Past

Even though most of us tie our shoelaces "wrongly," knots in ropes and filaments have been used as functional structures for millennia, from sailing and climbing to dewing and surgery. However, knowledge of the mechanics of physical knots is largely empirical, and there is much need for physics-based predictive models. Tight knots exhibit highly nonlinear and coupled behavior due to their intricate 3D geometry, large deformations, self-contact, friction, and even elasto-plasticity. Additionally, tight knots do not show separation of the relevant length scales, preventing the use of centerline-based rod models. In this talk, I will present an overview of recent work from our research group, combining precision experiments, Finite Element simulations, and theoretical analyses. First, we study the mechanics of two elastic fibers in frictional contact. Second, we explore several different knotted structures, including the overhand, figure-8, clove-hitch, and bowline knots. These knots serve various functions in practical settings, from shoelaces to climbing and sailing. Lastly, we focus on surgical knots, with a particularly high risk of failure in clinical settings, including complications such as massive bleeding or the unraveling of high-tension closures. Our research reveals a striking and robust power law, with a general exponent, between the mechanical strength of surgical knots, the applied pre-tension, and the number of throws, providing new insights into their operational and safety limits. These findings could have potential applications in the training of surgeons and enhanced control of robotic-assisted surgical devices.

In geophysical and astrophysical flows, we are often interested in understanding the impact of internal waves on the non-wavelike flow. For example, oceanic internal waves generated at the surface and the seafloor transfer energy from the large scale flow to dissipative scales, thereby influencing the global ocean state. A primary challenge in the study of wave-flow interactions is how to separate these processes – since waves and non-wavelike flows can vary on similar spatial and temporal scales in the Eulerian frame. However, in a Lagrangian flow-following frame, temporal filtering offers a convenient way to isolate waves. Here, I will discuss a recently developed method for evolving Lagrangian mean fields alongside the governing equations in a numerical simulation, and extend this theory to allow effective filtering of waves from non-wavelike processes.

Liquid crystal elastomers are rubbery solids containing molecular LC rods that align along a common director. On heating, the alignment is disrupted, leading to a substantial (~50%) contraction along the director.  In recent years, there has been a great deal of interest in fabrication LCE sheets with a bespoke alignment pattern. On heating, these patterns generate  corresponding patterns of contraction that can morph a sheet into a bespoke curved surface such as a cone or face. Moreover, LCEs can also be activated by light, either photothermally or photochemically, leading to similarly large contractions. Stimulation by light also introduces an important new possibility: using spatio-temporal patterns of illumination to morph a single LCE sample into a range of different surfaces. Such stimulation can enable non-reciprocal actuation for viscous swimming or pumping, and control over the whole path taken by the sheet through shape-space rather than just the final destination. In this talk, I will start by with an experimental example of a spatio-temporal pattern of illumination being used to actuate an LCE peristaltic pump. I will then introduce a second set of experiments, in which a monodomain sheet morphs first into a cone, an anti-cone and then an array of cones upon exposure to different patterns of illumination. Finally, I will then discuss the general problem of how to choose a pattern of illumination to morph a director-patterned sheet into an arbitrary surface, first analytically for axisymmetric cases, then numerically for low symmetry cases. This last study exceeds our current experimental capacity, but highlights how, with full spatio-temporal control over the stimulation magnitude, one can choreograph an LCE sheet to undergo almost any pattern of morphing.

applied mathematics.

Mathematical modelling can support decontamination processes in a variety of ways.  In this talk, we focus on the contamination step: understanding how much of a chemical spill has seeped into the Earth or a building material, and how far it has travelled, are essential for making good decisions about how to clean it up.  

We consider an infiltration problem in which a chemical is poured on an initially unsaturated porous medium, and seeps into it via capillary action. Capillarity-driven flow through partially-saturated porous media is often modelled using Richards’ equation, which is a simplification of the Buckingham-Darcy equation in the limit where the infiltrating phase is much more viscous than the receding phase.  In this talk, I will explore the limitations of Richards equation, and discuss some scenarios in which predictions for small-but-finite viscosity ratios are very different to the Richards simplification.

