Faraday Instability Research Articles

EFFECT OF EXTERNAL HARMONIC VIBRATION ON THE FORMATION OF SOLITARY WAVES IN FALLING TWO-LAYER LIQUID FILMS

Abstract We study the influence of a low-frequency harmonic vibration on the formation of the two-dimensional rolling solitary waves in vertically co-flowing two-layer liquid films. The system consists of two adjacent layers of immiscible fluids with the first layer being sandwiched between a vertical solid plate and the second fluid layer. The solid plate oscillates harmonically in the horizontal direction inducing Faraday waves at the liquid–liquid and liquid–air interfaces. We use a reduced hydrodynamic model derived from the Navier–Stokes equations in the long-wave approximation. Linear stability of the base flow in a flat two-layer film is determined semi-analytically using Floquet theory. We consider sub-millimetre-thick films and focus on the competition between the long-wavelength gravity-driven and finite wavelength Faraday instabilities. In the linear regime, the range of unstable wave vectors associated with the gravity-driven instability broadens at low and shrinks at high vibration frequencies. In nonlinear regimes, we find multiple metastable states characterized by solitary-like travelling waves and short pulsating waves. In particular, we find the range of the vibration parameters at which the system is multistable. In this regime, depending on the initial conditions, the long-time dynamics is dominated either by the fully developed solitary-like waves or by the shorter pulsating Faraday waves.

Faraday instability of viscous liquid films on a heated substrate with Maxwell–Cattaneo heat flux

Faraday instability of viscous liquid films with Maxwell–Cattaneo (MC) heat flux on an infinite, heated horizontal substrate subject to vertical time-varying periodic vibration is investigated theoretically. The MC effect means that the response of the heat flux to a temperature gradient obeys a relaxation time law rather than a classical Fourier time law. Applying the classic Floquet theory to linear analysis, a recursive relation is obtained. When considering the MC effect, a new phenomenon appears at a large wave number k. The neutral stability curves branch new tongues that turn left rather than right as before, but the tongues still move up and right as the wave number increases. Furthermore, typical harmonic (H) and subharmonic (SH) alternation behavior continues to exist. Interestingly, the first tongue of a branch is H or SH, implying that there is a transition following the branches. However, near the critical wave number kc of a branch, the SH and H almost overlap. As Cattaneo number C increases, the tongue-like unstable zones of branches become wider, and the critical wave number kc of the appeared branch becomes small. As the driving frequency ω decreases, the branch tongues become elongated and the critical wave number kc of the appeared branch becomes small.

Effect of a deep corrugated wall on the natural frequencies and the Faraday instability of a fluid interface

Effect of a deep corrugated wall on the natural frequencies and the Faraday instability of a fluid interface

Experimental observation of topological transition in linear and nonlinear parametric oscillators.

Parametric oscillators are examples of externally driven systems that can exhibit two stable states with opposite phase depending on the initial conditions. In this work, we propose to study what happens when the external forcing is perturbed by a continuously parametrized defect. Initially in one of its stable states, the oscillator will be perturbed by the defect and finally reach another stable state, which can be its initial one or the other one. For some critical value of the defect parameter, the final state changes abruptly. We theoretically and experimentally investigate such transition both in the linear and nonlinear cases, and the effect of nonlinearities is discussed. A topological interpretation in terms of winding number is proposed, and we show that winding changes correspond to singularities in the temporal dynamics. An experimental observation of such transition is performed using parametric Faraday instability at the surface of a vibrated fluid.

Experimental investigation of walking drops: Wave field and interaction with slit structures.

While bouncing walking silicone oil droplets (walkers) do show many quantumlike phenomena, the original, most intriguing, double-slit experiment with walkers has been shown to be an overinterpretation of data. Several experiments and numerical simulations have proven that for at least some parameter region there is no randomness. Still, true randomness was claimed to be observed in an experiment on chaotically bouncing walkers. Also, most of the available phase space has not been investigated. The main goal of this paper is therefore to look for true interference and chaos in the entire phase space. Recently, we made an extensive investigation of drops interacting with slits, but still in a limited range. However, the outcome was always deterministic and only incidentally mimicked the statistics of the corresponding quantum case. We also showed that the extra interference, already seen by others, in the double-slit case was caused by reflection of waves from the outlet of the unused slit after passage and thus was not a true double-slit effect. A new theoretical treatment of bouncing drop dynamics has since given analytic relations for the associated wave field, leading to a proposal for criteria for the possible occurrence of true interference in the double-slit experiment. Satisfying these criteria, requires working close to the onset of the Faraday instability, with two spatial conditions favoring slow walkers, and a temporal condition favoring fast walkers. The regions of high velocity, where the walkers bounce periodically, and of very low velocity, with chaotically bouncing walkers, have not been comprehensively investigated so far. We have therefore looked at these regions, probing the limits for interaction with slits. Furthermore, noting that a short transit time is essential to fulfill the criteria, experiments were done using double-slit barriers only 0.5 and 2mm broad. Nowhere was true interference or a chaotic response found. As the theory has implications for many of the observed quantumlike phenomena involving walkers as, e.g., tunneling and interaction between drops, we have measured the spatial and temporal decay of the wave field. A comparison with the theory shows very good agreement but leads to a reformulation of the above-mentioned criteria.

