The Applied Maths groups at UEA host regular research seminars, inviting leading mathematicians from around the world to present and discuss their latest research. The seminars are usually held on Mondays from 1-2pm, and all are welcome to attend.

### Spring 2020

#### 20 January 2020

Speaker: Ricardo Lopes-Barros (Loughborough University)

Large amplitude mode-2 internal solitary waves in three-layer flows

Abstract: We consider a strongly nonlinear long wave model for large amplitude internal waves in a three-layer flow bounded above and below by rigid boundaries. The model extends the two-layer Miyata-Choi-Camassa (MCC) model (Miyata 1988; Choi & Camassa 1999) and is able to describe the propagation of long internal waves of both the first and second baroclinic modes. Solitary-wave solutions of the model are shown to be governed by a Hamiltonian system with two degrees of freedom. Emphasis is given to the solitary waves of the second baroclinic mode (mode-2) and their strongly nonlinear characteristics that fail to be captured by weakly nonlinear models. In asymptotic limits relevant to oceanic applications and previous laboratory experiments, it is shown that, after choosing relevant physical parameters, large amplitude mode-2 waves with single-hump profiles can be described by the solitary wave solutions of the MCC model, originally developed for mode-1 waves in a two-layer system. As a result of the richness of the dynamical system with two degrees of freedom, in the case when the density stratification is weak and the density transition layer is thin, new classes of mode-2 solutions, characterised by multi- humped wave profiles of large amplitude, are also found. In contrast with the classical solitary-wave solutions described by the MCC equation, such multi-humped solutions cannot exist for a continuum set of wave speeds for a given layer configuration. Our analytical predictions based on asymptotic theory are then corroborated by a numerical study of the full dynamical system.

#### 27 January 2020

Speaker: Nazile B. Dişibüyük (Dokuz Eylul University, Izmir, Turkey)

Diffraction of flexural-gravity waves by a vertical cylinder of non-circular cross section

Abstract: The linear three-dimensional problem of hydro-elastic wave diffraction by a vertical cylinder of an arbitrary smooth cross section is studied using an asymptotic approach combined with the vertical mode method for water of finite depth. The surface of the water is covered by an infinite, continuous elastic ice plate. The rigid cylinder extends from the sea bottom to the ice surface. The ice plate is frozen to the cylinder. The ice deflection is described by the equation of a thin elastic plate of constant thickness with clamped edge conditions at the cylinder. The flow under the ice is described by the linear theory of potential flows. The coupled problem of wave diffraction is solved in two steps. First, the problem is solved without evanescent waves similar to the problem of water waves diffracted by a vertical cylinder. This solution does not satisfy the edge conditions. Second, a radiation problem with a prescribed motion of the ice plate edge is solved by the vertical mode method. The sum of these two solutions solve the original problem. Both solutions are obtained by an asymptotic method with a small parameter quantifying a small deviation of the cylinder cross section from a circular one. Fourth order asymptotic solutions are obtained by solving a set of two- dimensional boundary problems for Helmholtz equations in the exterior of a circle. Strains along the edge, where the ice plate is frozen to the cylinder, are investigated for nearly square and elliptic cross sections of the vertical cylinders depending on the characteristics of ice and incident wave. The strains are shown to be highest in the places of high curvatures of the cross sections. The derived asymptotic formulae can be used in design of vertical columns in ice. They directly relate the strains in ice plate to the shape of the column.

Authors: Nazile B. Dişibüyüka, A. A. Korobkinb, O. Yılmazc

- Department of Mathematics, Dokuz Eylul University, Izmir, Turkey
- School of Mathematics, University of East Anglia, Norwich, UK
- Department of Mathematics, Izmir Institute of Technology, Izmir, Turkey

#### 3 February 2020

Speaker: Paul Hammerton (UEA)

