The asymptotic analysis of steady azimuthally invariant electromagnetically driven flows occurring in a shallow annular layer of electrolyte undertaken in Part 1 of this study (McCloughan & Suslov, J. Fluid Mech., vol. 980, 2024, A59) predicted the existence of a two-tori flow state that has not been detected previously. In Part 2 of the study we confirm its existence by numerical time integration of the governing equations. We observe a hysteresis, where the type of solution obtained for the same set of governing parameters depends on the choice of the initial conditions and the way the governing parameters change, which is fully consistent with the analytic results of Part 1. Subsequently, we perform a linear stability analysis of the newly obtained steady state and deduce that the experimentally observed anti-cyclonic free-surface vortices appear on its background as a result of a centrifugal (Rayleigh-type) instability of the interface separating two counter-rotating toroidal structures that form the newly found flow solution. The quantitative characteristics of such instability structures are determined. It is shown that such structures can only exist in sufficiently thin layers with the depth not exceeding a certain critical value.

When liquid metal batteries are charged or discharged, strong electrical currents are passing through the three liquid layers that we find in their interior. This may result in the metal pad roll instability that drives gravity waves on the interfaces between the layers. In this paper, we investigate theoretically metal pad roll instability in idealised cylindrical liquid metal batteries that were simulated previously by Weber et al. (Phys. Fluids, vol. 29, no. 5, 2017b, 054101) and Horstmann et al. (J. Fluid Mech., vol. 845, 2018, pp. 1–35). Near the instability threshold, we expect weakly destabilised gravity waves, and in this parameter regime, we can use perturbation methods to find explicit formulas for the growth rate of all possible waves. This perturbative approach also allows us to include dissipative effects, hence we can locate the instability threshold with good precision. We show that our theoretical growth rates are in quantitative agreement with previous and new direct numerical simulations. We explain how our theory can be used to estimate a lower bound on cell size beneath which metal pad roll instability is unlikely.

We study the influence of the shape of the plasma container on the dynamics of the reversed-field pinch (RFP). The geometries we consider are periodic cylinders with elliptical and circular-shaped cross-sections. Numerical simulations of fully nonlinear viscoresistive magnetohydrodynamics are carried out to illustrate how the plasma dynamics is affected by shaping. It is shown that independent of the plasma shape, the quantity $\beta$, comparing the hydrodynamic pressure to the magnetic pressure, decreases for increasing values of the Lundquist number, but the pressure gradient fluctuations remain roughly constant, when compared to the Lorentz force. Different elliptical shapes of the cross-section of the domain lead to the excitation of different toroidal (or axial) magnetic and dynamic modes. Furthermore, it is found that in a geometry with circular cross-section, a significant local poloidal angular momentum is observed, absent in the geometries with elliptical cross-section. Because the confinement is dominantly determined by plasma movement, and the dynamics of the velocity and magnetic field are modified by the modification of the geometry, shaping can thus affect the performance of RFP devices.

Oscillatory motions of charged particles inside a liquid medium have been explored under the influence of an electric field emulating field-induced particle-laden fluid flows. The properties of the surrounding fluid are found to play key roles in the kinetics of such a particle aggregation process. While the weakly conducting or insulating liquids promote high-frequency oscillations of charged particles followed by a quick assemblage, the viscosity and relative permittivity of the liquid play significant roles in modulating the time scale. In fact, the origin of such motions in a multi-particle system is very similar to a system with a single charged particle wherein the particle gathers charge from one of the electrodes before moving towards the other of opposite polarity. Interestingly, in the multi-particle system, an unprecedented charge reversal is observed wherein a charged particle reverses its direction of motion after colliding with another particle of opposite polarity. Experiments together with simulations further reveal that, while the equal-sized particles undergo an electric field driven ‘elastic’ collision and show synchronized motions with nearly similar speeds of approach and separation, the motions of unequal-sized particles are rather non-uniform after undergoing an ‘inelastic’ collision. Importantly, the simulations with two-particle systems uncover the presence of counter-rotating vortices surrounding the charged particles. The results reported not only usher the genesis of the chain-like assemblage in the multi-particle systems but also open up the possibility of the generation of on-demand power-law liquid properties through ‘chaining’ or ‘layering’ of the charged particles.

