Applied Mathematics Seminars Spring 2022

Seminars (unless otherwise stated) will take place online on Tuesdays at 3.00pm on Zoom via the link https://ucl.zoom.us/j/99614222402. If you require any more information on the Applied seminars please contact Prof Jean-Marc Vanden-Broeck (e-mail: j.vanden-broeck AT ucl.ac.uk or tel: 020-7679-2835) or Prof Ilia Kamotski (e-mail: i.kamotski AT ucl.ac.uk or tel: 020-7679-3937).

18 January 2022

Speaker: Jonathan Marshall (UCL)

Title: Constructing arrays of equally-strong holes

For the purpose of designing a perforated thin plate that is to be subjected to a given external loading, it is helpful to be able to determine a priori for which shapes and positions of the plate's holes the stress distribution that is induced in the plate by the loading is in some sense optimised. For instance, it is desirable to determine sets of holes along whose boundaries the induced stresses are uniformly distributed, as localised peaks of these boundary stresses are commonly a source of structural failure. In this talk, we consider an inverse problem of this type. Using complex variable methods, we construct a parameterisation of a class of arrays of so-called `equally-strong’ holes. 

25 January 2022

Speaker: John Ockenden (Oxford University)

Title: Solar Reflectors and Caustics in 3 Dimensions

This talk will outline how high-frequency solutions of Helmholtz equation can be used to determine the efficiency of solar reflectors. Within the ray theory approximation, ideal reflectors concentrate incoming radiation at a focus, but small geometric imperfections often transform the focus into a caustic surface which may have singularities such as umbilics.   Brief  mention will be made of the local solutions of Helmholtz equation that smooth the wavefields near foci and  caustics.

01 February 2022

Speaker: Alex Doak (University of Bath)

Title: Three-layer internal mode-2 solitary waves

Internal waves occur in stratified fluids, such as the world's oceans. They are responsible for the transport of momentum, heat, and organic material, as well as inducing turbulent mixing. Rather than attempting to model a continuously stratified fluid, a common simplification is to assume several homogenous layers of immiscible fluids, separated by infinitesimal interfaces. Two-layer flows have been explored extensively, and such models are used to predict wave properties of 'mode-1' waves. To explore higher modes, one needs additional interfaces.

In this talk we shall discuss travelling wave solutions to a three-layer model. We will be presenting numerical solutions to both the full Euler system, and a reduced model called the three-layer Miyata-Choi-Camassa (MCC3) equations. Mode-2 waves (typically) occur within the linear spectrum, and are hence associated with a resonant mode-1 oscillatory tail. However, as was found for the MCC3 system by Barros et. al (2020), we will present numerical evidence that these oscillations can be found to have zero amplitude, resulting in a truly localised structure known as an embedded solitary wave. We also find mode-2 waves which travel faster than the maximum linear mode-1 wave speed, and are hence outside the linear spectrum. We relate large amplitude solutions to the so-called 'conjugate states' of the system, where the limiting solutions of many of the solution branches are a heteroclinic orbit between conjugate states (i.e. wavefront solutions).

08 February 2022

Speaker: Emilian Parau (University of East Anglia)

Title: Flexural-gravity waves generated by moving loads on floating ice plates

Three-dimensional waves generated by moving loads on top of floating ice plates are investigated. The ice plates are modelled using the thin plate theory.  Viscoelastic models for the ice will be presented and different wave patterns will be calculated. Nonlinear steady waves will be computed using boundary integral methods. Fully dispersive weakly nonlinear equations will also be derived and comparisons of solutions with field observations will be provided.

15 February 2022

Reading Week - No Seminar


22 February 2022

Speaker: Phil Trinh (University of Bath)

Title: Mysteries of waveless objects and the challenges of essential singularities in asymptotics

Since the early 2000s, beyond-all-orders or exponential asymptotic theories have been enormously successful in developing a fairly complete understanding of the free-surface waves produced by steady low-speed flow over an obstruction. In this talk, I demonstrate an object that clearly produces waves, but where the conventional asymptotic theory fails to detect. This motivates several open questions about the extension of exponential asymptotics to singular ODEs with essential singularities.

01 March 2022

Speaker: Linda Cummings (New Jersey Institute of Technology, USA)

Title: Asymptotic thermal modeling of droplet assembly in nanoscale molten metal films

We consider a thin metal film on a thermally conductive substrate exposed to an external heat source in a setup where the heat absorption depends on the local film thickness. Our focus is on modeling film evolution while the film is molten. The film geometry modifies local heat flow, which in turn may influence the film surface evolution through thermal variation of material properties. We use asymptotic analysis to develop a thermal model that is accurate, computationally efficient, and that accounts for the heat flow in both the in-plane and out-of-plane directions. We apply this model to describe metal films of nanoscale thickness exposed to heating and melting by laser pulses, a setup commonly used for self and directed assembly of various metal geometries via dewetting while the films are in the liquid phase. We find that thermal effects play an important role, and in particular that the inclusion of temperature dependence in the metal viscosity modifies the time scale of the evolution significantly. The thickness, thermal conductivity, and rate of heat loss of the underlying substrate are shown to be crucial in accurately modeling film temperatures and subsequent phase changes in the film. Since in many cases the substrate cools the film, modifications to the substrate temperature may induce different dewetting speeds via temperature dependent viscosity of the film. We show via 3D GPU simulations that this may result in various frozen film patterns since full dewetting may not occur while the film is in the liquid phase. This research was supported by NSF CBET-1604351, NSF-DMS-1815613 and by CNMS2020-A-00110.

