Capillary Waves Research Articles

Internal Doubly Periodic Gravity-Capillary Waves with Vorticity

Internal Doubly Periodic Gravity-Capillary Waves with Vorticity

Isotropic ↔ anisotropic surface geometry transitions induced by adsorbed surfactants at water/vapor interfaces.

Adsorbates at a water/vapor interface change the surface geometry through altered surface tension, yet detailed theoretical studies are relatively sparse, and many applications focus on ensemble average characteristics. Here, we demonstrate that different interpretations of surface geometry emerge when considering the distributions of surface curvature and orientation as a function of adsorbed surfactant concentration and sterics. At low surface densities, the tributyl phosphate (TBP) sorbed water/vapor surface has an increased presence of ridges that are defined by principal curvatures κ1 and κ2 of opposite signs yet close in magnitude. As the TBP surface density increases, the difference in principal curvatures slowly increases. There is a distinct transition of the surface geometry, where the ridge-like features become much more pronounced, having sides whose orientation is normal to a flat interfacial plane. Thus, as the TBP surfactant is added to the surface, the surface curvatures become anisotropic in terms of the difference in magnitude of κ1 and κ2. We label this an isotropic → anisotropic geometric transition. Comparing the surface geometry as a function of the carbon tail length of the alkyl phosphate surfactant reveals that smaller surfactants also anisotropically enhance surface curvatures and that adsorbed alkyl tails to the surface stabilize and increase the symmetry of surface waves along the two principal curvature axes. We label this an anisotropic → isotropic geometric transition. These results reflect the opportunity to incorporate more realistic distributions of surface geometry within the collective understanding of statistical theories of surfaces, including capillary wave theory.

Simulation of time-invariant three-dimensional Freak waves and their scattering identification based on the JONSWAP spectrum and Donelan direction function

Abstract The results of recent maritime accident investigations show that many maritime accidents are related to the attack of freak waves, which can sink ships instantaneously with colossal energy. To identify and forecast freak waves accurately and avoid the danger of freak waves to marine structures and people, we have conducted further research on freak waves. In this paper, we propose an efficient time-invariant three-dimensional (3-D) freak wave computational model to study the scattering characteristics of time-invarian 3-D freak waves and obtain a more explicit identification of their scattering differences. First, this paper adopts linear superposition method to simulate a 3-D random rough sea surface based on the JONSWAP wave spectrum and Donelan directional distribution function. Moreover, we analyze the effect of different angular frequencies on the simulated sea surface. At the same time, the Monte Carlo method is used to further simulate the capillary waves based on the random sea surface, and this method vastly improves the computational accuracy of the random sea surface. Then, based on the two-wave train superposition wave energy focusing mode, a numerical calculation method of time-invariant 3-D freak waves is proposed, and the time-series evolution process of 3-D freak waves is simulated. Finally, the backscattering coefficients of the 3-D freak wave surface are calculated using the two-scale method, and the electromagnetic scattering characteristics of the freak wave surface under different evolution stages, wind speeds, and radar incidence angles are analyzed. By this method, we conclude that the recognition of freak waves is more accurate under medium-high wind speed and significant incidence angle conditions. The experiments show that the research in this paper can identify and predict freak waves more effectively, further improve the forecast accuracy of future freak waves, and also have some application value for freak wave risk warnings.

Hamiltonian Birkhoff Normal Form for Gravity-Capillary Water Waves with Constant Vorticity: Almost Global Existence

We prove an almost global existence result for space periodic solutions of the 1D gravity-capillary water waves equations with constant vorticity. The result holds for any value of gravity, vorticity and depth, a full measure set of surface tensions, and any small and smooth enough initial datum. The proof demands a novel approach—that we call paradifferential Hamiltonian Birkhoff normal form for quasi-linear PDEs—in presence of resonant wave interactions: the normal form is not integrable but it preserves the Sobolev norms thanks to its Hamiltonian nature. A major difficulty is that paradifferential calculus used to prove local well posedness (as the celebrated Alinhac good unknown) breaks the Hamiltonian structure. A major achievement of this paper is to correct (possibly) unbounded paradifferential transformations to symplectic maps, up to an arbitrary degree of homogeneity. Thanks to a deep cancellation, our symplectic correctors are smoothing perturbations of the identity. Thus we are able to preserve both the paradifferential structure and the Hamiltonian nature of the equations. Such Darboux procedure is written in an abstract functional setting applicable also in other contexts.

