Nicolas Plihon

N Taberlet

J Ferrand

M Hibat

B Isnard

Acoustic emission from successive impacts on elastic membranes: the physics of the screaming balloon

Keywords: 46, 40, -f -Vibrations and mechanical waves 43, 20, +g -General linear acoustics 46, 25, -y -Static elasticity

Acoustic emission from successive impacts on elastic membranes: The physics of the screaming balloon

Introduction. -The physics of musical instruments is a rich and fascinating subject involving complex and highly coupled physical phenomena [START_REF] Fletcher | The Physics of Musical Instruments 2nd Edition[END_REF]. Linear theories only constitute a first-order model [START_REF] Fletcher | [END_REF] as a complete understanding of most instruments require to include non-linear effects (e.g. the interplay between stick-slip and the natural modes of vibration of a bowed string [START_REF] Cremer | The physics of the violin[END_REF][START_REF] Woodhouse | [END_REF], the exact design of the bell of a trumpet [5], or the equal temperament of a piano [6], the trapping of acoustic waves in musical saws due to the bending of the metal strip [7,8]). Bidimensional geometries such as drums exhibit sophisticated normal modes whose frequencies are not commensurable (as in Chladni figures [9]), producing a sound sometimes regarded as dissonant. However, by locally adjusting the thickness (or the mass per unit surface) of a membrane, on may turn tabla drums are turned into harmonic instruments [START_REF] Raman | Proc. Indian Acad. Sci.A[END_REF]. In the singing wine glasses [START_REF] Rossing | [END_REF][12][13] as well as in the tibetan singing bowl [14], sound emission emerges from the coupling between the mechanical resonances of tri-dimensional structures and fluid dynamics. In all cases the frequency of the acoustic emission is determined by the resonant eigenfrequencies of strings, tubes, plates, or tridimensional structures -the sound level being frequently amplified through a sound board, a sound box or a bell. Sound emission could also originate from repetitive pulses on structures. Natural examples include stridulations [15] where sound is created by insects rubbing their plectrum (or scraper) on the ridged pars stridens, as heard for cicadas or crickets [START_REF] Haskell | Insect Sounds 1st Edition[END_REF]. A similar process is used for some percussion instruments, such as the guiro or the washboard where sound is generated by rubbing a stick (or nails) on a ridged wooden or metallic surface [START_REF] White | Physics and Music: The Science of Musical Sound Dover Books on Physics[END_REF]: periodic pulses thus excite the structure and the periodicity is solely set by the velocity of the stick (or of the hand). From the musical point of view, the sound perception is related to the pitch. While for resonant percussions the pitch is given by the resonance features, it strongly depends upon the temporal properties of the impulses in the case of repetitive pulses [START_REF] Schouten | [END_REF]. In the case of periodic pulses at high audible frequencies -typically above one hundred Hertz -the pitch was shown to be unambiguously determined by the periodicity of the pulses [19,20]. In this article we investigate the sound produced by a hex nut (or a ridged-edge coin) spinning inside an inflated rubber balloon (see Fig. 1 and Supplementary Video).

On the high-speed video (parts 2 and 4), one can see the edges of the nut protruding out of the balloon and setting the membrane into motion. As it is common for drum instruments, the entire membrane vibrates and the deformation is not limited to the vicinity of the nut. However, we demonstrate here that the frequency of the sound emission follows the rate of successive impacts of the spinning object on the membrane, contrary to the classical musical drums and similarly to the washboard or the guiro precussions. This article is organized as follows. The frequency spectrum of the sound is investigated and the physical origin of the dominant frequency, of its overtones and sub-harmonics is elucidated. The influence of the tension within the membrane on the sound intensity is examined and an energetic approach of the acoustic emission is proposed.

Spectral properties of the sound emitted by the screaming balloon. -In this paper, transparent latex balloons(Transparent Size 9 latex balloons, Bishara) were used as they allow one to visualize and track the motion of the nut or coin. These balloons conveniently remain spherical over a large range of volumes. In order to avoid condensation of water inside, the balloons were inflated with dry compressed air. The microphone used for the investigation of the sound spectral characteristics (Microphone of an Apple iPhone 5c) has a bandwidth corresponding to the audible range (100 Hz -20 kHz) and acoustic signals are aquired at 24 kHz. In order to track the motion of the nut/coin, the balloons were filmed at 300 FPS (Ximea xiQ model MQ013MG-ON) and high-speed videos were recorded up to 10,000 FPS (Phantom -v2511).

Fundamental frequency of the sound. Fig. 2a) shows a typical spectrum of the sound emitted by a hex nut (M5, steel). A clear dominant frequency is visible (f 0 1320 Hz in this case). Higher commensurable harmonics, with decaying amplitude, are also clearly visible and are due to the anharmonicity of the signal. Indeed, the nature of the impacts is expected to create a non-sinusoidal waveform and their Fourier transform contains multiple frequencies.

The resonances observed in Fig. 2 are somewhat broad due to the strong dissipation within the rubber balloona typical quality factor of 20 was measured and presented in the Supplementary Fig. 2. In order to study the dependence of f 0 on the relevant parameters, a nut/coin is manually given a high initial spin. The balloon is then firmly held from the top and the bottom as the nut/coin rolls and gradually loses its kinetic energy while emitting an acoustic wave (see supplementary video, parts 1 and 5). As a side note, the sound emitted while the balloon is held down does not significantly differ from that of a free balloon. No assumptions are made regarding the trajectory of the nut/coin (elliptical, helical, horizontal) and its linear velocity, v, is computed from particle tracking (at 300 FPS). Measurements were performed for a variety of hex nuts (M2 and M8) made of steel (density 8) or nylon (density 1.24) as well as using several ridged-edge coins, whose characteristics are given in the Supplementary Table 1. For hex nuts, d is the length of one face (i.e. the distance between two edges), while for coins, d is the distance between two consecutive ridges. The dominant frequency was measured from the power spectrum with an accuracy given by the width of the distribution, and the uncertainties on the velocity v due to the particle tracking algorithm were estimated to remain below 5%.

Fig. 3 displays the dominant frequency, f 0 , as a function of the velocity-to-step ratio, v/d and shows that the two quantities are identical within experimental uncertainties. This holds for all nut/coin size, material, mass as well as for any value of the volume (and hence overpressure inside) of the balloon, and for frequencies ranging from 100 Hz to almost 10 kHz. Unlike drum instruments whose pitch depends on the tension in the membrane, the pitch is clearly only determined by the linear velocity v of the nut/coin along its trajectory, and by the step d of the ridged exciter, regardless of any other quantity: This demonstrates that the nut/coin undergoes a rollingwithout-slipping motion on the membrane (caused by the high friction coefficient of rubber [21]), as also clearly shown in the time lapse sequence of Fig. 1c). Indeed, between two consecutive impacts, the nut travels a distance d and the resulting frequency of impacts is given by v/d. These results rule out the relevance of other phenomena as a potential source of the emitted sound, such as any acoustic or mechanical resonance within the balloon (which can still exist, see Supplementary Fig. 1, although it does not set the frequency of the sound) or any hypothetical stickslip motion (as in bowed strings).

f 0 = v d (1)
Similarly to the washboard or the guiro percussions, the pitch depends linearly on the velocity v of the impacting object (the stick for the percussions and the nut for the screaming balloon) -there is however an extra degree of freedom for the screaming balloon. While, for washboards and guiros, the pitch of the sound is solely given by the velocity of the stick impacting the ridged structure, of step d, which radiates the sound, it is set here by the velocity and the geometrical properties of the impacting solid (i.e. the step d of the nut or the coin), independently of the balloon which radiates the sound.

Sub-harmonics.

A close examination of the frequency spectrum reveals that it also contains lower harmonics (see Fig. 2). The frequency of the highest amplitude lower harmonic, f s is equal to one sixth of the fundamental frequency f 0 (f s 220 Hz in this case). The ratio f 0 /f s = 6 was confirmed using a M2 hex nut for various values of the fundamental frequency, obtained for various velocities and balloon sizes (see n = 6, orange points in Fig. 4a). This indicates again that the frequency content of the signal depends solely of the step, d, and the velocity, v, and is independent of the other specification of the nut (mass or diameter) and balloon (Young's modulus, thickness, diameter, pressure). The ratio f 0 /f s = 6 for a hexagonal nut indicates that the sub-harmonics are caused by small asymmetries of the nut. Indeed, if one side of the nut is heavier or lighter, or if an edge is sharper of smoother than others, then the amplitude of the corresponding impact would differ, superimposing a 6-times-smaller frequency to the signal. Thus, the low frequency content of the spectrum does not result from non-linear sub-harmonics generation.

The same procedure was repeated for 3D-printed polygonal nuts, from squares (n=4) to decagons (n=10) (of typical size 10 mm). The results are shown in Fig. 4a) where the dashed lines correspond to f 0 = nf s . The same data is plotted in Fig. 4b) as the ratio f s /f 0 versus the number of sides of the polygonal nut n. Within experimental uncertainties, the sub-harmonic is clearly given by f 0 /n. It is also worth mentioning that sub-harmonics are also visible in the Fourier spectrum of the sound produced by coins (see Fig. 7) but due to the larger number of ridges, f s is considerably lower than f 0 and for clarity, the data is not shown in Fig. 4. Sound intensity. -We have shown that the fundamental frequency (along with the overtones and undertones) is given by the rate of collisions and does not directly depend on the mass and material of the nut, or on the exact properties of the balloon. However, these physical quantities are expected to have a clear effect on the sound intensity (i.e. level or power). More specifically, the tension T within the elastic membrane might have a strong effect, as it is the case in all string and drum instruments [START_REF] Fletcher | The Physics of Musical Instruments 2nd Edition[END_REF] The sound intensity was measured using a commercial sound level meter (TES-1350A, TES Corp.) placed 1 m away (horizontally) from a balloon in which a hex spins around the equator. The sound level was recorded for several balloon sizes (up to 44 cm in diameter, corresponding to a volume of 45 L) for a M5 steel hex nut producing an acoustic wave at f 0 = 750 ± 50 Hz, leading to an uncertainty of ±1 dB. The results are shown in Fig. 5c). The sound level is indeed greater (and reaches 65 dB) for large volumes but the data shows a counter-intuitive plateau for small volumes. For an extended range of radii between 6 cm and 14 cm (see dashed lines), the acoustic intensity remains constant (around 55 dB) although the volume of the balloons increases by a factor 10 (from 0.9 L to 11.5 L).

