Faraday Instability Research Articles

Smoothed particles hydrodynamics numerical simulations of droplets walking on viscous vibrating liquid

We study the phenomenon of the “walking droplet”, by means of numerical fluid dynamics simulations using the smoothed particle hydrodynamics numerical method. This phenomenon occurs when a millimetric drop is released on the surface of an oil of the same composition, contained in a tank and subjected to vertical oscillations of frequency and amplitude very close to the Faraday instability threshold. At appropriate values of the parameters of the system under study, the oil droplet jumps permanently on the surface of the vibrating liquid forming a localized wave-particle system, reminding the behaviour of a wave particle quantum system as suggested by de Broglie. In our study, we made relevant simplifying assumptions, however we observe that the wave-drop coupling is surely obtained. Moreover, the droplet and the wave travel at nearly constant velocity, as observed in experiments. These facts suggest that the phenomenon may occur in many contexts and opens the possibility to study it in an extremely wide range of physical configurations.

Modulation instability in the weak dispersion regime of a dispersion modulated passive fiber-ring cavity.

We present a theoretical and experimental study of the modulation instability process in a dispersion oscillating passive fiber-ring resonator in the low dispersion region. Generally, the modulation of the dispersion along the cavity length is responsible for the emergence of a regime characterised by multiple parametric resonances (or Faraday instabilities). We show that, under weak dispersion conditions, a huge number of Faraday sidebands can grow under the influence of fourth order dispersion. We specifically designed a piecewise uniform fiber-ring cavity and report on experiments that confirm our theoretical predictions. We recorded the dynamics of this system revealing strong interactions between the different sidebands in agreement with numerical simulations.

Instabilities in passive dispersion oscillating fiber ring cavities

We investigate theoretically and experimentally the development of instabilities in passive ring cavities with stepwise longitudinal variation of the dispersion. We derive an extended version of the Lugiato-Lefever equation that permits to model dispersion oscillating cavities and we demonstrate that this equation is valid well beyond the mean field approximation. We review the theory of Turing (modulational) and Faraday (parametric) instability in inhomogeneous fiber cavities. We report the experimental demonstration of the generation of stable Turing and Faraday temporal patterns in the same device, which can be controlled by changing the detuning and/or the input power. Moreover, we experimentally record the round-trip-to-round-trip dynamics of the spectrum, which shows that Turing and Faraday instabilities not only differ by their characteristic frequency but also by their dynamical behavior.

Rayleigh-Taylor instability in thin liquid films subjected to harmonic vibration

The dynamics of the Rayleigh-Taylor instability of a two-dimensional thin liquid film placed on the underside of a planar substrate subjected to either normal or tangential harmonic forcing is investigated here in the framework of a set of long-wave evolution equations accounting for inertial effects derived earlier by Bestehorn, Han, and Oron [“Nonlinear pattern formation in thin liquid films under external vibrations,” Phys. Rev. E 88, 023025 (2013)]. In the case of tangential vibration, the linear stability analysis of the time-periodic base state with a flat interface shows the existence of the domain of wavenumbers where the film is unstable. In the case of normal vibration, the linear stability analysis of the quiescent base state reveals that the instability threshold of the system is depicted by a combination of distinct thresholds separate for the Rayleigh-Taylor and Faraday instabilities. The nonlinear dynamics of the film interface in the case of the static substrate results in film rupture. However, in the presence of the substrate vibration in the lateral direction, the film interface saturates in certain domains in the parameter space via the mechanism of advection induced by forcing, so that the continuity of the film interface is preserved even in the domains of linear instability while undergoing the time-periodic harmonic evolution. On the other hand, sufficiently strong forcing introduces a new inertial mode of rupture. In the case of the normal vibration, the film evolution may exhibit time-periodic, harmonic or subharmonic saturated waves apart of rupture. The enhancement of the frequency or amplitude of the substrate forcing promotes the destabilization of the system and a tendency to film rupture at the nonlinear stage of its evolution. A possibility of saturation of the Rayleigh-Taylor instability by either normal or unidirectional tangential forcing in three dimensions is also demonstrated.

