Dr Sofia Eriksson (TU Darmstadt)
Abstract: We study the numerical solutions of time-dependent systems of partial differential equations, focusing on the implementation of boundary conditions. The numerical method considered is a finite difference scheme constructed by high order summation by parts operators, combined with a boundary procedure using penalties (SBP-SAT). Recently it was shown that SBP-SAT finite difference methods can yield super-convergent functional output if the boundary conditions are imposed such that the discretisation is dual consistent. We generalise these results so that they include a broader range of boundary conditions and penalty parameters. The results are also generalised to hold for narrow-stencil second derivative operators. The derivations are supported by numerical experiments.

Title: Chemical front propagation in periodic flows: the role of large deviations

Date: Monday 9th October, 2pm, (SCI 0.31)
Speaker: Dr Alexandra Tzella (University of Birmingham)
Abstract: We discuss the propagation of chemical fronts arising in Fisher-Kolmogorov-Petrovskii-Piskunov (FKPP) type models in the presence of a steady cellular flow. In the long-time limit, a pulsating front is established. Its speed, on which we focus, can be obtained by solving an eigenvalue problem closely related to large-deviation theory. We employ asymptotic methods to solve this eigenvalue problem in the limit of small molecular diffusivity (large Peclet number, $\mathit{Pe}$) and arbitrary reaction rate (arbitrary Damkohler number, $\mathit{Da}$). We identify three regimes corresponding to the distinguished limits $\mathit{Da} = O(1/\mathit{Pe})$, $\mathit{Da} = O(1/\log\mathit{Pe})$ and $\mathit{Da} = O(\mathit{Pe})$ and, in each regime, obtain the front speed in terms of a different non-trivial function of the relevant combination of $\mathit{Pe}$ and $\mathit{Da}$, determined by solving a ($\mathit{Pe}$-independent) one-dimensional problem. Our results are contrasted against front speed values obtained from the so-called G equation: a level-set approximation that is commonly used when $\mathit{Da} = O(\mathit{Pe})$.

(Joint work with J. Vanneste, Edinburgh).

Title: Acoustic-gravity waves, theory & applications

Date: Monday 16th October, 2pm (SCI 0.31)
Speaker: Dr Usama Kadri (Cardiff University)
Abstract: Acoustic–gravity waves (AGWs) are compression-type waves generated as a response to a sudden change in the water pressure, e.g. due to nonlinear interaction of surface waves, submarine earthquakes, landslides, falling meteorites and objects impacting the sea surface. AGWs can travel at near the speed of sound in water (ca. 1500 m/s), but can also penetrate through the sea-floor surface amplifying their speed, which turns them into excellent precursors. “Acoustic–gravity waves” is an emerging field that is rapidly gaining popularity among the scientific community, as it finds broad utility in physical oceanography, marine biology, geophysics, water engineering, and quantum analogues. This talk is an overview on AGWs, with emphasis on the propagation under elastic ice sheets.

Title: Modulation of multiphase wavetrains and nonlinear reductions

Date: Monday 23rd October, 2pm, (SCI 0.31)
Speaker: Dr Daniel Ratliff (Loughborough University)
Abstract: The modulation of single phase wavetrains (and more recently their generalisation, relative equilibria), instigated by Whitham, is a field that has been developed over the last 50 years. The procedure generates a set of dispersionless nonlinear PDEs that govern the local wavenumber and frequency of the wave. When these degenerate, it has been shown that dispersion emerges at such points leading to equations such as the Korteweg-de Vries (KdV) equation. Remarkably, such reductions possess coefficients that may be related to the conservation laws of the original system which can be calculated in advance. This property is known as ‘universal form’.

This talk concerns itself with taking these ideas and applying them to solutions that have more than one phase. The questions are now this – which nonlinear PDEs arise? Do these nonlinear reductions still emerge with universal form? It will be shown that yes, these properties generalise quite nicely to the multiphase problem and recover many of the same equations derived from single phased solutions (like the KdV). Unsurprisingly, the increase in the number of system parameters allows one to derive further nonlinear PDEs (and even some new ones).