In order to elucidate the origin of this plateau, the overpressure, ∆P , in the balloon was measured using a U-tube manometer (with an accuracy of 100 Pa) as its volume was varied. Fig. 5a) shows ∆P as a function of the balloon radius and three classical regimes [22][23][24] can be observed: an initial sharp increase (radius ≥ 6 cm), followed by a slow drop (between 6 cm and 14 cm), and finally a slow increase for larger volumes (radius ≥ 14 cm). The intermediate regime is reminiscent of Young-Laplace equations for bubbles and droplets governed by surface tension [START_REF] Triiton | Physical Fluid Dynamics 1st Edition[END_REF]. For an elastic material however, there exist two limitations: at high deformations the elastomer molecules are over-stretched and the material stiffens [START_REF] Treloar | The Physics of Rubber Elasticity 1st Edition[END_REF], leading to an increase in ∆P , whereas at small deformations the material recoils to its original length with vanishing tension.

For a sphere of radius R, the mechanical equilibrium imposes ∆P = 2T /R, where T is the tension within the membrane. Note that this equation holds for any spherical membrane and that no assumptions are made regarding T . In particular, T is not a priori expected to be a constant (as it would be for a gas/liquid interface) but in general, it should depend on the radius of the sphere, since the elastic material is stretched as the volume of the balloon varies. The membrane tension is shown in Fig. 5b) which reveals that T remains constant for 6 cm ≤ R ≤ 14 cm, which precisely corresponds to the range of constant sound level. Therefore, the plateau in the sound level can be attributed to an unexpected regime in which the tension within the membrane remains constant while the balloon is stretched by a factor 3.

Conversion of kinetic to acoustic energy. -Evidently, the membrane serves as an amplifier for the sound but the acoustic energy primarily originates from the kinetic energy of the object spinning in the balloon. The kinetic energy K (including the rotational energy) of a ridged-edge solid of mass m, rolling without slipping at a velocity v, and emitting a sound at a frequency f 0 reads

K = 3/4mv 2 = 3/4md 2 f 2 0 .
Every time a ridge impacts on the membrane (at a rate f 0 ), a fraction, α, of this kinetic energy is converted into acoustic energy. The acoustic power (or sound level), must therefore follow:

P ac = α × 3/4 md 2 f 3 0 (2)
For the study of the frequency dependence, coins are preferably used over hex nut as they allow one to explore a wider range of frequencies and emit a sound far above the natural resonant modes of the balloon (see Supplementary Fig. 1). Indeed, for frequencies corresponding to the first resonant modes of the balloon, the sound intensity can typically increase by a factor 2. Note however that due to the rapid frequency sweep (due to the velocity decrease) and oscillations (due to non-perfectly horizontal coin/nut trajectory) along with the quality factor of the membrane (Q = 20, see Supplementary material), the eigenmodes of the balloon are hardly excited. This justifies the assumption that the conversion factor (or efficiency) α is taken to be a constant. Again, in the low frequency range (0-500 Hz), this assumption might not hold as the eigenmodes might resonate, but the experimental data shown in Fig. 6 and 7 clearly show that for higher frequencies (¿500 Hz) the fraction of kinetic energy converted to acoustics remains constant. Fig. 6 shows the acoustic power as a function of the frequency on a log-log scale for a variety of coins. The sound level in dB (recorded 1m away from a balloon of radius 10 cm) was converted into acoustic power (in Watts) assuming a spherical waveform (right yaxis). For clarity, the experimental uncertainties are not shown but are of the order of 1 dB. The first conclusion that can be drawn from Fig. 6 a is that the predicted 3 rdpower-law for the frequency dependence is valid for all coins, and that, for a given frequency, the sound intensity is higher for larger d. Interestingly, all data sets collapse on a unique master curve when plotted as a function of md 2 f 3 0 in Fig. 6 b) that validates [START_REF] Fletcher | [END_REF]. This allows one to measure the energy ratio α = (5.6 ± 0.5).10 -7 . This very small value indicates that only a slight fraction of the available energy is converted into an acoustic emission at every impact.

It would be however more relevant to compare the acoustic power to the loss of energy per impact (rather than to the absolute value of the energy). Fig. 7 shows the spectrogram obtained with a coin, initially emitting a sound at f 0 (t = 0) = 15 kHz (as a side note the overtones and sub-harmonics are clearly visible). As its initial energy is slowly converted into acoustic energy as well as dissipated in the membrane, its velocity decreases along with the emitted frequency. Assuming that a fraction β of the kinetic energy is dissipated at every impact, one can find the time-dependence of the fundamental frequency (see Supplementary, part IV):

f 0 (t) = f 0 (t = 0) 1 + βtf 0 (t = 0)/2 (3)
The red dashed line shows a fit (β = 14.2 .10 -6 ) of the data using this prediction, and the very good agreement justifies the assumption that the relative energy loss per impact, β, is constant. This small value explains the long duration of a typical run: it takes τ = 2/(βf 0 (t = 0)) 9.25 s for the frequency to be halved (see supplementary video part 5). The ratio α/β 4% appears to be the relevant parameter to characterize the efficiency of the energy conversion: only 4% of the energy actually available is used to emit sound while the rest is dissipated (in all likelihood in the membrane). The high damping in the membrane is consistent with the low quality factor of the free oscillations of the balloons, Q 20, (see Supplementary Fig. 2), which indicates that most of the energy dissipates within a few vibrations. However, this seemingly low efficiency is in reality rather high for a drum instrument or an electro-acoustic loudspeaker [START_REF] Mccarthy | Sound systems: design and optimization: modern techniques and tools for sound system design and alignment[END_REF].

Let us finally mention that the energy conversion of hex nuts (whose edges are more prominent than coins) is considerably higher, leading to a louder sound (typically +30 dB at any given frequency), associated with a faster decay rate (no more than a few seconds vs. a few minutes for coins, see supplementary video).

Conclusion. -We have shown that the fundamental frequency of the screaming balloon is given solely by the rate of successive impacts of the edges of a nut/coin rolling without sliding on the membrane. The richness of the spectrum is due to the anharmonicity of the signal (for higher harmonics) and to irregularities in the shape of the nut (for sub-harmonics). The tension within the membrane was found to remain constant over a wide range of radii, which explains the plateau in the acoustic intensity of the sound when the volume of the balloon is increased. Finally we showed that the acoustic power is directly related to the kinetic energy of the nut/coin.

One aspect of the sound would probably deserve further investigation: as nuts/coins remain close in general to the equator, the emitted sound might show a pronounced anisotropy. Moreover, a detailed analysis of the effect of the shape of the ridges on a coin (whether sharp or smooth) could shed some light on the conversion ratio from kinetic to acoustic energy. Furthermore, a study of the interplay between the frequency imposed by the nut/coin and the resonant modes of the balloon itself would probably be interesting. Finally, in our study, the nut/coin is given an initial velocity by manually spinning or shaking the balloon but mechanisms by which the nut/coin is set into motion was not explored.

Fig. 1 :

 1 Fig. 1: a) Sketch of the phenomenon. b) Traces left on a transparent balloon by a M6 hex-nut, showing successive impacts of the edges on the membrane. c) Image sequence from high speed video showing the rolling without slipping motion decomposed in time. The red and blue dashed lines indicate the locations of two successive ridges impacts.

Fig. 2 :

 2 Fig. 2: transform of a typical acoustic signal emitted by a hex nut spinning in a balloon (a: entire frequency range, b: low frequencies). The insets show the data on a semi-log scale.

Fig. 3 :

 3 Fig.3: Dominant frequency of the sound as a function of the speed-to-step ratio (v/d) for a variety of hex nuts and ridgededge coins. The inset is a zoom of the same data set at lower frequencies.

Fig. 4 :

 4 Fig.4: Fundamental f0 versus first sub-harmonic for a variety of regular n-polygonal nuts, from squares (n=4) to decagons (n=10). a) f0 vs. fs showing a linear dependence of slope n (dashed lines). b) Frequency ratio f0/fs as a function of the number of sides of the polygonal nuts.

Fig. 5 :

 5 Fig. 5: a) Overpressure inside the balloon, b) tension in the membrane and c) sound amplitude of the acoustic wave emitted by a M5 nut at f0 750 Hz, as a function of balloon radius.

Fig. 6 :

 6 Fig. 6: Sound level (in dB at 1 m, and in Watts) as a function of the fundamental frequency for a series of coin (a). The same data rescaled as a function of 3 4 md 2 f 3 0 (b).

Fig. 7 :

 7 Fig. 7: Spectrogram of the sound emitted by a coin (1/2 Swiss franc). Dashed line: theoretical prediction using Eq. 3.

Acknowledgments. -This work was supported by the École Normale Supérieure de Lyon and University Claude Bernard, Lyon, France. The use of the high speed camera Phantom v2511 was made possible thanks to financial support from CNRS INP. The authors are grateful to A. Marquet, C. Pacary, C. Praud, H. Remise-Charlot and C. Rouvière for fruitful discussions.


Nicolas Plihon

Mickael Bourgoin

Jean-François Pinton

Sophie Miralles

Stochastic reversal dynamics of two interacting magnetic dipoles: a simple model experiment

We report an experimental study of the dynamics of two coupled magnetic dipoles. The experiment consists in two coplanar permanent disk magnets separated by a distance d, each allowed to rotate on fixed parallel axis -each magnets axis being perpendicular to its dipolar moment vector. A torque of adjustable strength can be externally applied to one of the magnets, the other magnet being free. The driving torque may be time independent or temporally fluctuating. We study the influence of the parameters of the driving torque on the dynamics of the coupled system, in particular the emergence of dynamical regimes like stochastic reversals. We report transitions between stationary and stochastic reversal regimes. All the observed features can be understood by a simple mechanical dynamical model. The transition between statistically stationary regimes and reversals is explained introducing an effective potential energy incorporating both the coupling between magnets and the external driving. Relations between this simple experimental model with macroscopic models of magnetic spins coupling, as well as with chaotic reversals of turbulent dynamos are discussed.

