Transport and instability in driven two-dimensional magnetohydrodynamic flows

This paper concerns the generation of large scale flows in forced two-dimensional systems. A Kolmogorov flow with a sinusoidal profile in one direction (driven by a body force) is known to become unstable to a large scale flow in the perpendicular direction at a critical Reynolds number. This can occur in the presence of a beta-effect and has important implications for flows observed in geophysical and astrophysical systems. It has recently been termed ‘zonostrophic instability’ and studied in a variety of settings, both numerically and analytically. The goal of the present paper is to determine the effect of magnetic field on such instabilities using the quasi-linear approximation, in which the full fluid system is decoupled into a mean flow and waves of one scale. The waves are driven externally by a given, random body force and move on a fast time scale, while their stress on the mean flow causes this to evolve on a slow time scale. Spatial scale separation between waves and mean flow is also assumed, to allow analytical progress. The paper first discusses purely hydrodynamic transport of vorticity including zonostrophic instability, the effect of uniform background shear, and calculation of equilibrium profiles in which the effective viscosity varies spatially, through the mean flow. After brief consideration of passive scalar transport or equivalently kinematic magnetic field evolution, the paper then proceeds to study the full MHD system and to determine effective diffusivities and other transport coefficients using a mixture of analytical and numerical methods. This leads to results on the effect of magnetic field, background shear and beta-effect on zonostrophic instability and magnetically driven instabilitiesWe are grateful to the EPSRC for funding SD via a DTG research studentship

Eroding dipoles and vorticity growth for Euler flows in R3: The hairpin geometry as a model for finite-time blowup

This is the author accepted manuscript. The final version is available from IOP Publishing via the DOI in this record.A theory of an eroding ``hairpin'' dipole vortex structure in three dimensions is developed, extending our previous study of an axisymmetric eroding dipole without swirl. The hairpin is proposed as a model of maximal ``self-stretching'' of vorticity. We derive a system of partial differential equations of ``generalized'' form, involving contour averaging of a locally two-dimensional Euler flow. The hairpin is proposed as a structure favouring a rapid stretching of vortex lines, and it is conjectured that an initial condition based upon the hairpin could lead to a a blowup of vorticity in finite time in R<sup>3</sup>. We do not attempt here to solve the system exactly, but point out that non-existence 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 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

Eroding dipoles and vorticity growth for Euler flows in R 3 : Axisymmetric flow without swirl

A review of analyses based upon anti-parallel vortex structures suggests that structurally stable dipoles with eroding circulation may offer a path to the study of vorticity growth in solutions of Euler’s equations in R3 . We examine here the possible formation of such a structure in axisymmetric flow without swirl, leading to maximal growth of vorticity as t 4/3 . Our study suggests that the optimizing flow giving the t 4/3 growth mimics an exact solution of Euler’s equations representing an eroding toroidal vortex dipole which locally conserves kinetic energy. The dipole cross-section is a perturbation of the classical Sadovskii dipole having piecewise constant vorticity, which breaks the symmetry of closed streamlines. The structure of this perturbed Sadovskii dipole is analyzed asymptotically at large times, and its predicted properties are verified numerically. We also show numerically that if mirror symmetry of the dipole is not imposed but axial symmetry maintained, an instability leads to breakup into smaller vortical structures

A new class of magnetically actuated pumps and valves for microfluidic applications

This is the final version of the article. Available from Springer Nature via the DOI in this record.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.This project has received funding from the European Union’s Horizon 2020 research and innovation programme under grant agreement No. 665440. We also acknowledge support via the EPSRC Centre for Doctoral Training in Metamaterials (Grant No. EP/L015331/1)

A Geometric Look at Momentum Flux and Stress in Fluid Mechanics

This is the final version. Available on open access from Springer via the DOI in this recordData Access: No data were created or analysed in this study.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-α 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.Engineering and Physical Sciences Research Council (EPSRC)Leverhulme Trus

Area waves on a slender vortex revisited

This is the author accepted manuscript. The final version is available from IOP Publishing via the DOI in this recordThis 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 & Ashurst (1989), J. Fluid Mech. 200, 283–307, averages the momentum equation. Another averaging method, due to Leonard (1994), Phys. Fluids 6, 765– 777, 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.Leverhulme TrustEngineering and Physical Sciences Research Council (EPSRC

Magnetically controlled ferromagnetic swimmers

This is the final version of the article. Available from Springer Nature via the DOI in this record.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.This project has received funding from the European Union’s Horizon 2020 research and innovation programme under grant agreement No. 665440. We also acknowledge support via the EPSRC Centre for Doctoral Training in Metamaterials (Grant No. EP/L015331/1)

An analytical study of the MHD clamshell instability on a sphere

This is the final version. Available on open access from Cambridge University Press via the DOI in this recordData access statement: No data were created or analysed in this study.This paper studies the instability of two-dimensional magnetohydrodynamic (MHD) 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 2001, Sol. Phys. vol. 199, 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.Engineering and Physical Sciences Research Council (EPSRC

Network Analysis of Host-Virus Communities in Bats and Rodents Reveals Determinants of Cross-Species Transmission

Bats are natural reservoirs of several important emerging viruses. Cross-species transmission appears to be quite common among bats, which may contribute to their unique reservoir potential. Therefore, understanding the importance of bats as reservoirs requires examining them in a community context rather than concentrating on individual species. Here, we use a network approach to identify ecological and biological correlates of cross-species virus transmission in bats and rodents, another important host group. We show that given our current knowledge the bat viral sharing network is more connected than the rodent network, suggesting viruses may pass more easily between bat species. We identify host traits associated with important reservoir species: gregarious bats are more likely to share more viruses and bats which migrate regionally are important for spreading viruses through the network. We identify multiple communities of viral sharing within bats and rodents and highlight potential species traits that can help guide studies of novel pathogen emergence

Atlas-based ventricular shape analysis for understanding congenital heart disease

Congenital heart disease is associated with abnormal ventricular shape that can affect wall mechanics and may be predictive of long-term adverse outcomes. Atlas-based parametric shape analysis was used to analyze ventricular geometries of eight adolescent or adult single-ventricle CHD patients with tricuspid atresia and Fontans. These patients were compared with an “atlas” of non-congenital asymptomatic volunteers, resulting in a set of Z-scores which quantify deviations from the control population distribution on a patient-by-patient basis. We examined the potential of these scores to: (1) quantify abnormalities of ventricular geometry in single ventricle physiologies relative to the normal population; (2) comprehensively quantify wall motion in CHD patients; and (3) identify possible relationships between ventricular shape and wall motion that may reflect underlying functional defects or remodeling in CHD patients. CHD ventricular geometries at end-diastole and end-systole were individually compared with statistical shape properties of an asymptomatic population from the Cardiac Atlas Project. Shape analysis-derived model properties, and myocardial wall motions between end-diastole and end-systole, were compared with physician observations of clinical functional parameters. Relationships between altered shape and altered function were evaluated via correlations between atlas-based shape and wall motion scores. Atlas-based shape analysis identified a diverse set of specific quantifiable abnormalities in ventricular geometry or myocardial wall motion in all subjects. Moreover, this initial cohort displayed significant relationships between specific shape abnormalities such as increased ventricular sphericity and functional defects in myocardial deformation, such as decreased long-axis wall motion. These findings suggest that atlas-based ventricular shape analysis may be a useful new tool in the management of patients with CHD who are at risk of impaired ventricular wall mechanics and chamber remodeling