The talk concludes (hopefully, time permitting) by discussing two applications of the theory. The first is a stratified shallow water system and the second is a set of coupled Nonlinear Schrodinger equations (which model ocean wave envelopes, Bose-Einstein condensates and electromagnetic waves), showing the possible reductions and how the conditions for each equation can be met.

This work is in collaboration with Tom Bridges (University of Surrey).

Title: How to make a splash: from high speed droplet impact to a novel methodology for calculating water catch on aircraft surfaces

Date: Monday 6th November, 2pm, (SCI 0.31)
Speaker: Dr Radu Cimpeanu (University of Oxford)
Abstract: A new methodology for the calculation of water collection efficiency on aircraft surfaces is discussed. The approach incorporates the detailed fluid dynamical processes often ignored in this setting, such as the drop interaction with the surrounding air flow, drop deformation, rupture and coalescence, as well as the motion of the ejected microdrops in the computational domain. Direct numerical simulations using the volume-of-fluid technique are performed using modelling assumptions which enable us to take advantage of the disparity of lengthscales in the system. Comparisons are performed in the pre-impact regime with available experimental data, while the early stages of the impact are validated using the analytical framework provided by Wagner theory, context in which recent developments are also presented. We then focus on quantifying useful information on the liquid movement on longer timescales. The analysis shows a high degree of flexibility and can be used efficiently when considering changes in geometry (aircraft design), flow conditions and cloud characteristics. The interaction with our industrial partners will also be a point of focus, in particular in the context of developing a framework that incorporates the above analysis in an industrial work pipeline with no additional computational cost, thus making direct use of several hundreds of thousands of hours of CPU time on local supercomputing facilities. The methodology is finally applied to representative test geometries in collaboration with our partners.

Title: Multilayer networks: a new framework for complex systems

Date: Monday 20th November, 2pm, (SCI 0.31)
Speaker: Dr Ginestra Bianconi (Queen Mary University London)
Abstract: Multilayer networks describe interacting complex systems formed by different interacting networks. Multilayer networks are ubiquitous and include social networks, financial markets, multimodal transportation systems, infrastructures, the brain and the cell. Multilayer networks cannot be reduced to a large single network. In this talk I will present first recent results showing how we can extract from multilayer networks more relevant information than from its single layer taken in isolation. Secondly I will provide evidence that dynamical processes on multilayer networks can display very novel properties that reflect the rich interplay between structure and multiplexity.

Title: When is the effective wavefield a useful tool to predict wave attenuation over long distances?

Date: Monday 27th November, 2pm, (SCI 0.31)
Speaker: Dr Luke Bennetts (University of Adelaide)
Abstract: Anderson localisation theory tells us that waves attenuate through disordered media, and this talk is motivated by seeking efficient ways to calculate the attenuation rate as a function of the incident wave properties (frequency) and properties of the given medium, including the statistical properties of the disorder. Effective media theory is an appealing way to approach the problem, as it provides analytical insight, circumventing the need to repeatedly compute individual wave fields for different realisations of the disorder, as well providing the opportunity for elegant mathematical analysis. I will discuss the usefulness of effective media theory in the setting of two canonical, linear 1D problems, along the way outlining some associated effective media methods. The findings are in some respects entirely intuitive, but in other respects surprising.

Title: Feedback control of thin film flows

Date: Monday 4th December, 2pm, (SCI 0.31)
Speaker: Dr Susana Gomes (Imperial College London)
Abstract: The flow of a thin film down an inclined plane is unstable when the Reynolds number is larger than a critical value depending on the slope angle. These flows are important for many industrial applications, including coating and heat transfer. While some applications benefit from a flat film, in many cases one wishes to explore the flow’s instabilities and drive the system towards a non-uniform state.

