An experimental investigation was carried out to understand the kinematics and dynamics of flexible panels having pure pitching motion in a quiescent fluid. Simultaneous measurements of force and angular position of a panel were performed for various panel configurations. A high-speed camera imaging technique was used to find the instantaneous position of the trailing edge. The wake vortex flow was also quantified using the particle image velocimetry technique (PIV). We established a generic correlation between a time-averaged thrust coefficient $C_T$ and an effective flexural stiffness $\varPi$ of the panel. The results in terms of both time and phase-evolution of hydrodynamic forces and torque are exhibited through the complete kinematics and dynamics of the pitching panels, perhaps for the first time. The elastic deformation of the flexible panel at stroke reversals is found to affect the longitudinal force generation significantly when compared with the rigid panel. We have described many interesting features related to the elastic deformation of the panel. During stroke reversal, the motion of the pitching for the flexible panels reduces when compared to the pitching motion imposed by the forcing function. Reduced panel motion is found to aid thrust generation for a longer time in a cycle for which magnitude is indirectly governed by the flexural stiffness of the panel. The onset of instantaneous thrust and trailing-edge vortex generation is found to match precisely with the spatio-temporal location of the panel inflection. We noted that both the maximum phase lag $\phi$ and corresponding hydrodynamic torque $T_{hyd}$ are strongly correlated to the panel's effective flexural stiffness $\varPi$.

In this paper we investigate the statics and the dynamics of large-viscosity ligaments attached to a rod and drawn out of a pure-liquid bath. Following the similar work of films pulling out of a bath (Champougny et al., J. Fluid Mech., vol. 811, 2017, pp. 499–524), a one-dimensional model is applied to describe the ligaments drawn at constant velocities. We focus on the whole drawing dynamics of the ligament up to breakup, for which the breakup height is determined. The breakup height coincides with the maximum static meniscus height for very slow drawing, whose process can be described by quasi-static solutions. We present the numerical results of the static menisci and analytically unravel the mechanism in the low gravity case. Starting from a stable static meniscus, the breakup height of faster drawing depends separately on the rod radius and the drawing velocity, the latter dependency being fully determined by considering the agravic limit. Next, it is shown that the entire lifetime of the ligament drawing can be sequenced into a ductility stage, a capillarity stage and a pinch-off stage, the latter being shown to be almost instantaneous. The ductility and capillarity stages are decorrelated with the help of an approximate solution of the ductility stage, and the transition between the two stages corresponds to the time at which the capillarity-induced contraction velocity exceeds the ductility-induced one. The one-dimensional predictions of the breakup height and the entrained liquid volume attached to the rod quantitatively agree with experimental results of silicone oil ligaments, and the deviations are rationalized in comparison with a two-dimensional model.

Jets in crossflow are a recurring flow configuration in many engineering applications and have been the subject of many computational and experimental investigations. While these studies have identified the jet-to-crossflow velocity ratio $R$ as a critical parameter for the global stability and mixing behaviour which ultimately leads to breakdown into turbulence, a far less studied, but equally determining, factor is the presence of chemical reactions. Understanding and quantifying the nature of intrinsic instabilities in reacting and non-reacting flows and their dependence on the governing flow parameters are paramount to devising optimal designs or effective control strategies. In this study, we concentrate on the effect of reactions on the flow behaviour and extract optimal forcing and response functions for a range of driving frequencies, juxtaposing the reactive and non-reactive configurations. Simulations are performed using a compressible framework with a free-stream Mach number of $0.2$, and a constant jet-to-crossflow ratio of three. The flow configuration is kept identical for the reactive and non-reactive cases in order to isolate the effect of combustion on the resulting optimal forcing and response solutions. The frequency response is extracted using an adjoint-based optimization formalism. The presence of reactions markedly impacts the dominant frequencies of the flow. The inert case shows forcing and response modes, whose support aligns with the shear layer – as would be expected from a globally unstable jet. However, the analogous forcing functions for the reactive case concentrate around the flame, in accordance with combustion instabilities.

