# Past Industrial and Applied Mathematics Seminar

31 January 2013
16:00
Yi Bin Fu
Abstract
When a rubber membrane tube is inflated, a localized bulge will initiate when the internal pressure reaches a certain value known as the initiation pressure. As inflation continues, the bulge will grow in diameter until it reaches a maximum size, after which the bulge will spread in both directions. This simple phenomenon has previously been studied both experimentally, numerically, and analytically, but surprisingly it is only recently that the character of the initiation pressure has been fully understood. In this talk, I shall first show how the entire inflation process can be described analytically, and then apply the ideas to the mathematical modelling of aneurysm initiation in human arteries.
• Industrial and Applied Mathematics Seminar
24 January 2013
16:00
Elie Raphael
Abstract
It is generally believed that in order to generate waves, a small object (like an insect) moving at the air-water surface must exceed the minimum wave speed (about 23 centimeters per second). We show that this result is only valid for a rectilinear uniform motion, an assumption often overlooked in the literature. In the case of a steady circular motion (a situation of particular importance for the study of whirligig beetles), we demonstrate that no such velocity threshold exists and that even at small velocities a finite wave drag is experienced by the object. This wave drag originates from the emission of a spiral-like wave pattern. The results presented should be important for a better understanding of the propulsion of water-walking insects. For example, it would be very interesting to know if whirligig beetles can take advantage of such spirals for echolocation purposes.
• Industrial and Applied Mathematics Seminar
17 January 2013
16:00
Jared Tanner
Abstract
The essential information contained in most large data sets is small when compared to the size of the data set. That is, the data can be well approximated using relatively few terms in a suitable transformation. Compressed sensing and matrix completion show that this simplicity in the data can be exploited to reduce the number of measurements. For instance, if a vector of length $N$ can be represented exactly using $k$ terms of a known basis then $2k\log(N/k)$ measurements is typically sufficient to recover the vector exactly. This can result in dramatic time savings when k << N, which is typical in many applications such as medical imaging. As another example consider an $m \times n$ matrix of rank $r$. This class of matrices has $r(m+n-r)$ degrees of freedom. Computationally simple and efficient algorithms are able to recover random rank $r$ matrices from only about 10% more measurements than the number of degrees of freedom.
• Industrial and Applied Mathematics Seminar
29 November 2012
16:00
Abstract
The periodic orbits of a discrete dynamical system can be described as permutations. We derive the composition law for such permutations. When the composition law is given in matrix form the composition of different periodic orbits becomes remarkably simple. Composition of orbits in bifurcation diagrams and decomposition law of composed orbits follow directly from that matrix representation.
• Industrial and Applied Mathematics Seminar
22 November 2012
16:00
Abstract
We propose a model to reproduce qualitatively and quantitatively the experimental behavior obtained by the AFM techniques for the titin. Via an energetic based minimization approach we are able to deduce a simple analytical formulations for the description of the mechanical behavior of multidomain proteins, giving a physically base description of the unfolding mechanism. We also point out that our model can be inscribed in the led of the pseudo-elastic variational damage model with internal variable and fracture energy criteria of the continuum mechanics. The proposed model permits simple analytical calculations and to reproduce hard-device experimental AFM procedures. The proposed model also permits the continuum limit approximation which maybe useful to the development of a three-dimensional multiscale constitutive model for biological tissues.
• Industrial and Applied Mathematics Seminar
15 November 2012
16:00
Abstract
Ultracold atomic gases have recently proven to be enormously rich systems from the perspective of a condensed matter physicist. With the advent of optical lattices, such systems can now realise idealised model Hamiltonians used to investigate strongly correlated materials. Conversely, ultracold atomic gases can exhibit quantum phases and dynamics with no counterpart in the solid state due to their extra degrees of freedom and unique environments virtually free of dissipation. In this talk, I will discuss examples of such behaviour arising from spinor degrees of freedom on which my recent research has focused. Examples will include bosons with artificially induced spin-orbit coupling and the non-equilibrium dynamics of spinor condensates.