Effect of vertical quasi-periodic vibrations on the stability of the free surface of a fluid layer

In this paper we examine the effect of vertical quasi-periodic oscillations on the stability of the free surface of a horizontal liquid layer. The quasi-periodic motion considered here is characterized by two incommensurate frequencies, $ \Omega_{1}$ and $ \Omega_{2}$ , i.e. the ratio $ \omega=\Omega_{2}/\Omega_{1}$ is irrational. The linear quasi-periodic oscillator, corresponding to the governing equations of the Faraday instability, is treated using the harmonic balance method developed by Rand et al. and Zounes and Rand, and by numerical methods using the Floquet analysis and knowing that an irrational number can be approximated by a rational number. We determine the marginal stability curves in terms of reduced amplitude forcing and wave number for inviscid and viscous liquids. For both cases, we show that the neutral stability curves depend strongly on the frequency ratio of oscillations, $ \omega$ , when this parameter is below $ \sqrt{2}$ . Beyond this value, there is no effect of $ \omega$ and the first resonance occurs always at the wave number corresponding to the value of the natural frequency squared, $ \delta=0.25$ . Below the value $ \omega=\sqrt{2}$ , variations of the wave number as a function of $ \omega$ and $ \Omega_{1}^{}$ are presented for the inviscid case. However, for the viscous case, we show the existence of the bicritical points and we present the instability threshold versus $ \Omega_{1}^{}$ for different values of $ \omega$ .

Streaming patterns in Faraday waves

Wave patterns in the Faraday instability have been studied for decades. Besides the rich wave dynamics observed at the interface, Faraday waves hide elusive flow patterns in the bulk – streaming patterns – which have not been studied experimentally. The streaming patterns are responsible for a net circulation in the flow, which is reminiscent of the circulation in convection cells. In this article, we analyse these streaming flows by conducting experiments in a Faraday-wave set-up using particle image velocimetry. To visualise the flows, we perform stroboscopic measurements to both generate trajectory maps and probe the streaming velocity field. We identify three types of patterns and experimentally show that identical Faraday waves can mask streaming patterns that are qualitatively very different. Next, we consider a three-dimensional model for streaming flows in quasi-inviscid fluids, whose key is the complex coupling occurring at all of the viscous boundary layers. This coupling yields modified boundary conditions in a three-dimensional Navier–Stokes formulation of the streaming flow. Numerical simulations based on this framework show reasonably good agreement, both qualitative and quantitative, with the velocity fields of our experiments. The model highlights the relevance of three-dimensional effects in the streaming patterns. Our simulations also reveal that the variety of streaming patterns is deeply linked to the boundary condition at the top interface, which may be strongly affected by the presence of contaminants.

Faraday wave patterns on a square cell network

We present the experimental observations of the Faraday instability when the vibrated liquid is contained in a network of small square cells for exciting frequencies in the range $$10\le F\le 24$$ Hz. A sweep of the parameter space has been performed to investigate the amplitudes and frequencies of the driving force for which different patterns form over the network. Regular patterns in the form of square lattices are observed for driving frequencies in the range $$10\le F<14$$ Hz, while ordered matrices of oscillons are formed for $$14<F\le 23$$ Hz. At $$F>23$$ Hz, disordered periodic patterns appear within individual cells for a small range of amplitudes. In this case, the wave field is dominated by oscillating blobs that interact on the capillary–gravity scale. A Pearson correlation analysis of the recorded videos shows that for all ordered patterns, the surface waves are periodic and correspond to Faraday waves of dominant frequency equal to half the excitation frequency (i.e., $$f=F/2$$ ). In contrast, the oscillons formed for $$14<F\le 23$$ Hz are at the first subharmonic ( $$f=F/2$$ ) and first harmonic ( $$f=F$$ ) response frequencies, with higher harmonics being negligible or absent as in most cases. The disordered wave fields forming at $$F>23$$ Hz are not subharmonic and correspond to periodic harmonic waves with $$f=nF/2$$ (for $$n=2,4,\ldots $$ ). We find that the experimentally determined minimum forcing necessary to destabilize the rest state and generate surface waves is consistent with a recent stability analysis of stationary solutions as derived from a new dispersion relation for time-periodic waves with nonzero forcing and dissipation.

Dynamics of Turing and Faraday instabilities in a longitudinally modulated fiber-ring cavity.

