The motion of a slender vortex in ideal incompressible fluid is a classic problem in hydrodynamics. Formulae for the velocity of a vortex ring go back to work of Kelvin, Helmholtz, Hicks and Dyson in the nineteenth century, while more recently a number of models have been created for simulating the motion of slender tubes of vorticity of general shape. For a vortex tube or filament localised near to a curve C(t), the relevant small parameter to measure slenderness is the tube radius divided by the radius of curvature of C. The present paper revisits this range of classic problems by introducing a coordinate system closely linked to the geometry of vortex surfaces in a slender vortex. The motion of fluid elements in this coordinate system has an action–angle form, and with this the vorticity equation simplifies radically. At the same time, however, most aspects of the shape or evolution of a vortex are thrown into the description of the coordinates, and in particular the corresponding metric and volume form. As the coordinate system is non-orthogonal and time-dependent, tools of differential geometry are most easily used to describe the structure of both vorticity and coordinate system, and a general mathematical framework is set out. This resulting system of equations is taken as far as possible with only the assumption of vortex slenderness in place, but allowing arbitrary motions and distortions of the vortex core and of the curve C(t). The modelling is applied to calculate the motion of a slender vortex ring with arbitrary axial flow, solving directly for the shape of perturbed vorticity surfaces and giving results in agreement with earlier studies. The general framework set up in this paper is suitable for the development of simplified equations for vortex motion and interaction in future studies.

Lagrangian averaging theories, most notably the generalized Lagrangian mean (GLM) theory of Andrews and McIntyre, have been primarily developed in Euclidean space and Cartesian coordinates. We reinterpret these theories using a geometric, coordinate-free formulation. This gives central roles to the flow map, its decomposition into mean and perturbation maps, and the momentum 1-form dual to the velocity vector. In this interpretation, the Lagrangian mean of any tensorial quantity is obtained by averaging its pull-back to the mean configuration. Crucially, the mean velocity is not a Lagrangian mean in this sense. It can be defined in a variety of ways, leading to alternative Lagrangian mean formulations that include GLM and Soward and Roberts's volume-preserving version. These formulations share key features that the geometric approach uncovers. We derive governing equations both for the mean flow and for wave activities constraining the dynamics of the perturbations. The presentation focuses on the Boussinesq model for inviscid rotating stratified flows and reviews the necessary tools of differential geometry.

In our earlier paper [Childress, S. and Gilbert, A.D., Eroding dipoles and vorticity growth for Euler flows in $ \scriptstyle {\mathbb {R}}<^>3 $ R3: the hairpin geometry as a model for finite-time blowup. Fluid Dyn. Res. 2018, 50, 011418.], we derived equations of motion for a vortex dipole in the shape of a hairpin, and subject to erosion of vorticity, that is the shedding of vorticity at the rear stagnation point of the dipole and consequent loss of circulation. We applied these to the question of Euler blow-up of vorticity in $ \scriptstyle {\mathbb {R}}<^>3 $ R3. In this paper, we shall calculate the axial flow within the vortex tubes of the hairpin and evaluate the resulting vorticity structure of their cores. The model is unusual in that it is not evolved from simple initial conditions. Rather, the hairpin structure is constructed at a time prior to possible blowup. It consists of a "nose", where blow-up would occur, from which there extend two symmetric, quasi-two-dimensional "tails" of infinite length and infinitely large spatial scale. The quasi-self-similarity of the structure determines blow-up at the point of joining of the tails. During this growth, the dipole maintains a quasi-two-dimensional geometry. The analysis is believed to be the first study of blow-up incorporating both the deformation of the cores of the constituent vortex tubes and the axial flow within the tubes. The analysis raises problems which we will not be able to resolve fully here. Our results suggest that axial flow, coupled with erosion, may provide a mechanism preventing blow-up in finite time. The essential difficulty is that axial flow changes the local dipole structure and hence the dipole propagation speed. This expels the possibility of complete self-similarity. Possible ways to deal with this obstacle are discussed.

