Imperial College London

Through the advent of enhanced medical imaging computational modelling can now be applied to anatomically correct arterial geometries. However many flow feautures under physiological and pathological flow paraemeters, even in idealised problems, are relatively poorly understood. A commonly studied idealisation of an arterial blockage or stenosis, potentially generated by atherosclerosis, is a sinusoidally varying constricted tube. Although most physiological flow conditions are typically laminar, in this configuration turbulent flow states can arise due to the local increase in sectional Reynolds number. To examine the onset of turbulence in this geometry, under a simple non-reversing pulsatile flows, we have applied Floquet stability analysis and direct
numerical simulation.
As illustrated in the above figure, a period-doubling absolute instability mode associated with alternating tilting of the vortex rings that are ejected out of the stenosis/constriction during each pulse. This primary instability occurs for relatively large reduced velocities associated with long pulse periods (or low Womersley numbers). For lower reduced velocities the primary instability typically manifests itself as azimuthal waves (Widnall instability modes) of low wavenumber that grow on each vortex ring. We have also observed the shear layer of the steady axisymmetric flow is convectively unstable at still shorter temporal periods.
In this presentation we shall outline the challenges of modelling vascular flow problems with a particular focus on idealised stenotic flow. After briefly outlining the numerical analysis methods we shall discuss the flow investigations outlined above and their relation to more classical vortex instabilities.

In this talk we show that there exists a smooth classical solution to the HJB equation for a large class of constrained problems with utility functions that are not necessarily differentiable or strictly concave.

The value function is smooth if admissible controls satisfy an integrability condition or if it is continuous on the closure of its domain.

The key idea is to work on the dual control problem and the dual HJB equation. We construct a smooth, strictly convex solution to the dual HJB equation and show that its conjugate function is a smooth, strictly concave solution to the primal HJB equation satisfying the terminal and boundary conditions

Voltammetry is a powerful method for interrogating electrochemical systems. A voltage is applied to an electrode and the resulting current response analysed to determine features of the system under investigation, such as concentrations, diffusion coefficients, rate constants and thermodynamic potentials. Here we will focus on ac voltammetry, where the voltage signal consists of a high frequency sine-wave superimposed on a linear ramp. Using multiple scales analysis, we find analytical solutions for the harmonics of the current response and show how they can be used to determine the system parameters. We also include the effects of capacitance due to the double-layer at the electrode surface and show that even in the presence of large capacitance, the harmonics of the current response can still be isolated using the FFT and the Hanning window.

Flows involving immiscible liquids are encountered in a variety of industrial and natural processes. Recent applications in micro- and nano-fluidics have led to a significant scientific effort whose aim (among other aspects) is to enable theoretical predictions of the spatiotemporal dynamics of the interface(s) separating different flowing liquids. In such applications the scale of the system is small, and forces such as surface tension or externally imposed electrostatic forces compete and can, in many cases, surpass those of gravity and inertia. This talk will begin with a brief survey of applications where electrohydrodynamics have been used experimentally in micro-lithography, and experiments will be presented that demonstrate the use of electric fields in producing controlled encapsulated droplet formation in microchannels.

The main thrust of the talk will be theoretical and will mostly focus on the paradigm problem of the dynamics of electrified falling liquid films over topographically structured substrates.

Evolution equations will be developed asymptotically and their solutions will be compared to direct simulations in order to identify their practicality. The equations are rich mathematically and yield novel examples of dissipative evolutionary systems with additional effects (typically these are pseudo-differential operators) due to dispersion and external fields.

The models will be analysed (we have rigorous results concerning global existence of solutions, the existence of dissipative dynamics and an absorbing set, and analyticity), and accurate numerical solutions will be presented to describe the large time dynamics. It is found that electric fields and topography can be used to control the flow.Time permitting, I will present some recent results on transitions between convective to absolute instabilities for film flows over periodic topography.

There is a long-studied correspondence between intersections of two quadrics and hyperelliptic curves, first noticed by Weil and since used

as a testbed for many fashionable theories: Hodge theory, motives, and moduli of vector bundles in the '70s and '80s, derived categories in the '90s, non-commutative geometry and mirror symmetry today. The story generalizes to three, four, and more quadrics, exhibiting new geometric behaviour at each step. The case of four quadrics nicely illustrates the modern theory of flops and derivced categories and, as a special case, gives a pair of derived-equivalent Calabi-Yau 3-folds.

Many proteins cleave and reseal DNA molecules in precisely orchestrated
ways. Modelling these reactions has often relied on the axis of the DNA
double helix
being circular, so these cut-and-seal mechanisms can be
tracked by corresponding changes in the knot type of the DNA axis.
However, when the DNA molecule is linear, or the
protein action does not manifest itself as a change in knot type, or the
knots types are not 4-plats, these knot theoretic models are less germane.

We thus give a taxonomy of local DNA axis configurations. More precisely, we
characterise
all rational tangles obtained from a given rational tangle via a rational
subtangle
replacement (RSR). This builds on work of Berge and Gabai. 
We further determine the sites for these RSR of distance greater than 1.
Finally, we classify all knots in lens spaces whose exteriors are
generalised Seifert fibered spaces and their lens space surgeries, extending work of
Darcy-Sumners.

Biologically then, this classification is endowed with a distance that
determines how many protein reactions
of a particular type (corresponding to steps of a specified size) are
needed to proceed from one local conformation to another.
We conclude by discussing a variety of biological applications of this
categorisation.

Joint work with Ken Baker

Gravitational instantons are complete hyperkaehler 4-manifolds whose Riemann curvature tensor is square integrable. They can be viewed as Einstein geometry analogs of finite energy Yang-Mills instantons on Euclidean space. Classical examples include Kronheimer's ALE metrics on crepant resolutions of rational surface singularities and the ALF Riemannian Taub-NUT metric, but a classification has remained largely elusive. I will present a large, new connected family of gravitational instantons, based on removing fibers from rational elliptic surfaces, which contains ALG and ALH spaces as well as some unexpected geometries.

The presence of additives, which may or may not be surface-active, can have a dramatic influence on interfacial flows. The presence of surfactants alters the interfacial tension and drives Marangoni flow that leads to fingering instabilities in drops spreading on ultra-thin films. Surfactants also play a major role in coating flows, foam drainage, jet breakup and may be responsible for the so-called ``super-spreading" of drops on hydrophobic substrates. The addition of surface-inactive nano-particles to thin films and drops also influences the interfacial dynamics and has recently been shown to accelerate spreading and to modify the boiling characteristics of nanofluids. These findings have been attributed to the structural component of the disjoining pressure resulting from the ordered layering of nanoparticles in the region near the contact line. In this talk, we present a collection of results which demonstrate that the above-mentioned effects of surfactants and nano-particles can be captured using long-wave models.