• Industrial and Applied Mathematics Seminar
8 November 2012
16:00
Stephen Wilson
Abstract
In this talk I shall describe two rather different, but not entirely unrelated, problems involving thin-film flow of a viscous fluid which I have found of interest and which may have some application to a number of practical situations, including condensation in heat exchangers and microfluidics. The first problem, which is joint work with Adam Leslie and Brian Duffy at the University of Strathclyde, concerns the steady three-dimensional flow of a thin, slowly varying ring of fluid on either the outside or the inside of a uniformly rotating large horizontal cylinder. Specifically, we study full-ring'' solutions, corresponding to a ring of continuous, finite and non-zero thickness that extends all the way around the cylinder. These full-ring solutions may be thought of as a three-dimensional generalisation of the full-film'' solutions described by Moffatt (1977) for the corresponding two-dimensional problem. We describe the behaviour of both the critical and non-critical full-ring solutions. In particular, we show that, while for most values of the rotation speed and the load the azimuthal velocity is in the same direction as the rotation of the cylinder, there is a region of parameter space close to the critical solution for sufficiently small rotation speed in which backflow occurs in a small region on the upward-moving side of the cylinder. The second problem, which is joint work with Phil Trinh and Howard Stone at Princeton University, concerns a rigid plate moving steadily on the free surface of a thin film of fluid. Specifically, we study two problems involving a rigid flat (but not, in general, horizontal) plate: the pinned problem, in which the upstream end of plate is pinned at a fixed position, the fluid pressure at the upstream end of the plate takes a prescribed value and there is a free surface downstream of the plate, and the free problem, in which the plate is freely floating and there are free surfaces both upstream and downstream of the plate. For both problems, the motion of the fluid and the position of the plate (and, in particular, its angle of tilt to the horizontal) depend in a non-trivial manner on the competing effects of the relative motion of the plate and the substrate, the surface tension of the free surface, and of the viscosity of the fluid, together with the value of the prescribed pressure in the pinned case. Specifically, for the pinned problem we show that, depending on the value of an appropriately defined capillary number and on the value of the prescribed fluid pressure, there can be either none, one, two or three equilibrium solutions with non-zero tilt angle. Furthermore, for the free problem we show that the solutions with a horizontal plate (i.e.\ zero tilt angle) conjectured by Moriarty and Terrill (1996) do not, in general, exist, and in fact there is a unique equilibrium solution with, in general, a non-zero tilt angle for all values of the capillary number. Finally, if time permits some preliminary results for an elastic plate will be presented. Part of this work was undertaken while I was a Visiting Fellow in the Department of Mechanical and Aerospace Engineering in the School of Engineering and Applied Science at Princeton University, Princeton, USA. Another part of this work was undertaken while I was a Visiting Fellow in the Oxford Centre for Collaborative Applied Mathematics (OCCAM), University of Oxford, United Kingdom. This publication was based on work supported in part by Award No KUK-C1-013-04, made by King Abdullah University of Science and Technology (KAUST).
• Industrial and Applied Mathematics Seminar
1 November 2012
16:00
Peter Kramer
Abstract
Recent years have seen increasing attention to the subtle effects on intracellular transport caused when multiple molecular motors bind to a common cargo. We develop and examine a coarse-grained model which resolves the spatial configuration as well as the thermal fluctuations of the molecular motors and the cargo. This intermediate model can accept as inputs either common experimental quantities or the effective single-motor transport characterizations obtained through systematic analysis of detailed molecular motor models. Through stochastic asymptotic reductions, we derive the effective transport properties of the multiple-motor-cargo complex, and provide analytical explanations for why a cargo bound to two molecular motors moves more slowly at low applied forces but more rapidly at high applied forces than a cargo bound to a single molecular motor. We also discuss how our theoretical framework can help connect in vitro data with in vivo behavior.
• Industrial and Applied Mathematics Seminar
25 October 2012
16:00
John Hinch
Abstract
We study a thin liquid film on a vertical fibre. Without gravity, there is a Rayleigh-Plateau instability in which surface tension reduces the surface area of the initially cylindrical film. Spherical drops cannot form because of the fibre, and instead, the film forms bulges of roughly twice the initial thickness. Large bulges then grow very slowly through a ripening mechanism. A small non-dimensional gravity moves the bulges. They leave behind a thinner film than that in front of them, and so grow. As they grow into large drops, they move faster and grow faster. When gravity is stronger, the bulges grow only to finite amplitude solitary waves, with equal film thickness behind and in front. We study these solitary waves, and the effect of shear-thinning and shear-thickening of the fluid. In particular, we will be interested in solitary waves of large amplitudes, which occur near the boundary between large and small gravity. Frustratingly, the speed is only determined at the third term in an asymptotic expansion. The case of Newtonian fluids requires four terms.
• Industrial and Applied Mathematics Seminar
18 October 2012
16:00
Richard Craster
Abstract
Some striking, and potentially useful, effects in electrokinetics occur for bipolar membranes: applications are in medical diagnostics amongst other areas. The purpose of this talk is to describe the experiments, the dominant features observed and then model the phenomena: This uncovers the physics that control this process. Time-periodic reverse voltage bias across a bipolar membrane is shown to exhibit transient hysteresis. This is due to the incomplete depletion of mobile ions, at the junction between the membranes, within two adjoining polarized layers; the layer thickness depends on the applied voltage and the surface charge densities. Experiments show that the hysteresis consists of an Ohmic linear rise in the total current with respect to the voltage, followed by a decay of the current. A limiting current is established for a long period when all the mobile ions are depleted from the polarized layer. If the resulting high field within the two polarized layers is sufficiently large, water dissociation occurs to produce proton and hydroxyl travelling wave fronts which contribute to another large jump in the current. We use numerical simulation and asymptotic analysis to interpret the experimental results and to estimate the amplitude of the transient hysteresis and the water-dissociation current.
• Industrial and Applied Mathematics Seminar