I. INTRODUCTION

Multi-stable stochastic processes play a crucial role in many fundamental problems in all fields of science (physics, chemistry, geophysics, biology, etc.) [START_REF] Van Kampen | Stochastic Processes in Physics and Chemistry[END_REF]. In such processes a dynamical system can transit between stable deterministic states separated by an energy barrier; the transition is generally activated by natural or external fluctuations (stochastic noise) promoting the escape of the system from one state to the other. The investigation of the probability of such escape events is a classical and complex problem (known as Kramers problem [START_REF] Kramers | Brownian motion in a field of force and the diffusion model[END_REF]), and an active research field of multi-stable stochastic systems. It depends on many aspects, such as the amplitude of the energy barrier between stable states, the type of activating noise (additive, multiplicative), the correlation time and the colour of the noise. In particular, multi-stable stochastic systems are omnipresent in spin systems or magnetic dipole-dipole interaction. For instance, finite size ensembles of magnetic dipoles exhibit transitions between ferromagnetic phases and chaotic regimes (zero average magnetisation) at zero temperature as shown in [START_REF] Borgonovi | Dynamics of random dipoles: chaos versus ferromagnetism[END_REF], the transition being dependent on the number of dipoles and on the strength of the coupling [START_REF] Celardo | Time scale for magnetic reversal and the topoligical nonconnectivity threshold[END_REF]. Similar transitions are often observed in fluid mechanics systems. For instance, closed turbulent flows generated between counter-rotating impellers (so called von Kármán flows) are known to exhibit hysteresis and multi-stability [START_REF] Ravelet | Multistability and Memory Effect in a Highly Turbulent Flow: Experimental Evidence for a Global Bifurcation[END_REF] with transitions between states activated by turbulent fluctuations. Recent experiments by de la Torre & Burguete [START_REF] De La Torre | Slow dynamics in a turbulent von Kármán swirling flow[END_REF] have shown for instance that a perfectly sym-metric von Kármán flow can exhibit such transitions between two asymmetric flow configurations (the global symmetry being then statistically recovered). Another example is given by the long term Lagrangian dynamics of a large particle in symmetric von Kármán flows [START_REF] Machicoane | Large sphere motion in a nonhomogeneous turbulent flow[END_REF], which have been shown to be preferentially trapped in two stable attractors (in the close vicinity of each impeller), with rare and sudden transitions between both sides of the flow activated by turbulent fluctuations. Such transitions have also been reported in dynamo experiments. In these systems involving electrically conducting fluids, an instability converting kinetic energy into magnetic energy is at the origin of the growth of a large-scale magnetic field. Reversals between two opposite polarities of the magnetic field have for instance been observed in the homogeneous von Kármán Sodium (VKS) experiment [START_REF] Monchaux | The von Kármán Sodium experiment: Turbulent dynamical dynamos[END_REF], as well in the semi-synthetic Bullard-von Kármán dynamo setup [START_REF] Miralles | Lorentz force effects in the Bullard von Kármán dynamo: saturation, energy balance and subcriticality[END_REF].

The present study explores the possibility to revisit some generic properties of the influence of noise on such multistable systems by considering a simple table-top experiment with two coupled magnetic dipoles. The free dynamics of two coupled, identical, frictionless magnetic dipoles has already been described by [START_REF] Pollack | Two magnets oscillating in each other's fields[END_REF] with analytical solutions depending on initial conditions. We consider here a similar geometry but with coupled dipoles having distinct moments of inertia and we incorporate losses. Moreover, in the present study, the angular position of one of the dipoles is forced by a driving torque (which could be the realization of a stochastic noise), the other dipole being free. The study is in the same vein as the one reported in [START_REF] Mellado | Macroscopic magnetic frustration[END_REF] which highlighted magnetic frustration at a macroscopic scale, with an ensemble of macroscopic magnetic dipoles. The article is organised as follows: first, in section 2, we describe the setup of the table-top experiment. In section 3, the computation of the fixed points of the system are detailed. The different dynamical regimes obtained for specific parameters are explained in section 4 and a phenomenological model is given in section 5 to explain qualitatively the observed regimes. Arguments are given in section 6 to compare our model experiment with the dynamical regimes observed in dynamo experiments. In particular, our tabletop experiment shares similar statistical properties with the reversals observed in the VKS experiment, such as transitions between stationary dynamos and reversals, distribution of the waiting time between reversals, as well as the relative influence of the noise on the dynamics.

II. EXPERIMENTAL SETUP

The experimental setup is schematically displayed in figure 1. Two Neodynum-Iron-Bore dipolar magnets are fixed at the tip of two rotating parallel shafts. These magnets are very thin circular plates (of diameter 30 mm, thickness 1 mm) magnetised along one diameter, with dipolar moments M 1 and M 2 respectively. Both magnets are set in the same vertical plane (x, y) (perpendicular to their rotation axis) and separated by a distance d. Note that the distance d sets the amplitude of the coupling between the two magnets and can be modified at will. The angular positions of each dipolar magnet are denoted θ 1 and θ 2 , relative to the reference horizontal axis x (see figure 1-(a)). When magnets are left interacting without external driving, the stable fixed points are θ 1 = θ 2 = 0 (mod π) and the unstable fixed points are θ i = ±π/2 (mod π). When an external torque is applied to shaft 1 such that θ 1 > 0, it leads to a stable equilibrium angle θ 2 < 0, as displayed in figure 1. Such an external torque is applied by a magnetic driving system added to the opposite end of shaft 1 (on which M 1 is set) -the magnetic nature of the drive was designed to avoid additional mechanical friction. The driving system is also schematically displayed in figure 1 and consists of a quadrupolar stator/rotorlike system. The rotor is composed of small permanent magnets while the stator is composed of eight coils. The currents in these coils impose the strength of the maximum torque applied to shaft 1 as well as preferential orientations for θ 1 . This is achieved by driving currents I i in coil i as I 1 = I 5 = -I 3 = -I 7 = I 0 cos α and I 2 = I 6 = -I 4 = -I 8 = I 0 sin α: the maximum strength of the externally applied torque is thus proportional to I 0 and the preferential orientation of θ 1 is α/2 (mod π) (reached when the driving torque overcomes the coupling between the magnets). Figure 1 displays the situation with α = π/2 (with the external driving torque dominating over the dipolar magnets interaction) for which θ 1 = α/2 = π/4. The choice of this quadrupolar forcing has been favored since (i) the strength and direction of the driving torques can be independently set, (ii) the modulus π of fixed points is preserved (see next section). Note that both I 0 and α may explicitly depend on time.

The present study focuses on the experimental observation of the relative influence of the coupling between the two magnets and of the external driving torque on the dynamics of the system -the coupling being controlled by the distance d for a fixed couple of magnets and the driving by the current in the external coils (both the amplitude I 0 and the angle α). Currents in the coils are driven by two bipolar current amplifiers Maxon LSC30/2, feeding the odd and even coils up to I max 0 = 2 A. These amplifiers are controlled by analog outputs of a National Instruments PCIe 6321 digitised at 400 Hz. Around each dipolar magnets, the projections of the magnetic field created by the dipolar moments M 1 and M 2 along the x horizontal axis (at locations H 1 and H 2 ) and along the y vertical axis (at locations V 1 and V 2 ) are measured using AnalogDevices 22151 Hall effect sensors, sampled at a rate of 400 Hz by National Instruments PCIe 6321 analog digitisers. The angular positions θ 1 and θ 2 of the interacting magnets are then computed from these measurements after a calibration procedure. Details about the experiment are provided as in appendix A.

The dynamics of the system is described by a set of 2 coupled differential equations (see appendix A for de-tailed calculations) in their normalised form:

J * 1 d 2 θ 1 dt 2 = A sin (α -2θ 1 ) -sin θ 2 cos θ 1 -2 sin θ 1 cos θ 2 -F * 1 dθ 1 dt -K * 1 sign dθ 1 dt , (1)
J * 2 d 2 θ 2 dt 2 = -sin θ 1 cos θ 2 -2 sin θ 2 cos θ 1 -F * 2 dθ 2 dt -K * 2 sign dθ 2 dt , (2)
with J * i , F * i and K * i respectively denoting the moments of inertia of shaft i, the fluid friction coefficient and the solid friction coefficient acting on shaft i, normalised by the coupling strength µ 0 M 1 M 2 /4πd 3 . Introducing D as an effective driving efficiency of the magnetic drive, A denotes the normalised driving torque as DI 0 µ 0 M 1 M 2 /4πd 3 -in other words A is the ratio between the maximum driving torque DI 0 and the coupling between the two magnets.

When the system is uncoupled (infinite d) or when one of the shafts is kept static, the dynamics simplifies as a second order differential equation with damping. In that case, each of the parameters of equations A1 and A2 can be determined by fitting experimental data with a second order model system. These measurements are straightforward and are described in appendix B.

III. COMPUTATION AND MEASUREMENT OF STATIONARY FIXED POINTS

Stationary fixed points of the system (A1, A2) can be computed as solutions of the system:

A 3 sin (α -2θ 1 ) + sin θ 2 cos θ 1 = 0 (3) sin θ 1 cos θ 2 + 2 sin θ 2 cos θ 1 = 0 (4)
and fixed points depend on the values of A (or equivalently on I 0 and d) and α.

It is also instructive to investigate the nature of these stationary fixed points from an energy point of view. It is convenient to introduce an effective potential energy from which the conservative dynamics of the system can be derived. The interaction potential, normalised by µ 0 M 1 M 2 /4πd 3 reads:

V int = -3 cos θ 1 cos θ 2 + cos(θ 2 -θ 1 ),
from which the normalised torque applied by magnet 2 on magnet 1 is computed as Γ 21 = -∂V int ∂θ 1 and the normalised torque applied by magnet 1 on magnet 2 as

Γ 12 = - ∂V int ∂θ 2
Equivalently, the normalised potential accounting for the driving torque applied to magnet 1 by the driving system reads:

V drive = - A 2 cos(α -2θ 1 ),
from which the normalised driving torque acting on magnet 1 is computed Γ drive = -∂V drive ∂θ 1 . We thus introduce a normalised effective potential energy V accounting for both the coupling between magnets and the external drive as:

V = -3 cos θ 1 cos θ 2 + cos(θ 2 -θ 1 ) + A 2 cos(α -2θ 1 ) + C
where the constant C is adjusted for the minimum value of V to be zero.

From system (3)-( 4), it is readily checked that eight fixed points exist for (θ 1 , θ 2 ) and that if θ 0 1 and θ 0 2 are solution of the above system, then θ 0 1 + π and θ 0 2 + π are also solutions. The quadrupolar driving scheme thus preserves the modulus π of the stationary fixed points. However, in the present setup, only four fixed points are easily accessible to the dynamics of the system. This can be understood by investigating the evolution of potential V in the configuration space (θ 1 , θ 2 ) which is displayed in figure 2 without driving (i.e. A = 0). Two stable fixed points correspond to global minima of potential V and are displayed as white circles (θ 1 = θ 2 = 0 (mod 2π) and θ 1 = θ 2 = ±π (mod 2π)). The two reachable unstable fixed points, which are saddle points in the configuration space (θ 1 , θ 2 ), are displayed as white crosses. Four other unstable fixed points (either saddle points or global maxima) are displayed as black diamonds, with potential values respectively 3 and 4 ( i.e. 3 and 4 times larger than the potential of the two saddle points displayed as white crosses). In the context of our study, the dynamics will be confined in the dark energy valley of figure 2 -along the diagonal θ 2 ∼ -θ 1 when A ∼ 0 -and the four fixed points at higher potential (black diamonds) are never observed (see figure 5). From the initial eight fixed points, only four of them (two stable and two unstable ones) are thus accessible. The situation remains similar when considering cases for which A = 0; as shown below, the potential landscape is modified, however the dynamics is still confined to low energy valleys of the potential landscape.

