### 1990s

68. Bajer, K. & Moffatt, H.K. 1990 On a class of steady confined Stokes flows with chaotic streamlines. J. Fluid Mech. 212, 337-364.

The phenomenon of Lagrangian chaos in the streamlines of a steady flow dates back to the work of Hénon (1966). It can be recognised by the scatter of points in a Poincaré section. Aref (1984) constructed an explicit example of chaos in an unsteady time-periodic two-dimensional flow in a finite domain (the 'blinking vortex' model). Konrad Bajer and I constructed what we believe was the first explicit example of a steady three-dimensional flow in a bounded region exhibiting this property, in fact a wide family of such flows, all exact solutions of the Stokes equations of slow viscous flow in a sphere, driven by tangential motion at the boundary.

This picture shows a Poincaré section of one such flow, i.e. the points where a single streamline crosses a diametral plane of section (40,000 times in this figure, limited only by the finite time of computation).

In a chaotic flow, adjacent streamlines diverge exponentially on average, and therefore promote rapid mixing of any passive scalar contaminant.

Aref, H. 1984 Stirring by chaotic advection. J.Fluid Mech. 143, 1-21.

Hénon, M. 1966 Sur la topologie des lignes de courant dans un cas particulier. C.R. Acad. Sci. Paris 262, 312-314.

77. Moffatt, H.K. 1990 The energy spectrum of knots and links. Nature 347, 367-369

This paper is a natural continuation of paper [53]. When the magnetic field is confined to a knotted flux tube, relaxation ‘tightens’ the knot, while conserving flux Φ, volume V, and helicity h Φ^2 . The energy decreases to a minimum

M = m(h) Φ^2 V^{-1/3} ,

where the dimensionless function m(h) depends only on the knot type; this function was computed some years later for a number of torus knots by Chui & Moffatt (1995).

This minimum-energy knot is closely related to the ‘ideal knot’ (Stasiak et al 1998) which essentially captures the familiar tightening process in mathematical terms (but without the complication of internal twist). The ‘energy’ is then simply the minimum length of a closed knot in a rope of unit cross-sectional radius. There is growing interest in this field of applied-mathematical research which has applications in chemistry and biology, as well as in physics.

Ashton, T., Cantarella, J.,Piatek, M. & Rawdon,E. 2011 Knot tightening by constrained gradient descent. Experim. Math. 20, 57.

Cantarella, J., Kusner, R.B. & Sullivan, J.M. 2002 On the minimum ropelength of knots and links. Invent. Math., 150, 257-286.

Chui, A.Y.K. & Moffatt, H.K. 1995 The energy and helicity of knotted magnetic flux tubes. Proc. Roy. Soc. Lond. A 451, 609-629.

Stasiak, A., Katrich, V. & Kauffman, L.H. (Eds) 1998 Ideal Knots. World Scientific.

82. Jeong, J.-T. & Moffatt, H.K. 1992 Free surface cusps associated with flow at low Reynolds number. J. Fluid Mech. 241, 1-22.

Here is a problem that attracted Royal attention!

HRH Prince Philip, then Chancellor of the University of Cambridge, at the official opening of the Isaac Newton Institute, 30 October 1992. I am on the left, Sir Michael Atiyah, Director of the Institute, on the right; and Andrew Soward and Uriel Frisch are behind.

Any two-dimensional strongly converging flow of a viscous fluid like syrup tends to form a cusp as in this photo. Here the flow is driven by two counter-rotating submerged cylinders. Surface tension tries to smooth the cusp, but is remarkably ineffective in doing so. Jae-Tak Jeong (visiting at the time from Korea) and I provided an exact mathematical solution of an idealised version of this problem and showed that the radius of curvature R at the ‘cusp’ is exponentially small:

R/d = (256/3) exp {-32π C}, !!

where d is the imposed scale of the converging flow, and C is the capillary number (the ratio of surface tension stress to viscous stress) -- a physical though not a mathematical singularity!

If a bubble is injected below the cusp it becomes lens-shaped, and is held steady by the downward flow, while itself forming a cusped rim from which smaller bubbles escape into the flow. The strong elliptic deformation of these bubbles is caused by the straining action of the flow.

Here, rising in a glycerine solution, is a similar ‘wavy skirted bubble’ , so-called because it sheds a ‘skirt’ of air from its cusped rim. This skirt breaks up through a wavy instability into small bubbles that are entrained into the wake.

Photo © Michael Manni Photographic, Cambridge

See also section 3.4 of: Moffatt, H. K. 2021b Some topological aspects of fluid dynamics. J. Fluid Mech. 914, P-1-- P1-56.

#### 89. Moffatt, H.K. & Ricca, R.L. 1992 Helicity and the Cãlugãreanu invariant. Proc. R. Soc. Lond. A 439, 411-429.

In this paper, Renzo Ricca and I established the relation between the helicity H of a knotted flux tube carrying flux Φ and the twist Tw and writhe Wr of the tube:

H = h Φ^2, where h = Tw + Wr

The sum of twist and writhe had been known to be an invariant for a twisted ribbon since the work of Cãlugãreanu (1961). This equation therefore provided a further satisfying bridge between fluid mechanics and topology.

Cãlugãreanu, G. 1961 Sur les classes d'isotopie des noeuds tridimensionnels et leurs invariants.

Czechoslovak Math. J . 11, 588-625.

see also

White. J. H. 1969 Self-linking and the Gauss integral in higher dimensions.Am. J.Math. 9 1,693-728.

#### 96. Moffatt, H.K., Kida, S. & Ohkitani, K. 1994 Stretched vortices: the sinews of turbulence; high Reynolds number asymptotics. J. Fluid Mech. 259, 241-264.

This paper was stimulated by Kida & Ohkitani (1992), who had detected in direct numerical simulation of turbulence a curious structure of the field of viscous dissipation. In the 1990s, there was a return to the simplistic view of turbulence as a random superposition of vortex tubes, each subject to the straining velocity field induced by all the others. This view was stimulated by the detection of such tubes by Douady et al (1991), and by direct numerical simulations (DNS) such as those of Vincent & Meneguzzi (1991). Küchemann's (1965) description of vortices as the 'sinews' of turbulence seemed increasingly apt.

In 1993, I spent three months at RIMS, Kyoto, and joined forces with Shigeo Kida and Koji Ohkitani to analyse the structure of such vortices when subjected to strain that is (generically) non-axisymmetric. We were able to explain the cross-sectional structure of the dissipation field.

Kida, S. & Ohkitani, K. 1992 Spatio-temporal intermittency and instability of a forced turbulence. Phys. Fluids A 4, 1018-1027

Douady, S., Couder, Y. & Brachet, M.E. 1991 Direct observation of the intermittency of intense vorticity filaments in turbulence. Phys. Rev. Lett. 67, 983-986.

Vincent, A. & Meneguzzi, M. 1991 The spatial structure and statistical properties of homogeneous turbulence. J. Fluid Mech. 225, 1-25.

Küchemann, D. 1965 Report on the IUTAM Symposium on concentrated vortex motions in fluids. J. Fluid Mech. 21, 1-20.