When surface tension is included in the classical formulation of a steadily travelling gravity wave (a Stokes wave), it is possible to obtain solutions that exhibit parasitic ripples: small capillary waves riding on the surface of steep gravity waves. However, it is not clear whether the singular small surface tension limit is well posed. That is, is it possible for an appropriate travelling gravity-capillary wave to be continuously deformed to the classic Stokes wave in the limit of vanishing surface tension? The work of Chen & Saffman (Stud. Appl. Maths, vol. 62, issue 1, 1980, pp. 1–21) had suggested smooth continuation was not possible, while the numerical study of Schwartz & Vanden-Broeck (J. Fluid Mech., vol. 95, issue 1, 1979, pp. 119–139) used an amplitude parameter that made it difficult to understand the structure of solutions for small values of the surface tension. In this paper we numerically explore the low surface tension limit of the steep gravity-capillary travelling-wave problem. Our results allow for a classification of the bifurcation structure that arises, and serve to unify a number of previous numerical studies. Crucially, we demonstrate that different choices of solution amplitude can lead to subtle restrictions on the continuation procedure. When wave energy is used as a continuation parameter, solution branches can be continuously deformed to the zero surface tension limit of a travelling Stokes wave.

Taylor–Couette (TC) flow, the flow between two independently rotating and co-axial cylinders, is commonly used as a canonical model for shear flows. Unlike plane Couette flow, pinned secondary flows can be found in TC flow. These are known as Taylor rolls and drastically affect the flow behaviour. We study the possibility of modifying these secondary structures using patterns of stress-free and no-slip boundary conditions on the inner cylinder. For this, we perform direct numerical simulations of narrow-gap TC flow with pure inner-cylinder rotation at four different shear Reynolds numbers up to $Re_s=3\times 10^4$. We find that one-dimensional azimuthal patterns do not have a significant effect on the flow topology, and that the resulting torque is a large fraction ($\sim$80 %–90 %) of torque in the fully no-slip case. One-dimensional axial patterns decrease the torque more, and for certain pattern frequency disrupt the rolls by interfering with the existing Reynolds stresses that generate secondary structures. For $Re\geq 10^4$, this disruption leads to a smaller torque than what would be expected from simple boundary layer effects and the resulting effective slip length and slip velocity. We find that two-dimensional checkerboard patterns have similar behaviour to azimuthal patterns and do not affect the flow or the torque substantially, but two-dimensional spiral inhomogeneities can move around the pinned secondary flows as they induce persistent axial velocities. We quantify the roll's movement for various angles and the widths of the spiral pattern, and find a non-monotonic behaviour as a function of pattern angle and pattern frequency.

Representing complex flows by evolving vortex structures is an important principle in many investigations of wall-bounded turbulence. The practice of this principle benefits from the bi-directional transformation between the velocity field and the corresponding vortex field. While the velocity-to-vortex transformation could be implemented by various vortex identification criteria, few efforts have been devoted to the inverse process. This work develops a linear reconstruction method, which allows an effective reconstruction for the velocity field of wall turbulence based on a given vortex field. The vortex field is defined as a vector field by combining the swirl strength and the real eigenvector of the velocity gradient tensor. The reconstructed velocity fields are calculated by convolution operations on the vortex fields, with the kernel functions derived by the field-based linear stochastic estimation. The method can effectively recover the turbulent motions in a large scale range, showing clear advantages over the Biot–Savart formula in the near-wall region. The method is also employed to investigate the inducing effects of vortices at different heights. The wall-bounding effect on the induced motions is observed from the contribution spectra of vortices. The higher-order moments of the reconstructed streamwise velocity component present larger deviations from the original data, which is discussed and explained reasonably. At last, the vortex fields filtered by prescribed thresholds are employed to reconstruct the velocity fields. It is found that the strongest vortex components occupying 5 % of the total volume can reasonably recover the main flow features including both the near-wall streaks and the large-scale motions.

