The ε expansion was invented more than 50 years ago and has been used extensively ever since to study aspects of renormalization group flows and critical phenomena. Its most famous applications are found in theories involving scalar fields in 4−ε dimensions. In this talk, we will discuss the structure of the ε expansion and the fixed points that can be obtained within it. We will mostly focus on scalar theories, but we will also discuss theories with fermions as well as line defects. Our motivation is based on the goal of classifying conformal field theories in d=3 dimensions. We will describe recently discovered universal constraints obtained within the framework of the ε expansion and show that a “heavy handed" quest for fixed points yields a plethora of new ones. These fixed points reveal aspects of the structure of the ε expansion and suggest that a classification of conformal field theories in d=3 is likely to be highly non-trivial.

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.