Interface coupling effect and multi-mode Faraday instabilities in a three-layer fluid system

We investigate the Faraday instabilities of a three-layer fluid system in a cylindrical container containing low-viscosity liquid metal, sodium hydroxide solution and air by establishing the Mathieu equations with considering the viscous model derived by Labrador et al. (J. Phys.: Conf. Ser., vol. 2090, 2021, 012088). The Floquet analysis, asymptotic analysis, direct numerical simulation and experimental method are adopted in the present study. We obtain the dispersion relations and critical oscillation amplitudes of zigzag and varicose modes from the analysis of the Mathieu equations, which agree well with the experimental result. Furthermore, considering the coupling strength of two interfaces, besides zigzag and varicose modes, we find a beating instability mode that contains two primary frequencies, with its average frequency equalling half of the external excitation frequency in the strongly coupled system. In the weakly coupled system, the $A$ -interface instability, $B$ -interface instability and $A$ & $B$ -interface instability are defined. Finally, we obtain a critical wavenumber $k_c$ that can determine the transition from zigzag or varicose modes to the corresponding $A$ -interface or $B$ -interface instability.

Effect of depth ratio on Faraday instability in a binary liquid system

The response of two incompressible and immiscible liquids of comparable densities at different depth ratios in a rectangular container subject to parametric sinusoidal forcing, similar to Faraday waves, has been considered in the present study. The effect of various depth ratios on the instability threshold and the flow field are investigated for (0, 2) and (0, 4) modes. Numerical simulation has been performed using OpenFOAM to solve Navier-Stokes equations. A source term is added in the interFoam module of OpenFOAM to take the effect of the oscillating container without using any dynamic mesh function. The threshold amplitude and wavenumbers are validated following the linear Floquet analysis of Kumar and Tuckerman (1994 J. Fluid Mech. 279 49–68). The Floquet analysis showed a substantial shift in natural frequency (reduction) and an increase in threshold amplitude when the interface is in close proximity to the wall. The response amplitude follows a square law with the forcing amplitude for all the depth ratios considered in the present study. The wave amplitude response above the threshold curve is found to be an exponential function of liquid depths and forcing parameters.

Influence of parametric forcing on Marangoni instability

We study a thin, laterally confined heated liquid layer subjected to mechanical parametric forcing without gravity. In the absence of parametric forcing, the liquid layer exhibits the Marangoni instability, provided the temperature difference across the layer exceeds a threshold. This threshold varies with the perturbation wavenumber, according to a curve with two minima, which correspond to long- and short-wave instability modes. The most unstable mode depends on the lateral confinement of the liquid layer. In wide containers, the long-wave mode is typically observed, and this can lead to the formation of dry spots. We focus on this mode, as the short-wave mode is found to be unaffected by parametric forcing. We use linear stability analysis and nonlinear computations based on a reduced-order model to investigate how parametric forcing can prevent the formation of dry spots. At low forcing frequencies, the liquid film can be rendered linearly stable within a finite range of forcing amplitudes, which decreases with increasing frequency and ultimately disappears at a cutoff frequency. Outside this range, the flow becomes unstable to either the Marangoni instability (for small amplitudes) or the Faraday instability (for large amplitudes). At high frequencies, beyond the cutoff frequency, linear stabilization through parametric forcing is not possible. However, a nonlinear saturation mechanism, occurring at forcing amplitudes below the Faraday instability threshold, can greatly reduce the film surface deformation and therefore prevent dry spots. Although dry spots can also be avoided at larger forcing amplitudes, this comes at the expense of generating large-amplitude Faraday waves.

