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 maximum vorticity $\unicode[STIX]{x1D714}_{max}$ as a function of vortex Reynolds number $R_{\unicode[STIX]{x1D6E4}}$ in the range $2000\leqslant R_{\unicode[STIX]{x1D6E4}}\leqslant 3400$, and deduce a compatible behaviour $\unicode[STIX]{x1D714}_{max}\sim \unicode[STIX]{x1D714}_{0}\exp [1+220(\log [R_{\unicode[STIX]{x1D6E4}}/2000])^{2}]$ as $R_{\unicode[STIX]{x1D6E4}}\rightarrow \infty$. This may be described as a physical (although not strictly mathematical) singularity, for all $R_{\unicode[STIX]{x1D6E4}}\gtrsim 4000$.

The evolution towards a finite-time singularity of the Navier–Stokes equations for flow of an incompressible fluid of kinematic viscosity $\unicode[STIX]{x1D708}$ is studied, starting from a finite-energy configuration of two vortex rings of circulation $\pm \unicode[STIX]{x1D6E4}$ and radius $R$, symmetrically placed on two planes at angles $\pm \unicode[STIX]{x1D6FC}$ to a plane of symmetry $x=0$. The minimum separation of the vortices, $2s$, and the scale of the core cross-section, $\unicode[STIX]{x1D6FF}$, are supposed to satisfy the initial inequalities $\unicode[STIX]{x1D6FF}\ll s\ll R$, and the vortex Reynolds number $R_{\unicode[STIX]{x1D6E4}}=\unicode[STIX]{x1D6E4}/\unicode[STIX]{x1D708}$ is supposed very large. It is argued that in the subsequent evolution, the behaviour near the points of closest approach of the vortices (the ‘tipping points’) is determined solely by the curvature $\unicode[STIX]{x1D705}(\unicode[STIX]{x1D70F})$ at the tipping points and by $s(\unicode[STIX]{x1D70F})$ and $\unicode[STIX]{x1D6FF}(\unicode[STIX]{x1D70F})$, where $\unicode[STIX]{x1D70F}=(\unicode[STIX]{x1D6E4}/R^{2})t$ is a dimensionless time variable. The Biot–Savart law is used to obtain analytical expressions for the rate of change of these three variables, and a nonlinear dynamical system relating them is thereby obtained. The solution shows a finite-time singularity, but the Biot–Savart law breaks down just before this singularity is realised, when $\unicode[STIX]{x1D705}s$ and $\unicode[STIX]{x1D6FF}/\!s$ become of order unity. The dynamical system admits ‘partial Leray scaling’ of just $s$ and $\unicode[STIX]{x1D705}$, and ultimately full Leray scaling of $s,\unicode[STIX]{x1D705}$ and $\unicode[STIX]{x1D6FF}$, conditions for which are obtained. The tipping point trajectories are determined; these meet at the singularity point at a finite angle. An alternative model is briefly considered, in which the initial vortices are ovoidal in shape, approximately hyperbolic near the tipping points, for which there is no restriction on the initial value of the parameter $\unicode[STIX]{x1D705}$; however, it is still the circles of curvature at the tipping points that determine the local evolution, so the same dynamical system is obtained, with breakdown again of the Biot–Savart approach just before the incipient singularity is realised. The Euler flow situation ($\unicode[STIX]{x1D708}=0$) is considered, and it is conjectured on the basis of the above dynamical system that a finite-time singularity can indeed occur in this case.

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 they do so to form a finite-time singularity at the apex of the tent. The minimum separation of the vortices and the maximum velocity and axial strain rate exhibit nearly self-similar Leray scaling, but the exponents of the velocity and strain rate deviate slightly from their respective self-similar values of $-1/2$ and $-1$; this deviation is associated with the appearance of distinct minima of curvature leading to cusp structures at the tips. The writhe and twist of each vortex are both zero at all times up to the instant of reconnection. By way of validation of the model, the structure of the eigenvalues and eigenvectors of the rate-of-strain tensor is investigated: it is shown that the second eigenvalue $\unicode[STIX]{x1D706}_{2}$ has dipole structure around the vortex filaments. At the tips, it is observed that $\unicode[STIX]{x1D706}_{2}$ is positive and the corresponding eigenvector is tangent to the filament, implying persistent stretching of the vortex.

