Autumn 2014 Autumn 2014

Effective Rates in Very Dilute Reaction-Advection Systems

Date: Monday 29th September, 2pm, (JSC 3.02)
Speaker: Dr Giorgio Krstulovic (Observatoire de la Cote d'Azur)
Abstract: Many natural and industrial processes involve the reaction or collision of diffusing species transported by an outer flow. Such systems are typically modelled in terms of the reaction-diffusion-advection equation. In a very dilute system of reacting particles, the basic hypothesis leading to the nonlinear terms of such equations are not satisfied and fluctuations due to a finite number of particles (reactants) must be taken into account.

In this talk I consider a very dilute system of reacting particles transported by fluid flows. The particles react as $A+A \rightarrow \emptyset$ with a given rate when they are within a finite radius of interaction. The system is described in terms of the joint n-point number spatial density that it is shown to obey a hierarchy of transport equations. An analytic solution is obtained in the dilute (or equivalently the long-time limit) by using a Lagrangian approach where statistical averages are performed along non-reacting trajectories. In this limit, I show that the moments of the number of particles have an exponential decay rather than the algebraic prediction of standard mean-field approaches. The effective reaction rate is then related to Lagrangian pair statistics by a large-deviation principle. Then, I introduce phenomenological model to study the qualitative behaviour of the effective rate as a function of the interaction length, the degree of chaoticity of the dynamics and the compressibility of the carrier flow. Exact computations, obtained via a Feynman–Kac approach, in a smooth, compressible, random delta-correlated-in-time Gaussian velocity field support the proposed heuristic approach.

A Fast Solver for the Napier-Stokes Equations

Date: Monday 6 October, 2pm, (TPSC 0.1)
Speaker: Prof Kees Vuik (Delft University of Technology)
Abstract: After linearization and discretization of the incompressible Navier Stokes equations one has to solve block-structured indefinite linear systems. The successful design of robust, scalable, and efficient preconditioners is intimately connected with an understanding of the structure of the resulting block matrix system. Effective preconditioners are often based on an approximate block decomposition of the discretized incompressible Navier Stokes equations. This requires a careful consideration of the spectral properties of the component block operators and their Schur complement operators. Through this purely algebraic view of preconditioning, a simplified system of block component equations is developed. Inclusion of "physics based" preconditioners of the various parts can lead to effective preconditioners with optimal or nearly optimal convergence rates for academic and industrial problems.

CFD applications in maritime industry, for example hull resistance prediction, involve high Reynolds number flows modelled by the incompressible Reynolds-averaged Navier-Stokes equations. The system of equations is discretized with a cell-centered finite-volume method with colocated variables. After linearization, various SIMPLE-type preconditioners can be applied to solve the discrete system. In this presentation, we discuss their performance for flows with Reynolds number up to $10^9$ and cell aspect ratio up to $10^6.$

Anomalous Scalings in Nonstationary Turbulence

Date: Monday 20 October, 2pm, (TPSC 0.1)
Speaker: Prof Sergey Nazarenko (The University of Warwick)
Abstract: We will discuss nonstationary turbulence at the stage prior (and leading to) formation of Kolmogorov cascade. We will consider several basic models of turbulence in which the energy spectrum obeys a nonlinear PDE or an integral equation. We will show that the spectrum has a power-law asymptotic with an anomalous exponent which is less than the Kolmogorov value -5/3. By comparison with turbulence in Burgers equation and with numerical simulations of Navier-Stokes equations, we will speculate that the anomalous scaling is related to formation of a singularity, and that this scenario may be generic for turbulence systems.

Pure and Applied Joint PhD Maths Seminars

Date: Monday 27 October, 2pm, (ARTS 3.03)
Speaker: Ruari Walker (UEA)
Abstract: KLR algebras and VV algebras.

A new family of graded algebras have been introduced by Khovanov, Lauda and independently by Rouquier, the representation theory of which is closely related to that of the affine Hecke algebra of type A. They are often called KLR algebras. More recently, Varagnolo and Vasserot have defined a new family of graded algebras whose representation theory is related to the representation theory of the affine Hecke algebras of type B. These algebras can be thought of as type B analogues of the KLR algebras. I plan to explain this in a little more detail, show how the KLR algebras relate to the VV algebras and compare their module categories via Morita equivalence.

Speaker: Davide Maestrini (UEA)
Abstract: Vortex Clustering and Negative Temperature States in a 2D Bose-Einstein Condensate.

In this work we investigate the question of clustering of like signed vortices in a two-dimensional Bose-Einstein condensate. Such clustering can be understood in terms of negative temperature states of a vortex gas. Due to the long-range nature of the Coloumb-like interactions in point vortex flows, these negative temperature states strongly depend on the shape of the geometry in which this clustering phenomena is considered. We analyze the problem of clustering of portices in a number of different regions. We present a theory to uncover the regimes for which clustering of like signed vortices can occur and compare our predictions with numerical simulations of a point vortex gas. We also extend our results to the Gross-Pitaevskii model of a Bose gas by performing numerical simulations for a range of vortex configurations using parameters that are relevant to current experiments.

