## Spring 2023

Seminars (unless otherwise stated) usually 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).

## Tuesday 17 January 2023

### Speaker: Jean-Luc Thiffeault (University of Wisconsin - Madison)

##### Title: The mathematics of burger flipping

**Abstract: **

Ever since the dawn of time people have (literally) asked the question -- what is the most effective way to grill food? Timing is everything, since only one surface is exposed to heat at a given time. Should we flip only once, or many times? I will show a simple model of cooking by flipping, and some interesting mathematics will emerge. The rate of cooking depends on the spectrum of a linear operator, and on the fixed point of a map. If the system is symmetric, the rate of cooking becomes independent of the sequence of flips, as long as the last point to be cooked is the midpoint. This toy problem has some characteristics reminiscent of more realistic scenarios, such as thermal convection and heat exchangers.

## Tuesday 24 January 2023

### Speaker: Jean-Marc Vanden-Broeck (UCL)

##### Title: Capillary waves, interfacial waves and waves with vorticity

**Abstract: **

Water waves have been studied for over 200 years and many important results have been obtained. However there are still open questions for nonlinear waves. In this talk we will follow branches of solutions and focus on the limiting configurations (i.e. the waves of maximum amplitude which can be reached on each branch). We shall assume the fluids to be incompressible and inviscid. We will include various effects in the dynamic boundary condition and show how the limiting configurations differ for capillary waves, gravity waves interfacial waves (i.e. waves travelling at the interface between two layers of fluid of different constant densities) and waves with constant vorticity. These problems are mathematically difficult because of the nonlineartity and the presence of free surfaces. They are solved by a combination of analytical and numerical methods. These include boundary integral equation methods and series truncation methods. Both periodic and solitary waves are considered in frames of reference moving with the waves. New branches of solutions for interfacial waves will be presented. They usually emerge as branches bifurcating from known branches but they do not have an equivalent in the classical problem of pure gravity surface waves. Some unexpected connections between capillary waves and waves with constant vorticity will also be explored in the talk. As time permits asymmetric waves and other types of waves will be discussed. Some of these recent works are joint with Xin Guan, Zhan Wang, Vera Hur, Frédéric Dias, Tao Gao and Alex Doak.

## Tuesday 31 January 2023

### Speaker: Gunnar Peng (Imperial College)

##### Title: weakly nonlinear dynamics of self-propelling active particles

**Abstract: **

A submerged isotropic active particle (or droplet) that emits/consumes a chemical and interacts with it to drive flow via diffusio-osmotic slip (or Marangoni effects) can exhibit symmetry-breaking spontaneous motion. We derive a reduced-order model for the slow dynamics of the particle near the threshold for spontaneous motion using a weakly nonlinear expansion, which involves matching a quasi-steady particle-scale solution to an unsteady diffusive remote region. The resulting amplitude equation for the particle velocity includes a term representing the interaction of the particle with its own wake in the remote region, which can be expressed as a time integral over the history of the particle motion, allowing theoretical analysis and efficient numerical simulation of fully three-dimensional problems. We study various cases, including the particle interacting with a force, a wall, other particles and/or other weak perturbations, resulting in linear motion, circular motion, and more exotic dynamics.

## Tuesday 07 February 2023

### Speaker: Frédéric Dias (Centre Borelli, ENS-Saclay, France)

##### Title: On extreme ocean waves

**Abstract: **

The study of extreme ocean waves is a rapidly expanding area of research worldwide. Although much work in this area is based on modelling and experiments in controlled wave tanks, the starting point of all studies is wave observation in the natural world. During this talk, we will provide some evidence of extreme wave events, describe the main mechanisms for their generation and conclude with what we believe makes ocean waves go rogue in the real world.

## Tuesday 14 February 2023

##### READING WEEK - NO SEMINAR

## Tuesday 21 February 2023

### Speaker: Valery Smyshlyaev (UCL)

##### Title: whispering gallery wave scattering by boundary inflection: a fundamental problem in asymptotics

**Abstract: **

The problem of interest is that of a whispering gallery high-frequency asymptotic mode propagating along a concave part of a boundary and approaching a boundary inflection point. The related inner problem leads to an arguably as fundamental canonical boundary-value problem for a special PDE describing transition from a "modal" to a "scattered" asymptotic behaviour, as Airy functions are for transition from oscillatory to exponentially decaying asymptotic patterns. Still, despite considerable progress over the last nearly 50 years the problem remains largely open. We give a brief overview of its history, resent progress (some joint with Dave Hewett, John Ockendon, Ilia Kamotski, and Shiza Naqvi), and current status.

## Tuesday 28 February 2023

### Speaker: Ted Johnson (UCL)

##### Title: the long-wave dynamics of coastal flows

**Abstract: **

Coastal or boundary currents are an integral part of global ocean circulation. For example, currents may respond to external forcing or intrinsic instability by expelling vortex filaments or larger eddies into the ocean, with implications for the mixing of coastal and ocean waters; and currents driven by outflows are important for the transport of freshwater, pollutants and land-derived nutrients. There is also much interest in the behaviour of ‘free’ fronts, i.e. those that are far from the coast, which can be used to model western boundary currents such as the Gulf Stream or the Kuroshio Extension. In the limit of rapid rotation the governing equations reduce to the quasi-geostrophic equations - a modified form of the two-dimensional Euler equations. For the problems considered here the vorticity of the flow is unity within the current and zero elsewhere. The unapproximated solution can thus be obtained numerically to high accuracy by applying the method of Contour Dynamics to the development of the current-ocean interface. These solutions provide comparisons for estimating the accuracy of asymptotic solutions.