Swinging Cricket Balls — Why Boundary Layer Transition Is Important

Abstract: This will be a talk in two parts — hopefully with something for everyone in the department. If a flat plate is placed in a high-speed wind tunnel, at some distance downstream of the leading edge the flow in the layer close to the plate can be observed to change in character from laminar flow (smooth flow almost parallel to the surface) to turbulent flow. The point at which transition occurs is quite sensitive to the particular wind-tunnel being used. This is known as the receptivity problem — the point of transition depends not only on the position of the point of neutral stability in the boundary layer, but on free-stream disturbances and how they interact with the surface. In the first part of the talk I will discuss the receptivity process, looking at how the shape and surface of a body can affect the level of receptivity. I will also discuss when transition is important be it drag reduction on plane wings or the importance in sport. The second part will focus more on the mathematical details. I will discuss work in progress, starting during my period of study leave. Looking at receptivity due to the leading edge of the body I will describe how understanding the eigenfunction decomposition of the solution in the boundary is important, and progress made in identifying these solutions.

#### 10 February 2020

Speaker: Demetrios Papageorgiou (Imperial College London)

Fluid dynamics in superhydrophobic channels structured with micro scale grooves

Abstract: Pressure-driven flows in channels are considered when one or both walls are structured with a large number of parallel micro grooves that are aligned longitudinally with the flow. The grooves have finite depth and are typically filled with gas, enabling the liquid in the channel to flow in an almost shear-free configuration known as a ``plastron". This leads to some interesting two-phase mathematical problems with mixed boundary conditions - no slip on solid surfaces and zero shear at liquid gas menisci that are allowed to have arbitrary curvature. Due to the imposed pressure gradient the meniscus curvature changes as we traverse the channel and in general a fully 3D flow ensues. This talk will be in three parts: (i) a complete description of solutions in the 2D case assuming the flow to be parallel; (ii) allowing 3D effects through slow longitudinal variations of the meniscus curvature leading to a semi-analytical construction; (iii) the hydrodynamic bi-global instability characteristics of such flows at arbitrary Reynolds numbers. All numerical work is constructed to be spectrally accurate and is coupled with singularity removal at boundary discontinuities. The stability results will also be compared with experimental observations.

#### 2 March 2020

Speaker: Stephen Wilson (University of Strathclyde)

Competitive Evaporation of Multiple Droplets

Abstract: The evaporation of sessile droplets is currently the subject of a great deal of international experimental, numerical and theoretical research activity, but most of the work thus far has, for understandable reasons, focused on the case of a single droplets. However, in practice, droplets rarely occur in isolation and so there is considerable interested in understanding the interactions between multiple evaporating droplets. To this end I shall present recent results on the interactions between two droplets in two dimensions [1] and between multiple thin droplets in three dimensions [2], and, in particular, investigate the effect of these interactions on droplet evolutions and lifetimes, as well as on the famous coffee-ring effect.

[1] Schofield, F.G.H., Wray, A.W., Pritchard, D., Wilson, S.K., The shielding effect extends the lifetimes of two-dimensional sessile droplets, to appear in J. Eng. Maths. (2020)

[2] Wray, A.W., Duffy, B.R., Wilson, S.K., Competitive evaporation of multiple sessile droplets, J. Fluid Mech. 884 A45 (2020)

#### 9 March 2020

Tao Gao (University of Greenwich)

Hydroelastic waves on a linear shear current at finite depth

Abstract: This work is concerned with waves propagating on water of finite depth with a constant-vorticity current under a deformable flexible sheet. The pressure exerted by the sheet is modelled by using the Cosserat thin shell theory. By means of multi-scale analysis, small amplitude nonlinear modulation equations in several regimes are considered, including the nonlinear Schrödinger equation (NLS) which is used to predict the existence of small-amplitude wavepacket solitary waves in the full Euler equations and to study the modulational instability of quasi-monochromatic wavetrains. Guided by these weakly nonlinear results, fully nonlinear steady and time-dependent computations are performed by employing a conformal mapping technique. Bifurcation mechanisms and typical profiles of solitary waves for different underlying shear currents are presented in detail. It is shown that even when small-amplitude solitary waves are not predicted by the weakly nonlinear theory, we can numerically find large-amplitude solitary waves in the fully nonlinear equations. Time-dependent simulations are carried out to confirm the modulational stability results and illustrate possible outcomes of the nonlinear evolution in unstable cases.