Let us now focus on the stable fixed points in the more general situation for which α, the preferential orientation of magnet 1 occurring from the driving torque, and A, the strength of the driving torque, are non zero. When α = 0, two asymptotic behaviours are identified:

• A 1: when the coupling between the two interacting magnets dominates over the driving, θ 1 → 0 (mod π) (in the experiment, for vanishing values of I 0 and finite d value).

• A

1: when the driving dominates over the coupling between the two interacting magnets, θ 1 → α/2 (mod π) (in the experiment, for large I 0 values at finite d).

θ 1 -π/2 0 π/2 π θ 2 -π -π/2 0 π/2 M 2 M 1 M 2 M 1 M 2 M 1 M 2 M 1 M 2 M 1 M 2 M 1 M 2 M 1 M 2 M 1 a c c
For a given value of the interaction length (d = 45 mm), the stationary stable fixed points θ i have been measured as a function of A and systematically compared to the solutions computed for system [START_REF] Borgonovi | Dynamics of random dipoles: chaos versus ferromagnetism[END_REF][START_REF] Celardo | Time scale for magnetic reversal and the topoligical nonconnectivity threshold[END_REF] for several values of α -three of which are displayed in This evolution of the stable fixed points as a function of A can also be understood from the evolution of the potential landscape V (θ 1 , θ 2 ). Figure 4 shows the evo-lution of of V for 6 values of A and α/2 = 1.56 (corresponding to the experimental points black circles in figure 3). Both the amplitude of the potential and the shape of the potential landscape as a function of θ 1 and θ 2 are modified. In particular global minima are observed around θ 1 ∼ θ 2 ∼ 0 mod(π) at low values of A (below 0.5), which correspond to the stable fixed points of figure 2. When A increases, a shift of the global minima towards θ 1 ∼ α/2 (mod π) and tan(θ 2 ) =tan(θ 1 )/2 is observed. In particular, for A > 2 in figure 4, the energy landscape drastically changed as compared to the A = 0 case: the low energy valley is no more along the diagonal θ 1 ∼ -θ 2 , but has shifted toward vertical lines around θ 1 ∼ α/2 (mod π). The potential barrier between stable fixed points increases as A increases (note the colorscales in figure 4). This can be understood in the following way: when A 1, the driving torque is dominant and magnet 1 barely rotates around its axis even if magnet 2 rotates. In this configuration, the unstable fixed point with the lowest energy is now θ 1 ∼ θ 2 ; this unstable fixed point is now easily accessible to the system.

IV. DYNAMICS WITH TIME-VARYING DRIVING

Let us now investigate the dynamics of the coupled system (A1,A2) with a time-varying driving. Several choices may be made for the time-dependent variation of the driving torque through time-variation of the driving current I 0 or the driving angle α. In the remaining of the paper α is kept constant and time-dependence arises from the amplitude of the driving current I 0 -or equivalently A at constant d. Since similar dynamics have been obtained for time-dependent phase and/or timedependent magnitude of the driving torque, we will focus on cases with time dependent magnitude of the driving torque without any loss of generality. A(t) was chosen as the digital realisation of a delta-correlated gaussian process, with average A and standard deviation Ã. This choice was motivated by the observation of non-trivial dynamics even in the presence of such simple noise realisations. The choice of the nature of the noise is further discussed in the conclusion, in particular when discussing the possible application of this analogue model to turbulent systems, for which the fluctuations are much more complex than gaussian delta-correlated noise. The angle α/2 = 1.3 was chosen as a typical value (quantitatively identical regimes were obtained for values of α/2 in the range [π/8; π/2]).

We first consider the influence of the standard deviation of the driving à for a fixed time averaged value A and fixed interacting distance (d = 45 mm). Let us first investigate the case where the driving fluctuations are kept very small (i.e. small values of Ã). The interacting magnets then fluctuate around their stable stationary fixed points and typical time series of θ 1 (t) and θ 2 (t) are given in figure 5(a-b). The associated (θ 1 (t), θ 2 (t)) configuration space is displayed in figure 5(c), and shows that the dynamics is confined to the potential well centered on (θ 1 ∼ 0.6, θ 2 ∼ -0.33). The trajectory of the fluctuating system is confined to an island in the configuration space and no reversals are observed. This results in a constant large scale dipolar magnetisation of the system, with fluctuations around the time-average dipole strength. The amplitude of the noise is large enough to explore locations in the configuration space where the potential is non zero close to stable fixed point (black circle), but it is too weak to drive the system toward the nearest unstable fixed points (white crosses). For higher values of the driving noise amplitude Ã, the dynamics is more complex and reversals of the magnets orientation can be observed. Typical time series of θ 1 (t) and θ 2 (t) in this regime are given in figure 5(d-e). The system now explores regions close to the unstable fixed points. This is evidenced when investigating the dynamics of the reversals displayed in figure 5(d-e) in the (θ 1 (t), θ 2 (t)) configuration space as given in figure 5(f). In this situation, the noise amplitude is strong enough to drive the system close to the unstable fixed points (white crosses), eventually leading to reversals of the magnets orientations. The trajectory is no more confined to an island. The system explores potential wells around the stable fixed points (black circles) during long intervals, leading to dense trajectories in the vicinity of these stable fixed points. However, the noise is strong enough to drive the system close to the unstable fixed points for short periods of time, before the trajectory is confined again close to a new stable fixed point. From figures 5(c) and (f), it is also clear that the dynamics is confined to valleys of low potential valuesdarker regions -as schematically explained in figure 2(b). This dynamics results in the occurrence of two opposite large-scale dipolar magnetisations (i.e. the dipolar structure can be oriented towards the x direction or the -x direction), with stochastic reversals between these two states. A video displaying the dynamics of such regimes is provided as a supplementary material at [URL will be inserted by publisher]. It shows the view of the magnets (where the red dot locates the north pole of the magnet), together with the time-evolution of angular positions θ 1 and θ 2 and of the x-component of the magnetic field, measured with a Hall effect sensor halfway between the two magnets axes. Let us now investigate in greater details the evolution of the dynamics as a function of both the time-averaged value A and the standard deviation à of the normalized driving torque. The experimental evolution of the timeaveraged angular position θ 1 of magnet 1 as a function of A is displayed as dots in figure 6(a) -an evolution in very good agreement with the stable fixed points computed from system (3)-( 4) plotted as a full black line. The evolution of θ1 , the standard deviation of the fluctuations of the angular position of magnet 1 around θ 1 is displayed in figure 6(b) as a function of A for several values of the driving fluctuations Ã. Let us first examine the case for which the amplitude of the noise is weak enough so that the dynamics is confined to potential wells around the fixed points in the (θ 1 , θ 2 ) configuration space (as in figure 5(c)). This is for instance the case for the green (vertical) crosses in figure 6(b), for which à = 0.35. The standard deviation θ1 first increases when A is increased up to 0.66 and then decreases for higher values of A . Similar behaviours are observed for higher values of Ã: the evolution of θ1 normalised by à nearly collapse onto a single curve as shown in the inset of figure 6(b). We provide a phenomenological explanation of these features in next section. In this regime, the interacting magnets fluctuate around their mean angular positions θ 1 and θ 2 without reversals. For higher values of the driving noise amplitude Ã, we already mentioned more com-plex dynamics, with magnets orientation reversals. For sufficiently large driving noise, the system may explore regions close to the unstable fixed points (displayed in figure 6(a) as a dashed line). These reversals are only observed for a narrow range of A values, displayed as arrows in figure 6(b). We will describe in details the case for which à = 0.79 -purple squares in figure 6(b). For values of A up to 0.7, the fluctuations of the angular position θ1 remain below 3π/16 and the regimes correspond to statistically stationary regimes (with magnets fluctuating around their mean angular positions). For higher values of A (between 0.75 and 1.2), stochastic reversals of the magnets orientation are observed. In this regime, the angular positions switches randomly between two angular positions -with a large level of fluctuations. These positions correspond to the stationary fixed points and each reversal consist in a rapid rotation of angle nπ, n being an unsigned integer. The extent of these regimes is depicted as a full arrow in figure 6(b) (since reversals of the magnets occur, the standard deviation is no longer an illustrative variable to consider for these regimes). Then, as A is further increased, the system do not exhibit reversals any longer and the system recovers a statistically stationary dynamics, with a decreasing evolution of θ1 with A . The first observation is that stochastic reversals can be observed with a gaussian white noise driving if the noise amplitude is sufficiently large. A second observation, is that the extent of the occurrence of these reversals depends on the driving noise amplitude.

A third observation is that, close to the onset of existence of stochastic regimes -i.e. for the smallest noise amplitude where reversals have been observed -these regimes are bounded by statistically stationary regimes as a function of A when à is kept constant.