Polarized vortical structures (i.e. with axial flow, thus coiled vortex lines) are generic to turbulent flows – hence the importance of their dynamics, interactions and cascade. Direct numerical simulations of two anti-parallel polarized vortex tubes are performed for vortex Reynolds numbers $Re$ ($\equiv \varGamma /\nu$) up to $9000$ and initial polarization strength $q$ (ratio of peak axial to azimuthal velocities) between $0$ and $4/3$. For both counter- and co-polarized cases, although the reconnection is delayed as $q$ increases – mainly due to weakened self-induction – it is more rapid and more complete for small $q$. Enstrophy growth and energy cascade are suppressed for weak polarization ($q < 1/2$) due to depleted nonlinearity, but are enhanced for strong polarization ($q > 1/2$) due to instability and/or transient growth. When counter-polarized, numerous structures with both positive and negative helicity densities (i.e. $\pm h$) are generated. For large $q$, strong axial flows opposite to the initial flows occur – causing polarization reversals. For the co-polarized cases, although $+h$ predominates, $-h$ structures also form and interact with positive ones – leading to helicity cascade to small scales. As $Re$ increases, small scales are more numerous: for counter-polarized cases, the threads undergo successive reconnections in a cascade – akin to the unpolarized case; for co-polarized cases, the newly formed vortex ring breaks up with numerous hairpin vortices wrapping around it. Increasing $q$ alters the energy spectrum in the inertial range with a scaling varying from $k^{-5/3}$ for the unpolarized case to $k^{-7/3}$ for the strongly polarized case, which seems to be associated with the enhanced vortex spiralling. In addition, for the strongly co-polarized cases, a $k^{-4/3}$ helicity spectrum develops. Furthermore, most of the energy and helicity in the inertial range with scale $L$ transfer to scales between $0.3L$ and $0.4L$. Therefore, polarization can significantly alter the dynamics of vortex reconnection as well as turbulence cascade.

We investigate deterministic and stochastic bifurcations in electroconvecitve flows of a dielectric liquid confined between two parallel plates subjected to a strong unipolar injection by direct numerical simulations. A long-standing discrepancy of linear instability criteria between the experiment and theory exists in this flow. We here test the hypothesis that the discrepancy may be related to the inhomogeneity in ion-exchange membranes used in experiments, contrasted by the homogeneous ion injection assumed in theoretical and numerical analyses. To study this effect, we consider stochastic boundary conditions around linear criticality and first bifurcations in this flow. For a complete presentation of flow bifurcations, deterministic bifurcation analysis (without stochasticity) is first performed to investigate primary bifurcations in this flow by progressively increasing the strength of electric field. Lyapunov spectrum and dimension are calculated and probed to characterise the chaotic motion therein. Our results confirm the high dimensionality of chaos in electroconvective flows and reveal for the first time that its chaos is extensive in a range of finite-sized systems. We then conduct stochastic bifurcation analyses by considering random perturbations in the boundary conditions of charge density and electric potential. Owing to the subcritical nature of electroconvective flows, the linear instability criteria under stochastic boundaries are closer to the experimental values than former theoretical and numerical results (assuming the homogeneous charge injection) for different levels of stochasticity, which confirms the hypothesis aforementioned. Furthermore, stochasticity can also enhance the efficiency of ionic transport.

Using computational modelling, we probe the hydrodynamics of a bio-inspired elastic propulsor with hybrid actuation that oscillates at resonance in a Newtonian fluid. The propulsor is actuated by a heaving motion at the base and by an internal bending moment distributed along the propulsor length. The simulations reveal that by tuning the phase difference between the external and internal actuation, the propulsor thrust and free-swimming velocity can be regulated in a wide range while maintaining high efficiency. Furthermore, the hybrid propulsor outperforms propulsors with either of the actuation methods. The enhanced performance is associated with the emerging bending pattern maintaining large tip displacement with reduced centre-of-mass displacement. The results are useful for developing highly efficient robotic swimmers utilizing smart materials as propulsors with simplified design and operation.

Cavitating flows include vapour structures with a wide range of different length scales, from micro-bubbles to large cavities. The correct estimation of small-scale cavities can be as important as that of large-scale structures, because cavitation inception as well as the resulting noise, erosion and strong vibrations occur at small time and length scales. In this study, a multi-scale cavitating flow around a sharp-edged bluff body is investigated. For numerical analysis, while popular homogeneous mixture models are practical options for large-scale applications, they are normally limited in the representation of small-scale cavities. Therefore, a hybrid cavitation model is developed by coupling a mixture model with a Lagrangian bubble model. The Lagrangian model is based on a four-way coupling approach, which includes new submodels, to consider various small-scale phenomena in cavitation dynamics. Additionally, the coupling of the mixture and the Lagrangian models is based on an improved algorithm that is compatible with the flow physics. The numerical analysis provides a detailed description of the multi-scale dynamics of cavities as well as the interactions between vapour structures of various scales and the continuous flow. The results, among others, show that small-scale cavities not only are important at the inception and collapse steps, but also influence the development of large-scale structures. Furthermore, a comparison of the results with those from experiment shows considerable improvements in both predicting the large cavities and capturing the small-scale structures using the hybrid model. More accurate results (compared with the traditional mixture model) can be achieved even with a lower mesh resolution.

