Faraday Instability Research Articles

Numerical study on the nonlinear dynamics of the ligament formation and its breakup of a spherical droplet induced by Faraday instability

Faraday instability is an important mechanism for the secondary atomization of a spherical liquid droplet. In this paper, a numerical simulation model was established to study the nonlinear dynamics of the ligament formation and its breakup of a spherical droplet caused by Faraday instability. A set of simulations were firstly carried out to find out the optimum grid size to accurately capture the gas-liquid interface. Then the simulation model was validated by comparing the simulated results with the experimental ones under the same conditions. Based on the simulation model built in this paper, a series of simulations were conducted to study the ligament formation mechanism of a spherical droplet placed on a vertically vibrating plate, which provides a sinusoidal inertial acceleration to the droplet. By analyzing the microscopic information such as the pressure field, velocity field and surface wave displacement, it was found that the ligament formation and breakup were actually the results of the interaction of inertial force and surface tension. Ligament formation and breakup can also be seen as a process of absorbing energy, storing energy, converting energy, delivering energy and erupting. When the inertial force does positive work on a certain surface wave, the kinetic energy of this surface wave increases. Under the action of surface tension, the kinetic energy is transformed into pressure potential energy. After a while, the pressure potential energy is converted back to kinetic energy, combined with the positive work imposed on the surface wave by inertial force. The kinetic energy of the surface wave further increases and part of the kinetic energy is passed to the adjacent surface waves. The adjacent surface waves have the similar process of energy conversion and transmission. After several cycles, more and more energy is absorbed by the surface wave from the inertial force, and the displacement amplitude of the surface wave becomes larger and larger. Finally, a liquid ligament is formed from the droplet surface. When the velocity difference between the top and bottom of the ligament is large enough, the ligament breaks and atomization occurs.

Faraday waves over a permeable rough substrate.

We report on an experimental study of the Faraday instability in a vibrated fluid layer situated over a permeable and rough substrate, consisting either of a flat solid plate or of woven meshes having different openings and wire diameters, open or closed (by a sealing paint). We measure the critical acceleration and the wavelength (on the images from top) at the onset of the instability for vibration frequencies between 28 and 42Hz. We observe that, in comparison with the flat plate, a mesh leads to an increase of the critical acceleration, whereas the wavelength is not significantly altered in none of the explored cases. In order to rationalize the observations, we use the linear theory written for the case of a flat bottom and a viscous fluid to define an effective thickness of the fluid layer. For the closed meshes the effective thickness is simply a linear function of the distance between wires constituting the mesh, whereas it exhibits a more complex behavior for the open meshes. We propose a qualitative understanding for the observed features.

Harmonic to subharmonic transition of the Faraday instability in miscible fluids

When a stable stratification between two miscible fluids is excited by a vertical and periodic forcing, a turbulent mixing zone can develop, triggered by the Faraday instability. The mixing zone grows and saturates to a recently predicted final value L sat [Grea and Ebo Adou, J. Fluid Mech. 837, 293 (2018)] when resonance conditions are no longer fulfilled. Notably, it is expected from the Mathieu stability diagram that the instability may evolve from a harmonic to a subharmonic regime for particular initial conditions. This transition is evidenced here in the full inhomogeneous system using direct numerical simulations with 1024 3 points: the analysis of one-point statistics and spectra reveals that turbulence is greatly enhanced after the transition, while the global anisotropy of both the velocity and concentration fields is significantly reduced. Furthermore, using the concept of sorted density field, we compute the background potential energy e b p of the flow, which increases only after the transition as a signature of irreversible mixing. While the gain in e b p strongly depends on the control parameters of the instability, the cumulative mixing efficiency is more robust. At saturation of the instability, available potential energy is partially released in the flow as background potential energy. Finally, it is shown numerically that for fixed parameters, a multiple-frequency forcing can modify the duration of the harmonic regime without significantly altering the asymptotic state.