The shallow-water equations are widely used to model interactions between horizontal shear flows and (rotating) gravity waves in thin planetary atmospheres. Their extension to allow for interactions with magnetic fields – the equations of shallow-water magnetohydrodynamics (SWMHD) – is often used to model waves and instabilities in thin stratified layers in stellar and planetary atmospheres, in the perfectly conducting limit. Here we consider how magnetic diffusion should be added to the equations of SWMHD. This is crucial for an accurate balance between advection and diffusion in the induction equation, and hence for modelling instabilities and turbulence. For the straightforward choice of Laplacian diffusion, we explain how fundamental mathematical and physical inconsistencies arise in the equations of SWMHD, and show that unphysical dynamo action can result. We then derive a physically consistent magnetic diffusion term by performing an asymptotic analysis of the three-dimensional equations of magnetohydrodynamics in the thin-layer limit, giving the resulting diffusion term explicitly in both planar and spherical coordinates. We show how this magnetic diffusion term, which allows for a horizontally varying diffusivity, is consistent with the standard shallow-water solenoidal constraint, and leads to negative semidefinite Ohmic dissipation. We also establish a basic type of antidynamo theorem.

Zonal flows are mean flows in the east-west direction, which are ubiquitous on planets, and can be formed through 'zonostrophic instability': within turbulence or random waves, a weak large-scale zonal flow can grow exponentially to become prominent. In this paper, we study the statistical behaviour of the zonostrophic instability and the effect of magnetic fields. We use a stochastic white noise forcing to drive random waves, and study the growth of a mean flow in this random system. The dispersion relation for the growth rate of the expectation of the mean flow is derived, and properties of the instability are discussed. In the limits of weak and strong magnetic diffusivity, the dispersion relation reduces to manageable expressions, which provide clear insights into the effect of the magnetic field and scaling laws for the threshold of instability. The magnetic field mainly plays a stabilising role and thus impedes the formation of the zonal flow, but under certain conditions it can also have destabilising effects. Numerical simulation of the stochastic flow is performed to confirm the theory. Results indicate that the magnetic field can significantly increase the randomness of the zonal flow. It is found that the zonal flow of an individual realisation may behave very differently from the expectation. For weak magnetic diffusivity and moderate magnetic field strengths, this leads to considerable variation of the outcome, that is whether zonostrophic instability takes place or not in individual realisations.

We develop a geometric formulation of fluid dynamics, valid on arbitrary Riemannian manifolds, that regards the momentum-flux and stress tensors as 1-form-valued 2-forms, and their divergence as a covariant exterior derivative. We review the necessary tools of differential geometry and obtain the corresponding coordinate-free form of the equations of motion for a variety of inviscid fluid models—compressible and incompressible Euler equations, Lagrangian-averaged Euler- $$\alpha $$ equations, magnetohydrodynamics and shallow-water models—using a variational derivation which automatically yields a symmetric momentum flux. We also consider dissipative effects and discuss the geometric form of the Navier–Stokes equations for viscous fluids and of the Oldroyd-B model for visco-elastic fluids.

This paper concerns the stability of Kolmogorov flow u = (0, sin x) in the infinite (x,y)-plane. A mean magnetic field of strength B0 is introduced and the MHD linear stability problem studied for modes with wave-number k in the y-direction, and Bloch wavenumber l in the x-direction. The parameters governing the problem are Reynolds number 1/nu, magnetic Prandtl number P, and dimensionless magnetic field strength B0. The mean magnetic field can be taken to have an arbitrary direction in the (x,y)-plane and a mean x-directed flow U0 can be incorporated. First the paper considers Kolmogorov flow with y-directed mean magnetic field, referred to as vertical. Taking l=0, the suppression of the pure hydrodynamic instability is observed with increasing field strength B0. A branch of strong-field instabilities occurs for magnetic Prandtl number P less than unity, as found by A.E. Fraser, I.G. Cresser and P. Garaud (J. Fluid Mech. 949, A43, 2022). Analytical results using eigenvalue perturbation theory in the limit k->0 support the numerics for both weak- and strong-field instabilities, and originate in the coupling of large-scale modes with x-wavenumber n=0, to smaller-scale modes. The paper considers the case of horizontal or x-directed mean magnetic field. The unperturbed state consists of steady, wavey magnetic field lines. As the magnetic field is increased, the purely hydrodynamic instability is suppressed again, but for stronger fields a new branch of instabilities appears. Allowing a non-zero Bloch wavenumber l allows further instability, and in some circumstances when the system is hydrodynamically stable, arbitrarily weak magnetic fields can give growing modes. Numerical results are presented together with eigenvalue perturbation theory in the limits k,l->0. The theory gives analytical approximations for growth rates and thresholds in good agreement with those computed.