The determination of the extent of the regimes where reversals are observed as depicted in figure 6(b) is somewhat abritrary. This can be best understood from the analysis of figure 7(a) which displays the evolution of the mean waiting time T r between reversals as a function of A for a fixed value of à = 1.14 and d = 45 mm. For these parameters, the situation is similar to the one presented in figure 6(b): reversals are bounded by statistically stationary regimes. Data presented in figure 7(a) where obtained from 86400 seconds (24 hours) long time series. The first observation is that reversals are more frequent in the center of the occurrence region than at its boundary. The second observation is that the mean waiting time between reversals rapidly increases as A is decreased or increased to values corresponding to the statistically stationary regimes. Whether the mean waiting time between reversals diverges or not as A approaches the values for which no reversals occur cannot be stated definitely from these data (this would require extremely long time series), but divergence is compatible with the evolution depicted in figure 7(a). Boundaries between regimes should thus be defined as when the mean time between reversals diverges. However, given the evolution provided in figure 7(a), a precise determination of boundaries between regimes in the parameter space ( A , Ã) for various values of d is a rather difficult task (which would require a very large number of extremely long time series -a task out of reach of this study). We provide in figure 7(b) an estimate of these boundaries from the analysis of finite length time series (at least two reversals over 480 seconds long time series) for three values of the coupling distance d and a fine scan in the ( A , Ã) parameter space. Let us first focus on the case d = 45 mm, which corresponds to the (blue) squares. In this figure, the lower and upper threshold in A for the occurrence of stochastic reversals are plotted as a function of à -this corresponds to the extent of the full arrows in figure 6(b). Schematically, three distinct regimes are observed for a given amplitude of the driving noise (equivalently Ã): (i) for the lowest values of the driving noise, only statistically stationary regimes are observed (left hand side of figure 7(b)), (ii) for moderate amplitude of the driving noise, reversals are bounded by statistically stationary regimes as A increases, (iii) for large amplitude of the driving noise, reversals are observed when A is below a threshold, while statistically stationary regimes are observed above. We provide a phenomenological explanation of this sequence in next section. The influence of the coupling strength on the occurrence of the stochastic regimes is also displayed in figure 7(b), for two other values of the coupling distance d, namely 40 mm (black stars) and 55 mm (empty red circles). A first striking observation is that the onset of stochastic regimes is observed for lower values of the noise when the coupling is stronger. A second observation is that for a given value of the noise amplitude, the window of occurence for theses regimes is smaller for stronger coupling. Finally, the time statistics of waiting time between reversals is studied through their probability distribution functions (pdf) displayed in figure 7(c). These pdf have been computed from 259200 seconds long (72 hours) time series for A = 0.97 and 324000 seconds long (90 hours) for the other two values of A . These pdf are compatible with an exponential distribution on a first approximation, with characteristic times (or equivalently meanwaiting times between reversals) τ = 56.2; 12.1 and 26.3 s respectively for A = 0.44, 0.97, 1.36.

V. PHENOMENOLOGICAL MODEL

In this section, we analyze the features of the coupled stochastic system in a simplified framework which explains qualitatively most of the experimental observations. This analysis relies on the interpretation of the potential energy V introduced earlier. In the presence of fluctuations of the driving torque, the potential V can then be written as the sum of two contributions: a time averaged potential V and a noise η.

V (t) = V + η(t), where V = -3 cos θ 1 cos θ 2 +cos(θ 2 -θ 1 )+ A 2 cos(α-2θ 1 )+C, and
η(t) = A(t) -A 2 cos(α -2θ 1 ),
Let us now make the approximation that the energy landscape is set by the time-average value of the potential V , while the fluctuations of the driving act as a noise which allows to reach locations of the configuration space where the potential is positive. In the real system, fluctuations of the driving indeed modify the instantaneous energy landscape, but, for the sake of simplicity, we neglect these modifications at leading order. Under our approximation, trajectories in the (θ 1 , θ 2 ) configuration space evolve within the time average potential V . We assume that excursions in the (θ 1 , θ 2 ) configuration space are restricted to locations where the value of the mean potential is lower than a maximal value proportional to η where η is the standard deviation of the noise. The evolution of isocontours of the potential as a function of A thus provide useful insight of the expected dynamics of the system; these are displayed in figure 8. For each value of A , three isolines are represented, respectively 0.5 (full line), 0.66 (dash-dotted line) and 1.01 (dotted line). We will assume that these isocontours represent the accessible locations in the (θ 1 , θ 2 ) configuration space for increasing values of the driving noise. The analysis of each of the three cases allows for providing a phenomenological understanding of the stationary and reversals regimes, their evolutions and the transitions between them as described in figure 7(b).

We will first focus on the evolution of isocontours for potential V = 0.5 (full lines). For all values of A , isocontours separate islands around a stable fixed point. The noise is not strong enough to drive the system close to an unstable fixed point: the dynamics of the system is restricted to fluctuations around one of the stable fixed points. This is the situation encountered when à < 0.7 in figure 6: no reversals are observed whatever the value of A . The trends observed in figure 6(b) for the evolution of θ1 (namely increasing with A at low values and decreasing with A at large values) can also be understood from the evolution of the isocontours of the potential landscape. Indeed, the extent of the islands -and thus the accessible range of θ 1 values -first increases with A (at low values), and then decreases with A (for larger values). Let us now consider the isocontours for potential 0.66 (dash-dotted lines). For A values below 0.4 and above 1.3, the potential isocontours separate islands around stable fixed points: no reversals are observed (similarly to the evolution depicted in figure 6(b) for à = 0.88 at the lowest and highest values of A ). On the contrary, when 0.4 ≤ A ≤ 1.3, isocontours encompass both stable fixed points. In this A interval, the noise amplitude is strong enough to explore vicinities of the unstable fixed points and leads to reversals of the magnets orientation. Once again, the evolution of θ1 as a function of A for the stationary regimes can be understood from the extent of the isocontour islands. Finally, isocontours for potential 1.01 (dotted lines) show a connexion between stable fixed points as long as A is below 1.7, allowing for reversals of the magnets orientations. For values of A above 1.7, the isocontours separate islands around stable fixed points, and statistically stationary regimes are recovered. While the direct computation of the relation between the amplitude of the fluctuations of the driving current Ĩ0 and the maximum potential value accessible to the system is a difficult task, the above rationale gives an accurate understanding of the observed dynamics and reproduces the three regimes described above, as well as the evolution of θ1 in the ( A , Ã) parameter space. Regime (ii) described above (namely that for moderate values of the driving noise, reversals are bounded by statistically stationary regimes when modifying the time-averaged value of the driving) can be understood the following way: for intermediate values of A , adding a driving torque to magnet 1 strongly modifies the energy landscape and lowers the energy gap between stable and unstable fixed points and eases the observation of reversals.

VI. DISCUSSION AND CONCLUSION

We reported the rich dynamics displayed by a simple table-top experiment involving the coupling of two macroscopic magnetic dipoles in presence of a driving torque on one of the magnet -which preserves a πrotation symmetry of the fixed points of the system. The accessible fixed points are accurately determined introducing an effective potential accounting for both the coupling and the driving. In the presence of fluctuations of the driving torque, two distinct behaviours are observed: the dipoles fluctuate around their stable fixed points (typically for low amplitudes of the fluctuations) or they can display stochastic reversals between stable fixed points (when fluctuations are strong enough to drive the system close to unstable fixed points). The mapping of these two regimes in the parameter space is accurately described when considering an energy argument: the dynamics is then restricted to locations in the configurationspace for which the effective potential energy is lower than a threshold (related to the energy of the fluctuations). For low energy of the fluctuations, the dynamics is restricted to disjoint basins of attraction close to the stable fixed points. For higher fluctuation energies, these basins become connected (now including an unstable fixed point) and the dynamics displays stochastic reversals with Poisson-distributed waiting time between reversals.

Our table-top experiment was built somehow in the spirit of the one in [START_REF] Mellado | Macroscopic magnetic frustration[END_REF]. It could be considered as a building block of a macroscopic experiment of a finite number of interacting dipoles, as reported numerically in [START_REF] Borgonovi | Dynamics of random dipoles: chaos versus ferromagnetism[END_REF][START_REF] Celardo | Time scale for magnetic reversal and the topoligical nonconnectivity threshold[END_REF]. Our driving scheme preserves symmetries and could be used to mimic defects in the lattice and/or noise mimicking thermal fluctuations. It could also be used as a macroscopic benchmark for the dynamical studies of slow-relaxing low-dimensional magnetic systems, such as one dimensional single-chain magnets [START_REF] Zhang | Single-chain magnets: beyond the Glauber model[END_REF].

The present experiment has also been designed to develop a simple model of the dynamics of chaotic reversals of the turbulent liquid metal dynamo experiment VKS (Von Kármán Sodium) -in which liquid sodium is set into motion in a cylindrical container by the rotation of coaxial impellers fitted with blades. Under specific parameters, the observed magnetic field can exhibit complex dynamics, including random and sudden reversals between opposite polarities [START_REF] Ravelet | Pétrélis "Chaotic Dynamos Generated by a Turbulent Flow of Liquid Sodium[END_REF]. A model has been proposed [START_REF] Pétrélis | Chaotic dynamics of the magnetic field generated by dynamo action in a turbulent flow[END_REF] which explains the occurrence of such reversals as a stochastic transition (activated by the turbulent fluctuations of the flow) between two large scale magnetic modes. In turn, these large scale modes (dipole and quadrupoles) and their stability properties can be understood from the interaction between dipoles localised in the vicinity of each ferromagnetic impeller (such as two parallel or anti-parallel localised dipoles) [START_REF] Verhille | Induction in a von Kármán flow driven by ferromagnetic impellers[END_REF]. The tabletop experiment described in the present article was inspired by this interpretation: each of the magnet dipole represents one of the localised dynamo, and the timedependent driving models the influence of large-scale turbulent fluctuations (which are modeled by gaussian white noise). The present experiment shares very similar behaviours with the ones observed in the VKS experiment, such as (i) existence of pockets of dynamical regimes bounded by stationary regimes [START_REF] Berhanu | Dynamo regimes and transitions in the VKS experiment[END_REF], (ii) Poisson distribution of the waiting time between reversals [START_REF] Monchaux | The von Kármán Sodium experiment: Turbulent dynamical dynamos[END_REF], (iii) divergence of the waiting time between reversals at the transition between stationary and dynamical regimes. This setup thus allows to understand, in a very simplified framework, the relative influence of the coupling between the two localised modes and the fluctuations of the driving on the occurrence of reversals. In particular, we showed a counter-intuitive result: a higher level of fluctuations is required to obtain reversals when the coupling strength between the two magnets decreases (i.e. the coupling distance increases). However, keeping in mind the comparison between the present work with the actual MHD system, several points are worth being mentioned : (i) in the present system, the localised dipoles have constant amplitude and are only allowed to rotate, whereas in the case of the actual MHD dynamo, the amplitude also fluctuates; (ii) the noise mimicking turbulent fluctuations is imposed here as gaussian white noise and only on one of the dipoles. Our results show that in spite of its simplicity and apparent naiveness, this simple system reproduces correctly some basic dynamical features of the magnetic field in VKS.

Evolution of this simple experiment could also be implemented in views of providing quantitive understanding of other stochastic phenomena. For instance, a spontaneous bifurcation between symmetric and and asymmetric flow configurations have been reported in fully turbulent vonKarman flows subject to multistability [START_REF] Ravelet | Multistability and Memory Effect in a Highly Turbulent Flow: Experimental Evidence for a Global Bifurcation[END_REF]. Similar bifurcations might be observed in our simple table-top experiment when driving both magnets with independent stochastic torques and would open ways to provide simple analysis of the dynamics of complex systems. The evolution of the amplitude of the response ∆θ 1 is typical of a resonant second-order system, with a static -low frequency -response ∆θ 1 equal to A and a resonance giving access to the resonance frequency and to the damping. Typical evolutions of ∆θ 1 as a function of f / √ I 0 are displayed in figure 9(c). From these methods, the following values for the coefficient have been measured:

• M 1 = M 2 = 0.775 A.m 2 .