Phase-conjugate mirror for water waves driven by the Faraday instability

The Faraday instability appears on liquid baths submitted to vertical oscillations above a critical value. The pattern of standing ripples at half the vibrating frequency that results from this parametric forcing is usually shaped by the boundary conditions imposed by the enclosing receptacle. Here, we show that the time modulation of the medium involved in the Faraday instability can act as a phase-conjugate mirror--a fact which is hidden in the extensively studied case of the boundary-driven regime. We first demonstrate the complete analogy with the equations governing its optical counterpart. We then use water baths combining shallow and deep areas of arbitrary shapes to spatially localize the Faraday instability. We give experimental evidence of the ability of the Faraday instability to generate counterpropagating phase-conjugated waves for any propagating signal wave. The canonical geometries of a point and plane source are implemented. We also verify that Faraday-based phase-conjugate mirrors hold the genuine property of being shape independent. These results show that a periodic modulation of the effective gravity can perform time-reversal operations on monochromatic propagating water waves, with a remarkable efficiency compared with wave manipulation in other fields of physics.

Size distributions of droplets produced by ultrasonic nebulizers

In many applications where small, similar-sized droplets are needed, ultrasonic nebulizers are employed. Little is known about the mechanism of nebulization, for example about what determines the median droplet size. Even less understood, is the droplet size distribution, which is often simply fitted with a log-normal distribution or assumed to be very narrow. We perform the first systematic study of droplet size distributions for different nebulizer technologies, showing that these distributions can be very well fitted with distributions found for sprays, where the size distribution is completely determined by the corrugation of ligaments and the distribution of ligament sizes. In our case, breakup is believed to be due to pinch-off of Faraday instabilities. The droplet size distribution is then set by the distribution of wavelengths of the standing capillary waves and the roughness of the pinch-off ligaments. We show that different nebulizer technologies produce different size distributions, which we relate to (variation in) wavelengths of the waves that contribute to the droplet formation. We further show that the median droplet size scales with the capillary wavelength, with a proportionality constant that depends only slightly on the type of nebulizer, despite order-of-magnitude differences in other parameters.

Faraday instability in double-interface fluid layers

Three-fluid systems subjected to periodic mechanical oscillations are theoretically and experimentally examined. The addition of the third fluid can trigger both stabilizing or destabilizing effects on the interfaces, and additional codimension points are obtainable.

Quasipatterns versus superlattices resulting from the superposition of two hexagonal patterns

We present a short review of the experimental observations and mechanisms related to the generation of quasipatterns and superlattices by the Faraday instability with two-frequency forcing. We show how two-frequency forcing makes possible triad interactions that generate hexagonal patterns, twelvefold quasipatterns or superlattices that consist of two hexagonal patterns rotated by an angle α relative to each other. We then consider which patterns could be observed when α does not belong to the set of prescribed values that give rise to periodic superlattices. Using the Swift–Hohenberg equation as a model, we find that quasipattern solutions exist for nearly all values of α. However, these quasipatterns have not been observed in experiments with the Faraday instability for α≠π/6. We discuss possible reasons and mention a simpler framework that could give some hint about this problem.

Faraday instability in small vessels under vertical vibration

The formation of Faraday waves in a liquid inside a cylindrical vessel under the influence of vertical vibration is studied. The stability thresholds and its mode decomposition are obtained using a linear stability analysis. The stability model is validated with a vibration experiment in a vertical vibration table. The Faraday instability threshold is found for accelerations ranging from 0.1 to 1.0 times the gravitational acceleration. The confinement effect by the vessel introduces cut-off the low frequency modes and the allowed frequencies are discretized. The resulting acceleration stability threshold is high at low frequencies and it is the lowest at medium frequencies, , where the discretization of the mode -momenta introduces low stability regions delimited by more stable frequency ranges. The relevance of these characteristics for the agitation of liquids will be discussed.