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 phases. The first is a rapid initial phase in which the magnetic energy drops sharply and the magnetic pressure becomes approximately uniform; the helicity density is redistributed during this phase but remains non-uniform, and although the total helicity remains relatively constant, a Taylor state is not established. The second phase is one of slow diffusion, in which the velocity is weak, though still driven by persistent weak non-uniformity of magnetic pressure; during this phase, magnetic energy and helicity decay slowly and at constant ratio through the combined effects of pressure equalisation and finite resistivity. The density field, initially uniform, develops rapidly (in association with the magnetic field) during the initial phase, and continues to evolve, developing sharp maxima, throughout the diffusive stage. Finally it is proved that, if the resistivity is zero, the spatial mean $\langle (\boldsymbol{b}\boldsymbol{\cdot }\boldsymbol{{\rm\nabla}}\times \boldsymbol{b})/b^{2}\rangle$ is an invariant of the governing one-dimensional induction equation.

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 antiparallel, they are annihilated on the strain time-scale, independent of kinematic viscosity $\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}\nu $ in the limit $\nu \rightarrow 0$. When the vortices are skew to each other, they are annihilated under this action over a local extent that increases exponentially in the stretching direction, with clear evidence of reconnection on the same strain time-scale. The initial helicity associated with the skewed geometry is eliminated during the process of reconnection. The model applies equally to the reconnection of weak magnetic flux tubes under the action of a strain field, when Lorentz forces are negligible.

Triad interactions, involving a set of wave-vectors $\{\pm \boldsymbol {k}, \pm \boldsymbol {p}, \pm \boldsymbol {q}\}$, with $ \boldsymbol {k} + \boldsymbol {p}+ \boldsymbol {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: a nonlinear two-dimensional flow problem in the triad plane, and a linear problem of ‘passive scalar’ type for the evolution of the component of velocity perpendicular to this plane. Several examples of triad evolution are presented in detail, and the marked contrast with Euler evolution is demonstrated. It is known that energy and helicity are conserved under triad truncation; it is shown that the ‘in-plane’ energy and enstrophy are also conserved. However, it is also shown that, in general, the evolution of the vorticity under triad truncation cannot be represented as transport by any divergence-free velocity field, with the consequence that the detailed topology of the vorticity field is not conserved under this truncation.

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 this viscosity may be. A major numerical attack on this problem for the case in which the object is a thin circular disc is presented by Auguste, Magnaudet & Fabre (J. Fluid Mech., vol. 719, 2013, pp. 388–405).

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 contribution to the continuing success of the Journal, to which it is appropriate and fitting that we pay grateful tribute.

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 of the Navier–Stokes equations, having a boundary-layer character analogous to that of the classical steady front stagnation point flow. The velocity profiles are obtained by numerical integration of a nonlinear ordinary differential equation. For the front-flow situation, the solution is unique for the accelerating case, but bifurcates for modest deceleration, while for sufficient rapid deceleration there exists a one-parameter family of solutions. For the rear-flow situation, a unique solution exists (remarkably!) for sufficiently strong acceleration, and a one-parameter family again exists for sufficient strong deceleration. Analytic results, which are consistent with the numerical results, are obtained in the limits of strong acceleration or deceleration, and for the asymptotic behaviour far from the boundary.

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 models such tidal deformations in the simplest way. Using analytical techniques developed by Lebovitz & Zweibel (Astrophys. J., vol. 609, 2004, pp. 301–312) in the limit of small elliptic (tidal) deformations, we find the EMF associated with each resonant instability described by Mizerski & Bajer (J. Fluid Mech., vol. 632, 2009, pp. 401–430), and for arbitrary ellipticity the EMF associated with unstable horizontal modes. Mixed resonance between unstable hydrodynamic and magnetic modes and resonance between unstable and oscillatory horizontal modes both lead to a non-vanishing mean EMF which grows exponentially in time. The essential conclusion is that interactions between unstable eigenmodes with the same wave-vector can lead to a non-vanishing mean EMF, without any need for viscous or magnetic dissipation. This applies generally (and not only to the elliptic instabilities considered here).