• J 1 = 1.64 10 -6 kg.m 2 , and J 2 = 8.3 10 -7 kg.m 2 , values compatible with rough computation from geometrical considerations.

• D = 1.45 10 -3 T.m 2 .

• F 1 /J 1 ∼ F 2 /J 2 ∼ 2.2 s -1 .

• K 1 /J 1 ∼ 2.8 s -2 and K 2 /J 2 ∼ 7.5 s -2 .

FIG. 1 .

 1 FIG. 1. (Color online) (a) 2D schematic view of the interaction of the two magnets studied in this article (b) Schematic of the experimental setup

FIG. 2 .

 2 FIG. 2. (a)Evolution of potential V in the configuration space (θ1, θ2) when A = 0; contour lines for 0.2, 0.6, 1, 2, 3 and 3.6 energy values. (b) Fixed points as in (a) and associated magnets M1, M2 configurations: stable fixed points (white circles), unstable saddle points (white crosses) and non-accessible unstable fixed points (black diamonds).

Figure 3 .FIG. 3 .

 33 FIG. 3. Color online. Evolution of experimental and theoretical fixed points as a function of the normalised driving amplitude A for three values of α: θ1 (experiment: full black symbols) and θ2 (experiment: open red symbols).

12 FIG. 4 .

 124 FIG. 4. Evolution of potential V in the configuration space (θ1, θ2), for increasing values of A from top left to bottom right and α/2 = 1.56.

  a,d) θ1(t) time series and (b,e) associated θ2(t) time series for α/2 = 1.3. (c,f) Corresponding trajectory in configuration space (gray scale codes for the amplitude of potential V ). Left column: statistically stationary regime, Right column: reversals. See text for details.

FIG. 6 .

 6 FIG. 6. (a) Evolution of the time average value θ1 : (full line) stable fixed points and (dash-dotted) line unstable fixed points computed from equations 3-4 and experiments (circles) and (b) evolution of the standard deviation θ1 as a function of A for various à values. The distance is kept constant d = 45 mm and α/2 = 1.3.

FIG. 7 .

 7 FIG. 7. (a) Evolution of the mean waiting time between reversals Tr as a function of A , for à = 1.14. (b) Estimated boundaries between regimes in the ( Ã, A ) parameter space for α/2 = 1.3. (c) Probability density function of waiting times between reversals for à = 1.14, and d = 45 mm for the three values of A . See text for details.

FIG. 8 .

 8 FIG.8. Isolines of V for increasing values of A in the configuration space (θ1, θ2) from top left to bottom right. Isopotentials 0.5 (full lines), 0.66 (dash-dotted lines) and 1.01 (dotted lines). See text for details.

FIG. 9 .

 9 FIG. 9. a) Dynamics of the relaxation for shaft 1 (uncoupled and in the presence of an external uniform magnetic field Bout = 144 G). Experimental point and non-linear fit for the estimate of parameters. Inset: same graph in logarithmic scale. See text for details. b) Evolution of the resonant pulsation ω0 as a function of Bout for both shafts. See text for details. c) ∆θ1/A as a function of f / √ I0 for 10 values of I0 (from 0.5 to 1.4 A in steps of 0.1 A) and 2 values of A (0.05 and 0.1 rad)

ACKNOWLEDGMENTS

We thank M. Moulin for his technical assistance, D. Micard for her participation in preliminary experiments, and G. Verhille for fruitful discussions.

Appendix A: System Differential equation

Torque applied to shaft 1 by the drive

In this appendix, the system of differential equations describing the coupled dynamics of the rotating shafts is determined. Let us first describe the driving scheme. It consists of a quadrupolar permanent magnet fixed on shaft 1 upon which a torque is imposed by an external quadrupolar magnetic field created by current circulation in 8 coils arranged around the quadrupolar permanent magnet (see Figure 1 of the mauscript). The quadrupolar permanent magnet, of magnetic moment M q is aligned with the dipolar moment M 1 (if θ 1 = 0, the outgoing directions of the magnetic field-lines from the permanent quadrupolar magnet are 0 and π, relative to axis x). It is assembled from 4 small dipolar cylindrical segments magnets of moment 0.354 Am 2 . The 8 coils create a magnetic field Q coils , of amplitude Q coils and whose outgoing field lines are oriented at an angle α/2 (modulus π) relative to the x axis . This is achieved by driving currents I i in coil i as I 1 = I 5 = -I 3 = -I 7 = I 0 cos α and I 2 = I 6 = -I 4 = -I 8 = I 0 sin α. The strength of the externally applied magnetic field Q coils is then proportional to the current I 0 . The angle α/2 sets a preferential orientation for magnet 1 (reached when I 0 → ∞ for finite d). The driving torque applied by the external magnetic quadrupolar field Q coils on the quadrupolar permanent magnet reads

With D an effective driving efficiency accounting for the coupling between the quadrupolar permanent magnetic field and the applied quadrupolar magnetic field.

Torque applied to shaft 1 by the drive

The full mechanical system is obtained when applying Newton's laws to each of the two shafts. Torque applied by magnet 2 on magnet 1 reads

Similarly, the torque applied by magnet 1 on magnet 2 reads

With J i , F i and K i respectively the moments of inertia of shaft i, the fluid friction coefficient and the solid friction coefficient acting on shaft i, the coupled dynamics of the system is then described by the set of 2 differential equations (A1, A2)

Appendix B: Parameters identification

All parameters of the differential system (A1,A2) have been directly measured or identified from the dynamics of the uncoupled system (i.e. with an infinite d or with one of the shaft kept static), which simplifies as the dynamics of a damped second order system. a. Measurement of M i : M 1 and M 2 have been computed by fitting the evolution of the magnetic field created by each small dipolar magnet as a function of the distance z along the axis of magnetisation from the center of the magnet. The theoretical calculation gives

The logarithmic evolution is linear for z > 25 mm and gives M 1 = M 2 = 0.775 Am 2 . b. Measurement of J i , F i , K i and D: Mechanical parameters were estimated from the relaxation dynamics of uncoupled or coupled shafts in the presence of a magnetic field. 1) A first set of measurements can be performed in the presence of a uniform, static magnetic field. Three different configurations were studied:

• A uniform, static, dipolar magnetic field B out is applied from external large electrical coils in a Helmholtz configuration along the reference horizontal axis around the small permanent magnet, for one single shaft. The uncoupled dynamics of shaft i is then

• When shaft j is maintained at a fixed angular position θ j = 0, a static dipolar magnetic field B j (created by the permanent magnet j) is applied to the small permanent magnet on shaft i. The system considered here is the one displayed in figure 1, for which one of the two shaft is set static. The dynamics of shaft i is then

• A static driving quadrupolar magnetic field is applied from current circulation in the 8 driving coils. When α = 0, the dynamical equation for the uncoupled dynamics of shaft 1 is then

For all these configurations, the dynamics of a single shaft is described by a second order differential equation with damping. The temporal evolution of the relaxation from θ i = 0 is then recorded for various values of the magnetic field setting the resonance frequency (i.e. varying either B out , I 0 or d for the three cases specified above). An estimate of the linear relaxation pulsations

is computed from the temporal evolution -see for instance figure 9(a) for a typical relaxation of uncoupled shaft 1 with an applied B out around magnet 1. An estimate of the fluid friction coefficient F i is provided by the temporal evolution of ln(θ i ) (whose envelope is linear with time at initial stages -see the inset figure 9(a)). Eventually, the solid friction coefficient K i is estimated from the latest stage of the temporal evolution of the envelope of θ i , which is linearly decreasing at latest times -see figure 9(a). From these first-guess estimates, a least square fitting method is applied between the experimental relaxation and numerical integration of equations ( 3), ( 4) or [START_REF] Ravelet | Multistability and Memory Effect in a Highly Turbulent Flow: Experimental Evidence for a Global Bifurcation[END_REF]. An example of the comparison between the experimental data and the best fit obtained is given in figure 9(a).

2) The determination of the coefficients is also made possible when driving uncoupled shaft 1 with a constant driving current I 0 and an harmonic variation of α = 2A cos(2πf t) (shaft 1 has to be uncoupled from shaft 2, i.e. d → ∞). In that case the dynamics is governed by


Guillaume Bousselin

Marius Tanase

Marc Moulin

Nicolas Plihon
email: nicolas.plihon@ens-lyon.fr

Flowrate measurements of conducting

may come

Introduction

Velocity measurements in electrically conducting liquids is an important issue for control and characterisation of flows in various situations, ranging from the control of liquid metals used as coolant in fast breeder fission reactor to processes in the chemical and food-processing industries [START_REF] Schulenberg | Flow measurement techniques in heavy liquid metals[END_REF]. However, this is usually a challenging task because of the opacity, high chemical reactivity and usually high ambient temperature of the liquid metals or chemical mixes [START_REF] Eckert | Velocity measurement techniques for liquid metal flows[END_REF][START_REF] Eckert | Some recent developments in the field of measuring techniques and instrumentation for liquid metal flows[END_REF]. Non invasive techniques developed for neutral fluids, such as ultrasonic methods [START_REF] Takeda | Measurement of velocity profile of mercury flow by ultrasound Doppler-shift method[END_REF][START_REF] Brito | Ultrasonic Doppler velocimetry in liquid gallium[END_REF] or neutron radiography [START_REF] Saito | Measurements of liquidmetal two-phase flow by using neutron radiography and electrical conductivity probe[END_REF] have been adapted for liquid metals. These techniques are indirect flow measurements techniques since they rely on the scattering of waves or particles by tracer impurities or bubbles and have major drawbacks such as difficult implementation or costly operation. Several methods relying on the interpretation of the interaction between a moving electrically conducting fluid with externally applied magnetic fields have also been proposed. An eddy-current flow meter, proposed as early as 1948 [START_REF] Lehde | Device for measuring rate of fluid flow US Patent 2435043[END_REF] is based on the measurement of the advection of a magnetic field, created by two coils fed by an AC current. This technique is still currently in use [START_REF] Pavlinov | Eddy current flowmeter for sodium flow[END_REF]. Magnetic phase shifts introduced by the flow in similar configurations may also be used to probe velocities or flow rates [START_REF] Priede | Contactless electromagnetic phase-shift flowmeter for liquid metals[END_REF]. A global tomographic method was demonstrated to reconstruct 2D or 3D features of the flows from the interpretation of induced magnetic field distributions from the velocity field distribution [START_REF] Stefani | A contactless method for velocity reconstruction in electrically conducting fluids[END_REF][START_REF] Ratajczak | Contactless Inductive Flow Tomography: Brief History and Recent Developments in Its Application to Continuous Casting[END_REF]. More recently, two techniques relying on the interaction of magnetic fields with flowing conducting fluids have been introduced. The first one interprets the force acting on a permanent magnet in the vicinity of an electrically conducting fluid flow, due to the interaction of the induced magnetic field -created by currents induced in the flow -with the magnetic moment of the permanent magnet. Several configurations have been tested: the Lorentz Force Velocimetry technique probes flow rates in pipes [START_REF] Thess | Lorentz force velocimetry[END_REF], while the Local Lorentz Flow Velocimetry probes the velocity at one point thanks to the use of a miniature permanent magnet [START_REF] Heinicke | Spatially resolved measurements in a liquid metal flow with Lorentz force velocimetry[END_REF]. On the other hand the magnetic distortion probe [START_REF] Miralles | The magnetic-distortion probe: velocimetry in conducting fluids[END_REF][START_REF] Pinton | Velocimetre and method for measuring the speed of flow of an electrically conducting fluid[END_REF], probes the induced magnetic field in the vicinity of the permanent magnet. The latter technique allows to achieve very good temporal response and to include corrections from external spurious magnetic fields.