Quantifying the turbulent mixing driven by the Faraday instability in rotating miscible fluids

The effect of the rotation on the turbulent mixing of two miscible fluids of small contrasting density, induced by Faraday instability, is investigated using direct numerical simulations. We quantify the irreversible mixing, which depicts the conversion of the available potential energy (APE) to the background potential energy (BPE) through the irreversible mixing rate M. We demonstrate that at lower forcing amplitudes, the turbulent kinetic energy (t.k.e.) increases with an increase in the Coriolis frequency f till (f/ω)2<0.25, where ω is the forcing frequency, during the sub-harmonic instability phase. This enhancement of t.k.e. is attributed to the excitement of more unstable modes. The irreversible mixing sustains for an extended period with increasing (f/ω)2 till 0.25 owing to the prolonged sub-harmonic instability phase and eventually ceases with instability saturation. When (f/ω)2>0.25, the Coriolis force significantly delays the onset of the sub-harmonic instabilities. The strong rotational effects result in lower turbulence because the bulk of the APE expends to BPE, decreasing APE that converts back to t.k.e. reservoir for (f/ω)2>0.25. Therefore, in the subsequent oscillation, the t.k.e. available to contribute to the external energy input from periodic forcing is small. Since the instability never saturates for (f/ω)2>0.25, the conversion of APE to BPE via M continues, and we find prolonged irreversible mixing. At higher forcing amplitudes, the instability delaying effect of rotation is negligible, and the turbulence is less intense and short-lived. Therefore, the irreversible mixing phenomenon also ends quickly for (f/ω)2<0.25. However, when (f/ω)2>0.25, a continuous irreversible mixing is observed. We also examine the mixing efficiency in terms of M and find that the mixing is efficient at lower forcing amplitudes and rotation rates of (f/ω)2>0.25 because the major portion of APE expends to BPE.

Mixing mechanism of power-law non-Newtonian fluids in resonant acoustic mixing

Resonant acoustic mixing (RAM) is a widely applied technology that utilizes low-frequency vertical harmonic vibration for fluid transfer and mixing. However, the current research on the mixing mechanism of RAM technology primarily focuses on the initial mixing stages, neglecting the subsequent turbulent transition. This lack of understanding hinders the further improvement of RAM technology. This paper aims to investigate the mixing mechanism of power-law non-Newtonian fluids (NNF) in RAM using the phase field model and the spectral analysis. The study focuses on understanding the facilitating effect of turbulent transition in mixing and explores the influence of the power-law index and the excitation parameter on the mixing characteristics. The results indicate that the flow field experiences Faraday instability due to the intense perturbation during transient mixing. This leads to the fluid mixing through the development of large-scale vortex to small-scale vortex. During this process, the frequency components of the flow field are distributed around the working frequency, demonstrating transient and broad frequency characteristics. The steady state then dissipates energy through the viscous dissipation of small-scale vortices and ultimately relies on the single-frequency components such as submultiples and multiples excited by the nonlinear effect to complete the mixing. The mixing effects of NNF and Newtonian fluids (NF) are essentially the same, but they consume energy in different ways. The mixing uniformity and mixing efficiency of NNF increase with increasing vibration acceleration and decrease with increasing vibration frequency. These findings provide new insights into the RAM mechanism of power-law NNF.

Theoretical study on the interfacial instability of a spherical droplet subject to vertical vibration

Interfacial instability would be aroused on a spherical liquid droplet when it is subject to external vertical vibration. In this paper, a linear analysis was conducted on this instability problem. The polar-angle dependent acceleration in the spherical coordinate is strongly coupled with the temporal and spatial component of the surface deformation displacement, which gives a recursion equation that implicitly expresses the dispersion relation between the growth rate and the spherical mode numbers. The unstable regions (or unstable tongues) for the inviscid fluids considering the latitudinal mode (longitudinal mode number m=0) were derived and presented in the parameter space. Compared to the solution of the spherical Faraday instability under radial vibration acceleration, the regions of harmonic unstable tongues for the mono-directional vibration cases become much narrower, and the subharmonic unstable tongues almost approach straight lines. The analysis shows that the latitudinal waves emerging on the spherical droplet surface ought to oscillate harmonically instead of subharmonically, which is opposite to the results for the case under radial vibration acceleration. A corresponding experiment of a liquid droplet lying on a vertically vibrating plate was conducted, and the observations substantiate our theoretical predictions.