### Autumn 2019

#### 7 October 2019

Bartosz Protas (McMaster University, Ontario, Canada)

A Calculus of Shapes for Free-Boundary Problems: a Case Study in Vortex Dynamics

Many problems in science and engineering are described in terms of equilibrium shapes on which certain conditions are imposed and which separate regions where the solution may have different properties. A prototypical problem of this type involves inviscid vortex equilibria in 2D and axisymmetric 3D geometries characterised by compact vortex regions embedded in a potential flow. Computation of such equilibrium configurations is made difficult by the fact that it requires finding the shape of the boundary separating the two regions. Similarly, studying the linear stability of such free-boundary problems is also challenging as it requires characterization of the sensitivity of the equilibrium solutions with respect to suitable perturbations of the boundary. We will demonstrate that such questions can be in fact systematically addressed using techniques of "shape calculus" applied to the boundary-integral formulations of such problems, leading to elegant and accurate computational approaches. In the context of vortex dynamics we use these techniques to efficiently compute the family of inviscid vortex rings initially discovered by Norbury (1973). We also obtain an equation characterizing the stability of general vortex equilibria from which certain classical results of vortex stability can be derived as special cases. Finally, this approach is employed to solve open problems concerning the linear stability of Hill's and Norbury's vortices to 3D axisymmetric perturbations, which leads to some unexpected findings.

#### 14 October 2019

Duncan Hewitt (DAMTP, University of Cambridge)

Swimming in mud: viscoplastic locomotion and slender-body theory

Many natural fluids, suspensions, emulsions and foams are characterised by a plastic yield stress, above which they flow like a viscous fluid and below which they do not significantly deform. In this talk, a variety of mechanisms for translation and locomotion through such ‘visco-plastic’ materials are discussed, in the limit of slow motion or small spatial scales. Various classical models for micro-swimming are revisited in the case of a visco-plastic material, and a general slender-body analytical theory for such materials is developed. Numerical solutions are presented and discussed, with particular attention paid to the ‘plastic’ limit of very slow motion or large yield stress. Canonical flow structures and swimming gaits are explored. Implications for real swimmers, as well as comparison with some experimental results, are discussed.

#### 21 October 2019

Andrey Cherdantsev (Kutateladze Institute of Thermophysics, Novosibirsk, Russia)

Experimental study of air entrapment at oblique impact of a body at a free surface

In annular gas-liquid flow, droplets are torn from liquid film surface and entrained by turbulent gas stream. Impacts of the entrained droplets back onto the film occur at shallow angles and high impact velocities. In such a case, a droplet creates a long and narrow "furrow" on liquid surface, which is accompanied by massive entrapment of gas bubbles into the liquid film. This phenomenon is different from the known mechanisms of air entrapment such as air cushioning; its nature is not entirely clear and deserves intensive experimental and theoretical investigation.

First part of the presentation reports on experimental study of oblique high-speed droplet impact in "natural" conditions of annular two-phase flow in a horizontal rectangular duct using three-dimensional and stereoscopic Laser-Induced Fluorescence approaches. The results are focused on the effect of parameters of an individual droplet on the type and shape of liquid surface perturbation and intensity of bubbles entrapment.

The contribution of this phenomenon into total amount of bubbles inside the liquid film and its role in evolution of the whole ensemble of bubbles are analysed.

The second part present the results of model experiments on impact of a large (21.3 cm) solid sphere onto a stagnant layer of liquid. This study is focused on air-cushioning mechanism of entrapment and elucidates the very initial stage of impact, prior to the contact between the solid body and the liquid. Despite that the sphere may embed into liquid by 5-6 mm, it is still separated from the liquid by a thin air layer. Synthetic Schlieren method is employed for spatiotemporal measurements of the shape of the liquid surface and the air layer, including the dynamics of the crater and different kinds of waves produced by the impact.

Further experimental and theoretical studies will be aimed at clarification of the physical mechanism of air entrapment for oblique high-speed impact and at the influence of the impact angle on air-cushioning entrapment.