The aim of this paper is to experimentally and analytically study the solutocapillary flow induced in a waterbody due to the presence of a solute source on its surface and the mixing induced by this flow of the solutes and gases dissolved at and near the surface into the waterbody. According to the analytic solution, the induced flow is analogous to a doublet flow in the sense that the flow is directed towards (or away from) the source within a conical region with its vertex at the source, and outside the conical region, the flow is away from (or towards) the source. Our particle image velocimetry and planar laser-induced fluorescence data show that the actual flow is far more complex than the analytic solution because of the influence of other factors which are not accounted for in the analytic model. There is an approximate agreement, but only for the intermediate distances from the source needle. In experiments, the flow changes direction into the waterbody when the solute-induced surface tension gradient driving the flow becomes comparable to the surface tension gradients on the surface due to factors such as contamination and temperature gradients. This causes the mixing of gases dissolved near the surface into the waterbody. Also, as the solute gradient at the surface gives rise to the force that drives the flow, when the solute diffusion coefficient is smaller the flow persists longer because the gradient is maintained for a longer distance.

Direct numerical simulations of free round jets at a Reynolds number ($Re_{D}$) of $5000$, based on jet diameter ($D$) and jet-exit bulk velocity ($U_{e}$), are performed to study jet turbulence characteristics at supercritical pressures. The jet consists of nitrogen ($\mathrm {N}_{2}$) that is injected into $\mathrm {N}_{2}$ at the same temperature. To understand turbulent mixing, a passive scalar is transported with the flow at unity Schmidt number. Two sets of inflow conditions that model jets issuing from either a smooth contraction nozzle (laminar inflow) or a long pipe nozzle (turbulent inflow) are considered. By changing one parameter at a time, the simulations examine the jet-flow sensitivity to the thermodynamic condition (characterized in terms of the compressibility factor ($Z$) and the normalized isothermal compressibility), inflow condition and ambient pressure ($p_{\infty }$) spanning perfect- to real-gas conditions. The inflow affects flow statistics in the near field (containing the potential core closure and the transition region) as well as further downstream (containing fully developed flow with self-similar statistics) at both atmospheric and supercritical $p_{\infty }$. The sensitivity to inflow is larger in the transition region, where the laminar-inflow jets exhibit dominant coherent structures that produce higher mean strain rates and higher turbulent kinetic energy than in turbulent-inflow jets. Decreasing $Z$ at a fixed supercritical $p_{\infty }$ enhances pressure and density fluctuations (non-dimensionalized by local mean pressure and density, respectively), but the effect on velocity fluctuations depends also on the local flow dynamics. When $Z$ is reduced, large mean strain rates in the transition region of laminar-inflow jets significantly enhance velocity fluctuations (non-dimensionalized by local mean velocity) and scalar mixing, whereas the effects are minimal in jets from turbulent inflow.

Liquid foam exhibits surprisingly high viscosity, higher than each of its phases. This dissipation enhancement has been rationalized by invoking either a geometrical confinement of the shear in the liquid phase, or the influence of the interface viscosity. However, a precise localization of the dissipation, and its mechanism, at the bubble scale is still lacking. With this aim, we simultaneously monitored the evolution of the local flow velocity, film thickness and surface tension of a five-film assembly, induced by different controlled deformations. These measurements allow us to build local constitutive relations for this foam elementary building block. We first show that, for our millimetric foam films, the main part of the film has a purely elastic, reversible behaviour, thus ruling out the interface viscosity in explaining the observed dissipation. We then highlight a generic frustration at the menisci, controlling the interface transfer between neighbour films and resulting in the localization of a bulk shear flow close to the menisci. A model accounting for surfactant transport in these small sheared regions is developed. It is in good agreement with the experiment, and demonstrates that most of the dissipation is localized in these domains. The length of these sheared regions, determined by the physico-chemical properties of the solution, sets a transition between a large bubble regime, in which the films are mainly stretched and compressed, and a small bubble regime, in which they are sheared. Finally, we discuss the parameter range where a model of foam viscosity could be built on the basis of these local results.