Under a steady DC electric field of sufficient strength, a weakly conducting dielectric sphere in a dielectric solvent with higher conductivity can undergo spontaneous spinning (Quincke rotation) through a pitchfork bifurcation. We design an object composed of a dielectric sphere and an elastic filament. By solving an elasto-electro-hydrodynamic (EEH) problem numerically, we uncover an EEH instability exhibiting diverse dynamic responses. Varying the bending stiffness of the filament, the composite object displays three behaviours: a stationary state, undulatory swimming and steady spinning, where the swimming results from a self-oscillatory instability through a Hopf bifurcation. By conducting a linear stability analysis incorporating an elastohydrodynamic model, we theoretically predict the growth rates and critical conditions, which agree well with the numerical counterparts. We also propose a reduced model system consisting of a minimal elastic structure which reproduces the EEH instability. The elasto-viscous response of the composite structure is able to transform the pitchfork bifurcation into a Hopf bifurcation, leading to self-oscillation. Our results imply a new way of harnessing elastic media to engineer self-oscillations, and more generally, to manipulate and diversify the bifurcations and the corresponding instabilities. These ideas will be useful in designing soft, environmentally adaptive machines.

We present a study of absolute and convective instabilities in electrohydrodynamic flow subjected to a Poiseuille flow (EHD-Poiseuille). The electric field is imposed on two infinite flat plates filled with a non-conducting dielectric fluid with unipolar ion injection. Mathematically, the dispersion relation of the linearised problem is studied based on the asymptotic response of an impulse disturbance imposed on the base EHD-Poiseuille flow. Transverse, longitudinal and oblique rolls are investigated to identify the saddle point satisfying the pinching condition in the corresponding complex wavenumber space. It is found that when the ratio of Coulomb force to viscous force increases, the transverse rolls can transit from convective instability to absolute instability. The ratio of hydrodynamic mobility to electric mobility, which exerts negligible effect on the linear stability criterion when the cross-flow is small, has significant influence on the convective–absolute instability transition, especially when the ratio is small. As we change the value of the mobility ratio, a saddle point shift phenomenon occurs in the case of transverse rolls. The unstable longitudinal rolls are convectively unstable as long as there is a cross-flow, a result which is deduced from a one-mode Galerkin approximation. Longitudinal rolls have a larger growth rate than transverse rolls except for a small cross-flow. Finally, regarding the oblique rolls, a numerical search for the saddle point simultaneously in the complex streamwise and transverse wavenumber spaces always yields an absolute transverse wavenumber of zero, implying that oblique rolls give way to transverse rolls when the flow is unstable.

This study seeks to elucidate the linear transient growth mechanisms in a uniform duct with square cross-section applicable to flows of electrically conducting fluids under the influence of an external magnetic field. A particular focus is given to the question of whether at high magnetic fields purely two-dimensional mechanisms exist, and whether these can be described by a computationally inexpensive quasi-two-dimensional model. Two Reynolds numbers of $5000$ and $15\,000$ and an extensive range of Hartmann numbers $0\leqslant \mathit{Ha}\leqslant 800$ were investigated. Three broad regimes are identified in which optimal mode topology and non-modal growth mechanisms are distinct. These regimes, corresponding to low, moderate and high magnetic field strengths, are found to be governed by the independent parameters; Hartmann number, Reynolds number based on the Hartmann layer thickness $R_{H}$ and Reynolds number built upon the Shercliff layer thickness $R_{S}$, respectively. Transition between regimes respectively occurs at $\mathit{Ha}\approx 2$ and no lower than $R_{H}\approx 33.\dot{3}$. Notably for the high Hartmann number regime, quasi-two-dimensional magnetohydrodynamic models are shown to be excellent predictors of not only transient growth magnitudes, but also the fundamental growth mechanisms of linear disturbances. This paves the way for a precise analysis of transition to quasi-two-dimensional turbulence at much higher Hartmann numbers than is currently achievable.

In this paper we examine the role of weak magnetic fields in breaking Kelvin’s circulation theorem and in vortex breakup in two-dimensional magnetohydrodynamics for the physically important case of a fluid with low magnetic Prandtl number (low $Pm$). We consider three canonical inviscid solutions for the purely hydrodynamical problem, namely a Gaussian vortex, a circular vortex patch and an elliptical vortex patch. We examine how magnetic fields lead to an initial loss of circulation $\unicode[STIX]{x1D6E4}$ and attempt to derive scaling laws for the loss of circulation as a function of field strength and diffusion as measured by two non-dimensional parameters. We show that for all cases the loss of circulation depends on the integrated effects of the Lorentz force, with the patch cases leading to significantly greater circulation loss. For the case of the elliptical vortex, the loss of circulation depends on the total area swept out by the rotating vortex, and so this leads to more efficient circulation loss than for a circular vortex.