#### 28 October 2019

Wei Guo, National High Magnetic Field Laboratory, Florida, USA

Department of Mechanical Engineering, Florida State University, Florida, USA

Visualisation study of quantum fluid dynamics in superfluid He-4

Helium-4 in the superfluid phase (He II) is a two-fluid system that exhibits fascinating quantum fluid dynamics with important scientific and engineering applications. It supports the most efficient heat-transfer mechanism (i.e. thermal counterflow), and it also allows the generation of flows with extremely high Reynolds numbers for turbulence modelling. However, the lack of high-precision flow measurement tools in He II has impeded the progress in understanding and utilising its hydrodynamics. In recent years, there have been extensive efforts in developing quantitative flow visualisation techniques applicable to He II. Two types of techniques based on the use of either particle tracers (i.e. micron-sized frozen particles) or molecular tracers (i.e. He2* excimer molecules) have been developed. We will discuss the advantages and issues associated with these visualisation techniques and will highlight some recent progresses in our visualisation study of counterflow and quasiclassical turbulence in He II. We will also briefly introduce our on-going work on developing the next generation flow visualisation techniques and our effort on imaging quantized vortices in a levitated drop of He II.

#### 11 November 2019

Linda Cummings, New Jersey Institute of Technology, New Jersey, USA

Title: Modeling and large-scale simulation of thin film liquid flows

Thin film flows of nematic liquid crystal will be considered, using the Leslie-Eriksen formulation for nematics. Our model can account for variations in substrate anchoring, which may exert a strong influence on patterns that arise in the flow. A number of simulations will be presented using an "in-house" code, developed to run on a GPU. Current modeling directions involving flow over interlaced electrodes, so-called "dielectrowetting" will be discussed.

#### 20 November 2019

Michikazu Kobayashi, Kyoto University, Japan

Title: Energy and helicity cascades in non-Abelian quantum turbulence

Quantum turbulence is realised as a dynamically and temporally complicated structure of quantized vortices in quantum fluid such as superfluid helium and atomic Bose-Einstein condensates. In this seminar, I talk about non-Abelian quantum turbulence comprised of non-Abelian quantized vortices, the topological charge of which is classified by the non-Abelian group. Being different from reconnecting dynamics of Abelian vortices, non-Abelian vortices show the formation of rung vortices when they collide keeping their linking topology. As a result, non-Abelian quantum turbulence shows a large-scale networking structure of vortices in which almost all vortices are connected. We also find several kinds of cascading processes in the wave-number space: inverse and direct cascades of the mass kinetic energy and helicity, respectively, and direct cascade of the spin kinetic energy. I will show the expecting scenario connecting the dynamics of vortices and cascading processes. Our prediction can be tested in, for example, the cyclic phase of a spin-2 spinor Bose-Einstein condensate.

#### 25 November 2019

Scott McCue, Queensland University of Technology, Queensland, Australia

Title: Using time-frequency analysis to identify features of steady and unsteady ship wakes

The motivation here is to study how properties of a ship wake can be extracted from surface height data collected at a single point as the ship travels past. The tool we use is a spectrogram, which is a heat map that visualises the time-dependent frequency spectrum of the surface height signal. In this talk, the focus will be on presenting the theoretical framework which involves an idealised mathematical model with a pressure distribution applied to the surface. A geometric argument based on linear water wave theory provides encouraging results for a range of ship speeds. The effects of nonlinearity are also studied. We compare our theoretical predictions with experimental results from the field and from data collected at the Australian Maritime College. This type of work has the potential to inform ship design, the detection of irregular vessels, and how coastal damage is attributed to specific vessels in shipping channels.

#### 2 December 2019

Stefan Llewelyn Smith (University of California San Diego)

Title: Motion of vortices with buoyancy

In this talk I will discuss vortical flows with density variations in the presence of gravity. Explicit calculations of the motion of non-trivial vortices in the presence of density differences and surface tension are not as common as one might expect. Three topics are presented. An asymptotic model for the evolution of a thin-core vortex filament with density variations is discussed, using a formulation based on forces due to Moore and Saffman that allows buoyancy and surface tension forces to be incorporated in a natural manner. A contour dynamics method for axisymmetric vortex rings with density differences is presented, which requires following the evolution of a vortex sheet on the boundary generated by baroclinic torques. Finally a formulation of contour dynamics applicable to helical vortices is outlined.