Autumn 2012 Autumn 2012

Capillary Billiards and the Fluid-Mechanical Sewing Machine

Date: Monday 1st October (2pm, S0.31)
Speaker: John Lister (Cambridge) [RJW]
Abstract: Some apparently simple problems in slow viscous flow lead to surprisingly complex behaviour. A layer of fluid coating the underside of a horizontal surface is obviously gravitationally unstable. Less obvious is the dynamics of the drops thus formed. Anyone awake at breakfast-time can observe that a stream of honey falling onto toast buckles and coils on impact. Prediction of the coiling frequency is surprisingly complex. And what happens when you move the toast? This talk will present theory, asymptotics, numerics and experiments to unpack the physical mechanisms responsible for these intriguing phenomena.

Unification of Dynamic Density Functional Theory for Colloids

Date: Monday 8th October (2pm, S3.05)
Speaker: Ben Goddard (Imperial) [MB]
Abstract: In recent years, a number of dynamic density functional theories (DDFTs) have been developed to describe colloid particle dynamics. These DDFTs aim to overcome the high-dimensionality of systems with large numbers of particles by reducing to the dynamics of the one-body density, described by a PDE in only three spatial dimensions, independently of the number of particles. The standard derivations are via stochastic equations of motion, but there are fundamental differences in the underlying assumptions in each case. We focus particularly on the inclusion of inertia and hydrodynamic interactions, both of which strongly influence non-equilibrium properties of the system. We derive a general DDFT in phase space, including hydrodynamic interactions, the results of which are in very good agreement with the full underlying Langevin dynamics. We also show that, in suitable limits, three existing DDFTs are special cases of our formulation, and that close to local equilibrium we obtain a Navier-Stokes-like equation with additional non-local terms. Finally, we describe the rigorous passage to the high-friction limit, where the one-body density satisfies a nonlinear, non-local Smoluchowski-like equation with a novel diffusion tensor.

The stability of two-layer flows

Date: Monday 15th October (2pm, S3.05)
Speaker: Mark Blyth (UEA)
Abstract: Interfaces between viscous liquids can be susceptible to a number of different instabilities including density-driven gravitational instabilities, viscous shear instabilities, capillary instabilities, and Marangoni-forced instabilities. The latter arise when there is a variation in the interfacial tension between the fluids which may be caused by, for example, variation in temperature. The more interesting case arises when interfacial tension variations arise in the presence of an insoluble surfactant which is free to diffuse and to convect along the interface in accordance with the prevailing local fluid velocity. In this talk we will discuss interfacial instability in the presence of surfactant in two-layer channel flow and core-annular pipe flow. In the former case, a weakly-nonlinear formulation reveals a very rich set of dynamical possibilities for the flow evolution, including travelling wave solutions, periodic travelling wave solutions, and chaos, even at zero Reynolds number. For core-annular flow a similar weakly-nonlinear analysis uncovers the possibility of using surfactant to remove the classical Rayleigh capillary instability and stabilise the flow - a most unexpected result.

Vortex knots dynamics in Euler fluids and optimal kinematics of elastic filaments

Date: Monday 22nd October (2pm, S3.05)
Speaker: Francesca Maggioni (Bergamo, Italy) [HS]
Abstract: In this talk we examine the effect of geometric and topological aspects on the kinematics, dynamics and energetics of knotted and unknotted filamentary structures in the contexts of perfect fluids and elastic filaments. In the context of Euler fluids we shall be concerned with vortex filaments in the shape of torus knots/unknots obtained as solution of the Localized Induction Approximation LIA and their evolution is analysed by numerically integrating the Biot-Savart law. In the context of elastic filaments the analysis is based on the thin rod approximation of linear elastic theory, evaluated by means of bending and torsional influences. Generic behaviors of the energy (kinetic or elastic) are determined in both the contexts and compared for several structures with increasing complexity. Vortex complexity is parametrized by the winding number w given by the ratio of the number of meridian wraps to that of longitudinal wraps. We find that for w < 1 vortex knots and toroidal coils move faster and carry more energy than a reference vortex ring of same size and circulation, whereas for w > 1 knots and poloidal coils have approximately same speed and energy of the reference vortex ring. For elastic filaments, kinematics of supercoiling as solutions of the elastic energy minimization are proposed. The deformation energy of the system is required to be monotonically decreasing in time with fixed initial condition. Time evolution functions are described by means of piecewise polynomial transformations based on cubic B-spline functions whose parameters are considered as the unknowns in a non-linear optimization problem. These results add new information on the interplay of geometric and topological aspects on the dynamics of complex systems.