Bose-Einstein Condensation in a Box

Date: Monday 10 November, 2pm, (JSC 3.02)
Speaker: Nir Naven (University of Cambridge)
Abstract: I will present our recent work on the experimental realisation of Bose-Einstein condensation in a quasi-uniform box potential [1]. We characterized the critical point for BEC, and observed saturation of the thermal component in a partially condensed cloud, in agreement with Einstein's textbook picture of a purely statistical phase transition. We also observed the quantum Joule-Thomson effect, namely isoenthalpic cooling of a non-interacting gas [2]. In the limit of low temperatures, we measured via Bragg spectroscopy the Heisenberg-limited momentum distribution of the ground state, consistent with a fully coherent macroscopic BEC spanning the whole box [3].

Finally I will present our latest efforts to address the dynamics of Bose condensation following a rapid temperature quench through the phase transition. Using homodyne matter-wave interferometry we have observed the homogeneous Kibble-Zurek power-law scaling of coherence length with quench time, which allowed us to extract the dynamical critical exponent of that universality class [4].

[1] A.L. Gaunt, et. al., Phys. Rev. Lett. 110, 200406 (2013)
[2] T.F. Schmidutz, et. al., Phys. Rev. Lett. 112, 040403 (2014)
[3] I. Gotlibovych, et. al., Phys. Rev. A 89, 061604 (2014)
[4] NN et. al., arXiv:1410.8487 (2014)

Molecules, Photons and the Helmholtz Decomposition of Fields: Quantum Dynamics of Energy Flow in Molecular Aggregates

Date: Monday 17 November, 2pm, (SCI 0.31)
Speaker: Dr Garth Jones (UEA)
Abstract: According to the theory of quantum electrodynamics, transfer of electronic energy between molecules occurs through the exchange of photons. The nature of energy transfer is highly dependent on distances between molecular centres involved in the transfer process. At short distances photons have a virtual character and molecules are coupled by longitudinal as well as transverse components of the field (with an $R^{-3}$ dependence on the coupling). As the distance of separation increases, photons take on a more real character and coupling is dominated by transverse components of the field (in the far-zone limit the coupling has the usual Columbic $R^{-1}$ dependence on separation). Of particular interest is the intermediate- zone where the distance separating the molecular centres is approximately equal to the reduced wavelength of the mediating photon. In this regime, electronic coupling has a separation dependence of $R^{-2}$.

After a brief introduction to electronic energy transfer, quantum electrodynamics and numerical quantum dynamics in the Heisenberg picture, I will present some of our recent results. The talk will focus on investigating how the changing nature of the photon, as it travels from donor (source) to acceptor (sink), affects the rate and directionality of energy flow. Both analytical calculations and numerical quantum dynamical simulations are considered. In the case of numerical simulations the density matrix of the exciton is evolved using the Liouville von-Neumann equations of motion. I will present results that indicate under what circumstances the widely assumed Förster theory (the classical approximation) of energy transfer breaks down. In the last part of the talk, I will present new results where dephasing terms are introduced to the system Hamiltonian. This allows us incorporate temperature to the simulations which gives rise to decoherence within an open quantum system formalism.

Simulation and Modelling of Turbulent Flames and Fuels

Date: Monday 24 November, 2pm, (JSC 2.03)
Speaker: Prof. Stewart Cant (University of Cambridge)
Abstract: Direct Numerical Simulation (DNS) has become a standard research tool in the field of turbulent combustion.  DNS is used to gain insight into the flow physics, and also to extract statistical data for use in Large Eddy Simulation (LES) and Reynolds Averaged Navier Stokes (RANS) modelling of practical combustion systems.  Increased computing resources coupled with improved numerical methods have made it feasible to tackle new and more complex problems at all levels.  This has led to even greater demands on the accuracy and fidelity of the modelling, and in turn to even greater challenges for DNS.  The seminar will provide some examples of DNS aimed at model development, with a focus on partially-premixed combustion and the primary break-up of fuel sprays, as well as some recent results on thermoacoustic analysis of practical gas turbine combustors.

Pure and Applied Joint Faculty Seminar: Extremely Large Cardinals in the Absence of Choice

Date: Monday 8 December, 2pm, (TPSC 0.1)
Speaker: Dr David Aspero (UEA)
Abstract: We will stroll through the upper reaches of the large cardinal hierarchy, especially in a non-AC context.

