Autumn Semester 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 Semester 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.