Keith Moffatt

• ScD
• Professor at University of Cambridge

The low-Reynolds-number Stokes flow driven by rotation of two parallel cylinders of equal unit radius is investigated by both analytical and numerical techniques. In Part I, the case of counterrotating cylinders is considered. A numerical (finite-element) solution is obtained by enclosing the system in an outer cylinder of radius R0≫1, on which the...

An exact analytical solution is obtained for the dynamical system derived in Part 1 of this series (Moffatt & Kimura, J. Fluid Mech , vol. 861, 2019 a , pp. 930–967), which describes the approach of two initially circular vortices of finite but small cross-section symmetrically located on inclined planes. This exact solution, applicable in the invi...

A rotational version of the fluid-mechanical sewing machine (FMSM) is investigated experimentally. A thin thread of silicon oil was dispensed at a constant flow rate Q from a height H and fell on a table rotating at an angular speed ω, at a distance R from the axis. In all experimental runs, the values of Q and H were kept constant while the radius...

The behaviour of a viscous drop squeezed between two horizontal planes (a squeezed Hele-Shaw cell) is treated by both theory and experiment. When the squeezing force $F$ is constant and surface tension is neglected, the theory predicts ultimate growth of the radius $a\sim t^{1/8}$ with time $t$ . This theory is first reviewed and found to be in exc...

The behaviour of a viscous drop squeezed between two horizontal planes is treated by both theory and experiment. When the squeezing force F is constant and surface tension is neglected, the theory predicts ultimate growth of the radius a~ t^{1/8}, in excellent agreement with our experiment. Surface tension at the drop boundary is included in the an...

The Stokes flow driven by the rotation of two parallel cylinders of equal unit radius is determined by both numerical and analytical techniques. A numerical solution is obtained by enclosing the system in an outer cylinder of radius R_0>>1, on which different boundary conditions can be imposed. With a gap 2epsilon between the inner cylinders, atten...

Towards a finite-time singularity of the Navier–Stokes equations. Part 2. Vortex reconnection and singularity evasion – CORRIGENDUM - Volume 887 - H. K. Moffatt, Yoshifumi Kimura

Singularities of the Navier-Stokes equations occur when some derivative of the velocity field is infinite at any point of a field of flow (or, in an evolving flow, becomes infinite at any point within a finite time). Such singularities can be mathematical (as, e.g., in two-dimensional flow near a sharp corner, or the collapse of a Möbius-strip soap...

Suppose that viscous fluid is contained in the space between a fixed sphere S2 and an interior sphere S1 which moves with time-periodic velocity U(t) and angular velocity Ω(t), with U(t)=Ω(t)=0. It is shown that, provided this motion is chiral in character, it can drive a flow that exerts a nonzero torque on S2. Thus angular momentum can be transfe...

Singularities of the Navier-Stokes equations occur when some derivative of the velocity field is infinite at any point of a field of flow (or, in an evolving flow, becomes infinite at any point within a finite time). Such singularities can be mathematical (as e.g. in two-dimensional flow near a sharp corner, or the collapse of a Mobius-strip soap f...

In Part 1 of this work, we have derived a dynamical system describing the approach to a finite-time singularity of the Navier–Stokes equations. We now supplement this system with an equation describing the process of vortex reconnection at the apex of a pyramid, neglecting core deformation during the reconnection process. On this basis, we compute...

Cambridge Core - Space Science - Self-Exciting Fluid Dynamos - by Keith Moffatt

Suppose that viscous fluid is contained in the space between a fixed sphere $S_2$ and an interior sphere $S_1$ which moves with time-periodic velocity ${\bf U}(t)$ and angular velocity ${\bf \Omega}(t)$, with $ \left<{\bf U}(t)\right> = \left<{\bf \Omega}(t)\right> = 0$. It is shown that, provided this motion is chiral in character, it can drive a...

In Part 1 of this work, we have derived a dynamical system describing the approach to a finite-time singularity of the Navier-Stokes equations. We now supplement this system with an equation describing the process of vortex reconnection at the apex of a pyramid, neglecting core deformation during the reconnection process. On this basis, we compute...

The evolution towards a finite-time singularity of the Navier-Stokes equations for flow of an incompressible fluid of kinematic viscosity is studied, starting from a finite-energy configuration of two vortex rings of circulation and radius , symmetrically placed on two planes at angles to a plane of symmetry . The minimum separation of the vortices...