The Faraday instability in rectangular and annular geometries: comparison of experiments with theory

A longstanding question on the mechanically forced Faraday instability in rectangular geometries arises from a disparity between theory and experiments (Craik and Armitage in Fluid Dyn Res 15:129–143, 1995). It can be stated as: do corners in a rectangular geometry Faraday experiment cause the disparity between prediction and experiment? This study is an attempt to settle this question by comparing the Faraday instability for two fluids in rectangular geometries where corners must be present with equivalent annular geometries where such corners are necessarily absent. Non-idealities, i.e., damping, arising from a slight meniscus wave motion and induced sidewall damping are observed for thin gaps, causing discrepancies between the predicted instability thresholds and those determined experimentally. However, even for thin-gap geometries, experimental agreement between the tested rectangular geometry and its corresponding annular geometry remains excellent. This suggests that corner effects on the system stability are negligible for rectangular geometries and thus any disparity between theory and experiment is due principally to the damping caused by proximity of the lateral walls.

The electrostatically forced Faraday instability: theory and experiments

The onset of interfacial instability in two-fluid systems using a viscous, leaky dielectric model is studied. The instability arises as a result of resonance between the parametric frequency of an imposed electric field and the system’s natural frequency. In addition to a rigorous model that uses Floquet instability analysis, where both viscous and charge effects are considered, this study also provides convincing validating experiments. In other results, it is shown that (a) the imposition of a periodic electrostatic potential acts to counter gravity and this countering effect becomes more effective if a DC voltage is also added, (b) a critical DC voltage exists at which the interface becomes unstable such that no parametric frequency is required to completely destabilize the interface and (c) the leaky dielectric model approaches a model for a perfect dielectric/perfect conductor pair as the conductivity ratio becomes large. It is also shown via experiments that parametric resonant instability using electrostatic forcing may be reliably used to estimate interfacial tension to sufficient accuracy.

Faraday waves on a cylindrical fluid filament – generalised equation and simulations

We perform linear stability analysis of an interface separating two immiscible, inviscid, quiescent fluids subject to a time-periodic body force. In a generalised, orthogonal coordinate system, the time-dependent amplitude of interfacial perturbations, in the form of standing waves, is shown to be governed by a generalised Mathieu equation. For zero forcing, the Mathieu equation reduces to a (generalised) simple harmonic oscillator equation. The generalised Mathieu equation is shown to govern Faraday waves on four time-periodic base states. We use this equation to demonstrate that Faraday waves and instabilities can arise on an axially unbounded, cylindrical capillary fluid filament subject to radial, time-periodic body force. The stability chart for solutions to the Mathieu equation is obtained through numerical Floquet analysis. For small values of perturbation and forcing amplitude, results obtained from direct numerical simulations (DNS) of the incompressible Euler equation (with surface tension) show very good agreement with theoretical predictions. Linear theory predicts that unstable Rayleigh–Plateau modes can be stabilised through forcing. This prediction is borne out by DNS results at early times. Nonlinearity produces higher wavenumbers, some of which can be linearly unstable due to forcing and thus eventually destabilise the filament. We study axisymmetric as well as three-dimensional perturbations through DNS. For large forcing amplitude, localised sheet-like structures emanate from the filament, suffering subsequent fragmentation and breakup. Systematic parametric studies are conducted in a non-dimensional space of five parameters and comparison with linear theory is provided in each case. Our generalised analysis provides a framework for understanding free and (parametrically) forced capillary oscillations on quiescent base states of varying geometrical configurations.

Linear analysis on the interfacial instability of a spherical liquid droplet subject to a radial vibration

Precursory surface standing waves for liquid atomization occur on a spherical droplet subjected to a radial time-periodic force. In this paper, we carried out a linear stability analysis on the spherical Faraday instability. With the Floquet analysis, a derived difference equation gives the dispersion relation between the Floquet exponent and the spherical modes. For inviscid instability, the problem can also be reduced to the standard Mathieu equation as the same as its planar counterpart, but the parameters in the equation correspond to different quantities due to the spherical configuration. The analysis shows that increasing the density ratio of the ambient fluid to the droplet narrows the range of possibly excited spherical modes under the same forcing condition. For viscous instability, an additional parameter corresponding to the viscous effects was introduced into the difference equation. With increasing the droplet viscosity, the surface waves with large mode numbers are stabilized and hence a larger forcing amplitude is required to cause instability. Furthermore, the most-unstable spherical mode of the largest growth rate excited in the experimental condition is determined and discussed for its physical interpretation for droplet atomization caused by Faraday instability.