Mechanical Balance Laws for Boussinesq and KdV equations

Date: Monday 29th October (2pm, Arts 2.01)
Speaker: Henrik Kalisch (Bergen) [EP]
Abstract: The Boussinesq scaling regime appears prominently in the study of long-crested surface waves as a requirement on the relation between undisturbed depth, amplitude and wavelength. If the waves appearing in a modelling situation fall into this regime, then the wave evolution may be effectively studied using a number of well known Boussinesq systems. If the waves are travelling predominantly in a single direction, then the KdV equation may be used to describe the waves. In this presentation, the focus will be on connections between surface wave patterns in the Boussinesq scaling, and properties of the underlying fluid flow. In particular, it will be shown how the reconstruction of the velocity field from the principal dependent variables of the evolution equations yields information about the associated flow beneath the surface. Such an analysis can be used advantageously in the study of undular bores, and the description of particle trajectories.

Me, myself and i, a history of the square-root of −1

Date: Monday 12th November (2pm, S3.05)
Speaker: Robert Jenkins (UEA) [MJC]
Abstract: From battling Italians to the greatest unsolved problem in mathematics, the history of i in 50 minutes.

Elliptical flow instability triggered by magnetic field

Date: Monday 19th November (2pm, S3.05)
Speaker: Konrad Bajer (Warsaw) [HS]
Abstract: The flow of a fluid of high electrical conductivity is usually expected to be more stable when that fluid is penetrated by a magnetic field. The reason being that distorting the fields lines takes energy which could otherwise 'fuel' the instability. A well-know counter example is the magneto-rotational instability (MRI) which, for some time, attracts great interest in the astrophysical context. We show a new type of magnetically-triggered instability and thus demonstrate that the destabilising effect of the magnetic field may be much more ubiquitous than previously thought. Possible implications for accretion discs are discussed.

Wave dynamics on a liquid film sheared by a turbulent gas

Date: Monday 26th November (2pm, S3.05)
Speaker: Dmitri Tseluiko (Loughborough)
Abstract: The dynamics of a thin laminar liquid film flowing under gravity down the lower wall of an inclined channel when turbulent gas flows above the film will be discussed. The solution of the full system of equations describing the gas-liquid flow faces serious technical difficulties. However, a number of assumptions allow isolating the gas problem and solving it independently by treating the interface as a solid wall. This permits finding the perturbations to normal and tangential stresses at the interface imposed by the turbulent gas in closed form. Then the liquid film flow under the influence of these perturbations can be analysed by deriving and analysing a hierarchy of model equations describing the dynamics of the interface, i.e. boundary-layer equations, a long-wave model and a weakly nonlinear model, which turns out to be the Kuramoto-Sivashinsky equation with an additional term due to the presence of the turbulent gas. Also, by combining the long-wave approximation with a weighted-residual technique, an integral-boundary-layer approximation that is valid for moderately large values of the Reynolds number can be obtained. This model is then used for a systematic investigation of the flooding phenomenon observed in various experiments: as the gas flow rate is increased, the initially downward-falling film starts to travel upwards while just before the wave reversal the amplitude of the waves grows rapidly.

Spring 2013 Spring 2013

Statistical mechanics of a neutral point vortex gas

Date: Monday 14th January (3pm, Arts 3.07)
Speaker: Gavin Esler (UCL) [HS]
Abstract: The statistical mechanics of a neutral point vortex `gas', in which equal numbers of vortices with positive and negative circulations evolve in a bounded two-dimensional container, are re-examined. It is emphasised that the system can be studied in (at least) two asymptotic limits. In the hydrodynamic limit, relevant to large positive energies, theory predicts the asymptotic state to be dominated by coherent vortices and mean circulations. In the thermodynamic limit, relevant to low energies (positive and negative), mean circulations are absent and fluctuations or eddies dominate. The cumulant expansion method can be used to derive a governing equation describing the amplitude of vorticity fluctuations as a function of energy. Here, the general solution of this `vorticity fluctuation' equation is given for the first time, and its predictions are tested against both statistical estimates of ensemble averages, and direct numerical simulations.