We experimentally investigate the round-trip-to-round-trip dynamics of the modulation instability spectrum in a passive fiber-ring cavity presenting an inhomogeneous dispersion profile. By implementing a real-time spectroscopy technique, we are able to record successive single-shot spectra, which display the evolution of the system toward a stationary state. We find that the two instability regimes (Turing and Faraday) that compete in this kind of inhomogeneous cavity not only differ by their characteristic frequency but also by their dynamical behavior. The dynamic transition between those two regimes of instability is also presented.

English

This research presents an experimental study on the Faraday instability in one-dimensional cells filled with a mixture of water and glycerol, and for a range forcing frequency between 10 and 60 Hz. It showed that for a particular forcing frequency, whose value depends on the width of the cell, an anomalous surface oscillation arises, that appears as a large wavelength mode oscillating subharmonically with respect to the normal modes predicted by the linear stability analysis. Since all others studied forcing frequency, the observed modes are in agreement with the ones theoretically predicted. Key words: Faraday Instability, non-linear dynamics PACS: 47.20.Dr, 47.20.Gv, 47.35.+i, 47.54.+r

Walking droplets in linear channels

When gently placing a droplet onto a vertically vibrated bath, a drop can bounce without coalescing. Upon increasing the forcing acceleration, the droplet is propelled by the wave it generates and becomes a walker with a well defined speed. We investigate the confinement of a walker in different rectangular cavities, used as waveguides for the Faraday waves emitted by successive droplet bounces. By studying the walker velocities, we discover that 1d confinement is optimal for narrow channels of width of $D \simeq 1.5 \lambda_F $. We also propose an analogy with waveguide models based on the observation of the Faraday instability within the channels.

Spatially inhomogeneous pattern formation due to parametric modulation in large broad-area lasers

The article discusses the formation of the optical patterns by modulation of the pump parameter, which may give rise to the Faraday instability in the wide-aperture lasers. The dynamics of class-B lasers is considered in the area of stability of the steady on-axis laser generation. Conditions for the appearance of spatio-temporal structures including stripes and oscillating hexagons are obtained.

Scattering theory of walking droplets in the presence of obstacles

We aim to describe a droplet bouncing on a vibrating bath using a simple and highly versatile model inspired from quantum mechanics. Close to the Faraday instability, a long-lived surface wave is created at each bounce, which serves as a pilot wave for the droplet. This leads to so called walking droplets or walkers. Since the seminal experiment by Couder et al (2006 Phys. Rev. Lett. 97 154101) there have been many attempts to accurately reproduce the experimental results.We propose to describe the trajectories of a walker using a Green function approach. The Green function is related to the Helmholtz equation with Neumann boundary conditions on the obstacle(s) and outgoing boundary conditions at infinity. For a single-slit geometry our model is exactly solvable and reproduces some general features observed experimentally. It stands for a promising candidate to account for the presence of arbitrary boundaries in the walker’s dynamics.

Faraday instability and nonlinear pattern formation of a two-layer system: A reduced model

Pattern formation of a two-layer thin liquid film system subjected to vertical oscillations is studied with an integrated boundary layer model. Squares, hexagons, or quasiperiodic patterns, as well as localized states, are found. For a Rayleigh-Taylor unstable layer, vibrations can delay or completely suppress film rupture.

Faraday instability on a sphere: Floquet analysis

Standing waves appear at the surface of a spherical viscous liquid drop subjected to radial parametric oscillation. This is the spherical analogue of the Faraday instability. Modifying the Kumar &amp; Tuckerman (J. Fluid Mech., vol. 279, 1994, pp. 49–68) planar solution to a spherical interface, we linearize the governing equations about the state of rest and solve the resulting equations by using a spherical harmonic decomposition for the angular dependence, spherical Bessel functions for the radial dependence and a Floquet form for the temporal dependence. Although the inviscid problem can, like the planar case, be mapped exactly onto the Mathieu equation, the spherical geometry introduces additional terms into the analysis. The dependence of the threshold on viscosity is studied and scaling laws are found. It is shown that the spherical thresholds are similar to the planar infinite-depth thresholds, even for small wavenumbers for which the curvature is high. A representative time-dependent Floquet mode is displayed.

Mode-locking via dissipative Faraday instability.