A model is developed describing the approach to a finite-time singularity of the Navier-Stokes equations for two interacting vortices. The model is derived from a combination of the Biot-Savart law and an equation describing the evolution of the vortex core cross-sections. The maximum vorticity is attained within a finite time and increases as the...

Random waves in a uniformly rotating plasma in the presence of a locally uniform seed magnetic field and subject to weak kinematic viscosity ν and resistivity η are considered. These “Lehnert” waves may have either positive or negative helicity, and it is supposed that waves of a single sign of helicity are preferentially excited by a symmetry-brea...

Vortex reconnection under Biot–Savart evolution is investigated geometrically and numerically using a tent model consisting of vortex filaments initially in the form of two tilted hyperbolic branches; the vortices are antiparallel at their points of nearest approach. It is shown that the tips of these vortices approach each other, accelerating as t...

This short review is a contribution to an issue of Comptes Rendus Mécanique commemorating the scientific work of Jean-Jacques Moreau (1923–2014). His main contribution to fluid mechanics appeared in a brief paper in the Comptes Rendus à l'Académie des Sciences in 1961, but was not recognised till much later. It may now be seen as a significant mile...

The relaxation of a helical magnetic field in a high-conductivity plasma contained in the annulus between two perfectly conducting coaxial cylinders is considered. The plasma is of low density and its pressure is negligible compared with the magnetic pressure; the flow of the plasma is driven by the Lorentz force and energy is dissipated primarily...

Significance The dynamics of a rolling ball activated by an internal battery mechanism are analyzed by theoretical and numerical techniques. The problem involves four independent dimensionless parameters and is governed by a six-dimensional nonholonomic nonautonomous dynamical system with cubic nonlinearity. It can serve as a prototype for rolling...

The degree of knottedness of tangled vortex lines – CORRIGENDUM - Volume 830 - H. K. Moffatt

Observing and probing writhe and twist in vortex dynamics

Reconnection of a vortex filament under the Biot-Savart law is investigated numerically using a vortex ring twisted in the form of a figure-of-eight. For the numerical method, the vortex ring is divided into piecewise linear segments, and the Biot-Savart integral is approximated by a summation over the segments with a cut-off method to deal with th...

The origins of the Journal of Fluid Mechanics, of which the first volume was published in 1956, are discussed, with reference to editorial correspondence during the early years of the Journal. This paper is based on a lecture given at the colloquium: A Century of Fluid Mechanics, 1870–1970, IMFT, Toulouse, France, 19–21 October 2016.

The relaxation of a helical magnetic field ${\bf B}({\bf x}, t)$ in a high-conductivity plasma contained in the annulus between two perfectly conducting coaxial cylinders is considered. The plasma is of low density and its pressure is negligible compared with the magnetic pressure; the flow of the plasma is driven by the Lorentz force and and energ...

The response of a soap film to the continuous deformation of its wire boundary is considered, with particular attention to the topological transitions that can occur at critical stages of the deformation process. Two well-known examples that have been studied by both theory and experiment are the catenoid suspended between circular wires in paralle...

Book Review - Singularities: Formation, Structure, and Propagation. Eggers J. & Fontelos M. A. . Cambridge Texts in Applied Mathematics, Cambridge University Press, 2015. Paperback, 453+xvi pp. ISBN 9781107485495. £39.99. - Volume 804 - H. K. Moffatt

This informal article discusses the central role of magnetic and kinetic helicity in relation to the evolution of magnetic fields in geophysical and astrophysical contexts. It is argued that the very existence of magnetic fields of the intensity and scale observed is attributable in large part to the chirality of the background turbulence or random...

A one-dimensional model of magnetic relaxation in a pressureless low-resistivity plasma is considered. The initial two-component magnetic field $\boldsymbol{b}(\boldsymbol{x},t)$ is strongly helical, with non-uniform helicity density. The magnetic pressure gradient drives a velocity field that is dissipated by viscosity. Relaxation occurs in two...

A small electrically conducting sphere contains a dynamo source of magnetic field which externally has quadrupolar symmetry. It is surrounded by a force-free medium of finite electrical conductivity. Differential rotation of the sphere, together with a ‘solar wind’ blowing off it, can maintain a twisted force-free external field despite the ohmic d...

We describe the first analytically tractable example of an instability of a nonorientable minimal surface under parametric variation of its boundary. A one-parameter family of incomplete Meeks Möbius surfaces is defined and shown to exhibit an instability threshold as the bounding curve is opened up from a double-covering of the circle. Numerical a...