Dip coating is a common technique used to cover a solid surface with a thin liquid film, the thickness of which was successfully predicted by the theory developed in the 1940s by Landau & Levich (Acta Physicochem. URSS, vol. 17, 1942, pp. 141–153) and Derjaguin (Acta Physicochem. URSS, vol. 20, 1943, pp. 349–352). In this work, we present an extension of their theory to the case where the dipping bath contains two immiscible liquids, one lighter than the other, resulting in the entrainment of two thin films on the substrate. We report how the thicknesses of the coated films depend on the capillary number, on the ratios of the properties of the two liquids and on the relative thickness of the upper fluid layer in the bath. We also show that the liquid/liquid and liquid/gas interfaces evolve independently from each other as if only one liquid were coated, except for a very small region where their separation falls quickly to its asymptotic value and the shear stresses at the two interfaces peak. Interestingly, we find that the final coated thicknesses are determined by the values of these maximum shear stresses.

Fish schools are ubiquitous in marine life. Although flow interactions are thought to be beneficial for schooling, their exact effects on the speed, energetics and stability of the group remain elusive. Recent numerical simulations and experimental models suggest that flow interactions stabilize in-tandem formations of flapping foils. Here, we employ a minimal vortex sheet model that captures salient features of the flow interactions among flapping swimmers, and we study the free swimming of a pair of in-line swimmers driven with identical heaving or pitching motions. We find that, independent of the flapping mode, heaving or pitching, the follower passively stabilizes at discrete locations in the wake of the leader, consistent with the heaving foil experiments, but pitching swimmers exhibit tighter and more cohesive formations. Further, in comparison to swimming alone, pitching motions increase the energetic efficiency of the group while heaving motions result in a slight increase in the swimming speed. A deeper analysis of the wake of a single swimmer sheds light on the hydrodynamic mechanisms underlying pairwise formations. These results recapitulate that flow interactions provide a passive mechanism that promotes school cohesion, and afford novel insights into the role of the flapping mode in controlling the emergent properties of the school.

The streamwise vortex instability of boundary layers caused by wall roughness in the form of surface undulations is investigated. The instability is characterised by a roughness parameter $\varGamma$ depending on the geometry and fluid properties. At $O(1)$ values of $\varGamma$ disturbances develop on the same length scale as the basic boundary layer flow. The instability is driven by a boundary condition relating the disturbance wall shears in the streamwise and normal directions. The undulations have a wavelength comparable with the boundary layer depth and the amplitude is asymptotically small compared with the depth. If the roughness parameter is large then, apart from a narrow window of vortex wavenumbers, the instability responds in a quasi-parallel manner. Falkner–Skan boundary layers are considered in detail and the dependence on the angle of the wedge associated with the flows investigated. A particular susceptibility to roughness instabilities of flows past $90^{\circ }$ wedges is uncovered. The limits of small and large wavenumbers are considered and universal results given for the critical roughness height $h$ and wavelength $b$ needed for instability.

Oscillatory boundary layer (OBL) flows over a smooth surface are studied using laser Doppler velocimetry in a large experimental oscillatory flow tunnel. The experiments cover a range of Reynolds numbers in the transitional regime ($ {\textit {Re}}_{\delta }=254 - 1315$). Motivated by inconsistencies in the literature, the focus is to shed light regarding the phase shift ${\rm \Delta} \phi$ between the bed shear stress and the free stream velocity maxima. Details of the mean flow structure and turbulence characteristics in transitional OBL flows indicate the emergence of a logarithmic profile, which for $ {\textit {Re}}_{\delta }=763$ appears at the middle of the deceleration and as the $ {\textit {Re}}_{\delta }$ increases, it appears for a longer part of the period and for a larger region of the boundary layer. Turbulence statistics profiles approach those of equilibrium, unidirectional boundary layer flows with similar $ {\textit {Re}}_{\theta }$, defined using the local free stream velocity and momentum thickness $\theta$. Analysis of the ensemble-average bed shear stress variation reveals that for $ {\textit {Re}}_{\delta } < 552$ a single peak, associated with the laminar regime, occurs during the acceleration phase. For $ {\textit {Re}}_{\delta }=552$ a second peak, associated with the transition to turbulence, appears towards the middle of the deceleration phase. This turbulence peak becomes larger than the ‘laminar’ one for $ {\textit {Re}}_{\delta } \sim 763$ and lags with respect to the free stream velocity maximum. For $ {\textit {Re}}_{\delta }>1036$ the laminar peak disappears under the effect of the turbulence peak. The presence of the phase lag is discussed using data from this study and the literature, and a revised ${\rm \Delta} \phi$ diagram is introduced for the whole range of flows, from laminar to fully turbulent.

