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.

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.

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.

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.

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.

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.