### 2010-2020

172. Goldstein, R.E., Moffatt, H.K., Pesci, A.I. & Ricca, R.L. 2010 Soap-film Möbius strip changes topology with a twist.

Proc. Nat. Acad. Sc. USA, 107, 21979-21984.

If a circular wire is twisted and folded back on itself, then dipped in soap solution and withdrawn, a one-sided soap film in the form of a Möbius strip can be easily formed (Courant 1940). The question then is: what happens if the wire is now unfolded and untwisted back to its original circular form? At some stage in this process, the film jumps to a two-sided 'disc' spanning the wire. This jump takes place in about one hundredth of a second. The linking number of the Plateau border with the wire frame has to jump from 2 to zero when the surface becomes two-sided, a mechanism controlled by viscosity at the wire surface.

We have analysed this phenomenon both theoretically, and experimentally by high-speed photography. This is the start of a more comprehensive study of dynamical processes in soap films (minimum-area surfaces) spanning flexible wires that may be knotted and/or linked.

.

I

Courant, R. 1940 Soap film experiments with minimal surfaces. Am. Math. Mon. 47, 167–174.

Douglas, J. 1932 One-sided minimal surfaces with a given boundary. T. Am. Math. Soc. 34, 731–756..

For subsequent developments, see

Goldstein, R. E., McTavish, J., Moffatt, H. K. & Pesci, A. I. 2014 Boundary singularities produced by the motion of soap films. PNAS, 111 (23), 8339–8344

176. Moffatt, H. K. & Shuckburgh, E. (ed.) 2011 Environmental Hazards: The Fluid Dynamics and Geophysics of Extreme Events. World Scientific.

In April 2009, together with Emily Shuckburgh, I organised a Spring School at the Institute for Mathematical Sciences of the National University of Singapore."This volume contains the content of the nine short lecture courses given at this School, with a focus mainly on tropical cyclones, tsunamis, monsoon flooding and atmospheric pollution, all within the context of climate variability and change.The book provides an introduction to these topics from both mathematical and geophysical points of view, and will be invaluable for graduate students in applied mathematics, geophysics and engineering with an interest in this broad field of

study, as well as for seasoned researchers in geophysical fluid mechanics."

Practical classes were held each afternoon, when the students carried out experimental or computational projects, here an experiment to examine the spread of an inverted turbulent plume of 'pollutant' of slightly greater density than its surroundings.

178. Kolomenskiy, D., Moffatt, H. K., Farge, M. & Schneider, K. 2011 The Lighthill-Weis-Fogh clap-fling-sweep mechanism revisited. J. Fluid Mech. 676, 572--606.

James Lighthill.analysed the Weis-Fogh clap-fling-sweep lift-producing mechanism (supposed inviscid) employed by the tiny wasp 'encarsia formosa', as illustrated here. We analysed the flow near the 'hinge' where viscosity actually dominates over inertia, the lift being thereby augmented.

Clap (a), fling (b,c) and sweep (d) of E. formosa; schematic adapted from Weis-Fogh (1973)

Weis-Fogh, T. 1973 Quick estimates of flight fitness in hovering animals, including novel mechanisms for lift production. J. Exp. Biol. 59, 169–230

Lighthill, M. J. 1973 On the Weis-Fogh mechanism of lift generation. J. Fluid Mech. 60 (1), 1–17.

see also:

Kolomenskiy, D., Moffatt, H. K., Farge, M. & Schneider, K. 2010 Vorticity generation during the clap-fllng-sweep of some hovering insects. Theor. Comp. Fluid Dyn.

24 (1-4, SI), 209-215.

186. Moffatt, P.G. & Moffatt, H.K. (2014) Giffen goods and their reflexion property. The Manchester School, 82, 129-142.

My son Peter is Professor of Econometrics at the University of East Anglia. Occasionally he consults me about mathematical problems that he runs into. This paper concerned Giffen goods which have the unusual property that the demand for a Giffen good increases as its price increases, other prices being held fixed, in contrast to 'normal' response. Consider, by way of example, a hypothetical Scot who spends all his disposable income on oatmeal and malt whisky, the latter being a relatively expensive luxury. Suppose that the price of oatmeal increases due to persistent rain and a poor harvest; our Scot can then afford less whisky, but, needing to maintain his calory-intake, must compensate by buying more oatmeal, which is in these circumstances a Giffen good.

Our first problem was to devise a family of Utility Functions U corresponding to goods that have this Giffen property. The figure shows surfaces U(x,y,z) = cst. for a three-good situation, for which each good has the Giffen property in some region of the 'goods-space'

210. lin, K. I., Moffatt, H. K. & Vladimirov, V. A. 2017 Dynamics of a rolling robot. Proc. Nat. Acad. Sci. 114 (49), 12858-12863

This paper was motivated by a toy known as the 'Beaver Ball' which can roll on a horizontal surface in an apparently chaotic manner. The motion is driven by an internal battery mechanism which causes a rotor to rotate about the axis of symmetry of the ball on which it is mounted. The motion is governed by "a six-dimensional, non-holonomic, non-autonomous dynamical system with cubic nonlinearity". This figure shows two possible trajectories, one quasi-periodic, the other chaotic.

216. Moffatt, H. K. & Dormy, E. 2019 Self-Exciting Fluid Dynamos. Cambridge Texts in Applied Mathematics, CUP, 520+xviii pp.

"Exploring the origins and evolution of magnetic fields in planets, stars and galaxies, this book gives a basic introduction to magnetohydrodynamics and surveys the observational data, with particular focus on geomagnetism and solar magnetism. Pioneering laboratory experiments that seek to replicate particular aspects of fluid dynamo action are also described. The authors provide a complete treatment of laminar dynamo theory, and of the mean-field electrodynamics that incorporates the effects of random waves and turbulence. Both dynamo theory and its counterpart, the theory of magnetic relaxation, are covered. Topological constraints associated with conservation of magnetic helicity are thoroughly explored and major challenges are addressed in areas such as fast-dynamo theory, accretion-disc dynamo theory and the theory of magnetostrophic turbulence. The book is aimed at graduate-level students in mathematics, physics, Earth sciences and astrophysics, and will be a valuable resource for researchers at all levels."

217. Moffatt, H. K. & Kimura, Y. 2019 Towards a finite-time singularity of the Navier-Stokes equations. Part 1. Derivation and analysis of dynamical system. J. Fluid Mech. 861, 930-967

This paper represents ongoing work that seeks to determine the maximum vorticity that can be generated in the 'collision' of two vortex tubes approaching each other at a finite angle. Biot-Savart analysis, as shown in this figure, is very suggestive of a 'finite-time singularity'; but how does this behave under true Euler or Navier-Stokes evolution? Much current numerical work is (as at December 2021) being devoted to this problem.

see particularly: Yau, J. & Hussain, F. 2020 On singularity formation via viscous vortex reconnection, J. Fluid Mech. 888, R2.

[Fazle Hussain was one of the graduate students who attended my course on Turbulence, April-June 1965, at Stanford University; he has made man outstanding contributions to research on turbulence and vortex dynamics, and is a leading authority in this broad field of study.]