The flow of an electrified liquid film down an inclined plane wall is investigated with the focus on coherent structures in the form of travelling waves on the film surface, in particular, single-hump solitary pulses and their interactions. The flow structures are analysed first using a long-wave model, which is valid in the presence of weak inertia, and second using the Stokes equations. For obtuse angles, gravity is destabilising and solitary pulses exist even in the absence of an electric field. For acute angles, spatially non-uniform solutions exist only beyond a critical value of the electric field strength; moreover, solitary-pulse solutions are present only at sufficiently high supercritical electric-field strengths. The electric field increases the amplitude of the pulses, can generate recirculation zones in the humps and alters the far-field decay of the pulse tails from exponential to algebraic with a significant impact on pulse interactions. A weak-interaction theory predicts an infinite sequence of bound-state solutions for non-electrified flow, and a finite set for electrified flow. The existence of single-hump pulse solutions and two-pulse bound states is confirmed for the Stokes equations via boundary-element computations. In addition, the electric field is shown to trigger a switch from absolute to convective instability, thereby regularising the dynamics, and this is confirmed by time-dependent simulations of the long-wave model.

We present hydrodynamic and magnetohydrodynamic (MHD) simulations of liquid sodium flows in the von Kármán sodium (VKS) set-up. The counter-rotating impellers made of soft iron that were used in the successful 2006 experiment are represented by means of a pseudo-penalty method. Hydrodynamic simulations are performed at high kinetic Reynolds numbers using a large eddy simulation technique. The results compare well with the experimental data: the flow is laminar and steady or slightly fluctuating at small angular frequencies; small scales fill the bulk and a Kolmogorov-like spectrum is obtained at large angular frequencies. Near the tips of the blades the flow is expelled and takes the form of intense helical vortices. The equatorial shear layer acquires a wavy shape due to three coherent co-rotating radial vortices as observed in hydrodynamic experiments. MHD computations are performed: at fixed kinetic Reynolds number, increasing the magnetic permeability of the impellers reduces the critical magnetic Reynolds number for dynamo action; at fixed magnetic permeability, increasing the kinetic Reynolds number also decreases the dynamo threshold. Our results support the conjecture that the critical magnetic Reynolds number tends to a constant as the kinetic Reynolds number tends to infinity. The resulting dynamo is a mostly axisymmetric axial dipole with an azimuthal component concentrated near the impellers as observed in the VKS experiment. A speculative mechanism for dynamo action in the VKS experiment is proposed.

Vortex-current filaments have been used to study phenomena such as coronal loops and solar flares as well as tokamaks, and recent experimental work has demonstrated dynamics akin to vortex-current filaments on a table-top plasma focus device. While MHD vortex dynamics and related applications to turbulence have attracted consideration in the literature due to a wide variety of applications, not much analytical progress has been made in this area, and the analysis of such vortex-current filament solutions under various geometries may motivate further experimental efforts. To this end, we consider the motion of open, isolated vortex-current filaments in the presence of magnetohydrodynamic (MHD) as well as the standard hydrodynamic effects. We begin with the vortex-current model of Yatsuyanagi, Hatori & Kato (J. Phys. Soc. Japan, vol. 65, 1996, pp. 745–759) giving the self-induced motion of a vortex-current filament. We give the ‘cutoff’ formulation of the Biot–Savart integrals used in this model, to avoid the singularity at the vortex core. We then study the motion of a variety of vortex-current filaments, including helical, planar and self-similar filament structures. In the case where MHD effects are weak relative to hydrodynamic effects, the filaments behave as expected from the pure hydrodynamic theory. However, when MHD effects are strong enough to dominate, then we observe structural changes to the filaments in all cases considered. The most common finding is reversal of vortex-current filament orientation for strong enough MHD effects. Kelvin waves along a vortex filament (as seen for helical and self-similar structures) will reverse their translational and rotational motion under strong MHD effects. Our findings support the view that vortex-current filaments can be studied in a manner similar to classical hydrodynamic vortex filaments, with the primary role of MHD effects being to change the filament motion, while preserving the overall geometric structure of such filaments.