Emergence of coherent structures and patterns at the nonlinear stage of modulation instability of a uniform state is an inherent feature of many biological, physical and engineering systems. There are several well-studied classical modulation instabilities, such as Benjamin–Feir, Turing and Faraday instability, which play a critical role in the self-organization of energy and matter in non-equilibrium physical, chemical and biological systems. Here we experimentally demonstrate the dissipative Faraday instability induced by spatially periodic zig-zag modulation of a dissipative parameter of the system—spectrally dependent losses—achieving generation of temporal patterns and high-harmonic mode-locking in a fibre laser. We demonstrate features of this instability that distinguish it from both the Benjamin–Feir and the purely dispersive Faraday instability. Our results open the possibilities for new designs of mode-locked lasers and can be extended to other fields of physics and engineering.

Faraday instability of a two-layer liquid film with a free upper surface

The classic Faraday instability in a one-layer liquid film is extended to a two-layer film consisting of a lower liquid layer, supported by a vibrating solid plate, overlaid by a second layer of immiscible fluid with a free upper surface. A detailed analytical and numerical analysis of the linear stability of the film, and the interplay between the Faraday and the Rayleigh-Taylor instabilities, is presented.

Faraday wave lattice as an elastic metamaterial.

Metamaterials enable the emergence of novel physical properties due to the existence of an underlying subwavelength structure. Here, we use the Faraday instability to shape the fluid-air interface with a regular pattern. This pattern undergoes an oscillating secondary instability and exhibits spontaneous vibrations that are analogous to transverse elastic waves. By locally forcing these waves, we fully characterize their dispersion relation and show that a Faraday pattern presents an effective shear elasticity. We propose a physical mechanism combining surface tension with the Faraday structured interface that quantitatively predicts the elastic wave phase speed, revealing that the liquid interface behaves as an elastic metamaterial.

Experimental investigation of the Faraday instability on a patterned surface

The effect of patterned substrates on the onset and behavior of the Faraday instability is studied experimentally. We show that the onset of Faraday standing waves in a vertically oscillating layer of liquid can be delayed due to the topography of the underlying two-dimensional patterned substrate. The magnitude of this stabilization effect can be predicted by existing linear stability theories, and we provide an additional physical explanation for the behavior. These observations suggest the feasibility of exploiting the Faraday instability in thin liquid layers in practical engineering systems.

Patterns beyond Faraday waves: observation of parametric crossover from Faraday instabilities to the formation of vortex lattices in open dual fluid strata

Faraday first characterised the behaviour of a fluid in a container subjected to vertical periodic oscillations. His study pertaining to hydrodynamic instability, the ‘Faraday instability’, has catalysed a myriad of experimental, theoretical, and numerical studies shedding light on the mechanisms responsible for the transition of a system at rest to a new state of well-ordered vibrational patterns at fixed frequencies. Here we study dual strata in a shallow vessel containing distilled water and high-viscosity lubrication oil on top of it. At elevated driving power, beyond the Faraday instability, the top stratum is found to ‘freeze’ into a rigid pattern with maxima and minima. At the same time there is a dynamic crossover into a new state in the form of a lattice of recirculating vortices in the lower layer containing the water. Instrumentation and the physics behind are analysed in a phenomenological way together with a basic heuristic modelling of the wave field. The study, which is based on relatively low-budget equipment, stems from related art projects that have evolved over the years. The study is of value within basic research as well as in education, especially as more advanced collective project work in e.g. engineering physics, where it invites further studies of pattern formation, the emergence of vortex lattices and complexity.

Competing Turing and Faraday Instabilities in Longitudinally Modulated Passive Resonators.

We experimentally investigate the interplay of Turing (modulational) and Faraday (parametric) instabilities in a bistable passive nonlinear resonator. The Faraday branch is induced via parametric resonance owing to a periodic modulation of the resonator dispersion. We show that the bistable switching dynamics is dramatically affected by the competition between the two instability mechanisms, which dictates two completely novel scenarios. At low detunings from resonance, switching occurs between the stable stationary lower branch and the Faraday-unstable upper branch, whereas at high detunings we observe the crossover between the Turing and Faraday periodic structures. The results are well explained in terms of the universal Lugiato-Lefever model.