Alongshore variations in the flows take place over scales large compared to offshore scales and so analysis of the flows leads naturally to a long-wave equation for the current- ocean interface. Two examples will be discussed in depth. First, the development of the flow when fluid is discharged from a source on the coast to turn and form an alongshore current and, second, the Riemann problem for the subsequent development of a step change in width of a coastal flow.

The flux function appearing in the long-wave equation is non-convex and this leads to a wide variety of behaviours. Many of these are well-described following the method of El (2005) but some discrepancies remain.

## Tuesday 07 March 2023

### Speaker: Alessandro Manacorda (Universite du Luxembourg)

##### Title: Pulsating with discrete symmetry - lattice dynamics of deformable active particles

**Abstract: **

Cells in epithelial tissues can drastically deform their shapes and volume giving rise to collective behavior such as size oscillation and wave propagation [1]. These phenomena have a strong impact in many biological contexts such as embryonic development, cardiac arrhythmias and uterine contraction [2]. Following the recent formulation of active deformable particles [3], we introduce a lattice model of pulsating particles whose activity is given by the ability to change an internal degree of freedom at the single-particle level. The system then corresponds to a spatially-extended version of the periodically driven Potts model, exhibiting complex behavior already at fully-connected level [4]; we show how the interplay between pulsation and synchronization gives rise to emergent behavior such as wave propagation.

Fluctuating hydrodynamic equations are derived from microscopic dynamics and highlight the role of symmetry breaking. The latter proves to be a fundamental ingredient for wave propagation because of the competition between pulsating and arrested phases and its relation with reaction-di↵usion systems.

The model introduced thus stands as a suitable candidate to understand the targeted phenomenology and paves the way to the analysis of symmetry breaking in nonequilibrium field theories and many-body energetics, two of the main future directions in the growing field of pulsating active matter.

References

[1] J. Solon, A. Kaya-C¸ opur, J. Colombelli and D. Brunner, Cell 137, 7, 1331-1342 (2009) ; S. M.Zehnder, M. Suaris, M. M. Bellaire and T. E. Angelini, Biophys. J. 108, 2, 247-250 (2015).

[2] C.-P. Heisenberg and Y. Bella¨ıche, Cell 153, 948 (2013) ; A. Karma, Annu. Rev. Condens. Matter Phys. 4, 313 (2013) ; K. M. Myers and D. Elad, WIREs Syst. Biol. Med. 9, e1388 (2017).

[3] Y. Zhang and ´ E. Fodor, arXiv:2208.06831 (2022)

[4] T. Herpich, J. Thingna, and M. Esposito, Phys. Rev. X 8, 031056 (2018) ; T. Herpich and M.Esposito, Phys. Rev. E 99, 022135 (2019)

## Tuesday 14 March 2023

### Speaker: Sasha Korobkin (University of East Anglia)

##### Title: TBC

**Abstract: **

TBC

## Tuesday 21 March 2023

### Speaker: Bernhard Scheichl (Technische Universitat Wien, Vienna, Austria) & Robert I. Bowles (UCL)

##### Title: PRedicting capillary ripples and nonlinear squire-taylor modes by viscous-inviscid interaction past a trailing edge

**Abstract: **

We consider the steady developed flow of a liquid layer having just passed the trailing edge of horizontal flat plate in the asymptotic limit of large Reynolds number. The body/interface forces at play are gravity acting vertically and constant surface tension on both sides of the layer. Two-tiered (internal) viscous-inviscid interaction accounts rigorously for the interplay of these and the viscous forces in the long-wave limit. Here a suitably formed reciprocal Weber number (T) and reciprocal Froude number (G) form the sole control parameters. The solutions of the nonlinear interaction problem disclose a surprising richness of phenomena, distinguished by the range of values of T: antisymmetric (0 ≤ T ≤ 1/2) and symmetric (T > 1) capillary wave trains represent a nonlinear extension of classical Squire modes, where the latter ones can be seen as planar Rayleigh–Plateau modes; if 1/2 < T < 1, either gross flow reversal far downstream or a blow-up at some finite distance from the edge terminates the interactive flow description. Both scenarios are found to be separated by an unstable flow represented by a curve in the (T,G)-plane. Amongst others, we present a weakly nonlinear WKBJ analysis capturing the waves and briefly address the types of choking for T being close to 1/2 or 1 as well as the surprising intricacies associated with the Euler stage regularising the blow-up. Finally, the analogies and differences to the impact of capillarity on the developed flow through the orifice of a circular pipe are highlighted.

Parts of this work were carried out jointly with Samuel Harris, Shiza Naqvi, Michael Nguyen (all UCL) and Georgios Pasias (Cyprus University of Technology).