One of the simplest geometries in which to study fluid flow between two soap films connected by a Plateau border is provided by a catenoid with a secondary film at its narrowest point. Dynamic variations in the spacing between the two rings supporting the catenoid lead to fluid flow between the primary and secondary films. When the rings are moved...

Based on experimental evidence that vortex reconnection commences with the approach of nearly antiparallel segments of vorticity, a linearised model is developed in which two Burgers-type vortices are driven together and stretched by an ambient irrotational strain field induced by more remote vorticity. When these Burgers vortices are exactly antip...

Significance The dynamics of topological rearrangements of surfaces are generically associated with singularities. It is not yet understood whether the location of a singularity (in the bulk or at boundaries) is a general consequence of the surface topology, nor how this dependence could arise from an equation of motion for the surface. Here we use...

The following ‘reflexion property’ of Giffen behaviour is proved: the two‐good direct utility function (DUF) obtained by reversing the sign of two‐good indirect utility function (IUF) displays Giffen behaviour with respect to one of the two goods if and only if the IUF itself displays Giffen behaviour with respect to the other good. A particular IU...

Significance This paper covers aspects of the dynamics of fluids that are of central importance for ( i ) the origin of planetary and astrophysical magnetism, and ( ii ) the determination of stable magnetic field configurations used in thermonuclear fusion reactors like the tokamak. The technique of magnetic relaxation also has implications for the...

Triad interactions, involving a set of wave-vectors {+/- k, +/- p, +/- q}, with k + p + q = 0, are considered, and the results of triad truncation are compared with the results of exact Euler evolution starting from the same initial conditions. The essential two-dimensionality of the triad interaction is used to separate the problem into two parts:...

The flow field over an accelerating rotating wing model at Reynolds numbers Re ranging from 250 to 2000 is investigated using particle image velocimetry, and compared with the flow obtained by three-dimensional time-dependent Navier-Stokes simulations. It is shown that the coherent leading-edge vortex that characterises the flow field at Re~200-300...

The process of relaxation of a unidirectional magnetic field in a highly conducting tenuous fluid medium is considered. Null points of the field play a critical role in this process. During an initial stage of relaxation, variations in magnetic pressure are eliminated, and current sheets build up in the immediate neighborhood of null points. This i...

By making simple, heuristic assumptions, a new method based on the derivation of the Jones polynomial invariant of knot theory to tackle and quantify structural complexity of vortex filaments in ideal fluids is presented. First, we show that the topology of a vortex tangle made by knots and links can be described by means of the Jones polynomial ex...

If, in a large expanse of fluid such as air or water, an object that is heavier than the fluid displaced is released from rest, it descends in a manner that can depend in a complex way on its geometry and density (relative to that of the fluid), and on the fluid viscosity, which, as in other fluid contexts, remains important no matter how small thi...

The method of magnetic relaxation for the determination of solutions of the Euler equations representing steadily propagating vortical structures is reviewed, and compared with alternative artificial relaxation procedures that conserve the topology of the vorticity field. Attention is fo-cussed first on axisymmetric vortex ring configurations, for...

ABSTRACT A class of similarity solutions for two-dimensional unsteady flow in the neighbourhood of a front or rear stagnation point on a plane boundary is considered, and a wide range of possible behaviour is revealed, depending on whether the flow in the far field is accelerating or decelerating. The solutions, when they exist, are exact solutions...

A variety of physical and biological systems exhibit dynamical behaviour that has some explicit or implicit topological features. Here, the term ‘topological’ is meant to convey the idea of structures, e.g. physical knots, links or braids, that have some measure of invariance under continuous deformation. Dynamical evolution is then subject to the...

A brief review of developments in the theory of homogeneous turbulence over the last 50 years is given, many of these developments stemming from lectures and discussions at the 1961 Marseille Colloquium Mécanique de la Turbulence. The following topics are discussed: Kolmogorov’s 1961 lecture, intermittency and the finite-time singularity problem, t...

The Lighthill-Weis-Fogh clap-fling-sweep mechanism is a movement used by some insects to improve their flight performance. As first suggested by Lighthill (1973), this mechanism allows large circulations around the wings to be established immediately as they start to move. Initially, the wings are clapped. Then they fling open like a book, and a no...

The mean electromotive force (EMF) associated with exponentially growing perturbations of an Euler flow with elliptic streamlines in a rotating frame of reference is studied. We are motivated by the possibility of dynamo action triggered by tidal deformation of astrophysical objects such as accretion discs, stars or planets. Ellipticity of the flow...