The onset and saturation of the Faraday instability in miscible fluids in a rotating environment

We investigate the influence of rotation on the onset and saturation of the Faraday instability in a vertically oscillating two-layer miscible fluid using a theoretical model and direct numerical simulations (DNS). Our analytical approach utilizes Floquet analysis to solve a set of the Mathieu equations obtained from the linear stability analysis. The solution of the Mathieu equations comprises stable and harmonic, and subharmonic unstable regions in a three-dimensional stability diagram. We find that the Coriolis force delays the onset of the subharmonic instability responsible for the growth of the mixing zone size at lower forcing amplitudes. However, at higher forcing amplitudes, the flow is energetic enough to mitigate the instability delaying effect of rotation, and the evolution of the mixing zone size is similar in both rotating and non-rotating environments. These results are corroborated by DNS at different Coriolis frequencies and forcing amplitudes. We also observe that for$(\,f/\omega )^2<0.25$, where$f$is the Coriolis frequency, and$\omega$is the forcing frequency, the instability and the turbulent mixing zone size-$L$saturates. When$(\,f/\omega )^2\geq 0.25$, the turbulent mixing zone size-$L$never saturates and continues to grow.

On the Influence of the Surface Electric Charge on the Regularities of the Formation of Faraday Ripples on the Surface of a Low-Viscosity Liquid

The purpose of the study is to analyze the effect of a surface electric charge on the formation conditions of the Faraday ripples on a horizontal surface of a low-viscosity liquid in a vibration field.Methods. The problem is solved analytically in the limit of small amplitude deformation of the free surface of the liquid. The final relation is derived under the condition that the dissipation is small. The liquid was considered ideally conductive with a surface-distributed electric charge. Results. A simple analytical expression is derived that quantitatively describes the effect of suppression of the Faraday ripple if the surface density of the electric charge increases. It is shown that the increase in the surface density of the electric charge significantly enlarge the threshold value of the vibration field amplitude, the excess of which leads to the formation of ripples. The threshold value of the vibration amplitude is proportional to the viscosity of the liquid and depends on its density, surface tension coefficient and the specific horizontal scale of the ripple.Conclusion. The Faraday’s ripple formed on the surface of a liquid in a vertically oscillating container is very sensitive to the value of the surface density of the electric charge. An increase of the surface charge density leads to suppression of the ripple formation. The effect can be used to prevent the appearance of parasitic convective flows that arise in liquid layers placed in vibration fields. The physical mechanism of Faraday ripple suppression is the rivalry between two qualitatively different types of flows near the liquid surface. Increasing the surface charge density changes the balance of surface forces in such a way as to promote the appearance of aperiodic motions and suppress oscillatory ones. In particular, oscillatory motions responsible for the development of Faraday instability caused by vertical vibrations of the liquid container are suppressed.

Electrostatically induced Faraday instability of thin film with spontaneous odd viscosity

Electrostatically induced Faraday instability of thin film with spontaneous odd viscosity

Faraday wave instability analog in vibrated gas-fluidized granular particles.

Granular materials are critical to many natural and industrial processes, yet the chaotic flow behavior makes granular dynamics difficult to understand, model, and control, causing difficulties for natural disaster mitigation as well as scale-up and optimization of industrial devices. Hydrodynamic instabilities in externally excited grains often resemble those in fluids, but have different underlying mechanisms, and these instabilities provide a pathway to understand geological flow patterns and control granular flows in industry. Granular particles subject to vibration have been shown to exhibit Faraday waves analogous to those in fluids; however, waves can only form at high vibration strengths and in shallow layers. Here, we demonstrate that combined gas flow and vibration induces granular waves without these limitations to enable structured, controllable granular flows at larger scale with lower energy consumption for potential industrial processes. Continuum simulations reveal that drag force under gas flow creates more coordinated particle motions to allow waves in taller layers as seen in liquids, bridging the gap between waves produced in conventional fluids and granular particles subject to vibration alone.

Instabilities and pattern formation in viscoelastic fluids

Instabilities and pattern formation in viscous fluids have been a major topic of non-linear fluid dynamics for several decades. The study of pattern formation in viscoelastic thin films offers the opportunity to find new fascinating structures that cannot be observed in viscous fluids. Rayleigh–Taylor and Faraday instabilities, such as the resulting patterns in thin films of viscoelastic fluids, are investigated. We use the long-wave approximation and a Karman–Pohlhausen approach to simplify the mass and momentum equations. The viscoelastic stress tensor is calculated applying the linear Maxwell model. Conditions for the Faraday instability have been found using Floquet’s theorem. It is shown that viscoelastic films can exhibit harmonic resonance under external vibration. Moreover, a simulation of the non-linear problem in 2D and 3D is conducted with a finite difference method. Unstable oscillating Rayleigh–Taylor modes occur in the 2D numerical solution. Furthermore, we find that the wavenumber changes with the relaxation time of the fluid. Faraday patterns in viscous films emerge as regular structures of the surface, like squares or hexagons. Numerical simulations of the viscoelastic fluid also show regular structures. However, they collapse into a chaotic stripe-like pattern after a certain time.