In this paper, the instability of shallow-water shear flow with a sheared parallel magnetic field is studied. Waves propagating in such magnetic shear flows encounter critical levels where the phase velocity relative to the basic flow,$c-U(y)$, matches the Alfvén wave velocities$\pm B(y)/\sqrt {\mu \rho }$, based on the local magnetic field$B(y)$, the magnetic permeability$\mu$, and the mass density of the fluid$\rho$. It is shown that when the two critical levels are close to each other, the critical layer can generate an instability. The instability problem is solved, combining asymptotic solutions at large wavenumbers and numerical solutions, and the mechanism of instability explained using the conservation of momentum. For the shallow-water magnetohydrodynamic system, the paper gives the general form of the local differential equation governing such coalescing critical layers for any generic field and flow profiles, and determines precisely how the magnetic field modifies the purely hydrodynamic stability criterion based on the potential vorticity gradient in the critical layer. The curvature of the magnetic field profile, or equivalently the electric current gradient$J' = - B''/\mu$in the critical layer, is found to play a complementary role in the instability.

This paper studies the instability of two-dimensional magnetohydrodynamic systems on a sphere using analytical methods. The underlying flow consists of a zonal differential rotation and a toroidal magnetic field is present. Semicircle rules that prescribe the possible domain of the wave velocity in the complex plane for general flow and field profiles are derived. The paper then sets out an analytical study of the ‘clamshell instability’, which features field lines on the two hemispheres tilting in opposite directions (Cally, Sol. Phys. , vol. 199, 2001, pp. 231–249). An asymptotic solution for the instability problem is derived for the limit of weak shear of the zonal flow, via the method of matched asymptotic expansions. It is shown that when the zonal flow is solid body rotation, there exists a neutral mode that tilts the magnetic field lines, referred to as the ‘tilting mode’. A weak shear of the zonal flow excites the critical layer of the tilting mode, which reverses the tilting direction to form the clamshell pattern and induces the instability. The asymptotic solution provides insights into properties of the instability for a range of flow and field profiles. A remarkable feature is that the magnetic field affects the instability only through its local behaviour in the critical layer.

Biological cilia generate fluid movement within viscosity-dominated environments using beating motions that break time-reversal symmetry. This creates a metachronal wave, which enhances flow efficiency. Artificially mimicking this behaviour could improve microfluidic point-of-care devices, since viscosity-dominated fluid dynamics impede fluid flow and mixing of reagents, limiting potential for multiplexing diagnostic tests. However, current biomimicry schemes require either variation in the hydrodynamic response across a cilia array or a complex magnetic anisotropy configuration to synchronise the actuation sequence with the driving field. Here, we show that simple modifications to the structural design introduce phase differences between individual actuators, leading to the spontaneous formation of metachronal waves. This generates flow speeds of up to 16 μm/s as far as 675 μm above the actuator plane. By introducing metachronal waves through lithographic structuring, large scale manufacture becomes feasible. Additionally, by demonstrating that metachronal waves emerge from non-uniformity in internal structural mechanics, we offer fresh insight into the mechanics of cilia coordination.

