Seminars (unless otherwise stated) will take place online on **Tuesdays at 3.00pm on Zoom (link).**** **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).

## 19 January 2021

### Speaker: Zhan Wang, Chinese Academy of Sciences

##### Title: Waves near Resonance: from Fast Train Track to Moving Loads on Very Large Floating Structures

**Abstract: **

The problem of forced unsteady water waves under an elastic sheet is a model for waves under ice or under very large floating structures. Even though small-amplitude solitary waves are not predicted to exist by standard perturbation analyses, we find large-amplitude solitary waves and explore their crucial role in the forced problem of a moving load on the elastic cover. This phenomenon is very much like vibrations of fast train tracks and meant to represent a model of the use of extended ice sheets as roads and aircraft runways. Both numerical and theoretical results will be discussed in the talk, and some unsolved problems will be mentioned in the end.

## 26 January 2021

### Speaker: Duncan Hewitt, UCL

##### Title: Clogging, yielding and burrowing: channel flows and locomotion in plastic fluids

**Abstract: **

This talk will consider two different problems associated with the mechanics of so-called ‘visco-plastic’ fluids. Such materials flow like a fluid under sufficient force, but exhibit plastic-like jamming if the applied stress is too low. Materials with this property are widespread in natural, industrial and everyday household settings (e.g. mud, magma, industrial waste, mucus, toothpaste, whipped cream). First, visco-plastic flow in channels will be considered, with particular attention paid to the influence of obstacles and geometrical forcing on the flow structures. The flow is controlled by a non-linear elliptic problem, which is both solved numerically, using an Augmented-Lagrangian iterative approach, and analysed using hodograph techniques and asymptotic analysis. Second, a brief overview of methods of locomotion through such materials will be presented, as exploited by various biological organisms. Some basic mechanisms for inertialess swimming are considered from a theoretical and numerical standpoint, and the speed and efficiency of locomotion are discussed. This study involves a generalisation of the classical slender-body theory of viscous fluids. Favourable comparison is made between theory and experiments with both analogue and real biological organisms.

## 02 February 2021

### Speaker: Tao Gao, University of Greenwich

##### Title: Continuation of ‘Gravity-Capillary Free-Surface Flows’

**Abstract: **

This talk is concerned with gravity-capillary surface waves. It is a one-layer free boundary problem of a potential flow in the presence of the gravitational force and the surface tension. A monograph by Vanden-Broeck in 2010 has summarised the major achievements. Since then, new results have been discovered, and will be discussed in this presentation. In the first part of the talk, a weakly nonlinear theory is introduced to illustrate the bifurcations of the gravity-capillary solitary waves under different configurations e.g. in the presence of normal electric fields or constant vorticity. Fully nonlinear computational results are compared to the theoretical prediction. In the second part of the talk, symmetry-breaking of periodic progressive waves is demonstrated in detail. The symmetric-breaking takes place at bifurcation points on the symmetric branches, where asymmetric solutions are discovered by a method of numerical continuation.

## 09 February 2021

### Speaker: Katie Oliveras, University of Seattle (USA)

##### Title: Conservation Laws and Multiple Scales Methods for Free-Boundary Problems in Water Waves

**Abstract: **

In this talk, I will consider a new nonlocal formulation of the water-wave problem for a free surface with an irrotational flow based on the work of Ablowitz, Fokas, and Musslimani. The formulation is also extended to constant vorticity and interfacial flows of different density fluids. I will also show how this formulation can be used to systematically derive Olver’s eight conservation laws not only for an irrotational fluid, but for constant vorticity and interfacial flows. This framework easily lends itself to computing the related conservation laws for various asymptotic models via a non-traditional approach to multiple-scales which will be discussed.

## 16 February 2021

**NO SEMINAR - READING WEEK**

## 23 February 2021

### Speaker: Vera Mikyoung Hur, University of Illinois at Urbana-Champaign (USA)

##### Title: Limiting Stokes waves in a constant vorticity flow

**Abstract: **

In an irrotational flow, it is well-known that the so-called wave of greatest height or extreme wave has a 120 degrees' angle at the crest. In a constant vorticity flow, by contrast, recent numerical computation suggests that a limiting configuration is either an extreme wave, like in an irrotational flow, or a touching wave, enclosing a bubble of air at the trough, particularly, the celebrated Crapper's touching wave in the zero gravity limit (even though there is no surface tension). I will explain how to perturb a Crapper wave by gravity to prove the existence of overturning Stokes waves. Also I will discuss a new numerical result that an almost extreme wave has a boundary layer at the crest, in which the angle increases sharply to ~30.3787032466 degrees (when the maximum angle of the extreme wave is 30 degrees), regardless of the vorticity, followed by oscillations like the Gibbs phenomenon. Joint with S. Dyachenko, D. Silantyev, and M. Wheeler.

## 02 March 2021

### Speaker: Olga Trichtchenko, Western University (Canada)

##### Title: Three-dimensional waves under ice in various regimes computed with novel preconditioning methods

**Abstract: **

In this talk, we present three-dimensional travelling wave solutions describing flexural-gravity waves. Reformulating the equations using a boundary-integral method, we solve the resulting equations with a Newton-Krylov method, employing a novel preconditioning technique. We examine solutions for a variety of conditions at the boundary such as waves generated by different pressure distributions and the effect varying flexural rigidity has on wave propagation. We show that even under these different conditions, our preconditioning methods are efficient and allow us to increase the grid refinement and decrease the run-time of our computations in comparison to previous methods.

## 09 March 2021

### Speaker: Didier Clamond, Université Côte d'Azur

##### Title: On dispersion-improved shallow water equations with uneven bottom

**Abstract: **

We show that some asymptotically consistent modifications of shallow water approximations, in order to improve their dispersive properties, can fail for uneven bottoms (i.e., the dispersion is actually not improved). We also show that these modifications can lead to ill-posed equations when the water depth is not constant. These drawbacks are illustrated with the (fully nonlinear, weakly dispersive) Serre equations. We then derive asymptotically consistent, well-posed, modified Serre equations with improved dispersive properties for constant slopes of the bottom.

## 16 March 2021

##### NO SEMINA**r**

## 23 March 2021

### Speaker: Shane Cooper, UCL

##### Title: A Homogenisation Theory For A General Class Of High-Contrast Problems; Asymptotics With Error Estimates

**Abstract: **

In the propagation of waves in composite materials, an interesting problem is to determine what `observable’ wave phenomena manifests from the microstructure of a given periodic composite.

Solving this problem is strongly connected to establishing the asymptotic behaviour of spectral properties of periodic non-uniformly elliptic operators with respect to a small (period) parameter.

In this talk, we present a framework to study the asymptotic behaviour of (a large class of) periodic non-uniformly elliptic systems with respect to a parameter.

We determine, for a small number of readily verifiable assumptions, the leading-order approximation of the solution and derive error estimates, uniform in right-hand-side. Spectral asymptotics with error estimates directly follow.

This is joint work with Ilia Kamotski and Valery Smyshlyaev.