In this talk, I will present a control methodology based on same-fluid blowing and suction at the wall. Given a desired interface shape, we apply controls which are proportional to the deviation between the current state of the system and the chosen solution. We study the effect of these controls on three partial differential equations which model the interfaces of thin film flows in different limits: two long-wave models (the Benney equation and a first-order weighted residuals model) and in the weakly nonlinear regime (the Kuramoto-Sivashinsky (KS) equation). We show that for the KS equation we can use a finite number of point-actuated controls based on observations of the full interface to stabilise both the flat solution and chosen nontrivial solutions, and investigate the robustness of the designed controls to uncertain observations and/or parameter values. Furthermore, we show robustness of the controls between the more general models and to limited observations.

Title: Mathematical models for the shape of the Eiffel tower: historical perspective and new results

Date: Wednesday 6th December, 3pm, (Queens 2.21)
Speaker: Prof Patrick Weidman (University of Colorado, Boulder)
Abstract: Equations modeling the shape of the Eiffel Tower are investigated. One model, based on equilibrium of moments, gives the wrong tower curvature. A second model, based on constancy of vertical axial stress, does provide a fair approximation to the tower's skyline profile of twenty-nine contiguous panels. However, neither model can be traced back to Eiffel's writings. Reported here is a new model embodying Eiffel's concern for wind loads on the tower, as documented in his communication to the French Civil Engineering Society on March 30, 1885. The result is a nonlinear, integro-differential equation which may be solved to yield an exponential profile. An analysis of actual panel coordinates reveals a profile closely approximated by two piecewise continuous exponentials with different growth rates. This is explained by specific safety factors for wind loading that Eiffel & Company incorporated in the design and construction of the free-standing tower.

Title: Drowning by numbers: mathematical design & implications of Wetropolis' flood demonstrator

Date: Monday 11th December, 2pm, (SCI 0.31)
Speaker: Prof Onno Bokhove (University of Leeds)
Abstract: The Wetropolis Flood Demonstrator will be introduced and analysed. Wetropolis commenced as outreach model for the public to let them experience rainfall events causing river flooding. Wetropolis is a table-top model with a conceptual river, flood plain, city, porous moor representing the upper catchment and groundwater flow, and an upland reservoir. Key is the rainfall, in terms of rain amount per Wetopolis day (a day is WD=10s), it rains either 1s, 2s, 4s or an extreme 9s in a WD, and rainfall location, either in the moor, in both reservoir and moor, in the reservoir or not. These 4x4=16 rain amount times rain location combinations are visually drawn daily (so every 10s) from two skewed Galton boards (as two steel balls fall down), with the most extreme rainfall event: 90% rainfall in both moor and reservoir with a probability of 7/256~3%, causing floods in the city every 5 minutes on average, by mathematical design.

Although it started as outreach tool, Wetropolis has also triggered the thinking about flood mitigation amongst flood practitioners. Wetropolis was inspired by the extreme 2015 Boxing Day Aire River floods in and around Leeds. Straightforward and more advanced analysis of extreme floods peaks of several rivers in Yorkshire shows that it is useful to introduce the excess flood volume, the volume of water above a certain river level threshold that caused the flooding. Given these excess volumes, I will show which flood mitigation measures are expected to be useful.

Flood storage via controlled and enhanced flooding of certain sections of flood plains and reservoir storage seems hitherto the only mechanisms to create the required volumes or "space for water". Natural flood management solutions, whilst seemingly appealing, generally contribute (far too) little to flood mitigation, as simple estimates can illustrate. To wit, the Boxing Day floods in Leeds, taking a flooding threshold of 3.9m at the Armley river level gauge, lea to a flood excess volume/lake of 2.1km by 2.1km and 2m depth. When one is able to partition this lake in parts on upstream flood plains, then the flood damage can possibly be minimised or prevented.