We computationally investigate the Richtmyer–Meshkov instability of a density interface with a single-mode perturbation in a two-fluid, ion–electron plasma with no initial magnetic field. Self-generated magnetic fields arise subsequently. We study the case where the density jump across the initial interface is due to a thermal discontinuity, and select plasma parameters for which two-fluid plasma effects are expected to be significant in order to elucidate how they alter the instability. The instability is driven via a Riemann problem generated precursor electron shock that impacts the density interface ahead of the ion shock. The resultant charge separation and motion generates electromagnetic fields that cause the electron shock to degenerate and periodically accelerate the electron and ion interfaces, driving Rayleigh–Taylor instability. This generates small-scale structures and substantially increases interfacial growth over the hydrodynamic case.

Experimental observations of an azimuthal electrolyte flow driven by Lorentz force in a thin annular fluid layer placed on top of a magnet show that it develops a robust vortical system near the outer cylindrical wall. It appears to be a result of instabilities developing on a background of steady axisymmetric flow. Therefore, the goal of this paper is to establish a scene for a future comprehensive stability analysis of such a flow. We discuss popular depth-averaged and quasi-two-dimensional approximate solutions that take advantage of the thin-layer assumption first, and argue that they cannot lead to the observed flow patterns. Thus, three-dimensional toroidal flows are considered. Their similarities to various other well-studied rotating flow configurations are outlined, but no close match is found. Multiple axisymmetric solutions are detected numerically for the same governing parameters, indicating the possibility of subcritical bifurcations, namely type 1, consisting of a single torus, and type 2, developing a second counter-rotating toroidal flow near the outer cylinder. It is suggested that the transition between these two axisymmetric solutions is likely to be caused by the centrifugal instability, while the shear-type instability of the type 2 solution may be responsible for the observed vortex structures. However, a dedicated stability analysis which is currently underway and will be reported in a separate publication is required to confirm these hypotheses.

We investigate single-fluid magnetohydrodynamics (MHD) with anisotropic viscosity, often referred to as Braginskii MHD, with a particular eye to solar coronal applications. First, we examine the full Braginskii viscous tensor in the single-fluid limit. We pay particular attention to how the Braginskii tensor behaves as the magnetic field strength vanishes. The solar corona contains a magnetic field with a complex and evolving topology, so the viscosity must revert to its isotropic form when the field strength is zero, e.g. at null points. We highlight that the standard form in which the Braginskii tensor is written is not suitable for inclusion in simulations as singularities in the individual terms can develop. Instead, an altered form, where the parallel and perpendicular tensors are combined, provides the required asymptotic behaviour in the weak-field limit. We implement this combined form of the tensor into the Lare3D code, which is widely used for coronal simulations. Since our main focus is the viscous heating of the solar corona, we drop the drift terms of the Braginskii tensor. In a stressed null point simulation, we discover that small-scale structures, which develop very close to the null, lead to anisotropic viscous heating at the null itself (that is, heating due to the anisotropic terms in the viscosity tensor). The null point simulation we present has a much higher resolution than many other simulations containing null points, so this excess heating is a practical concern in coronal simulations. To remedy this unwanted heating at the null point, we develop a model for the viscosity tensor that captures the most important physics of viscosity in the corona: parallel viscosity for strong fields and isotropic viscosity at null points. We derive a continuum model of viscosity where momentum transport, described by this viscosity model, has the magnetic field as its preferred orientation. When the field strength is zero, there is no preferred direction for momentum transport and viscosity reverts to the standard isotropic form. The most general viscous stress tensor of a (single-fluid) plasma satisfying these conditions is found. It is shown that the Braginskii model, without the drift terms, is a specialization of the general model. Performing the stressed null point simulation with this simplified model of viscosity reveals very similar heating profiles to those of the full Braginskii model. The new model, however, does not produce anisotropic heating at the null point, as required. Since the vast majority of coronal simulations use only isotropic viscosity, we perform the stressed null point simulation with isotropic viscosity and compare the heating profiles to those of the anisotropic models. It is shown that the fully isotropic viscosity can overestimate the viscous heating by an order of magnitude.