Turbulence is widely recognized as one of the outstanding problems of the physical sciences, but it still remains only partially understood despite having attracted the sustained efforts of many leading scientists for well over a century. In A Voyage Through Turbulence we are transported through a crucial period of the history of the subject via bi...

Turbulence is widely recognized as one of the outstanding problems of the physical sciences, but it still remains only partially understood despite having attracted the sustained efforts of many leading scientists for well over a century. In A Voyage Through Turbulence we are transported through a crucial period of the history of the subject via bi...

Turbulence is widely recognized as one of the outstanding problems of the physical sciences, but it still remains only partially understood despite having attracted the sustained efforts of many leading scientists for well over a century. In A Voyage Through Turbulence we are transported through a crucial period of the history of the subject via bi...

The importance of three-dimensional effects for flapping wings is addressed by means of numerical simulation. In particular, the clap–fling–sweep mechanism is examined. The flow at the beginning of the downstroke is shown to be in reasonable agreement with the two-dimensional approximation. After the wings move farther than one chord length apart,...

The emphasis in this short introductory chapter is on those fluid dynamical phenomena that are best understood in terms of convection and diffusion of vorticity, the curl of the velocity field. Vorticity is generated at fluid boundaries and diffuses into the fluid where it is subject to convection, stretching, and associated intensification. Far fr...

The Lighthill–Weis-Fogh ‘clap–fling–sweep’ mechanism for lift generation in insect flight is re-examined. The novelty of this mechanism lies in the change of topology (the ‘break’) that occurs at a critical instant tc when two wings separate at their ‘hinge’ point as ‘fling’ gives way to ‘sweep’, and the appearance of equal and opposite circulation...

The Lighthill–Weis-Fogh ‘clap–fling–sweep’ mechanism for lift generation in insect flight is re-examined. The novelty of this mechanism lies in the change of topology (the ‘break’) that occurs at a critical instant tc when two wings separate at their ‘hinge’ point as ‘fling’ gives way to ‘sweep’, and the appearance of equal and opposite circulation...

It is well-known that a soap film spanning a looped wire can have the topology of a Möbius strip and that deformations of the wire can induce a transformation to a two-sided film, but the process by which this transformation is achieved has remained unknown. Experimental studies presented here show that this process consists of a collapse of the fi...

Shear flow is prone to transient instability in which perturbations having little or no variation in the streamwise direction can grow linearly for a long time if the Reynolds number is large. This behaviour is known to provide a trigger for the development of secondary instabilities and transition to turbulence. It is shown by a simple analysis of...

Ten years have elapsed since the passing of George Keith Batchelor (8 March 1920–30 March 2000), formerly Professor of Fluid Dynamics at the University of Cambridge, and Founder Editor of the Journal of Fluid Mechanics. It is fitting to remind the readers of this Journal what a great scientist he was, both in respect of his own contributions to our...

Numerical simulations of the Lighthill–Weis-Fogh mechanism are performed using a Fourier pseudo-spectral method with volume penalization. Single-winged and double-winged configurations are compared, and the vortex shedding patterns are related to the lift generated in both cases. The computations of the lift coefficient are validated against the re...

We show that if two goods whose Indirect Utility Function V (p; q) exhibits the Giffen property for good 1 in some subdomain G(p; q) of the positive quadrant, and if U(x; y) is a Direct Utility Function given by U(x; y) = -V (x; y) and therefore having the same convex contours as V, then U also exhibits the Giffen property for good 2 rather than fo...

Three types of singularity that can arise in fluid dynamical prob-lems will be distinguished and discussed. These are: (i) singularities driven by boundary motion in conjunction with viscosity (e.g. corner singularities, or the Euler-disc finite-time singularity); (ii) free-surface (cusp) singularities associated with surface-tension and viscosity;...

The Lighthill--Weis-Fogh ``clap-fling-sweep'' description of insect flight involves a novel mechanism, which can apparently operate in a strictly inviscid fluid, of generation of circulation and lift through instantaneous change of topology. However, viscous effects substantially influence this mechanism, both near the sharp edges of the wings by t...

We show how the ideas of topology and variational principle, opened up by Euler, facilitate the calculation of motion of vortex rings. Kelvin–Benjamin’s principle, as generalised to three dimensions, states that a steady distribution of vorticity, relative to a moving frame, is the state that maximizes the total kinetic energy, under the constraint...

A physically transparent and mathematically streamlined derivation is presented for a third-order nonlinear dynamical system that describes the curious chiral reversals of a celt (rattleback). The system is integrable, and its solutions are periodic, showing an infinite succession of spin reversals. Inclusion of linear dissipation allows any given...