Modeling of the separation process of a two-phase liquid in a high-temperature gas environment

Obtaining a high-quality highly dispersed salt solution can be realized from a salt solution by the method of ultrasonic exposure. As you know, concentrated salt solutions can be presented in the form of a gas-liquid medium. The influence of ultrasound, in the case of wetting a surface vibrating with an ultrasonic frequency by a thin layer, leads to the formation of a cavitation layer in the liquid layer and capillary waves on its surface, from the ridges of which, at a certain intensity of oscillations, finely dispersed aerosol droplets are detached.The detachment of droplets from the vibrating surface leads to the formation of finely dispersed salt aerosol, which saturates the heated air, which is tangentially fed into the cylindrical working chamber with further separation of hydrodynamic processes. Part of the aerosol wets the heated inner surface of the cylindrical chamber with the formation of a thin film, which gravitationally flows down the vertical solid surface and is subjected to active evaporation with the removal of the salt phase, and the second part is subjected to additional grinding and active evaporation in the central turbulent flow and centrifugal turbulent flow of hot air with additional by removing the salt phase. In order to intensify the salt removal process, the surface on which the film is formed can be profiled. Ultrasonic dispersion of the salt solution to a finely dispersed state allows to significantly increase the surface of contact with the heated air, which allows to intensify the diffusion process.

Modulating outcomes of oil drops bursting at a water–air interface

Recent studies have shown that capillary waves generated by the bursting of an oil drop at the water–air interface produces a daughter droplet inside the bath, while a part of it floats above it. Successive bursting events produce next generations of daughter droplets, gradually diminishing in size until the entire volume of oil rests atop the water–air interface. In this work, we demonstrate two different ways to modulate this process by modifying the constitution of the drop. First, we introduce hydrophilic clay particles inside the parent oil drop and show that it arrests the cascade of daughter droplet generation preventing it from floating over the water–air interface. Second, we show that bursting behavior can be modified by a compound water–oil–air interface made of a film of oil with finite thickness and design a regime map, which displays each of these outcomes. We underpin both of these demonstrations by theoretical arguments providing criteria to predict outcomes resulting therein. Finally, all our scenarios have a direct relation to control of oil–water separation and stability of emulsified solutions in a wide variety of applications, which include drug delivery, enhanced oil recovery, oil spills, and food processing, where a dispersed oil phase tries to separate from a continuous phase.

Fluctuation-driven dynamics of liquid nano-threads with external hydrodynamic perturbations

Instability and rupture dynamics of a liquid nano-thread, subjected to external hydrodynamic perturbations, are captured by a stochastic lubrication equation (SLE) incorporating thermal fluctuations via Gaussian white noise. Linear instability analysis of the SLE is conducted to derive the spectra and distribution functions of thermal capillary waves influenced by external perturbations and thermal fluctuations. The SLE is also solved numerically using a second-order finite difference method with a correlated noise model. Both theoretical and numerical solutions, validated through molecular dynamics, indicate that surface tension forces due to specific external perturbations overcome the random effects of thermal fluctuations, determining both the thermal capillary waves and the evolution of perturbation growth. The results also show two distinct regimes: (i) the hydrodynamic regime, where external perturbations dominate, leading to uniform ruptures, and (ii) the thermal-fluctuation regime, where external perturbations are surpassed by thermal fluctuations, resulting in non-uniform ruptures. The transition between these regimes, modelled by a criterion developed from linear instability theory, exhibits a strong dependence on the amplitudes and wavenumbers of the external perturbations.

Optimal reconstruction of water-waves from noisy pressure measurements at the seabed

We consider the problem of recovering the surface wave profile from noisy bottom pressure measurements with (a priori unknown) arbitrary pressure at the surface. Without noise, the direct approach developed in Clamond and Labarbe (2023) provides an effective way to recover the sea surface. However, the assumption of analyticity for the measured pressure renders this method inefficient in the presence of noise. In order to address this issue, we introduce here an optimisation procedure based on the minimisation of a distance between a recovered bottom pressure and its measurement. Such method proves to be well-designed to handle perturbed signals. We illustrate the effectiveness of this approach in the recovery of gravity-capillary waves from unfiltered noisy data.