We analyse numerically the linear stability of fully developed liquid metal flow in a square duct with insulating side walls and thin, electrically conducting horizontal walls. The wall conductance ratio $c$ is in the range of 0.01 to 1 and the duct is subject to a vertical magnetic field with Hartmann numbers up to $\mathit{Ha}=10^{4}$. In a sufficiently strong magnetic field, the flow consists of two jets at the side walls and a near-stagnant core with relative velocity ${\sim}(c\mathit{Ha})^{-1}$. We find that for $\mathit{Ha}\gtrsim 300,$ the effect of wall conductivity on the stability of the flow is mainly determined by the effective Hartmann wall conductance ratio $c\mathit{Ha}.$ For $c\ll 1$, the increase of the magnetic field or that of the wall conductivity has a destabilizing effect on the flow. Maximal destabilization of the flow occurs at $\mathit{Ha}\approx 30/c$. In a stronger magnetic field with $c\mathit{Ha}\gtrsim 30$, the destabilizing effect vanishes and the asymptotic results of Priede et al. (J. Fluid Mech., vol. 649, 2010, pp. 115–134) for ideal Hunt’s flow with perfectly conducting Hartmann walls are recovered.

A mathematical model of rhythmic motion of a charged droplet between two parallel electrodes is developed in this study. The work is motivated by recent experimental findings that report oscillatory behaviour of water in oil droplets under a direct current electric field. The model considers the presence of a charged droplet placed in a dielectric medium. The droplet is immediately attracted to the electrode with the opposite polarity. When approaching the electrode, the electric charge is electrochemically reversed within the droplet, which is then repelled to the other electrode. The entire process can periodically repeat. The model is able to track a deformable liquid–liquid interface, the dynamics of the wetting process at the electrodes and the dynamics of electrochemical charge transfer between the droplet and the electrodes. The dependences of the oscillation frequency, charge acquired by the droplet and charging time on several model parameters (surface charge density on electrodes, kinetic parameter of charging, droplet–electrode contact angle, droplet size, liquid permittivity) are examined. Qualitative agreement of the model predictions with available experimental data is obtained, e.g. the oscillation frequency increases with growing electric field strength or droplet size. Our model represents the first successful attempt to predict oscillatory motion of aqueous droplets by a pseudo-three-dimensional two-phase approach. Our model also strongly supports the theory that the oscillatory motion relies on the combination of electrochemical charge injection at the electrodes and electrostatic attraction/repulsion processes.

We consider the full three-dimensional dynamics of a thin falling liquid film on a flat plate inclined at some non-zero angle to the horizontal. In addition to gravitational effects, the flow is driven by an electric field which is normal to the substrate far from the flow. This extends the work of Tseluiko & Papageorgiou (J. Fluid Mech., vol. 556, 2006b, pp. 361–386) by including transverse dynamics. We study both the cases of overlying and hanging films, where the liquid lies above or below the substrate, respectively. Starting with the Navier–Stokes equations coupled with electrostatics, a fully nonlinear two-dimensional Benney equation for the interfacial dynamics is derived, valid for waves that are long compared to the film thickness. The weakly nonlinear evolution is governed by a Kuramoto–Sivashinsky equation with a non-local term due to the electric field effect. The electric field term is linearly destabilising and produces growth rates proportional to $|\unicode[STIX]{x1D743}|^{3}$, where $\unicode[STIX]{x1D743}$ is the wavenumber vector of the perturbations. It is found that transverse gravitational instabilities are always present for hanging films, and this leads to unboundedness of nonlinear solutions even in the absence of electric fields – this is due to the anisotropy of the nonlinearity. For overlying films and a restriction on the strength of the electric field, the equation is well-posed in the sense that it possesses bounded solutions. This two-dimensional equation is studied numerically for the case of periodic boundary conditions in order to assess the effects of inertia, electric field strength and the size of the periodic domain. Rich dynamical behaviours are observed and reported. For subcritical Reynolds number flows, a sufficiently strong electric field can promote non-trivial dynamics for some choices of domain size, leading to fully two-dimensional evolutions of the interface. We also observe two-dimensional spatiotemporal chaos on sufficiently large domains. For supercritical flows, such two-dimensional chaotic dynamics emerges in the absence of a field, and its presence enhances the amplitude of the fluctuations and broadens their spectrum.