What do fiber polymers and ice sheets have in common? They both flow with a directionally dependent - anisotropic - viscosity. This behaviour occurs in other geophysical flows, such as the Earth's mantle, where a material's microstructure affects its large-scale flow. In ice, the alignment of crystal orientations can cause the viscosity to vary by an order of magnitude, consequently having a strong impact on the flow of ice sheets and glaciers. However, the effect of anisotropy on large-scale flow is not well understood, due to a lack of understanding of a) the best physical approximations to model crystal orientations, and b) how crystal orientations affect rheology. In this work, we aim to address both these questions by linking rheology to crystal orientation predictions, and testing a range of models against observations from the Greenland ice sheet. The results show assuming all grains experience approximately the same stress provides realistic predictions, and we suggest a set of equations and parameters which can be used in large-scale models of ice sheets. 

Traditionally, stochastic processes are modelled one of two ways: a continuum Fokker-Planck approach, where a PDE is solved to determine the time evolution of the probability density, or a Langevin approach, where the SDE describing the system is sampled, and multiple simulations are used to collect statistics. There is also a third way: the functional or path integral. Originally developed by Wiener in the 1920s to model Brownian motion, path integrals were famously applied to quantum mechanics by Feynman in the 1950s. However, they also have much to offer classical stochastic processes (and statistical physics).  

In this talk I will introduce the formalism at a physicist’s level of rigour, and focus on determining the dominant contribution to the path integral when the noise is weak. There exists a remarkable correspondence between the most-probable stochastic paths and Hamiltonian dynamics in an effective potential [1,2,3]. I will then discuss some applications, including reaction pathways conditioned on finite time [2]. We demonstrate that the most probable pathway at a finite time may be very different from the usual minimum energy path used to calculate the average reaction rate. If time permits, I will also discuss the extremely nonlinear crystal dislocation response to applied stress [4].  

[1] Ge, Hao, and Hong Qian. Int. J. Mod. Phys. B 26.24 1230012 (2012)     

[2] Fitzgerald, Steve, et al. J. Chem. Phys. 158.12 (2023).

[3] Honour, Tom and Fitzgerald, Steve. in press J. Phys. A (2024)

[4] Fitzgerald, Steve. Sci. Rep. 6 (1) 39708 (2016)

We develop a comprehensive geometrically-exact theory for an end-loaded elastic rod constrained to deform on a cylindrical surface. By viewing the rod-cylinder system as a special case of an elastic braid, we are able to obtain all forces and moments imparted by the deforming rod to the cylinder as well as all contact reactions. This framework allows us to give a complete treatment of static friction consistent with force and moment balance. In addition to the commonly considered model of hard frictionless contact, we analyse two friction models in which the rod, possibly with intrinsic curvature, experiences either lateral or tangential friction. As applications of the theory we study buckling of the constrained rod under compressive and torsional loads, finding critical loads to depend on Coulomb-like friction parameters, as well as the tendency of the rod to lift off the cylinder under further loading. The cylinder can also have arbitrary orientation relative to the direction of gravity. The cases of a horizontal and vertical cylinder, with gravity having only a lateral or axial component, are amenable to exact analysis, while numerical results map out the transition in buckling mechanism between the two extremes. Weight has a stabilising effect for near-horizontal cylinders, while for near-vertical cylinders it introduces the possibility of buckling purely due to self-weight. Our results are relevant for many engineering and medical applications in which a slender structure winds inside or outside a cylindrical boundary.

The Fokker-Planck equation is one of the major tools of statistical physics in the description of stochastic processes, with numerous applications in physics, chemistry and biology. In the case of heterogeneous diffusions, the formulation of the equation depends on the choice of the discretization of the stochastic integral in the underlying Langevin-equation due to the multiplicative noise. In the Fokker-Planck equation, the choice of discretization then enters as a parameter in the definition of drift and diffusion terms. I show how both short- and long-time limits are affected by this choice. In the long-time limit, the existence of normalizable probability distribution functions is not always guaranteed which can be remedied by invoking elements of infinite ergodic theory. 