Date: Monday 21st January (2pm, ARTS 3.07)
Seminar Postponed due to inclement weather

Poroelastic trailing edge noise and the silent flight of owls

Date: Monday 28th January (3pm, Arts 3.07)
Speaker: Justin Jaworski (Cambridge) [PWH]
Abstract: Many owl species rely on specialised plumage to mitigate the generation of aerodynamic noise to realise functionally-silent flight whilst hunting.  One such plumage feature, the arrangement of flexible trailing edge feathers, is idealised as a semi-infinite poroelastic plate to model the effects of edge compliance and flow seepage.  The interaction of the poroelastic edge with a turbulent eddy is examined analytically with respect to the efficiency of scattered aerodynamic noise.  The scattering problem is solved exactly using the Wiener-Hopf technique to identify the scaling dependence of the noise on the flight velocity, where special attention is paid to the limiting cases of rigid-porous and elastic-impermeable plate conditions.  Results from this analysis identify parameter spaces where the porous and/or elastic properties of a trailing edge may be tailored to diminish or even eliminate the edge scattering effect and contribute to the understanding of the owl hush-kit.

Pattern formation in exciton-polariton condensates

Date: Monday 4th February (3pm, S3.05)
Speaker: Natalia Berloff (Cambridge) [HS]
Abstract: I will discuss the phenomena observed in, and properties of, microcavity exciton-polariton condensates. These are condensates of mixed light and matter, consisting of superpositions of photons in semiconductor microcavities and excitons in quantum wells. Because of the imperfect confinement of the photon component, exciton-polaritons have a finite lifetime, and have to be continuously re-populated. Therefore, exciton-polariton condensates lie somewhere between equilibrium Bose-Einstein condensates and lasers. I review in particular the evidence for condensation, the coherence properties studied experimentally, and the wide variety of spatial structures either observed or predicted to exist in exciton-polariton condensates, including quantised vortices and other coherent structures.

Solidification of molten metallic foams

Date: Monday 11th February, 2pm Arts 3.07
Speaker: Peter Stewart (Oxford) [RJW]
Abstract: High-porosity metallic solids can be formed by solidification of the corresponding molten gas-liquid foam. However, molten metallic foams are thermally and dynamically unstable, so in the absence of solidification the thin liquid films drain rapidly toward the bubble vertices and eventually become unstable to interfacial instabilities, leading to film rupture and bubble coalescence. To explore the competition between coarsening and freezing we have constructed a large-scale network model to describe the dynamics and stability of a planar foam with low liquid fraction, incorporating a coupling between pressure and volume in the gas bubbles, surface tension forces on the gas- liquid interfaces, draining flow in the films, a criterion for film rupture, temperature variations and a solidification front. Initially, the foam is arranged in a regular array of approximately polygonal bubbles, held at a uniform temperature above the melting point of the material. The walls of the container are then cooled to a temperature well below the melting point, driving a solidification front inwards; numerical simulations of the model predict the structure of the resulting porous metal solid.

Coherent Structure Dynamics and Turbulence in Superfluids

Date: Monday 18th February, 3pm, S3.05
Speaker: Davide Proment (UEA)
Abstract: I will revise what is a Bose-Einstein condensate (BEC) and derive the easiest model to study its dynamics: the Gross-Pitaevskii equation (GPE). This is nothing but a particular case of a nonlinear Schrödinger equation which has been applied in many other physical systems like ocean waves and nonlinear optics. I will first discuss why the GPE models a superfluid and which type of coherent structures may be found depending on its spatial dimensionality: one-dimensional solitons and breathers, two-dimensional vortices, and three-dimensional vortex rings and knots. I will show how these structures interact and address the main unsolved questions arising in this field. I will then explain how turbulence arises in superfluids, discussing different turbulent regimes. By revising some concepts of classical turbulence, I will present a mathematical approach called weak wave turbulence that can analytically predicts some turbulent states in BECs.

Localised Three-Dimensional Capillary-Gravity waves: Solitons and Breathers

Date: Monday 25th February, 2pm S3.05
Speaker: Paul Milewski (Bath) [MGB,EP]
Abstract: The capillary gravity wave problem exhibits a large variety of phenomena, one of which is the combination of geometric and nonlinear self-interaction focussing of localised disturbances. While in the small amplitude limit this focussing leads to an instability with  finite-time blow-up of model equations, we show that in the full fluid equations, the focussing is arrested, and, instead, fully localised breathers are generated.

Ground States and Dynamics in a Two-component Condensate

Date: Monday 4th March, 2pm, SCI 3.05
Speaker: Peter Mason (Durham) [HS]
Abstract: A mixture of Bose-Einstein condensates (such as two different isotopes) display various ground state geometries that are dependent on the system parameters, like the inter- and intra-component coupling strengths. I will present an analysis, using the nonlinear sigma model in the mean-field limit, that will enable us to understand these ground states theoretically. After the theory have been developed, I will introduce various dynamical properties of vortices in a single species condensate before moving on to two-component condensates. A vortex in a two-component condensate creates a density depletion/peak in the other component. The profile of this depletion/peak will be analysed in a miscible condensate using a variational analysis which will in turn allow us to find the velocity of a vortex pair in such systems.