The magnetic distortion probe is based on the interpretation of the induced magnetic field B i in the vicinity of a localised externally applied magnetic field B 0 and an electrically conducting fluid flow at velocity u. This technique was initially developed to measure local fluid velocities (with a 2D resolution in Ref. [START_REF] Miralles | The magnetic-distortion probe: velocimetry in conducting fluids[END_REF]) implementing a permanent magnet as the localised applied magnetic field. In this article, we investigate its ability to measure flowrates in pipes. Using an Hallbach array of permanent magnet, experimental measurements of flow rates in an benchmark pipe flow are reported in section 2. A numerical model is introduced in section 2, and very good agreement between the experimental data and the simulations are observed. We show that this numerical model may be used to compute a calibration coefficient taking into account the exact geometry of the probe and of the pipe flow. This model may also be used to optimise the probe design in order to probe flowrates, as demonstrated in section 3. Finally, the issue of the spatial resolution of measurements made with the magnetic distortion probe in the configuration used in Ref. [START_REF] Miralles | The magnetic-distortion probe: velocimetry in conducting fluids[END_REF] is addressed in section 4. Concluding remarks are provided in section 5

Measurements in a liquid gallium pipe flow

Experimental setup

A mid-size gallium pipe flow is created between two tanks connected by a 1 m long, D = 10 mm inner diameter and 1 mm thick amagnetic stainless steel pipe. The upper tank may be pressurised at pressure P up to 3 bar with Argon and an electrically controlled valve allow liquid gallium, of electrical conductivity σ = 3.8 10 6 S.m -1 , density ρ = 6.095 g.cm -3 and temperature around 50 • C, to flow within the pipe to the lower tank. The velocity of the flow follows Torricelli's law and strain gauges placed on the upper tank allow the measurement of the mass m(t) of gallium exiting the tank. The loop contains 8 L of liquid gallium and steady regimes from 5 to 25 s are achieved depending on the value of P . Following the magnetic distortion probe sensing method proposed by Miralles et coworkers [START_REF] Miralles | The magnetic-distortion probe: velocimetry in conducting fluids[END_REF], a set of permanent magnet is placed in the vicinity of the pipe, creating an externally applied magnetic field B 0 . A commercial Hall sensor (GW Bell 7030) probes the value of the induced magnetic field B i in the direction of the flow. A sketch of the experimental setup is provided in Fig. 1 and shows an Hallbach type array implementing ten NdFeB magnets (10 mm × 10 mm × 40 mm, magnetised along one of the smallest direction) placed 7.5 mm away from the outer edge of pipe. The motivation to choose an Hallbach array [START_REF] Mallinson | One-sided fluxes -a magnetic curiosity?[END_REF] relies on (i) the fact that the amplitude of the magnetic distortion probe signal is proportional to the gradient of the applied magnetic field in the flow direction (Hallbach arrays allow to impose very strong gradients), (ii) the high spanwise homogeneity in the applied magnetic field, (iii) the very low value of applied magnetic field outside the array. Note that similar arrangements have been proposed to probe flowrates of very low electrical conductivity electrolytes using Lorentz force velocimetry [17]. The Hall sensor probes the induced magnetic field component B i x in the flow direction at 3 mm from the pipe and can be moved along the x axis from x = -50 to x = 50 mm (the origin of the x-axis lying in the center of the Hallbach array). Signals are sampled at a rate of 1000 Hz, with 16 bits resolution.

Numerical model

The finite element solver Comsol Multiphysics has been used to compute the magnetic field induced by the coupling between the liquid metal flow and an external magnetic field. The coupled set of Navier Stokes and Maxwell equations are solved using Comsol's Computational Fluid Dynamcis (CFD) and electromagnetic AC/DC modules. This technique was shown to accurately predict or reproduce features of MHD flows at low magnetic Reynolds numbers [START_REF] Cébron | Magnetohydrodynamic simulations of the elliptical instability in triaxial ellipsoids[END_REF][START_REF] Stelzer | Experimental and numerical study of electrically-driven MHD flow in a modified cylindrical annulus: (2) Instabilities[END_REF] and high magnetic Reynolds number [START_REF] Jackson | A spherical shell numerical dynamo benchmark with pseudo vacuum magnetic boundary conditions[END_REF]. The geometry implemented in the numerical simulations is identical to that of the experimental setup. Liquid gallium with electrical conductivity σ and density ρ is flowing inside a 10 mm inner diameter, 1 mm thick and 300 mm long stainless steel pipe. The externally applied magnetic field is generated by the Hallbach configuration introduced earlier (see Fig. 1). Each permanent magnet is a rectangular parallelepiped of dimension 10 x 10 x 40 mm, having a remanent flux density B r = 1.3 10 4 G according to the NdFeB N42 magnet grade used in the experimental arrangement, and is magnetised along one of the smallest direction (see Fig. 1), alternatively along the y and x directions. The Comsol CFD module is used to solve the Navier Stokes equation in the fluid domain for an incompressible and stationary gallium flow:

∇.u = 0 (1)
ρ(u.∇)u = ∇. -pI + µ(∇u + (∇u) T ) + F (2)
with u the liquid gallium velocity field, µ = 1.8 10 -3 Pa.s the dynamic viscosity, p the pressure and F the Lorentz force. The flowrate is imposed at the tube inlet and null pressure at the outlet.

The velocity field is then supplied to the AC/DC module which computes the induced current according to the generalised Ohm's law J = σ(-∇V + u × B) in the different domains of the model; V being the electric potential and B = µ 0 H + B r b where b is the magnetisation direction of each magnet (we recall that the remanent flux density B r is non-zero in the magnets domain only). This is solved imposing Ampere's law and the current conservation equation:

∇ × H = J ; ∇. J = 0
Continuity conditions are used for the internal boundaries between internal domains refereed to as α and β:

n × (H α -H β ) = 0 ; n × (J α -J β ) = 0
and magnetic insulation conditions are specified for the external boundaries of the model:

n × A = 0 ; V = 0
where A is the magnetic vector potential. The induced magnetic field is then computed from the flow-induced currents J.

The coupling between the CFD and the AC/DC modules is repeated until convergence is found. At first, the CFD module is run discarding the Lorentz force (i.e. F = 0) to solve for a purely hydrodynamic velocity profile. This purely hydrodynamic velocity field is then supplied to the AC/DC module which computes the induced current. The Lorentz force, computed from the interaction of the induced current with the applied magnetic field, is then fed into the CFD module for the subsequent computations.

The entire model has been meshed using free tetrahedral elements with a heterogeneous density. The mesh density has been increased in the fluid domain as well as in the air domain between the magnets assembly and the pipe edge to account for the strong gradients in both the velocity and magnetic fields and to increase the resolution in the region where the experimental measurements have been done. The size of the mesh elements has been chosen in order to obtain a good spatial resolution while keeping the computational time reasonable. Finally, the number of degrees of freedom is approximatively 2.6 millions.

Experimental validation of the numerical model

The profiles of the components of the applied magnetic field in the spanwise direction B 0 y and in the streamwise direction B 0 x are displayed in Fig. 2(a) and (b) respectively, for z = 0 mm, i.e. at the altitude of the center of the pipe. The leftmost panel shows very large values of the streamwise gradient of the spanwise component of the magnetic field, of the order of 4 10 5 Gm -1 . The very good agreements between the experimental values and numerical model, respectively displayed as blue circles and full red lines in Fig. 2, both for the components B 0

x and B 0 y , validates the numerical model. While the y (spanwise) component has even parity, the x (streamwise) component has odd parity. The (red) full line curve of Fig. 2(c) displays the values of the streamwise component of the induced magnetic field B i x , which is three order lower in magnitude than the applied magnetic field. Typical time series of the streamwise component of the induced magnetic field as the valve is opened are reported in Fig. 3(a) for different probing locations along the pipe. The three reported time traces have been acquired in successive shots and show that steady-state flows are obtained typically one second after triggering the valve opening. In the remaining of this section, one will report evolution of time-averaged values of the induced magnetic field B i

x in the stationary regime. As expected from Fig. 2(c), the values of the induced magnetic strongly depend on the probing location. We also note that the x-component of the induced magnetic field has even parity. As a consequence, the measurements of the induced magnetic field at locations where B 0

x ∼ 0 are relatively easy. At locations where |B 0

x | is maximal, very small B i x values are induced, and accurate measurements require an extremely large resolution of the magnetic sensor (of the order of 10 -5 ). The very good agreement between the experimental values and the numerical simulations of the x-component of the induced magnetic field B i

x reported in Fig. 2(c Let us now investigate how the maximum value of the time-averaged induced magnetic field B i x (i.e. at x = 0) evolves as a function of the pipe flowrate Q. The flowrate Q is controlled by the pressure in the upper Argon tank, ranging from 0.2 to 3 bars. The flowrate Q is computed from the time evolution of the upper tank gallium mass m(t) as Q(t) = dm(t)/dt . As expected from the linearity of the induction equation, B i

x scales linearly with Q, as shown in Fig. 3(b). Once again, a very good agreement is obtained between the experimental values and our numerical model (where Q is the CFD boundary condition at the pipe inlet). It is important to note here that, in the geometry relative to the numerical simulations, the effect of the Lorentz force is small.