This paper considers magnetohydrodynamics (MHD) and some of its applications from the perspective of differential geometry, considering the dynamics of an ideal fluid flow and magnetic field on a general three-dimensional manifold, equipped with a metric and an induced volume form. The benefit of this level of abstraction is that it clarifies basic aspects of fluid dynamics such as how certain quantities are transported, how they transform under the action of mappings (e.g. the flow map between Lagrangian labels and Eulerian positions), how conservation laws arise, and the origin of certain approximations that preserve the mathematical structure of classical mechanics. First, the governing equations for ideal MHD are derived in a general setting by means of an action principle and making use of Lie derivatives. The way in which these equations transform under a pull back by the map taking the position of a fluid parcel to a background location is detailed. This is then used to parameterise Alfven waves using concepts of pseudomomentum and pseudofield, in parallel with the development of Generalised Lagrangian Mean theory in hydrodynamics. Finally non-ideal MHD is considered with a sketch of the development of the Braginsky alpha,omega-dynamo in a general setting. Expressions for the alpha-tensor are obtained, including a novel geometric formulation in terms of connection coefficients, and related to formulae found elsewhere in the literature.

This paper considers the classic problem of the dynamics of axisymmetric waves on a rectilinear vortex, in the absence of viscosity. The waves alter the axial pressure distribution and thus generate axial flows which depend on the radial distribution of vorticity. To simplify this problem, models have been introduced which average over the cross-section and eliminate the radial dependence. One approach, pioneered by Lundgren and Ashurst (1989 J. Fluid Mech. 200 283–307), averages the momentum equation. Another averaging method, due to Leonard (1994 Phys. Fluids 6 765–77), focuses on the vorticity equation. The present paper takes a fresh look at the derivation of these two distinct models, which we refer to as the momentum wave model and vorticity wave model respectively, using the tools of differential geometry to develop a hybrid Eulerian–Lagrangian approach. We compare these models with area waves in the asymptotic limit of a slender vortex, with radial structure retained. Numerical calculations are presented to show the differences between waves in the full slender vortex system and those in the momentum and vorticity wave models. We also discuss modification of the vorticity wave model to allow an external irrotational flow, and simulations are presented where a vortex is subjected to uniform axial stretching. Our approach can also be developed to model more complicated configurations, such as occur during vortex collisions.

This paper develops a simple model of the inertial range of turbulent flow, based on a cascade of vortical filaments. The filaments are taken to be helical, one turn of the helix playing the role of a turbulent eddy. A binary branching structure is proposed, involving the splitting of filaments at each step into pairs of daughter filaments with differing properties, in effect two distinct simultaneous cascades. Neither of the cascades of this bimodal structure, in isolation, has the Richardson exponent of 1/3. If cascades are assumed to be initiated continuously and throughout space we obtain a model of the inertial range of stationary turbulence. We impose the constraint associated with Kolmogorov's four-fifths law and then adjust the splitting to achieve good agreement with the observed structure exponents zeta(p). The presence of two elements to the cascade is responsible for the non-linear dependence of zeta(p) upon p. The proposed binary branching cascade provides a model for the initial-value problem of the Navier-Stokes equations in the limit of vanishing viscosity. To simulate this limit we let the cascade continue indefinitely, energy removal occurring in the limit. We are thus able to compute the decay of energy in the model.

A theory of an eroding 'hairpin' vortex dipole structure in three-dimensions is developed, extending our previous study of an axisymmetric eroding dipole without swirl. The axisymmetric toroidal dipole was found to lead to maximal growth of vorticity, as t(4/3). The hairpin is here similarly proposed as a model to produce large 'self-stretching' of vorticity, with the possibility of finite-time blow-up. We derive a system of partial differential equations of 'generalized' form, involving contour averaging of a locally two-dimensional Euler flow. We do not attempt here to solve the system exactly, but point out that nonexistence of physically acceptable solutions would most probably be a result of the axial flow. Because of the axial flow the vorticity distribution within the dipole eddies is no longer of the simple Sadovskii type (vorticity constant over a cross-section) obtained in the axisymmetric problem. Thus the solution of the system depends upon the existence of a larger class of propagating two-dimensional dipoles. The hairpin model is obtained by formal asymptotic analysis. As in the axisymmetric problem a local transformation to 'shrinking' coordinates is introduced, but now in a self-similar form appropriate to the study of a possible finite-time singularity. We discuss some properties of the model, including a study of the helicity and a first step in iterating toward a solution from the Sadovskii structure. We also present examples of two-dimensional propagating dipoles not previously studied, which have a vorticity profile consistent with our model. Although no rigorous results can be given, and analysis of the system is only partial, the formal calculations are consistent with the possibility of a finite time blowup of vorticity at a point of vanishing circulation of the dipole eddies, but depending upon the existence of the necessary two-dimensional propagating dipole. Our results also suggest that conservation of kinetic energy as realized in the eroding hairpin excludes a finite time blowup for the corresponding Navier-Stokes model.