Experimental investigation on the atomization of a spherical droplet induced by Faraday instability

Faraday instability plays an important role to realize liquid atomization which has been widely applied in various fields. The spherical Faraday instability has different characteristics from the planar one. It is of great importance to understand the fundamental mechanisms of the surface deformation evolution and atomization of a spherical droplet induced by Faraday instability. In this paper, we first experimentally recorded the deformation and fragmentation process of a spherical droplet on a vertically vibrated plate with high speed camera. The results showed that the zonal, meridional and approximately circular standing waves in proper order exist on the surface of the spherical droplet with the increase of excitation amplitude. When Bo<3.0, the modes of low (zonal) and high (meridional) order m of the spherical harmonic occur more easily, and as Bo is increased, the order m becomes closer to the intermediate value of spherical mode number l. The mechanism of droplet atomization is that a spike forms first on the droplet surface due to the impingement of the liquid flowing from the neighboring trough portions, and then a neck appears because of the velocity difference between the head and bottom of the spike, and the velocity difference determines whether the head liquid can form a sub-droplet ejected from the surface of the parent droplet. The Mathieu equation was derived by a linear theoretical analysis with inviscid and incompressible assumptions for the spherical Faraday instability, in which the parameters have different definitions from its planar counterpart. In the instability diagram, the iso-curves of larger linear growth rate deviate further from the ordinate axis and finally disappear. It is also validated that Lang’s equation is applicable to the spherical Faraday instability in low frequency.

Nonlinear dynamics of electrostatic Faraday instability in thin films

The nonlinear evolution of an interface between a perfect conducting liquid and a perfect dielectric gas subject to periodic electrostatic forcing is studied under the long-wave approximation. It is shown that inertial thin films become unstable to finite-wavelength Faraday modes at the onset, prior to the long-wave pillaring instability reported in the lubrication limit. It is further shown that the pillaring-mode instability is subcritical in nature, with the interface approaching either the top or the bottom wall, depending on the liquid–gas holdup. On the other hand, the Faraday modes exhibit subharmonic or harmonic oscillations that nonlinearly saturate to standing waves at low forcing amplitudes. Unlike the pillaring mode, wherein the interface approaches the wall, Faraday modes may exhibit saturated standing waves when the instability is subcritical. At higher forcing amplitudes, the interface may approach either wall, again depending on the liquid–gas holdup. It is also shown that a gravitationally unstable configuration of such thin films, under the long-wave approximation, cannot be stabilized by periodic electrostatic forcing, unlike mechanical Faraday forcing. In this case, it is observed that the interface exhibits oscillatory sliding behaviour, approaching the wall in an ‘earthworm-like’ motion.

Selective generation of Kerr combs induced by asymmetrically phase-detuned dual pumping of a fiber ring cavity.

We numerically and experimentally investigate the asymmetrically phase-detuned dual pumping of a passive inhomogeneous fiber ring cavity. This configuration originates from the fine control of frequency mismatch between the frequency spacing of the bichromatic pump and the free spectral range of the cavity. Multicomb states at offset frequencies can be selectively generated by means of the mismatch parameter and the coexistence of Turing and Faraday instabilities.