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 circulations around the wings at this critical instant. Our primary aim is to elucidate the behaviour near the hinge point as time t passes through tc. First, Lighthill's inviscid potential flow theory is reconsidered. It is argued that provided the linear and angular accelerations of the wings are continuous, the velocity field varies continuously through the break, although the pressure field jumps instantaneously at t = tc. Then, effects of viscosity are considered. Near the hinge, the local Reynolds number is very small and local similarity solutions imply a logarithmic (integrable) singularity of the pressure jump across the hinge just before separation, in contrast to the ‘negligible pressure jump’ of inviscid theory invoked by Lighthill. We also present numerical simulations of the flow using a volume penalization technique to represent the motion of the wings. For Reynolds number equal to unity (based on wing chord), the results are in good agreement with the analytical solution. At a realistic Reynolds number of about 20, the flow near the hinge is influenced by leading-edge vortices, but local effects still persist. The lift coefficient is found to be much greater than that in the corresponding inviscid flow.

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 subject, and even more in respect of his inspirational influence on generations of research students and younger colleagues, and also more widely on the international stage, on which he was a revered, if sometimes controversial, personality.

We construct numerical examples of a ‘modon’ (counter-rotating vortices) in an Euler flow by exploiting the analogy between steady Euler flows and magnetostatic equilibria in a perfectly conducting fluid. A numerical modon solution can be found by determining its corresponding magnetostatic equilibrium, which we refer to as a ‘magnetic moclon’. Such an equilibrium is obtained numerically by a relaxation procedure that conserves the topology of the relaxing field. Our numerical results show how the shape of a magnetic modon depends on its ‘signature’ (magnetic flux profile), and that these magnetic modons are unexpectedly unstable to non-symmetric perturbations. Diffusion can change the topology of the field through a reconnection process and separate the two magnetic eddies. We further show that the analogous Euler flow (or modon) behaves similar to a perturbed Hill's vortex.

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 number of reversals, and a typical celt's observed behaviour is well captured by tuning the dissipation parameters.

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 found. These exact solutions are (like Hill's vortex) unstable, so that, under weak disturbance, a narrow spike of vorticity is shed from the neighbourhood of the rear stagnation point. Numerical simulation using a contour-dynamics formulation shows that, for general initial contour shape, a contour singularity appears at a finite time $t^*$, like that which appears on a disturbed vortex sheet. Techniques of regularization and sample-point redistribution are used so that the eddy contours can be tracked well beyond $t^*$. When the fluid is initially at rest, the magnetic eddy first contracts towards the axis of symmetry under the action of its Lorentz force distribution; then two spherical fronts form, which propagate in the two opposite directions along the axis of symmetry, in a manner captured well by the exact solution. The magnetic energy remains bounded away from zero despite the fact that there is no topological barrier to its further decrease. Magnetic eddy evolution and the possible existence of steady states under a uniform compressive strain field is also numerically investigated.

A large-Reynolds-number asymptotic theory is presented for the problem of a vortex tube of finite circulation [Gcy ] subjected to uniform non-axisymmetric irrotational strain, and aligned along an axis of positive rate of strain. It is shown that at leading order the vorticity field is determined by a solvability condition at first-order in ε = 1/R[Gcy ] where R[gcy ] = [gcy ]/ν. The first-order problem is solved completely, and contours of constant rate of energy dissipation are obtained and compared with the family of contour maps obtained in a previous numerical study of the problem. It is found that the region of large dissipation does not overlap the region of large enstrophy; in fact, the dissipation rate is maximal at a distance from the vortex axis at which the enstrophy has fallen to only 2.8% of its maximum value. The correlation between enstrophy and dissipation fields is found to be 0.19 + O(ε2). The solution reveals that the stretched vortex can survive for a long time even when two of the principal rates of strain are positive, provided R[gcy ] is large enough. The manner in which the theory may be extended to higher orders in ε is indicated. The results are discussed in relation to the high-vorticity regions (here described as ‘sinews’) observed in many direct numerical simulations of turbulence.