Theoretical Models of Blood Flow in Vascular Networks: Discrete and Continuum Approaches

Date: Monday 11th March, 3pm, S3.05
Speaker: Rebecca Shipley (UCL) [RJW]
Abstract: The vasculature comprises a hierarchy of vessels that is frequently categorized according to vessel size. Although the geometry and topology of the vasculature is organ-specific, blood flows into an organ from a feeding artery, through the arterioles into the microcirculation, and exits through the venules then veins. Gas exchange occurs primarily in the microcirculation and, indeed, the function of the vasculature is to provide oxygen for cellular metabolism. Understanding and predicting the flow of blood through these networks will play a crucial role in, for example, promoting angiogenesis and vascular remodelling to treat myocardial ischaemia, and in analyzing perfusion maps extracted using medical imaging techniques. Traditional modelling approaches employ a discrete approach by solving equations for blood flow in each vessel of a network. However, recent advancements in imaging methods have led to a wealth of data that describe vascular structure in a highly detailed way, and it is becoming too computationally intensive to simulate flow and mass transport in the complete vascular tree using a discrete approach. Continuum models must be developed that can be used alongside a discrete approach to capture the key functional properties of blood flow. Continuum multiscale models that describe blood flow in the microcirculation, derived using the mathematical process of asymptotic homogenization, will be discussed. A strategy for combining discrete and continuum models to simulate blood flow in large networks will be presented, and results of testing this strategy for explicit examples of rat mesentary networks will be demonstrated.

Student BAMC Practice Talks

Date: Monday 18th March, 2pm, SCI 3.05

Multi-layer Curtain Coating
Speaker: Julian Thompson
Abstract: We discuss experiments and theory for multi-layer curtain coating. Curtain coating is used in the manufacturing of photographic film, often needing
several layers for colour photographs. This requires an even coating, so minimising disturbances is essential. Previous experiments on curtain
coating have been done by manufacturers and as such are hidden in patents. We have conducted experiments for various parameter ranges to deduce an
acceptable parameter range for the curtain to remain stable. Our work extends this to three layers. It is found that the flow rate to create a
stable curtain in general is greater than the flow rate at which the curtain breaks up. This involves finding the velocity field for multiple layer flow
down an inclined plane for fixed layer thicknesses and perturbing by small quantities to analyse stability. A preliminary theoretical analysis
considers the stability of the layers flowing down an inclined plane before they reach the lip and form a curtain.

The Effect of Wall Inertia on Oscillations in an Elastic-Walled Tube
Speaker: Martin Walters
Abstract: Fluid flow through elastic-walled tubes has many biological applications, including to the cardiovascular and respiratory systems. It is found that in certain parameter regimes, steady flow  through such tubes is unstable to self-excited oscillations. Whittaker et al (2010, Proc Roy Soc A 466) solved an asymptotic model for the onset of self-excited oscillations in a long, thin-walled, flexible tube clamped between two rigid tubes, with a large axial tension. However, this work neglected the effects of wall inertia. Here, we add wall inertia terms to the governing equations for the wall mechanics and derive a new 'tube law' to describe the wall motion. Using this, along with a description of the fluid dynamics of the flow, we quantify the effect of wall inertia on the stability and growth rate of the oscillations. We find that wall inertia is a destabilising effect.

On contact line dynamics with mass transfer

Date: Monday 25th March, 11am
Speaker: Jim Oliver (University of Oxford)

Summer 2013 Summer 2013

The Prediction of Flows in Wind Farms and Tidal Stream Farms

Date: Monday 22nd April, 2pm, S3.05
Speaker: Iain Jones (Ansys) [NR]
Abstract: The renewable energy industry faces many challenges. Among them are the need to improve efficiency, reduce costs and increase the reliability of resource estimation. Methods based on a linearised flow analysis have been very successful in the prediction of flows on wind farm sites, but these methods have a number of well-known limitations, including complex terrain, forestry, and turbine wake interactions.
This talk will describe the use of Computational Fluid Dynamics (CFD) to provide predictive methods which overcome these limitations.  The presentation will outline the methods and sub-models used, and then describe the extension of the approach to include the effect of atmospheric stability. Throughout, the talk will be illustrated by case studies taken from real wind farm sites, both onshore and offshore, which highlight the significant benefits that can be obtained from the improved methodology. Finally the presentation will discuss the extension of the methods to predict flows and wake interactions in tidal stream farms.