Influence of capillarity and gravity on confined Faraday waves

Experiments characterizing the influence of gravity and interfacial tension on Faraday instability in immiscible, confined fluid layers are presented. The variation in interfacial tension was obtained by controlling the temperature of a suitable binary fluid pair while the influence of gravity was analyzed through a series of terrestrial and microgravity (parabolic flight) experiments. These experiments confirm the existence of a crossover frequency, on either side of which gravity plays opposing roles. The current experiments also reveal that the neutral stability curves under Earth's gravity drift toward lower frequencies as the temperature of the liquid pair approaches its upper consolutal value, i.e., the temperature of complete miscibility. Such drifts in the low frequency range are shown to occur primarily due to the reduction in density difference between the layers, whereas at very high frequencies they are controlled by the lowering of interfacial tension. In the absence of gravity, the Faraday waves are characterized by larger wave numbers, and, as in terrestrial conditions, the instability thresholds at high frequencies increase with an increase of temperature, i.e., reduction in interfacial tension value. This surprising stabilization originates from the lowering of the critical wavelength that leads to increased viscous dissipation.

Faraday instability and subthreshold Faraday waves: surface waves emitted by walkers

A walker is a fluid entity comprising a bouncing droplet coupled to the waves that it generates at the surface of a vibrated bath. Thanks to this coupling, walkers exhibit a series of wave–particle features formerly thought to be exclusive to the quantum realm. In this paper, we derive a model of the Faraday surface waves generated by an impact upon a vertically vibrated liquid surface. We then particularise this theoretical framework to the case of forcing slightly below the Faraday instability threshold. Among others, this theory yields a rationale for the cosine dependence of the wave amplitude to the phase shift between impact and forcing, as well as the characteristic time scale and length scale of viscous damping. The theory is validated with experiments of bead impact on a vibrated bath. We finally discuss implications of these results for the analogy between walkers and quantum particles.

Self-Induced Faraday Instability Laser.

We predict the onset of self-induced parametric or Faraday instabilities in a laser, spontaneously caused by the presence of pump depletion, which leads to a periodic gain landscape for light propagating in the cavity. As a result of the instability, continuous wave oscillation becomes unstable even in the normal dispersion regime of the cavity, and a periodic train of pulses with ultrahigh repetition rate is generated. Application to the case of Raman fiber lasers is described, in good quantitative agreement between our conceptual analysis and numerical modeling.

What is the final size of turbulent mixing zones driven by the Faraday instability?

Miscible fluids of different densities subjected to strong time-periodic accelerations normal to their interface can mix due to Faraday instability effects. Turbulent fluctuations generated by this mechanism lead to the emergence and the growth of a mixing layer. Its enlargement is gradually slowed down as the resonance conditions driving the instability cease to be fulfilled. The final state corresponds to a saturated mixing zone in which the turbulence intensity progressively decays. A new formalism based on second-order correlation spectra for the turbulent quantities is introduced for this problem. This method allows for the prediction of the final mixing zone size and extends results from classical stability analysis limited to weakly nonlinear regimes. We perform at various forcing frequencies and amplitudes a large set of homogeneous and inhomogeneous numerical simulations, extensively exploring the influence of initial conditions. The mixing zone widths, measured at the end of the simulations, are satisfactorily compared to the predictions, and bring a strong support to the proposed theory. The flow dynamics is also studied and reveals the presence of sub-harmonic as well as harmonic modes depending on the initial parameters in the Mathieu phase diagram. Important changes in the flow anisotropy, corresponding to the large scale structures of turbulence, occur. This phenomenon appears directly related to the orientation of the most amplified gravity waves excited in the system, evolving due to the enlargement of the mixing zone.

High-repetition-rate, multi-pulse all-normal-dispersion fiber laser.

In this Letter, a fiber laser that exploits the dissipative Faraday instability as a pulse-generating mechanism is presented, and its dynamics are studied numerically. The proposed laser operates in the all-normal-dispersion regime and produces a train of quasi-parabolic pulses, with a repetition rate that can be controlled depending on the cavity dispersion and nonlinearity, ranging from 10 to 50GHz. It exploits a lumped amplification scheme, which can be potentially realized with rare-earth gain media. The issues concerning the stability of the pulses are discussed, and the differences with similar pulsed lasers are highlighted. In particular, the transition from the ordered multi-pulse regime proposed here to the random pulse operation mode already studied in the literature is discussed.