The electrification of particles embedded in a turbulent flow may cause hazards such as spark discharges but is also exploited in several industrial applications. Nonetheless, due to its complexity and sensitivity to the initial conditions, the process of build-up of particle charge is currently not well understood. In order to gain a deeper understanding of this phenomenon, we performed fully resolved numerical simulations of particle charging. More specifically, our study concerned the charging process of particles dispersed in a turbulent channel flow at a friction Reynolds number of $Re_{\unicode[STIX]{x1D70F}}=180$. Emphasis was placed on the analysis of the interplay between the different physical mechanisms underlying particle electrification, such as fluid turbulence, particle dynamics and particle collisions. Further, we investigated the influence of some important physical parameters. According to our simulations the charge build-up depends strongly on the particle Stokes number, $Stk$. In particular, at small Stokes numbers, $Stk=0.2$, the turbopheretic drift inhibits particle charging. By contrast, at moderate Stokes numbers, $Stk=2$, and low particle number densities, the electric charge builds up but cannot escape the viscous sublayer due to limited particle migration. However, in the case of high particle number densities, the charge is transported away from the wall via inter-particle charge diffusion. A further increase to $Stk=20$ leads to strong charging and particle-bound charge transport towards the bulk of the channel.

We analyse numerically a pinch-type instability in a semi-infinite planar layer of inviscid conducting liquid bounded by solid walls and carrying a uniform electric current. Our model is as simple as possible but still captures the salient features of the instability which otherwise may be obscured by the technical details of more comprehensive numerical models and laboratory experiments. Firstly, we show the instability in liquid metals, which are relatively poor conductors, differs significantly from the astrophysically relevant Tayler instability. In liquid metals, the instability develops on the magnetic response time scale, which depends on the conductivity and is much longer than the Alfvén time scale, on which the Tayler instability develops in well conducting fluids. Secondly, we show that this instability is an edge effect caused by the curvature of the magnetic field, and its growth rate is determined by the linear current density and independent of the system size. Our results suggest that this instability may affect future liquid-metal batteries when their size reaches a few metres.

We investigated experimentally the flow driven by a Lorentz force induced by an axial magnetic field $\boldsymbol{B}$ and a radial electric current $I$ applied between two fixed concentric copper cylinders. The gap geometry corresponds to a rectangular section with an aspect ratio of $\unicode[STIX]{x1D702}=4$ and we probe the azimuthal and axial velocity profiles of the flow along the vertical axis by using ultrasonic Doppler velocimetry. We have performed several runs at moderate magnetic field strengths, corresponding to moderate Hartmann numbers $M\leqslant 300$. At these forcing parameters and because of the geometry of our experimental device, we show that the inertial terms are not negligible and an azimuthal velocity that depends on both $I$ and $B$ is induced. From measurements of the vertical velocity we focus on the characteristics of the secondary flow: the time-averaged velocity profiles are compatible with a secondary flow presenting two pairs of stable vortices, as pointed out by previous numerical studies. The flow exhibited a transition between two dynamical modes, a high- and a low-frequency one. The high-frequency mode, which emerges at low magnetic field forcing, corresponds to the propagation in the radial $r$-direction of tilted vortices. This mode is consistent with our previous experiments and with the instability described in Zhao et al. (Phys. Fluids, vol. 23 (8), 2011, 084103) taking place in an elongated duct geometry. The low-frequency mode, observed for high magnetic field forcing, consists of large excursions of the vortices. The dynamics of these modes matches the first axisymmetric instability described in Zhao & Zikanov (J. Fluid Mech., vol. 692, 2012, pp. 288–316) taking place in an square duct geometry. We demonstrated that this transition is controlled by the inertial magnetic thickness $H^{\prime }$ which is the characteristic length we introduce as a balance between the advection and the Lorentz force. The key point here is that when the inertial magnetic thickness $H^{\prime }$ is comparable to one geometric characteristic length ($H/2$ in the vertical or $\unicode[STIX]{x0394}r$ in the radial direction) the corresponding mode is favoured. Therefore, when $H^{\prime }/(H/2)\approx 1$ we observe the high-frequency mode taking place in an elongated duct geometry, and when $H^{\prime }/\unicode[STIX]{x0394}r\approx 1$ we observe the low-frequency mode taking place in square duct geometry and high magnetic field.