Annular two-phase flow in a small diameter tube: OpenFOAM simulations with turbulence damping vs optical measurements

In this work, numerical simulations are performed to predict two-phase annular flow of refrigerant R245fa inside a 3.4 mm diameter vertical channel. The VOF (Volume of Fluid) method implemented in an OpenFOAM solver is used to accurately track the vapor-liquid interface. A 2D axisymmetric domain is considered and the Adaptive Mesh Refinement (AMR) method is applied to the cells near the liquid/vapor interface. The Reynolds-Averaged Navier Stokes (RANS) equations are solved and the k-ω SST model is adopted for turbulence modelling in both the liquid and vapor phase. Simulations are used to calculate instantaneous and mean values of the liquid film thickness at mass flux G = 100 kg m-2 s-1 and vapor quality ranging between 0.2 and 0.85. Numerical results are compared against measurements of the liquid film thickness taken during vertical annular downflow. Previous works from the literature and the deviations observed between present numerical and experimental results suggest the need for turbulence damping at the vapor-liquid interface by adding a source term in the ω equation. The simulations show that a low value of the turbulence damping parameter (e.g. 1) causes the average liquid film thickness to increase by 25 %–52 % compared to the non-damped scenario. The interface presents large amplitude disturbance waves in the non-damped case, whereas small ripple waves are predicted when turbulence damping is introduced. Furthermore, the difference between the application of a symmetric and asymmetric treatment for the source term is analysed. From the comparison between experimental data and numerical simulations, it emerges that the value of the correct damping source term to be applied is strictly dependent on the vapor quality.

A model of thermodynamic stabilization of nanocrystalline grain boundaries in alloy systems

Nanocrystalline (NC) materials are intrinsically unstable against grain growth. Significant research efforts have been dedicated to suppressing the grain growth by solute segregation, including the pursuit of a special NC structure that minimizes the total free energy and completely eliminates the driving force for grain growth. This fully stabilized state has been predicted theoretically and by simulations but is yet to be confirmed experimentally. To better understand the nature of the full stabilization, we propose a simple two-dimensional model capturing the coupled processes of grain boundary (GB) migration and solute diffusion. Kinetic Monte Carlo simulations based on this model reproduce the fully stabilized polycrystalline state and link it to the condition of zero GB free energy. The simulations demonstrate the emergence of a fully stabilized state by the divergence of capillary wave amplitudes on planar GBs and by fragmentation of a large grain into a stable ensemble of smaller grains. The role of solute diffusion in the full stabilization is examined. Possible extensions of the model are discussed.

Wave resonances and the time-dependent capillary gravity wave motion

The interconnection of time and frequency domains for capillary gravity wave motion in the presence of current is discussed in this article. The general time-dependent problem is solved using Green’s function technique, and the asymptotic solution is derived using the method of stationary phase for large time and space. Also, the frequency domain solution is derived as a special case using the Cauchy Residue theorem. Different types of wave resonances like Trapping, Blocking and Bragg resonances are discussed. The existence of the trapped mode below the cutoff frequency is justified theoretically, and numerical results are obtained using the multipole expansion method. The blocking and Bragg resonances are analyzed above the cutoff frequency. It is found that in the presence of current, when the ripple wavenumber of the bottom undulation equals twice the cosine angle of incidence of wave times the wavenumber of the wave, Bragg resonance occurs. It is found that three propagating modes exist in the case of wave blocking, and the trapped modes exist only for the first propagating mode. Furthermore, because of the negative group velocity inside the blocking zone, the Bragg reflection increases while decreasing outside. The effect of current on the wave energy propagation in the form of group velocity is analyzed and the same is verified in the case of time-dependent problem.

2PP-Hydrogel Covered Electrodes to Compensate for Media Effects in the Determination of Biomass in a Capillary Wave Micro Bioreactor.

Microbioreactors increase information output in biopharmaceutical screening applications because they can be operated in parallel without consuming large quantities of the pharmaceutical formulations being tested. A capillary wave microbioreactor (cwMBR) has recently been reported, allowing cost-efficient parallelization in an array that can be activated for mixing as a whole. Although impedance spectroscopy can directly distinguish between dead and viable cells, the monitoring of cells in suspension within bioreactors is challenging because the signal is influenced by the potentially varying properties of the culture medium. In order to address this challenge, an impedance sensor consisting of two sets of microelectrodes in a cwMBR is presented. Only one set of electrodes was covered by a two-photon cross-linked hydrogel to become insensitive to the influence of cells while remaining sensitive to the culture medium. With this impedance sensor, the biomass of Saccharomyces cerevisiae could be measured in a range from 1 to 20 g L-1. In addition, the sensor can compensate for a change in the conductivity of the suspension of 5 to 15 mS cm-1. Moreover, the two-photon cross-linking of hydroxyethyl starch methacrylate hydrogel, which has been studied in detail, recommends itself for even much broader sensing applications in miniaturized bioreactors and biosensors.

