### Seminars will be a live meetings at **1.15pm on Tuesdays. **

### Live meetings in 4W 1.7 (Wolfson Room)

Week | Date | Name | Title | Abstract | Host |
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1 | 4 Oct | No seminar | |||

2 | 11 Oct | Sarah Waters (Oxford) | Fundamental and translational biomedical fluid dynamics | Mechanistic mathematical modelling in combination with state-of-the art experiments can generate fundamental insights into complex biomedical fluid dynamics systems. Embedding such predictive mathematical modelling approaches into preclinical toolkits has the potential to boost the clinical translation of preclinical research. I will discuss two examples where insights gained from fundamental biomedical fluid dynamics studies can be harnessed to advance clinical therapies and medical devices. I will illustrate how mathematical modelling can be guide the design and implementation of the next generation of medical devices used in urology, and facilitate the identification and optimisation of safe clinical cell therapies. | Beth Stokes (& Ruth/Phil) |

3 | 18 Oct | Ben Walker (Bath) | Multiscale methods and microswimmer models | Swimming on the microscale has long been the subject of intense research efforts, from experimental studies of bacteria, sperm, and algae through to varied theoretical questions of low-Reynolds-number fluid mechanics. The biological and biophysical settings that drive this ongoing research are often confoundingly complex, a fact that has driven the development and use of simple models of microswimmers. In this talk, we'll motivate and explore some of these models, building up our intuition for Stokesian fluid dynamics and the behaviours of microscale swimmers. Using these models, we will showcase how we can often exploit separated scales present in these problems to reveal surprisingly simple emergent dynamics, ranging from coarse-grained flow profiles to predictions of globally attracting, long-term behaviours. In doing so, we'll also uncover a surprising cautionary tale, the root of which is captured by a single, elementary statement that nevertheless calls into question much of the intuition gained from commonplace models of microswimming. In particular, we'll see that an elementary wave-of-the-hands, which I have been guilty of before, can drastically and qualitatively change the dynamics that simple models predict, and we'll see how such missteps can be addressed through systematic multiscale methods. | AS |

4 | 25 Oct | No talk | |||

5 | 1 Nov | Alex Mietke (Bristol) | Odd dynamics of living chiral crystals | The emergent dynamics exhibited by collections of living organisms often shows signatures of symmetries that are broken at the single-organism level. At the same time, organism development itself encompasses a well-coordinated sequence of symmetry breaking events that successively transform a single, nearly isotropic cell into an animal with well-defined body axis and various anatomical asymmetries. Combining these key aspects of collective phenomena and embryonic development, we describe here the spontaneous formation of hydrodynamically stabilized active crystals made of hundreds of starfish embryos that gather during early development near fluid surfaces. We describe a minimal hydrodynamic theory that is fully parameterized by experimental measurements of microscopic interactions among embryos. Using this theory, we can quantitatively describe the stability, formation and rotation of crystals and rationalize the emergence of mechanical properties that carry signatures of an odd elastic material. Our work thereby quantitatively connects developmental symmetry breaking events on the single-embryo level with remarkable macroscopic material properties of a novel living chiral crystal system. | AS (note PHT away) |

6 | 8 Nov | Scott McCue (QUT) | Burgers’ equations revisited, but this time in the complex plane | Many of you will know the viscous Burgers’ equation, a very well-studied parabolic pde which sets up a competition between nonlinear advection that tends to steepen the solution profile and linear diffusion that tends to smooth it out. This pde is a simple toy model for the Navier-Stokes equation in 1D. A simpler version is the inviscid Burgers’ equation, which is a first-order nonlinear pde that can be solved exactly by an undergraduate using the method of characteristics. For this version of Burgers’ equation, there is no diffusion and so nonlinear advection drives the solution to continue to steepen until the derivative blows up somewhere in finite time. We shall revisit these models, but instead of restricting ourselves to the real line, we shall continue the solution out to the complex plane. In this way, we observe directly the culprit for the finite-time blow-up in the inviscid Burgers’ equation, which is a branch point that moves towards the real axis and touches it at the blow-up time. In the viscous Burgers case, the singularity structure in the complex plane is much more complicated, but with some investigative tools we can track the motion of the singularities and show why blow-up does not occur. This fun story contains some exact solutions, nice plots of complex functions, matched asymptotic expansions, similarity solutions, special functions, the method of steepest descents, rational approximations, and more. | PHT |

7 | 15 Nov | Katie Oliveras, University of Seattle | Measuring Water Waves: Using Pressure to Reconstruct Wave Profiles | Euler's equations describe water-waves on the surface of an ideal fluid. In this talk, I will discuss an inverse problem related to measuring water-waves using pressure sensors placed inside the fluid. Using a non-local formulation of the water-wave problem, we can directly determine the pressure below both traveling-wave and time-dependent solutions of Euler's equations. This method requires the numerical solution of a nonlinear, nonlocal equation relating the pressure and the surface elevation which is obtained without approximation. From this formulation, a variety of different asymptotic formulas are derived and are compared with both numerical data and physical experiments. | PAM/PHT |