[1] S. Giordano, F. Cleri, R. Blossey, Phys Rev E 107, 044111 (2023)

[2] T. Dupont, S. Giordano, F. Cleri, R. Blossey, arXiv:2401.01765 (2024)

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Global scale ocean currents are strongly constrained by the Earth’s rotation, while this effect is generally negligible at small scales. In between, motions with scales from 1-10km are marginally affected by the Earth’s rotation. These intermediate scales, collectively termed the ocean submesoscale, have been hidden from view until recent years. Evidence from field measurements, numerical models, and satellite data have shown that submesoscales play a particularly important role in the upper ocean where they help to control the transport of material between the ocean surface and interior. In this talk I will review some recent work on submesoscale dynamics and their influence on biogeochemistry and accumulation of microplastics in the surface waters.

We study “bottom-up” models for the collective motion of large groups of animals. Similar models can be encoded into (micro)robotic matter, capable of sensing light and processing information. Agents are endowed only with visual sensing and information processing. We study a model in which moving agents reorientate to maximise the path-entropy of their visual environment over paths into the future. There are general arguments that principles like this that are based on retaining freedom in the future may confer fitness in an uncertain world. Alternative “top-down” models are more common in the literature. These typically encode coalignment and/or cohesion directly and are often motivated by models drawn from physics, e.g. describing spin systems. However, such models can usually give little insight into how co-alignment and cohesion emerge because these properties are encoded in the model at the outset, in a top-down manner. We discuss how our model leads to dynamics with striking similarities with animal systems, including the emergence of coalignment, cohesion, a characteristic density scaling anddifferent behavioural phenotypes. The dynamics also supports a very unusual order-disorder transition in which the order (coalignment) initially increases upon the addition of sensory or behavioural noise, before decreasing as the noise becomes larger.

During the last fifteen years, there has been a paradigm shift in the continuum modelling of granular materials; most notably with the development of rheological models, such as the μ(I)-rheology (where μ is the friction and I is the inertial number), but also with significant advances in theories for particle segregation. This talk details theoretical and numerical frameworks (based on OpenFOAM®) which unify these disconnected endeavours. Coupling the segregation with the flow, and vice versa, is not only vital for a complete theory of granular materials, but is also beneficial for developing numerical methods to handle evolving free surfaces. This general approach is based on the partially regularized incompressible μ(I)-rheology, which is coupled to a theory for gravity/shear-driven segregation (Gray & Ancey, J. Fluid Mech., vol. 678, 2011, pp. 353–588). These advection–diffusion–segregation equations describe the evolving concentrations of the constituents, which then couple back to the variable viscosity in the incompressible Navier–Stokes equations. A novel feature of this approach is that any number of differently sized phases may be included, which may have disparate frictional properties. The model is used to simulate the complex particle-size segregation patterns that form in a partially filled triangular rotating drum. There are many other applications of the theory to industrial granular flows, which are the second most common material used after fluids. The same processes also occur in geophysical flows, such as snow avalanches, debris flows and dense pyroclastic flows. Depth-averaged models, that go beyond the μ(I)-rheology, will also be derived to capture spontaneous self-channelization and levee formation, as well as complex segregation-induced flow fingering effects, which enhance the run-out distance of these hazardous flows.

Recent advances in experimental techniques have enabled remarkable discoveries and insight into how the dynamics of thin gas/vapour films can profoundly influence the behaviour of liquid droplets: drops impacting solids can “skate on a film of air” [1], so that they can “bounce off walls” [2,3]; reductions in ambient gas pressure can suppress splashing [4] and initiate the merging of colliding droplets [5]; and evaporating droplets can levitate on their own vapour film [7] (the Leidenfrost effect). Despite these advances, the precise physical mechanisms governing these phenomena remains a topic of debate.  A theoretical approach would shed light on these issues, but due to the strongly multiscale nature of these processes brute force computation is infeasible.  Furthermore, when films reach the scale of the mean free path in the gas (i.e. ~100nm) and below, new nanoscale physics appears that renders the classical Navier-Stokes paradigm inaccurate.