Many fluctuation-driven phenomena in fluids can be analysed effectively using the generalised Lagrangian-mean (GLM) theory of Andrews & McIntyre ( J. Fluid Mech. , vol. 89, 1978, pp. 609–646) This finite-amplitude theory relies on particle-following averaging to incorporate the constraints imposed by the material conservation of certain quantities in inviscid regimes. Its original formulation, in terms of Cartesian coordinates, relies implicitly on an assumed Euclidean structure; as a result, it does not have a geometrically intrinsic, coordinate-free interpretation on curved manifolds, and suffers from undesirable features. Motivated by this, we develop a geometric generalisation of GLM that we formulate intrinsically using coordinate-free notation. One benefit is that the theory applies to arbitrary Riemannian manifolds; another is that it establishes a clear distinction between results that stem directly from geometric consistency and those that depend on particular choices. Starting from a decomposition of an ensemble of flow maps into mean and perturbation, we define the Lagrangian-mean momentum as the average of the pull-back of the momentum one-form by the perturbation flow maps. We show that it obeys a simple equation which guarantees the conservation of Kelvin’s circulation, irrespective of the specific definition of the mean flow map. The Lagrangian-mean momentum is the integrand in Kelvin’s circulation and distinct from the mean velocity (the time derivative of the mean flow map) which advects the contour of integration. A pseudomomentum consistent with that in GLM can then be defined by subtracting the Lagrangian-mean momentum from the one-form obtained from the mean velocity using the manifold’s metric. The definition of the mean flow map is based on choices made for reasons of convenience or aesthetics. We discuss four possible definitions: a direct extension of standard GLM, a definition based on optimal transportation, a definition based on a geodesic distance in the group of volume-preserving diffeomorphisms, and the ‘glm’ definition proposed by Soward & Roberts ( J. Fluid Mech. , vol. 661, 2010, pp. 45–72). Assuming small-amplitude perturbations, we carry out order-by-order calculations to obtain explicit expressions for the mean velocity and Lagrangian-mean momentum at leading order. We also show how the wave-action conservation of GLM extends to the geometric setting. To make the paper self-contained, we introduce in some detail the tools of differential geometry and main ideas of geometric fluid dynamics on which we rely. These include variational formulations which we use for alternative derivations of some key results. We mostly focus on the Euler equations for incompressible inviscid fluids but sketch out extensions to the rotating–stratified Boussinesq, compressible Euler, and magnetohydrodynamic equations. We illustrate our results with an application to the interaction of inertia-gravity waves with balanced mean flows in rotating–stratified fluids.

We propose a new class of magnetically actuated pumps and valves that could be incorporated into microfluidic chips with no further external connections. The idea is to repurpose ferromagnetic low Reynolds number swimmers as devices capable of generating fluid flow, by restricting the swimmers' translational degrees of freedom. We experimentally investigate the flow structure generated by a pinned swimmer in different scenarios, such as unrestricted flow around it as well as flow generated in straight, cross-shaped, Y-shaped and circular channels. This demonstrates the feasibility of incorporating the device into a channel and its capability of acting as a pump, valve and flow splitter. Different regimes could be selected by tuning the frequency and amplitude of the external magnetic field driving the swimmer, or by changing the channel orientation with respect to the field. This versatility endows the device with varied functionality which, together with the robust remote control and reproducibility, makes it a promising candidate for several applications.