The motion of thin curved falling particles is ubiquitous in both nature and industry but is not yet widely examined. Here, we describe an experimental study on the dynamics of thin cylindrical shells resembling broken bottle fragments settling through quiescent fluid and homogeneous anisotropic turbulence. The particles have Archimedes numbers based on the mean descent velocity $0.75 \times 10^{4} \lesssim Ar \lesssim 2.75 \times 10^{4}$. Turbulence reaching a Reynolds number of $Re_\lambda \approx 100$ is generated in a water tank using random jet arrays mounted in a coplanar configuration. After the flow becomes statistically stationary, a particle is released and its three-dimensional motion is recorded using two orthogonally positioned high-speed cameras. We propose a simple pendulum model that accurately captures the velocity fluctuations of the particles in still fluid and find that differences in the falling style might be explained by a closer alignment between the particle's pitch angle and its velocity vector. By comparing the trajectories under background turbulence with the quiescent fluid cases, we measure a decrease in the mean descent velocity in turbulence for the conditions tested. We also study the secondary motion of the particles and identify descent events that are unique to turbulence such as ‘long gliding’ and ‘rapid rotation’ events. Lastly, we show an increase in the radial dispersion of the particles under background turbulence and correlate the time scale of descent events with the local settling velocity.

A small drop of a heavier fluid may float on the surface of a lighter fluid supported by surface tension forces. In equilibrium, the drop assumes a radially symmetric shape with a circular triple-phase contact line. We show that such a floating liquid drop with a sufficiently small volume has two distinct equilibrium shapes at terrestrial gravity: one with a larger and one with a smaller radius of the triple-phase contact line. Static stability analysis reveals that both shapes could be stable if the drop volume is below a certain critical value. Experiments conducted with $\mathrm {\mu }\textrm {L}$-sized water drops floating on commercial oil support the existence of multiple contact line radii for a drop with fixed volume. Next, we experimentally study the floatability of a less viscous water drop on the surface of a more viscous and less dense oil, subjected to a low-frequency (Hz-order) vertical vibration. We find that in a certain range of amplitudes, vibration helps heavy liquid drops to stay afloat. The physical mechanism of the increased floatability is explained by the horizontal elongation of the drop driven by subharmonic Faraday waves. The average length of the triple-phase contact line increases as the drop elongates that leads to a larger average lifting force produced by the surface tension.

The evolution of a buoyancy disturbance in a stratified incompressible fluid permeated by a uniform vertical magnetic field is investigated. Two regimes are considered in the absence of background rotation – that of strong stratification, where the internal gravity wave frequency $\omega _A$ is much higher in magnitude than the magnetic (Alfvén) wave frequency $\omega _M$, and that of strong magnetic field, where $\omega _M$ is dominant. For small but finite magnetic diffusion, perturbations that initially lie in the strong-field regime are shown to cross over to the regime of strong stratification, so that small-scale motions may exist as damped internal gravity waves at large times. The induced magnetic field propagates as damped Alfvén waves for a much longer time than the velocity before undergoing the above transition. With strong rotation, the unstably stratified system that satisfies the inequality $|\omega _C| > |\omega _M| \gg |\omega _A| \gg |\omega _\eta |$, where $\omega _C$ is the inertial wave frequency and $\omega _\eta$ is the diffusion frequency, is of relevance to convection-driven dynamos. Here, a parameter space with $|\omega _M/\omega _C| \sim 0.1$ is found wherein the flow intensity of the slow magnetic-Archimedean-Coriolis (MAC) waves is of the same order of magnitude as that of the fast MAC waves. Slow wave motions at horizontal length scales much smaller than the width of the fluid layer can therefore generate substantial helicity in rapidly rotating dynamos. The excitation of slow MAC waves at scales of $\sim$10 km in the Earth's core may play a crucial role in the generation of the axial dipole field.