In this talk, I will overview our development of efficient computational models for the aforementioned droplet dynamics in the presence of gas nanofilms into which gas-kinetic, van der Waals and/or evaporative effects can be easily incorporated [8,9].  It will be shown that these models can reproduce experimental observations – for example, the threshold between bouncing and wetting for drop impact on a solid is reproduced to within 5%, whilst a model excluding either gas-kinetic or van der Waals effects is ~170% off!  These models will then be exploited to make new experimentally-verifiable predictions, such as how we expect drops to behave in reduced pressure environments.  Finally, I will conclude with some exciting directions for future wor

[1] JM Kolinski et al, Phys. Rev. Lett.  108 (2012), 074503. [2] JM Kolinski et al, EPL.  108 (2014), 24001. [3] J de Ruiter et al, Nature Phys.  11 (2014), 48. [4] L Xu et al, Phys. Rev. Lett. 94 (2005), 184505. [5] J Qian & CK Law, J. Fluid. Mech. 331 (1997), 59.  [6] KL Pan J. Appl. Phys. 103 (2008), 064901. [7] D Quéré, Ann. Rev. Fluid Mech. 45 (2013), 197. [8] JE Sprittles, Phys. Rev. Lett.  118 (2017), 114502.  [9] MV Chubynsky et al, Phys. Rev. Lett.. 124 (2020), 084501.

Predictive scoring modelling is a common approach to measure financial health and credit worthiness in banking. Whilst the latter is a key factor in making decisions for lending, evaluating financial health helps to identify vulnerable customers trending towards financial hardship, who need support. The current macroeconomic uncertainties amplify the importance of extensive flexibility in modelling data solutions so that they can remain effective and adaptive to a volatile economic environment. This workshop is focused on discussing relevant techniques and mathematical methodologies which can help modernise traditional scoring models and accelerate innovation. In summary, the problem definition in the banking context is how automation and optimality can be achieved in a multiobjective problem where a subset of existing data features should be selected by relevance and uniqueness, assigned scoring weights by importance and how a pool of customers can be categorised accordingly using their individual scores and auto-adjusting thresholds of risk classification scales. The key challenge is imposed by the mutual dependency of the three sub-problems and their objectives. Introducing or removing constraints in any of them can change the feasibility and optimality of the others and the overall solution. It is common for traditional scoring models to be mainly focused on the predictive accuracy and their setup is often defined and revised manually, following ad-hoc exploratory data analysis and business-led decision making. An automated optimisation of the data features’ selection, scoring weights and classification thresholds definition can achieve respectively: ▪ Precise financial health evaluation and book classification under changing economic climate; ▪ Development of innovative data-driven solutions to enhance prevention from financial hardship and bankruptcies.

Capillary forces acting at the surface of a liquid drop can be strong enough to deform small objects and recent studies have provided several examples of elastic instabilities induced by surface tension. Inspired by the windlass mechanism in spider webs, we present a system where a liquid drop sits on a straight fiber and attracts the fiber which thereby coils inside the drop. We then introduce a 2D extension of the mechanism and build a membrane that can extend/contract by a factor of 20.

In order for artificial neural networks to learn a task, one must solve an inverse design problem. What network will produce the desired output? We have harnessed AI approaches to design physical systems to perform functions inspired by biology, such as protein allostery. But artificial neural networks require a computer in order to learn in top-down fashion by the global process of gradient descent on a cost function. By contrast, the brain learns by local rules on its own, with each neuron adjusting itself and its synapses without knowing what all the other neurons are doing, and without the aid of an external computer. But the brain is not the only biological system that learns by local rules; I will argue that the actin cortex and the amnioserosa during the dorsal closure stage of Drosophila development can also be viewed this way.

The key role of physical and mechanical interactions in cancer emerges from a very large variety of data sources and methods - from genomics to bioimaging, from proteomics to clinical records. Thus, learning physics-driven relational information is crucial to characterize its progression at different scales.

In this talk I will discuss how mathematical and computational tools allow for learning  and better understanding of  the mechano-biology of cancer,  thanks to the integration of  patient-specific data and physics-based models. I will present a few applications developed in the last decade in which the development of  digital twins,  empowered by ad-hoc learning tools,  allows us to test new hypotheses,  to assess the model predictions against biological and clinical data, and to aid decision-making in a clinical setting.