Numerical study of non-Newtonian power-law fluids under low-frequency vertical harmonic vibration

Resonance Acoustic MixingⓇ(RAM) technology applies an external low-frequency vertical harmonic vibration to mix ultrafine granular materials and subsequently non-Newtonian fluids. Although this system is used for various applications, its mechanism is yet not well understood, especially in the mixing of non-Newtonian fluids. To address this gap in knowledge, a phase model of the shear-thinning and shear-thickening non-Newtonian power-law fluid in a low-frequency vertical harmonic vibration container is established in this study, and the different power-law index is also considered. During the initial mixing process, there is Faraday instability at the gas–liquid interface, and Faraday waves are related to the power-law index. With the continuous input of external energy, the flow field is further destabilized, so that the uniform mixing is finally completed. In addition, the rheology of non-Newtonian fluids is consistent with the constitutive relation of power-law fluids. The dynamic viscosity of shear-thinning non-Newtonian fluid decreases rapidly with the increase of mixing time, while the shear-thickening non-Newtonian fluid decreases rapidly with the increase of mixing time. The variation of shear rate for Newtonian and non-Newtonian fluids is identical. Finally, a proper vibration parameter for the high mixing efficiency of RAM is proposed.

Pattern formation in Faraday instability-experimental validation of theoretical models.

Two types of resonance-derived interfacial instability are reviewed with a focus on recent work detailing the effect of side walls on interfacial mode discretization. The first type of resonance is the mechanical Faraday instability, and the second is electrostatic Faraday instability. Both types of resonance are discussed for the case of single-frequency forcing. In the case of mechanical Faraday instability, inviscid theory can forecast the modal forms that one might expect when viscosity is taken into account. Experiments show very favourable validation with theory for both modal forms and onset conditions. Lowering of gravity is predicted to shift smaller wavelengths or choppier modes to lower frequencies. This is also validated by experiments. Electrostatic resonant instability is shown to lead to a pillaring mode that occurs at low wavenumbers, which is akin to Rayleigh Taylor instability. As in the case of mechanical resonance, experiments show favourable validation with theoretical predictions of patterns. A stark difference between the two forms of resonance is the observation of a gradual rise in the negative detuning instability in the case of mechanical Faraday and a very sharp one in the case of electrostatic resonance. This article is part of the theme issue 'New trends in pattern formation and nonlinear dynamics of extended systems'.

Solitary-like Wave Dynamics in Thin Liquid Films over a Vibrated Inclined Plane

Solitary-like surface waves that originate from the spatio-temporal evolution of falling liquid films have been the subject of theoretical and experimental research due to their unique properties that are not readily observed in other physical systems. Here we investigate, experimentally and theoretically, the dynamics of solitary-like surface waves in a liquid layer on an inclined plane that is subjected to a harmonic low-frequency vibration in a range from 30 to 50 Hz. We employ a standard boundary layer model, which describes large-amplitude deformations of the film surface, assuming that it has a self-similar parabolic longitudinal flow velocity profile, to confirm the experimental results and to explain the interplay between the short-wavelength Faraday instability and long-wavelength gravitational instability. In particular, we demonstrate that the vibration results in a decrease in the average and peak amplitude of the long solitary-like surface waves. However, the speed of these waves remains largely unaffected by the vibration, implying that they may propagate over large distances almost without changing their amplitude, thus rendering them suitable for a number of practical applications, where the immunity of pulses that carry information to external vibrations is required.

Quartic soliton dynamics and pattern formations in passive driven fiber resonators with different quartic dispersion regimes

Solitons are particle-like localized structures which can sustain characteristic shape in various nonlinear systems. Very recently, the novel concept of quartic soliton with remarkable energy scaling law arising from the interaction of quartic dispersion and Kerr nonlinearity, provides innovations in nonlinear optics and applications. Here we theoretically investigate the quartic soliton dynamics in passive driven fiber resonators from negative to positive quartic dispersion driving. In the pure negative quartic dispersion regimes, the typical energy-duration scaling relationship is sufficiently demonstrated. Worth to note that, we further investigate the dissipative quartic soliton formation in the positive quartic dispersion regimes, achieving remarkable pulse energy increase. Interestingly, echoing our previous work on the influence of cavity group-velocity dispersion on dissipative Faraday instability formation, pattern formation dynamics in the quartic dispersion driving fiber resonators is also explored. Our investigations might benefit the fundamental understanding of the quartic soliton dynamics.