Spring 2015 Spring 2015

UEA Faculty Seminar: Axisymmetric Solitary Waves on a Ferrofluid Jet

Date: Monday 26th January, 3pm, (ARTS 01.01)
Speaker: Dr Mark Blyth
Abstract: Since they were first spotted by Scott Russell on the Edinburgh-Glasgow Union canal in 1834, there has been a great deal of interest in hydrodynamic solitary waves. Such waves are localised structures which propagate along the surface of a fluid at constant speed while preserving their form. In this talk, we will discuss the propagation of axisymmetric solitary waves on the surface of an otherwise cylindrical ferrofluid jet subjected to a magnetic field. The main interest here lies in the fact that the solitary waves are axisymmetric, and therefore a significant departure from the 2D and 3D waves on planar surfaces which have received the most attention in the literature.
We will begin with a brief introduction to the fundamental equations of inviscid fluid mechanics and classical potential flow theory. We will then show how solitary waves arise as solutions to the classical weakly-nonlinear model PDE, the so-called KdV equation, which emerges from a particular asymptotic limit of the classical water wave problem. We will then discuss wave propagation on a ferrofluid jet, both from the point of view of a KdV model, and through nonlinear computations of the full problem for large amplitude waves.

The Virtual  Element Method

Date: Monday 9th February, 3pm, (SCI 1.20)
Speaker: Dr Andrea Cangiani (University of Leicester)
Abstract: Can we extend the Finite Element Method (FEM) to general polytopic meshes while maintaining the ease of implementation and computational cost comparable to that of standard FEMs?
Within this talk, I will present an approach that achieve just that (and much more):
the Virtual Element Method (VEM).
The Virtual Element spaces are like the usual (polynomial) finite element spaces with the addition of suitable non-polynomial functions. This is far from being a novel idea.
The novelty of the VEM approach is that it avoids expensive evaluations of the non-polynomial "virtual" functions as all computations are solely based on the method's carefully chosen degrees of freedom. In doing that we can easily deal with complicated element geometries and/or higher continuity requirements (like C1, C2, etc.), while maintaining the computational complexity comparable to that of standard finite element computations.
As you might expect, the choice and number of the degrees of freedom depends on such continuity requirements. If mesh flexibility is the goal, while one is ready to  give up on regularity, other approaches can be considered. For instance, dG approaches are naturally suited to deal with polygonal/polyhedral meshes. Time permitting, I will also discuss recent results for hp-discontinuous Galerkin (dG) methods on general meshes.
The talk is based on joint work with: L. Beirao da Veiga, F. Brezzi, G. Manzini, L.D. Marini, A. Russo, O. Sutton (VEM), and P. Dong, E. Georgoulis, P. Houston (dG).

A Computational Model for Macroscale Simulations of Moving Contact Lines

Date: Monday 16th February, 2pm, (SCI 3.05)
Speaker: Dr Yi Sui (Queen Mary)
Abstract: Flows involving moving contact lines (MCLs) are widely found in nature and industry. A major challenge in modelling MCLs is that the conventional hydrodynamic theory combined with a no-slip boundary condition at the wall leads to a non-integrable stress singularity at the contact line. In this seminar I will first review the major theoretical models and computational methods for moving contact lines, and the main associated challenges. Then I will introduce a recent computational model for practical simulations of moving contact lines. The model borrows the idea from the large eddy simulation in turbulence modelling; it resolves the macroscale flows only while model the effect of MCLs using modified hydrodynamic theories.

Emergent Phenomena in Two-Dimensional Quantum Vortex Dynamics

Date: Monday 23rd February, 2pm, (SCI 1.20)
Speaker: Dr Tom Billiam (Durham University)
Abstract: Atomic Bose-Einstein condensates provide a highly controllable system with which to realise quantum vortices in quasi-two-dimensional (2D) geometries, and the study of superfluid turbulence in this reduced dimensionality --- so-called 2D quantum turbulence --- is now an experimental reality. While 2D quantum vortex dynamics is closely related to the motion of classical point vortices, vortex-phonon interactions and the finite size of vortex cores can play a crucial role in many regimes. In this talk I will discuss two key phenomena in turbulent quantum vortex dynamics: the emergence of negative temperature states, and the emergence of classical turbulent cascades. I will also discuss a key question regarding the transition to quantum turbulence: What is the Reynolds number in an inviscid, zero-temperature superfluid?

Dissipation in Turbulent Flows

Date: Monday 2nd March, 2pm, (ARTS 01.01)
Speaker: Prof. Christos Vassilicos (Imperial College)
Abstract: The talk will start with a presentation of the Kolmogorov (1941) theory of turbulence and the Richardson-Kolmogorov equilibrium cascade which, according to this theory, is the mechanism of turbulence dissipation away from walls. The dissipation law which is consistent with this theory is a cornerstone scaling for turbulence modeling and predictions. Recent laboratory experiments involving grid-generated turbulence (2007-2015) and turbulent wakes (2013-2015) have revealed a different universal dissipation law which characterises non-equilibrium turbulence and which underpins new non-equilibrium wake laws, also recently (2013) observed. Direct Numerical Simulations of idealised periodic turbulence to be published in March 2015 confirm this new dissipation scalings in yet another setting and shed light on the non-equilibirum interscale dynamics of the turbulence, in particular the scaling of the interscale energy flux.