The flow generated by a random buoyancy field in a rotating medium permeated by a dynamo-generated magnetic field is considered, under the assumptions that the Rossby number and the magnetic Reynolds number (based on the scale of the buoyancy fluctuations) are both small. This permits linearisation of the governing evolution equations. Provided ‘up...

The year 2007 will mark the centenary of the death of William Thomson (Lord Kelvin), one of the great nineteenth-century pioneers of vortex dynamics. Kelvin was inspired by Hermann von Helmholtz’s [7] famous paper “Ueber Integrale der hydrodynamischen Gleichungen, welche den Wirbelbewegungen entsprechen”, translated by P. G. Tait and published in E...

Howell Peregrine, who died on 20th March after a brief illness, had an exceptionally long and distinguished period of service for this Journal, first as Assistant Editor (1979–1980) then Associate Editor (1981–2007). In this capacity, he had editorial responsibility for some 50 papers every year, more than 1200 in total. This was a massive contribu...

The smoothness, or alternatively the finite-time singularity, of the Navier-Stokes equations offers a challenge that will continue to make great demands on both analytical ingenuity and computational power. If, as computer simulations continue to indicate (Kerr 1997), a finite-time singularity does occur and if this is generic behavior, then of cou...

This essay provides a personal account of the development of the subject of magnetohydrodynamic (MHD) turbulence from its birth in 1950 to its “coming-of-age” in 1971, following the development of mean-field electrodynamics, a major breakthrough of the 1960s. The discussion covers the early ideas based on the analogy with vorticity, the passive vec...

The magnetohydrodynamic evolution of axisymmetric magnetic eddies within which the magnetic field is purely toroidal with $B_\theta/r$ piecewise-constant, and the velocity field is poloidal, is studied both analytically and numerically. A family of exact solutions, generalizing Hill's spherical vortex to the case of non-zero magnetic field, is foun...

Stokes flow of a viscous fluid in a cylindrical container driven by time-periodic forcing, either at the boundary or through oscillation of the cylinder about an axis parallel to its generators, is considered. The behaviour is governed by a dimensionless frequency parameter $\eta$ and by the geometry of the cylinder cross-section. Various cross-sec...

Following parts I and II of this series, the geometry of steady states for a general convex axisymmetric rigid body spinning on a horizontal table is analysed. A general relationship between the pedal curve of the cross-section of the body and the height of its centre-of-mass above the table is obtained which allows for a straightforward determinat...

We present two classes of exact solutions of the Navier–Stokes equations, which describe steady vortex structures with two-dimensional symmetry in an infinite fluid. The first is a class of similarity solutions obtained by conformal mapping of the Burgers vortex sheet to produce wavy sheets, stars, flowers and other vorticity patterns. The second i...

Following part I of this series, the general spinning motion of an axisymmetric rigid body on a horizontal table is further analysed, allowing for slip and friction at the point of contact. Attention is focused on the case of spheroids whose density distribution is such that the centre-of-mass and centre-of-volume coincide. The governing dynamical...

The viscous interaction of two weakly curved and oppositely directed vortex tubes of flattened cross-section driven together by an imposed uniform strain is considered using a perturbation technique. The evolution of the cross-sectional vorticity distribution is computed, and the rate of reconnexion, as indicated by the decrease of circulation of e...

The general spinning motion of an axisymmetric rigid body on a horizontal table is analysed, allowing for slip and friction at the point of contact P. Attention is focused on the case of spheroids (prolate or oblate), and particularly on spheroids whose density distribution is such that the centre-of-mass and centre-of-volume coincide. Four classes...

A forced reconnection problem was modeled by two infinite wires that are embedded in a plasma and which carry parallel currents. They are brought together at a specified rate. The distance between the wires is taken as 2a(1-e/sup omega/). For small displacements, the hydromagnetic equations can be linearized and solved asymptotically. For larger di...

A general theoretical review of helicity in flows that are laminar or turbulent is presented with references to corresponding experimental investigations. The topological nature of the helicity invariant is examined, and the turbulent dynamo mechanism is discussed in relation to the phenomenon of helicity. Helicity is also discussed in terms of bei...

We study the stability of a perfectly conducting body in a magnetostatic equilibrium. The body is immersed in a fluid which is threaded by a three-dimensional magnetic field. The fluid may be perfectly conducting, non-conducting or have finite conductivity. We generalise the classical stability criterion of Bernstein et al. (Proc. Roy. Soc. London...