Microscopic swimming devices hold promise for radically new applications in lab-on-a-chip and microfluidic technology, including diagnostics and drug delivery. In this paper, we realize a macroscopic single particle ferromagnetic swimmer experimentally and investigate its swimming properties. The flagella-based swimmer is comprised of a hard ferromagnetic head attached to a flexible tail. We investigate the dynamic performance of the swimmer on the air-liquid interface as a function of the external magnetic field parameters (frequency and amplitude of an applied magnetic field). We show that the speed of the swimmer can be controlled by manipulating the strength and frequency of the external magnetic field (<3.5 mT) and that the propagation direction has a dependence on parameters of the external magnetic field. The experimental results are compared to a theoretical model based on three beads, one of which having a fixed magnetic moment and the other two non-magnetic, connected via elastic filaments. The model shows sufficient complexity to satisfy the “non-reciprocity” condition and gives good agreement with experiment. Via a simple conversion, we also demonstrate a fluid pump and investigate the induced flow. This investigation paves the way to the fabrication of such swimmers and fluid pump systems on a micro-scale, promising a variety of microfluidic applications.Microscopic swimming devices hold promise for radically new applications in lab-on-a-chip and microfluidic technology, including diagnostics and drug delivery. In this paper, we realize a macroscopic single particle ferromagnetic swimmer experimentally and investigate its swimming properties. The flagella-based swimmer is comprised of a hard ferromagnetic head attached to a flexible tail. We investigate the dynamic performance of the swimmer on the air-liquid interface as a function of the external magnetic field parameters (frequency and amplitude of an applied magnetic field). We show that the speed of the swimmer can be controlled by manipulating the strength and frequency of the external magnetic field (<3.5 mT) and that the propagation direction has a dependence on parameters of the external magnetic field. The experimental results are compared to a theoretical model based on three beads, one of which having a fixed magnetic moment and the other two non-magnetic, connected via elastic filaments. The ...

Microscopic swimming devices hold promise for radically new applications in lab-on-a-chip and microfluidic technology, diagnostics and drug delivery etc. In this paper, we demonstrate the experimental verification of a new class of autonomous ferromagnetic swimming devices, actuated and controlled solely by an oscillating magnetic field. These devices are based on a pair of interacting ferromagnetic particles of different size and different anisotropic properties joined by an elastic link and actuated by an external time-dependent magnetic field. The net motion is generated through a combination of dipolar interparticle gradient forces, time-dependent torque and hydrodynamic coupling. We investigate the dynamic performance of a prototype (3.6 mm) of the ferromagnetic swimmer in fluids of different viscosity as a function of the external field parameters (frequency and amplitude) and demonstrate stable propulsion over a wide range of Reynolds numbers. We show that the direction of swimming has a dependence on both the frequency and amplitude of the applied external magnetic field, resulting in robust control over the speed and direction of propulsion. This paves the way to fabricating microscale devices for a variety of technological applications requiring reliable actuation and high degree of control.

Fluid flow governed by the Navier–Stokes equation is considered in a domain bounded by two cones with the same axis. In the first, ‘non-parallel’ case the two cones have the same apex and different angles θ = α and β in spherical polar coordinates (r, θ, φ). In the second, ‘parallel’ case the two cones have the same opening angle α, parallel walls separated by a gap h and apices separated by a distance h/ sinα. Flows are driven by a source Q at the origin, the apex of the lower cone in the parallel case. The Stokes solution for the non-parallel case is discussed and the angles (α, β) identified where it breaks down, analogously to flow in a wedge geometry. For the case of convergent flow, Q < 0, solutions governed by the Navier–Stokes equation are discussed for both parallel and non-parallel geometries. At large distances the flow is in a viscous regime and takes a Poiseuille profile. As the origin is approached the inertial terms become important and a plug flow emerges, with constant radial velocity in the core, sandwiched between thin boundary layers. By systematic approximation, PDEs are derived that describe the transition from viscous to high Reynolds number flow, and solutions describing the plug flow and boundary layers are obtained using matched asymptotic expansions.