Due to the moderate values of the interaction parameter

N = σ (B 0 ) 2 D 2ρU ∼ 0.
1, the modifications of the flow profiles and of the induced magnetic field are of the order of 1% (not shown). This is still verified though the Hartmann number Ha = B 0 D/2 σ/µ is of the order of 50 since the length over which a strong magnetic field is imposed is small (of the order of the tube diameter D). As a partial conclusion, we showed the accuracy of our numerical model to predict the behaviour of the magnetic distortion experimentally observed in a gallium pipe flow.

Spatial resolution of the magnetic distortion probe using a Hallbach array

In order to characterise finely the ability of the magnetic distortion probe to measure flowrates of conducting fluids in pipes, the spatial resolution of the measurement is addressed using our numerical model. The magnetic distortion probe was demonstrated to be a local velocity sensor [START_REF] Miralles | The magnetic-distortion probe: velocimetry in conducting fluids[END_REF], and, since the velocity profile in the pipe does not evolve with velocity (nor the pipe section), the measurements presented above could be expressed equivalently as a function of the velocity or as a function of the flowrate Q. The use of the magnetic distortion probe as a flowmeter thus requires a uniform contribution to the induced magnetic field from each location in the pipe section. To address this issue, a slightly modified geometry, displayed in Fig. 4, with a pipe having a square cross-section of length 10 mm, has been used. The pipe has been divided into 36 equal cells (2 by 2 mm squares) and a set of numerical simulations was run where a uniform (solib-body like) flow is imposed successively in only one of the cells, the other cells being kept at rest. The magnetic configuration is that created by the Hallbach array presented in section 2. The electrical conductivity of all cells was kept constant and equal to the value of liquid gallium. The intensity of the induced magnetic field probed at location 1 when the liquid metal is flowing in only one of the the cells is displayed at that specific cell location in Fig. 4(b). These values are normalised to the highest value of all cells. It is important to note that the simulations have been led in three dimensions, but that the representation of Fig. 4(b) is in two dimensions. This is a representation of the spatial response function of the magnetic distortion probe in the spanwise direction, which shows that the configuration with one sensor at location 1 averages roughly one third of the pipe. A straightforward upgrade relies on the use of two sensors (at location 1 and location 2). The intensity of the sum of the normalised induced magnetic field is displayed in Fig. 4(c) for each of the cells. A much more homogeneous response function is obtained in this configuration. This nearly uniform response function may be used to probe the global flow rate even in the presence of flow inhomogeneities in the pipe. It is important to note that, in principle, this configuration also allow for the determination of flowrates in a pipe subject to material deposition if the electrical conductivity of the deposited material is the same as that of the fluid. In practice, this might not be verified since, for instance, deposited oxides would have a lower conductivity than the fluid. This would modify the details of how the induced currents loop back in the system, requiring a dedicated numerical study to determine how the accuracy of the flowrate measurement depends on the electrical conductivity and on the thickness of the deposited material. Note that more advanced and adapted configurations than the Hallbach array coupled to two magnetic sensors could be designed (using for instance a larger number of probes and optimised applied magnetic field configurations).

Spatial resolution of the magnetic distortion probe using a single magnet

The proof of concept of velocity measurements in electrically conducting fluids using a single permanent magnet and magnetic field sensors probing the induced magnetic field component in the flow direction, in the vicinity of the magnet, was demonstrated in Ref. [START_REF] Miralles | The magnetic-distortion probe: velocimetry in conducting fluids[END_REF] in various operational configurations. In this previous work, the spatial resolution of the measurement was not investigated in details. We now provide a set of numerical simulations addressing the issue of the spanwise spatial resolution of the magnetic distortion measurement, following the method introduced in the previous section, where the flow is decomposed into elementary cells to compute the response function. A configuration with one single magnet and one magnetic field sensor, sketched in Fig. 5 (leftmost panel), has been implemented with 98 cells (leading to 49 computations due to symmetry arguments) in a rectangular pipe with 7 × 14 mm inner dimensions and 1 mm thick stainless steel walls. The location of the magnetic sensor is set at 1 mm from the outer wall of the pipe, at z = 0. A small cylindrical permanent magnet of diameter 2 mm and length 2 mm is set at a distance D m from the wall. The panels of Fig. 5 show the normalised spatial response, in the spanwise directions, as D m increases from 1 to 5 mm (left to right). As D m increases, the volume probed by the sensor increases. This can be easily understood since the applied magnetic magnetic field is much more homogeneous in the pipe for larger values of D m (as the applied magnetic field scales as l -3 at a distance l from the center of the magnet). Fig. 6(a) displays the evolution of the z-component of the induced current J z i at z = 0 in each cell as a function of the distance l of the cell relative to the center of the magnet. We note that the induced current density at the center of the pipe scales as l -3 and is thus proportional to the amplitude of the applied magnetic field. The amplitude of the induced magnetic field at the location of the sensor, from each of the individual cell, may then be approximated from Biot-Savart law, scaling as the inverse of the square of the distance between the cell and the sensor, i.e. (l -D m + 1) -2 . This is evidenced in Fig. 6(b). Note that, as D m increases, the volume probed by the magnetic distortion probe increases, but the absolute amplitude of the signal decreases, as seen in Fig. 6(c), which could limit the accuracy of the measurement. A representation of the streamwise spatial resolution is less straightforward; however the extent of the flow probed in the streamwise direction may be estimated from the analysis of the vertical induced current density J z i in the (x, z) plane at y = 3 mm. observation is that the profile of the intensity of the vertical induced current (driven by the electromotive force (ux) × (B 0 y) since the applied magnetic field is mainly in the y-direction) is very similar to the intensity of the applied magnetic field. To gain further insight, the circles displayed in Fig. 7(a) and (c) as black dots show a region where the intensity of the magnetic field is 70% of the maximum in the plane. These circles are reproduced in Fig. 7(b) and (d) and match the region in which the induced current density is larger that 68% of the maximum. The conclusions drawn previously about the spanwise extent is thus valid in the streamwise direction: the extent of the flow probed by the magnetic distortion probe is set by the homogeneity of the applied magnetic field and may be computed from the magnetic field profile. As a partial conclusion, several features at leading order emerge from our numerical model in the simple configuration with a single magnet in the vicinity of the flow:

• The induced current in the flow is proportional to the intensity of the applied magnetic field.

• The extent of the probed flow is given by the extent of homogeneous amplitude of the applied magnetic field.

• The induced magnetic field at a given location is fairly well understood from the Biot-Savart law taking into account the induced current.

From these features, the basics for optimal configurations allowing for flowrate measurements are the following: (i) the most homogeneous profile of applied magnetic field throughout the pipe cross section, (ii) magnetic field sensors located at positions chosen such as to compensate for the inhomogeneity of the applied magnetic field (i.e. the sensing location has to be closer to pipe areas where the amplitude of the applied magnetic field are the lowest. It thus appear that, in the Hallbach array presented in sections 2 and 3, the optimal configuration requires to select an Hallbach array that imposes the most homogeneous B 0 within the cross-section and sensors located midway between the y = 0 plane and the magnet face to compensate for the spatial inhomogeneity of the induced current density. We emphasize that while the above mentioned features allow to identify the behavior of the magnetic distortion probe at leading order, the detailed analysis of the spatial resolution also depends on the way induced currents loop back within the flow or the pipe, and thus depends on the exact geometry and electromagnetic boundary conditions.

Conclusion

In this article, an accurate numerical model of the contactless magnetic distortion probe has been introduced. This numerical model accurately reproduces the stationary regimes observed in a dedicated experimental pipe flow benchmark for several flow and applied magnetic field configurations. The calibration procedure of the magnetic distortion probe, as for all of the electromagnetic flowmeter or velocimeter, usually require extensive experimental tests. In the context of the magnetic distortion probe, this task may now be simplified thanks to the use of our numerical model.

Moreover, our numerical model is a valuable tool used to investigate the spatial extent of the flow probed by the magnetic distortion probe. In particular, we discussed the features leading to local velocity measurements or global flowrate measurements. The optimal arrangement required to probe global flowrates in pipes was finally discussed based on the numerical model.

Figure 1 .

 1 Figure 1. Sketch of the experimental arrangement of the magnetic distortion probe using an Hallbach array in the vicinity of a gallium pipe flow. Liquid gallium is flowing along the x direction. Arrows indicate the magnetisation direction of each of the magnets of the Hallbach array.

Figure 2 .

 2 Figure 2. Profiles of experimental values and numerical simulations of (a) the ycomponent (spanwise) imposed magnetic field B 0 y , (b) the x-component (streamwise) imposed magnetic field B 0x field and (c) the x-component (streamwise) induced magnetic field B ix along the x axis for a Q = 0.2 L/s flowrate.

Figure 3 .

 3 Figure 3. (a) Typical times series of the streamwise component of the induced magnetic field for three probing locations along the pipe (10 Hz low-pass filtered), obtained for the flowrate Q = 0.2 L/s. (b) Evolution of B i x , at x = 0, as a function of the pipe flowrate Q.

Figure 4 .

 4 Figure 4. (a) Sketch of the geometry used to compute the spatial resolution using an Hallbach array around a square duct, see text for details. Normalised spatial resolution with (b) a single probe at location 1, (c) two sensors at locations 1 and 2.

Figure 5 .

 5 Figure 5. Leftmost panel: sketch of the geometry used to compute the spatial resolution using a single cylindrical permanent magnet facing a rectangular duct, see text for details. Normalised spatial resolution in the spanwise directions as the distance D m between the magnet and the pipe increases.

2 .G - 1 )Figure 6 .

 216 Figure 6. (a) Vertical induced current density in each cell (at z = 0) as a function of l, distance from the cell to the center of the magnet, (b) induced magnetic field from each cell normalised by the vertical induced current density as a function of l -D m + 1, (c) B i x induced from the whole pipe flow as a function of D m .

Fig. 7 Figure 7 .

 77 Figure 7. (a) and (c) Intensity of the applied magnetic field and (b) and (d) normalised vertical induced current density profiles in the (x, z) plane at y = 3 mm for (a) and (b) D m = 1 mm and (c) and (d) D m = 5 mm.

Acknowledgements

The authors thank D. Cébron for his valuable time and numerous advices when starting using COMSOL. The authors acknowledge financial support from PULSALYS under contract L700.