08 March 2022

Speaker: Mark Blyth (University of East Anglia)

Title: Towards a model of a deformable aerofoil

Aerofoils that include flexible components are designed to enhance aerodynamic capability and to promote fuel efficiency under a range of different flight conditions. Some small, unmanned aerial vehicles (UAVs) use entirely collapsible, elastic wings to maximise portability. In this talk we work toward a model of a deformable aerofoil by studying the behaviour of a thin-walled elastic cell in a uniform stream. We use a conformal mapping approach to determine the cell shape and the ambient flow simultaneously. Our initial analysis of a light cell in an oncoming flow can be viewed as a generalisation of Flaherty et al.'s 1972 work on the buckling of an elastic cell under a constant transmural pressure difference. Introducing cell mass and circulation/lift, we study equilibrium shapes at different flow speeds and for different transmural pressure jumps. A fixed-angle corner at the trailing edge is introduced by way of a Karman-Trefftz conformal mapping, and an internal strut is included, to more accurately mimic the shape and aerodynamic properties of a traditional, rigid aerofoil. DNS simulations are carried out to assess the performance of the equilibrium aerofoil shapes in a real flow

15 March 2022

Speaker: Anna Kalogirou (University of Nottingham)

Title: Theoretical and numerical investigations of extreme waves through oblique soliton interactions

Extreme water-wave motion is investigated analytically and numerically by considering two-soliton and three-soliton interactions on a horizontal plane. We successfully determine numerically that soliton solutions of the unidirectional Kadomtsev-Petviashvili equation (KPE), with equal far-field individual amplitudes, survive reasonably well in the bidirectional and higher-order Benney-Luke equations (BLE). A well-known exact two-soliton solution of the KPE on the infinite horizontal plane is used to seed the BLE at an initial time, and we confirm that the KPE-fourfold amplification approximately persists. More interestingly, a known three-soliton solution of the KPE is analysed further to assess its eight- or ninefold amplification, the latter of which exists in only a special and difficult to attain limit. This solution leads to an extreme splash at one point in space and time. Subsequently, we seed the BLE with this three-soliton solution at a suitable initial time to establish the maximum amplification: it is approximately 7.8 for a KPE amplification of 8.4. In our simulations, the computational domain and solutions are truncated approximately to a fully periodic or half-periodic channel geometry of sufficient size, essentially leading to cnoidal-wave solutions. Moreover, special geometric (finite-element) variational integrators in space and time have been used in order to eradicate artificial numerical damping of, in particular, wave amplitude.

22 March 2022

Speaker: Mohit Dalwadi (UCL)

Title: Levitation, tissue engineering, and cryopreservation: a medley of moving/free boundary problems with interesting asymptotic structures

In this talk I present three problems, linked via moving/free boundaries and the importance of small asymptotic regions on global outcomes.

In the first part, I discuss a problem of fundamental fluid mechanics: if one places a cylinder on a vertical belt covered in a thin layer of oil, is it possible to keep the cylinder centre at a fixed location by moving the belt upwards at a fixed speed? I will present the results of an experimental, asymptotic, and numerical study of this fluid-structure interaction, and show that the asymptotic structure is integral to understanding levitation.

In the second part, I explore a tissue engineering application that involves understanding the motion of a porous cylindrical cell scaffold in a nutrient-fluid-filled rotating bioreactor with a small aspect ratio. While this is ostensibly a problem of coupled lubrication flow, I will show how weak inertia can have a significant effect on the scaffold trajectories. Moreover, I will show how the asymptotic structure of the flow near the bulk-porous interface is key to capturing the trajectories of such scaffolds over timescales of interest.

Finally, in the third part, I investigate a problem of cryopreservation - the process of preserving biological constructs by cooling them to temperatures low enough to halt metabolic processes. In general, cooling too quickly results in the formation of lethal intracellular ice, while cooling too slowly amplifies the toxic effects of the cryoprotective agents (CPA) added to limit ice formation. I present a minimal mathematical model to understand and quantify these observations, resulting in a three-phase, six-variable system with two moving boundaries. Using a combination of numerical and asymptotic methods, I show how to use these to characterize the cell damage caused by freezing, accounting for supercooling and CPA toxicity, and hence how to predict optimal cooling rates.