Non Linear Physics Group- Eric Falcon

## Tuning the resonant frequencies of a drop by a magnetic field

Ferrofluids(papers by our group)

## Since Lord Rayleigh’s works in 1879, it is well-known that a drop of liquid displays oscillating lobes at its surface when shaken. The dynamical study of the resonant frequencies of such a drop occurs in various domains including astrophysics (to model stellar mass collisions), nuclear physics and biology (to model nuclei), metrology (to measure viscosity or surface tension) or industrial applications (droplet manipulation in microfluidic, optofluidic, and inkjet printer and pharmaceutical industries). A new challenge would be to accurately control and tune the natural oscillations of a drop in a non-intrusive way to be able, for instance, to shift its resonant frequencies to avoid some annoying frequency bands in some fundamental or applied situations.

Here, by adding ferromagnetic particles to a water drop and varying the strength of an applied magnetic field, we show for the first time that we are experimentally able to efficiently tune the resonant frequencies of such a drop. By energy conservation arguments, we show theoretically that the magnetic field contribution is equivalent to add an effective negative surface tension to the drop. Beyond the good agreement between the model and the experiments with no fitting parameter, our study could have some interest for potential applications. Indeed, the weakness of the magnetic field strength, the small size of ferromagnetic particles used are favorable to miniaturization to plan to control the oscillations of centimeter-to-micro-scale drop in a new non-intrusive way. MORE and papers

Magnetically tuned folded structures

## A drop of liquid deposited on a millimetric elastic membrane is wrapped by the membrane if the latter is flexible enough, a phenomenon called capillary origami. We renewed the interest for this phenomenon, by using a drop of ferrofluid - a liquid whose shape may be controlled by a magnetic field. The magnetic field thus becomes a control parameter to remotely drive the shape of the origami.

Magnetic turbulent waves

We use a triangular membrane, leading to a pyramidal-shaped origami. When the applied magnetic field is increased, the origami undergoes an overturn leading to a new configuration. This dynamic instability is explained by an interplay between the magnetic forces (which tend to stretch the ferrofluid along the magnetic field) and the gravitational forces (which tend to flatten the liquid). The overturn instability, controlled in a non intrusive way, is suitable for miniaturization as lower magnetic fields are required for smaller size membranes. It could therefore have interesting applications for the manufacturing process of micro-scale 3D structures, such as MEMs or photovoltaic cells, based on elastocapillary self-assembly. MORE and papers

## Wave turbulence describes the statistical behavior of a set of randomly interacting waves and therefore has been applied to a great variety of systems (ocean surface waves, plasma waves in solar wind, spin waves in solids…). We report the first observation of magnetic wave turbulence on the surface of a ferrofluid submitted to a magnetic field, a regime that has not yet been envisaged in theoretical studies.

When wave amplitudes are high enough, the wave turbulence theory predicts a nonlinear resonant process between waves which generates smaller wavelengths. In a ferrofluid (a liquid with a suspension of nanometric magnetic particles), the dispersion relation of surface waves was known to be tuned by applying a magnetic field. This leads the authors to the first observation of a magnetic wave turbulence regime. The existence domains of gravity and capillary wave turbulence are also documented as well as a triple point of coexistence of these three regimes: these new results are understood using dimensional analysis. Such an experimental system where the dispersion relation is tuned by the operator from a non dispersive to a dispersive system is thus of primary interest to test the wave turbulence theory. MORE and papers

## Two-dimensional melting

The transition from a solid to a liquid phase in 2D systems occurs in numerous domains including solid-state physics, thin colloidal suspensions, liquid films, vibrated granular monolayers, and vortex lattices in superconductors. However, melting of 2D solids is a much less understood phenomenon that for 3D solids. We have designed an unusual system to investigate the 2D melting of a macroscopic analogous of a crystalline lattice: we have observed the transition from an ordered solidlike phase to a disordered liquidlike phase of a lattice of spikes on a ferrofluid surface submitted to horizontal sinusoidal vibrations. The onset of the melting transition and the structural changes across the transition (solid, hexatic and liquid phases) are found in good agreement with the theoretical predictions of 2D melting (KTHNY theory) whose universality is controversial. Our dissipative out-of-equilibrium system exhibits strong similarities with 2D melting of equilibrium systems in solid-state physics. These results could thus contribute to develop a comprehensive statistical mechanics of non-equilibrium phase transitions. MORE and papers

Axisymmetric solitons

Solitary waves or solitons are localized nonlinear waves that propagate almost without deformation due to the balance between the nonlinearity and the dispersion. Solitons are ubiquitous in hydrodynamics, optics and condensed matter. However, most of them propagates within a quasi-one-dimensional plane system. Observation of axisymmetric solitary waves is much more scarce. We have designed an experiment to observe axisymmetric solitary waves on the surface of a cylindrical magnetic fluid layer. Generally, in a usual fluid, a cylindrical fluid layer is unstable and droplets appear (Rayleigh-Plateau instability). By using a ferrofluid (a colloidal suspension of magnetic nanoparticles), it is possible to stabilize this cylindrical fluid layer. To wit, a metallic tube carrying an electrical current creates a magnetic field which stabilizes the cylindrical layer of ferrofluid around the tube due to the magnetic centripetal force. Both the shape and the speed of the solitary waves are modified by the strength of the magnetic field. Such a system allows us to observe for the first time the axisymmetric magnetic solitary waves predicted theoretically in the 80s. The study of collisions between these new type of solitary waves should be of particular interest. MORE and papers

Interplay bewteen Rosensweig and Faraday instabilities:

We report an experimental study of the inhibition of the instability generated by a magnetic field applied perpendicularly to the surface of a magnetic fluid (the Rosensweig instability), by vertical vibrations of the fluid container. Our measurements are in quantitative agreement with a simple analytical model using the theory of Mathieu functions. Paper

7. T. Jamin, Y. Djama, J.-C. Bacri &

E. Falcon 2016

Tuning the resonant frequencies of a drop by a magnetic field

Physical Review Fluids 1, 021901(R) (2016) - Rapid Communication

6. T. Jamin, C. Py & E. Falcon 2011

Instability of the origami of a ferrofluid drop in a magnetic field

Phys. Rev. Lett. 107, 204503 (2011)

5. S. Dorbolo & E. Falcon 2011

Wave turbulence on the surface of a ferrofluid in a horizontal magnetic field

Phys. Rev. E 83, 046303 (2011)

4. E. Bourdin, J.-C. Bacri & E.Falcon 2010:Observation of axisymmetric solitary waves on the surface of a ferrofluid,

Physical Review Letters 104, 094502 (2010)2. F. Boyer &

- 3. F. Boyer & E. Falcon 2009

Two-Dimensional Melting of a Crystal of Ferrofluid Spikes

Physical Review Letters 103, 144501 (2009)

E. Falcon 2008Wave turbulence on the surface of a ferrofluid in a magnetic field

Phys. Rev. Lett. 101, 244502 (2008)

- 1. F. Pétrélis,
E. Falcon& S. Fauve 2000:- Parametric stabilization of the Rosensweig instability.
European Physical Journal B,15, 3 - 6 (2000)

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