8 | 22 Nov | Roberto Camassa, UNC | Fluid-boundary interaction: confinement effects, stratification and transport | Arguably some of the most interesting phenomena in fluid dynamics, both from a mathematical and a physical perspective, stem from the interplay between a fluid and its boundaries. This talk will present some examples of how boundary effects lead to remarkable outcomes. Singularities can form in finite time as a consequence of the continuum assumption when material surfaces are in smooth contact with horizontal boundaries of a fluid under gravity. For fluids with chemical solutes, the presence of boundaries impermeable to diffusion adds further dynamics which can give rise to self-induced flows and the formation of coherent structures out of scattered assemblies of immersed bodies. These effects can be analytically and numerically predicted by simple mathematical models and observed in “simple” experimental setups. | PAM/PHT |

9 | 29 Nov | No talk | No talk | ||

10 | 6 Dec | Ian Hewitt (Oxford) | Fluid-mechanical models of ice-sheet sliding and dynamics | The volume of ice stored in glaciers and ice sheets is the primary control on Earth’s sea level, so the question of how it responds to the changing climate is of profound interest. Fundamental to addressing this question is the fluid-like creep of the ice, and the rate at which it can slip over the underlying ‘bed'. In this talk we will first discuss basic mathematical models of ice-sheet volume and evolution, and how these are coupled to the climate. Secondly, we’ll consider the detailed problem of how slip at the ice-bed interface is facilitated by the formation of water-filled cavities, which results in a very interesting free-boundary problem. Finally, we’ll look at a simplified model for marine ice-sheet dynamics when a Coulomb-like plastic law is assumed for the slip. | PHT/Hui Tang/Timothy Peters |

11 | 13 Dec | Ananyo Maitra (CY Cergy Paris Universite) | Chiral active liquid crystals | Chiral molecules form a plethora of liquid-crystalline phases, even in equilibrium. However, while chirality is important for understanding the microstructure of these phases, it is cloaked in the long-time, large-scale properties of these passive states. Activity allows chirality to reveal its full dynamic potential. For instance, while the long-wavelength elastic and hydrodynamic properties of passive chiral layered and columnar states are equivalent to their achiral counterparts, activity leads to unique forms of odd elastic behaviours in active cholesteric and chiral columnar phases. The interplay of chirality and activity also leads to liquid-crystalline phases without any passive analogue, such as time liquid crystals. In this talk, I will summarise my work on chiral active liquid crystals, highlighting how the interplay of chirality and activity allows the former to mould the dynamic and static properties of ordered states, in direct contrast to passive matter. | AS (Note PHT away) |

12 | 10 Jan | Chris Lustri (Macquarie University) | The effect of discretization on exponentially small phenomena | In this talk I will introduce a higher-dimensional generalization of the NLS equation, known as the Karpman equation. These equations describe the behaviour of optical solitary waves that are used to describe laser behaviour. I will show that the waves present in solutions to this equation are generically "nanoptera", or nonlocal solitary waves with exponentially small oscillatory tails, whenever the equation parameter is positive. I will then consider the same equation after a finite-difference discretization has been applied in time and space. I will show that the same oscillations are present, but only when the oscillation parameter exceeds 1/4. The act of discretizing this equation changes the value at which the bifurcation occurs. Finally, I will show how the two behaviours can be connected together by increasing the order of the discretization. This is joint work with Aaron Moston-Duggan (Macquarie University) and Mason Porter (UCLA). | PHT/Cecilie Andersen |

Semester 2 2023 | |||||

1 | 7 Feb | Alan Champneys (Bristol) | Localised patterns - a dynamic phase space view of some beyond-all-orders problems | Many beyond-all-orders problems arise in the unfolding of bifurcation points in four-dimensional phase space | PHT |

2 | 14 Feb | ||||

3 | 21 Feb | Jack Binysh (Bath) | An introduction to soft matter | Postdoc from Anton Souslov's group will give a gentle introduction to mathematics of soft matter and current research | AS |

4 | 28 Feb | AS | |||

5 | 7 Mar | Jennifer Tweedy | |||

6 | 14 Mar | Carl Whitfield (Manchester) | Network models of the human airway tree and applications to lung ventilation and gas transport in health and disease | The airways in human lungs form a complex tree network that branches in a dyadic manner repeatedly to transport gas to and from the microscopic alveoli. In healthy lungs this elaborate branching structure acts to maximise the surface area available for gas exchange in the limited 3D space. However, in many obstructive lung conditions the airways can become blocked or constricted, causing uneven delivery of gas or ventilation heterogeneity (VH). In this talk I will present an overview of our network-based model of ventilation and gas transport in the lung and in particular discuss a novel method for dimensionality reduction in these systems using spectral graph theory. This has applications in reducing the computational complexity of simulations of the ventilation distribution in image-based models without compromising on details at the scale of the small airways. I will also present some of the wider applications of this network-based model to the deposition of aerosolised pharmaceuticals in healthy and diseased airways, which is the subject of ongoing work. | AS/Jack Binysh |