Funding from MUR - PRIN 2020, Progetto di Eccellenza 2023-2027 and Regione Lombardia (NEWMED Grant, ID: 117599, POR FESR 2014-2020) is gratefully acknowledged.

We formulate and solve a generalized inverse Navier–Stokes boundary value problem for velocity field reconstruction and simultaneous boundary segmentation of noisy Flow-MRI velocity images. We use a Bayesian framework that combines CFD, Gaussian processes, adjoint methods, and shape optimization in a unified and rigorous manner.
With this framework, we find the velocity field and flow boundaries (i.e. the digital twin) that are most likely to have produced a given noisy image. We also calculate the posterior covariances of the unknown parameters and thereby deduce the uncertainty in the reconstructed flow. First, we verify this method on synthetic noisy images of flows. Then we apply it to experimental phase contrast magnetic resonance (PC-MRI) images of an axisymmetric flow at low and high SNRs. We show that this method successfully reconstructs and segments the low SNR images, producing noiseless velocity fields that match the high SNR images, using 30 times less data.
This framework also provides additional flow information, such as the pressure field and wall shear stress, accurately and with known error bounds. We demonstrate this further on a 3-D in-vitro flow through a 3D-printed aorta and 3-D in-vivo flow through a carotid artery.

The classical model of evaporation of liquids hinges on Maxwell’s assumption that the air near the liquid’s surface is saturated. It allows one to find the evaporative flux without considering the interface separating liquid and air. Maxwell’s hypothesis is based on an implicit assumption that the vapour-emission capacity of the interface exceeds the throughput of air (i.e., its ability to pass the vapour on to infinity). If indeed so, the air adjacent to the liquid would get quickly saturated, justifying Maxwell’s hypothesis.

In the present paper, the so-called diffuse-interface model is used to account for the interfacial physics and, thus, derive a generalised version of Maxwell’s boundary condition for the near-interface vapour density. It is then applied to a spherical drop floating in air. It turns out that the vapour-emission capacity of the interface exceeds the throughput of air only if the drop’s radius is r_d ≳ 10μm, but for r_d ≈ 2μm, the two are comparable. For r_d ≲ 1μm, evaporation is interface-driven, and the resulting evaporation rate is noticeably smaller than that predicted by the classical model.

Over the years, I've often taught a first course in asymptotics and perturbation methods, even though I don't know much about the subject. In this talk, I'll discuss a textbook example of a singularly perturbed nonlinear boundary-value problem that has revealed delightful new surprises, every time I teach it. These include a pitchfork bifurcation in the number of solutions as one varies the small parameter, and transcendentally small terms in the solutions' initial conditions that can be calculated by elementary means.

I’m excited to share with everyone some new, unpublished work that we are just in the process of wrapping up and could use everyone’s reactions. It is a reconciliation of evolutionary game theory and ecological dynamics that I have wrestled with since I moved from an evolution program into an ecology-heavy department. It always seemed like, depending on the problem I was thinking about, I had to change my perspective and approach it as either an evolutionary game theorist, or an ecologist; and yet I had this nagging feeling that, at its core, the problem was often one and the same, and therefore one theoretical framework should suffice. So when should one write down an n-type replicator equation and when should one write down an n-species Lotka-Volterra system; and what does it mean mathematically and biologically when one has made such a choice? In the process of reconciling, I also got a deeper appreciation of what is and is not a proper game, such as a Prisoner’s Dilemma. These findings can help shed light on previously puzzling empirical findings.

Metal forming involves permanently deforming metal into a required shape.  Many forms of metal forming are used in industry: rolling, stamping, pressing, drawing, etc; for example, 99% of steel produced globally is first rolled before any subsequent processing.  Most theoretical studies of metal forming use Finite Elements, which is not fast enough for real-time control of metal forming processes, and gives little extra insight.  As an example of how little is known, it is currently unknown whether a sheet of metal that is squashed between a large and a small roller should curve towards the larger roller, or towards the smaller roller.  In this talk, I will give a brief overview of metal forming, and then some progress my group have been making on some very simplified models of cold sheet rolling in particular.  The mathematics involved will include some modelling and asymptotics, multiple scales, and possibly a matrix Wiener-Hopf problem if time permits.