Modelling Bladder Cancer: initiation, progression, treatment

Date: Monday 9th March, 2pm, (MED 2.02)
Speaker: Dr Eugene Kashdan (University College Dublin)
Abstract: Bladder cancer is the seventh most common cancer worldwide. According to existing statistics, 80% of BC patients had occupational exposure to chemicals (rubber, dye, textile, or plant industry) or/and were smoking regularly during long periods of time. The carcinogens accumulating in the bladder lumen affect umbrella cells of the urothelium (epithelial tissue surrounding bladder) first and then subsequently penetrate into the deeper layers of the tissue (intermediate and basal cells).  It is a years-long process until the carcinogenic substance will accumulate in the tissue in the quantity necessary to trigger DNA mutations leading to the development of urinary bladder carcinoma.
In my talk, I will give insight in the biological processes responsible for bladder cancer initiation and spread and present their mathematical formulation using a combination of discrete and continuous techniques.  I will show how the cellular automata rules are employed for simulation of the cell cycle (e.g. cell death or cell division), while the carcinogen penetration and oxygen diffusion are described by the nonlinear parabolic PDEs. I will also introduce the sub-models of angiogenesis and cancer invasion (high-grade tumour) that might be embedded into existing simulation framework. If the time allows, I will give a brief overview of our most recent work on bladder cancer treatment analysis and optimisation (personalisation).
Joint work with Dr Svetlana Bunimovich (Ariel College, Israel) and clinical urologists in Israel and in Russia.

UEA Joint PhD Research Seminars

Date: Monday 16th March, 3pm, (EFRY 01.02)

Shape Optimization of Spectral Functionals

Date: Monday 20th April, 2pm, (SCI 1.20)
Speaker: Dorin Bucur (Laboratoire de Mathématiques, Université de Savoie, France)
Abstract: Rayleigh conjectured that among all attached membranes of prescribed area, the disk is the one having minimal fundamental frequency. This result is known as the  Faber-Krahn inequality. In this talk we discuss the question of finding the shape of a membrane with minimal higher frequencies. I shall present some recent results about existence of such membranes and their regularity, in the context of free boundary problems. As well, I will show some numerical approximations obtained in the framework of shape optimization.

Stability of Periodic Gravity-Capillary Water Waves

Date: Monday 11th May, 2pm, (SCI 0.31)
Speaker: Dr Olga Trichtchenko (UCL).
Abstract: I will present results on the computation and stability of periodic surface gravity-capillary waves.  First, I will show how we solve Euler's equations to compute these waves. Then I will present the results of the stability analysis for these solutions by making use of Hill's method. Depending on the coefficient of surface tension, we see resonant effects called Wilton's ripples. These resonant solutions for gravity-capillary waves are found to have interesting instabilities. Since this stability analysis is general to all Hamiltonian systems, we can also use it to compare and contrast the results for different models for water waves.

Soap Film Dynamics and Topological Jumps Under Continuous Deformation

Date: Wednesday 20th May, 2pm, (SCI 3.05)
Speaker: Keith Moffatt (DAMTP, University of Cambridge)
Abstract: Consider the dynamics of a soap-film bounded by a flexible wire (or wires) which can be continuously and slowly deformed. At each instant the soap-film is relaxed in quasi-static manner to a minimum-area (i.e. minimum-energy) state compatible with the boundary configuration. This can however pass through a critical configuration at which a topological jump is inevitable. We have studied an interesting example of this behaviour: the jump of a one-sided (Mobius strip) soap-film to a two-sided film as the boundary is unfolded and untwisted from the double cover of a circle. The nature of this jump will be demonstrated and explained.
More generally, dynamical systems have a natural tendency to relax through dissipative processes to a minimum-energy state, subject to any relevant constraints. An example is provided by the relaxation of a magnetic field in a perfectly conducting but viscous fluid, subject to the constraint that the magnetic field lines are frozen in the fluid. One may infer the existence of magnetostatic equilibria (and analogous steady Euler flows) of arbitrary field-line topology. In general, discontinuities (current sheets) appear during this relaxation process, and this is where reconnection of field-lines (with associated change of topology) can occur. Just as for the soap film, slow change of boundary conditions can lead to critical conditions in which such topological jumps are inevitable.
(Work in collaboration with Ray Goldstein and Adriana Pesci)