7 | 21 Mar | Frank Smith (UCL) | Modelling fluid /body interactions | The interest here is in free motion of a body (particle) within fluid flow concerned with impacts and ice formation. Many interactions are involved. This mathematical-modelling research, motivated by observed icing of vehicles and by environmental and biomedical applications, aims to address several related problem areas. One area is the two-way fluid /body interaction occurring when a body is lifted of the ground by an oncoming fluid flow, with account taken of the small density and viscosity ratios present. Another area is on the impact of a body, passing through air, onto a solid surface. The effect of a water layer here is also of relevance. | Timothy Peters |

8 | 28 Mar | PhD students | Reserved for PhD student practice talks for BAMC | ||

9 | 18 Apr | Speaker cancelled | |||

10 | 25 Apr | Jeremy Worsfold and Rosa Kowalewski (Bath) |
Synchronisation of non-locally coupled oscillators has been extensively studied since Kuramoto’s model of oscillators was proposed in 1975. From fireflies to pedestrians on a bridge, this model is widely applicable in modelling the collective behaviour of large groups of individuals. This and other traditional models of synchronisation are predicated on deterministic forcing between individuals. Intrinsic randomness of oscillators has been incorporated into the Kuramoto model but this invariably causes a decrease in synchronization. In this talk we will show that, while noise usually creates disorder, systems can reach order through randomness. We propose and analyse a model that reproduces many of the common features of the Kuramoto model around the incoherent state but exhibits binary phase locking instead of full coherence. Using recently developed methods we are able to find exact solutions for the stationary, synchronized state. We also find approximate low dimensional dynamics for this model which qualitatively describes the full system of oscillators.
Geometric mechanics is a modern formulation of classical mechanics in the language of differential geometry. A geometric viewpoint highlights the underlying structure of equations of motion. Moreover, the general formulation on arbitrary manifolds allows to translate the equations into different coordinate systems. As is well known, Hamilton's principle is only applicable for conservative systems. By restoring the time-symmetry, Galley formulated a type of action principle for non-conservative systems, inspired by Hamilton's principle. In this talk I will give an introduction to geometric fluid mechanics, and explain the idea behind Galley's action principle. Then I will present my work on the derivation of Navier-Stokes equations on a fluid manifold by generalising Galley's non-conservative action principle to a geometric setting. | PHT | |

11 | 2 May | Speaker cancelled – seeking replacement (25 Apr 2023) | |||

12 | 9 May | Jack Keeler (UEA) | Termination points of a nonlinear non-autonomous ODE | PHT (Anton not here) | |

16 May | Mariia Dvoriashyna (Oxford) | Jennifer Tweedy and Ben Walker; note PHT away | |||

23 May | |||||

30 May | |||||

Sometime Jan 2024 onwards | Gibin Powathil (Swansea) | Ruth Bowness | |||

Held in reserve | Katerina Kaori (asked; seeking dates); Gibin (asked; seeking dates) | ||||

Rosti Readioff (Bath) | Applying FE Methods to solve partial differential equations in the study of knee joint structure | Mathematical approaches, such as Finite Element methods, can help us model the mechanics and material characteristics of biological systems and components, including human knee joints and soft tissues. Such mathematical models can guide preventative strategies for joint degeneration diseases by investigating the characteristics of the articulating tissues in the natural knee joint. In this session, I will discuss a few examples where finite element models have been used for (i) investigating the mechanics of biological tissues, including knee joint ligaments and menisci, (ii) studying the effect of subject-specific morphology on contact mechanics of articulating tissue in the knee joint, and (iii) assessing and guiding surgical repair techniques for the knee joint menisci. | |||

Dr Finn Box, Manchester: finn.box@manchester.ac.uk Dr Adam Lowe, Aston: a.lowe3@aston.ac.uk Dr Jonathan Skipp, Aston: j.skipp@aston.ac.uk Dr Joseph Pollard: joseph.pollard@durham.ac.uk Dr Carl Whitfield: carl.whitfield@manchester.ac.uk | |||||

Held in reserve | Eileen Russell and Kat Philips (Bath) |
A millimetric droplet of silicon oil will bounce periodically on a suitably vibrating bath. Increasing the amplitude of the vibrations, the droplet walks across the surface and as the amplitude increases further the droplet's motion becomes chaotic. This system is of interest to many mathematicians and physicists as it bears resemblance to the quantum realm. Indeed, the droplet has both a mass and a wavefield which is compared to quantum mechanic's wave-particle duality hypothesis. Many physicists have succeeded in creating analogues between this system and common features of quantum mechanics such as tunnelling, diffraction, interference, and quantised orbits. In this talk, we discuss a complex fluid dynamics model used to depict this system. We perform a number of reductions in order to simplify the mathematical complexity of the system whilst maintaining the principal dynamics of the system. We discuss the stability of the steady states that arise in our models and compare and contrast our results with the more in-depth models. |