Intracellular Signalling Cascades and Understanding the Cellular Response —  A Mathematical Approach

Date: Monday 29th April, 2pm, S0.31
Speaker: Marcus Tindall (University of Reading) [RJW]
Abstract: 40 years of combined experimental and theoretical research into understanding the cellular response and intracellular signalling cascades in Escherichia coli bacterial chemotaxis has become a paradigm for the recently emerged field of Systems Biology. In this talk I will consider the analysis of a recently published mathematical model of the signalling cascade within E. coli to answer a simple question regarding the cellular response. The results of the analysis indicate that variability in protein concentration and not the protein-protein interaction kinetics within the intracellular signalling cascade is a key determinant of the cell's behaviour, affecting both the form of the response and the timescale over which it occurs. Model reduction of the original model allows us to determine the key components of the signalling network which are required to ensure a robust cellular response.

Quantum Turbulence in Superfluid 4He in the Zero Temperature Limit

Date: Monday 13th May, 2pm, S0.31
Speaker: Paul Walmsley (University of Manchester) [HS]
Abstract: Turbulence in a pure superfluid, where the classical normal component is absent, consists entirely of a dynamic tangle of quantized vortex filaments. Different types of turbulence are possible depending on the large-scale correlations of the vortex lines. At scales larger than the typical intervortex spacing, classical-like behaviour is possible due to polarized bundles of vortices mimicking classical fluid eddies. At smaller scales the quantized nature of vorticity dominates. In this quantum regime, it is widely thought that energy is transferred to smaller scales due to various types of reconnections and a Kelvin wave cascade until dissipation due to sound emission becomes efficient. In recent years, we have developed new experimental techniques for generating and probing turbulence in superfluid 4He at very low temperatures. Our main achievement is using injected electrons to measure the rate of decay of quantum turbulence, allowing the observation of turbulence with different spectra (quasiclassical and ultraquantum). We have also investigated inertial waves and turbulence in a rotating container; the interactions within an anisotropic beam of micron-sized vortex rings; and the small scale quantum structure of vortex tangles due to the emission of small vortex rings. These various experiments will be outlined and the direction for new experiments currently being developed will be discussed.

Passive Control of Instabilities in Hypersonic Flow

Date: Monday 20th May, 2pm, S3.05
Speaker: Sharon Stephen (University of Birmingham) [MGB]
Abstract: Passive porous walls have been investigated as a means for delaying transition to turbulence in hypersonic flow. Here a theoretical linear stability analysis will be presented to consider the effect of a porous wall on the first (viscous) instability mode of a hypersonic boundary-layer flow over a sharp slender cone. The effect of curvature and of the attached shock is included for axisymmetric and non-axisymmetric disturbances. The flow in the hypersonic boundary layer is coupled to the flow in the porous layer. Asymptotic methods are used for large Reynolds number and large Mach number. The linear results for neutral stability and spatial instability will be presented for physical parameters and porous wall models, chosen to correspond to relevant experiments. The effects of varying the porous wall parameters will be shown. A weakly nonlinear stability analysis has been carried out allowing an equation for the amplitude of disturbances to be derived. The coefficients of the terms in the amplitude equation determine the effects of nonlinearity. The stabilising or destabilising effect of nonlinearity is found to depend on the cone radius. The presence of porous walls significantly influences the effect of nonlinearity.

Mathematical Modelling of Cell/Matrix Growth on 3D Structures

Date: Thursday 18th July, 3pm, S3.05
Speaker: Yann Guyot (University of Liege)
Abstract: The kinetics of in vitro cell growth has been shown to depend on the local surface curvature of the substrate, an observation that lead to the description of curvature-controlled cell growth. Additionally, Rumpler et al. (2008) proposed a 2D computational model capable of capturing in vitro cell growth with a curvature-driven velocity advecting the cell surface. Inspired by the results of Rumpler and co-workers, the work presented here aims to extend their model to 3 dimensions to capture 3D in vitro cell growth, in this case applied for cell-seeded open porous scaffolds cultured under static conditions. To be able to simulate cell/matrix growth in a dynamics bioreactor environment and study the effect of fluid flow on the cell/matrix growth, this study also addresses for the first time a coupling of the proposed growth model to a fluid model.