Capillary wave-assisted collapse of non-Newtonian droplets

Understanding the peripheral capillary wave propagation during droplet impact is crucial for comprehending the physics of wetting onset and droplet fragmentation. Although Newtonian droplets have been extensively studied, we show how capillary waves deform non-Newtonian droplets in such a way that rheological features, such as the critical concentrations for the overlap (c*) and entangled polymer molecules (c**), may be directly obtained from the deformation history. Determining these critical concentrations is essential as they mark transitions in the rheological behavior of aqueous polymeric solutions, influencing viscosity, elasticity, and associated fluid dynamics. We have also compared capillary waves among Newtonian, shear-thinning, and Boger fluid droplets and found that although the fluid kinematics appear to be purely biaxial extensional flow, the infinite-shear properties of the droplets dominate the physics of capillary wave formation and propagation.

Pinch-off of a regular bubble during the impact of droplets on a liquid pool

A droplet impacting a liquid pool can result in the pinch-off of a regular bubble. In this study, the cavity deformation and regular bubble pinch-off after droplets impacting a liquid pool were experimentally investigated. The results indicate that the phenomenon of regular bubble pinch-off results from the combined effects of cavity expansion and capillary wave propagation. The inertial force, viscous force, and droplet diameter are all crucial to the cavity evolution and the process of regular bubble pinch-off. The cavity depth increases with the droplet's inertial force but does not change with the droplet's viscous force. As the droplet's inertial force increases, the diameter of the regular bubble first increases and then decreases. The diameter of the regular bubble at We = 165 is almost four times that at We = 193. In addition, the peak size of the regular bubble decreases as the liquid viscosity increases, but increases as the droplet diameter increases. These variations in the regular bubble size are explained using a theoretical analysis. Finally, the effects of the inertial and viscous forces on the thresholds between no-bubble and regular-bubble pinch-off regimes are studied. As the Ohnesorge number increases from 1.37 × 10−2 to 2.03 × 10−2, the upper and lower critical Weber numbers for the occurrence of the regular bubble pinch-off can increase by 30 and 25, respectively. The theoretical boundaries of regular bubble pinch-off agree well with the boundaries observed in the experiments.

Surfactant-laden bubble bursting: Dynamics of capillary waves and Worthington jet at large Bond number

We present a numerical study of the main substages preceding aerosol formation via bursting bubbles: capillary wave propagation along the bubble, convergence at the bubble's apex, and the ascent of a Worthington jet and its breakup to release liquid drops. We focus on two crucial yet overlooked aspects of the system: the presence of surface-active agents and dynamics driven by non-negligible gravitational effects, quantified by the Bond number. Our results propose a mechanism explaining capillary wave retardation in the presence of surfactants, involving the transition from bi- to unidirectional Marangoni stresses, which pull the interface upwards, countering the motion of the waves. We also quantitatively elucidate the variable nature of the waves' velocity with various surfactant parameters, including surfactant solubility and elasticity, a departure from the constant behavior well documented in clean interfaces. Published by the American Physical Society 2024

Nonlinear flow phenomenon of a power-law non-Newtonian fluid falling down a cylinder surface

In this paper, we present a comprehensive study of the fingering phenomenon of a power-law non-Newtonian fluid falling down a cylinder surface. A theoretical analysis is firstly carried out and the governing equation describing the film thickness is established for the non-Newtonian fluid denoted by a power-law index n. Using the lubrication theory with dimensionless variables, the partial differential equation for the film thickness is derived, and both two-dimensional flow and three-dimensional flow are investigated. The traveling wave characteristic of the two-dimensional flow is displayed by using the finite difference scheme associated with a Newton iteration technique. The effect of changing the radius of the cylinder, precursor-layer thickness and power-law index is then considered through examining the variation of the capillary waves. Furthermore in three-dimensional complex flow, the fingering patterns for different parameters are simulated. The linear stability analysis based on the traveling wave solutions is given to elucidate the influence of different physical parameters, explaining the physical mechanism of nonlinear flow behaviors in two dimension and three dimension. Results from linear stability analysis show that the power-law index reinforces the fingering instability whether the fluid is shear-thinning or shear-thickening. Moreover, the numerical results illustrate that increasing the radii of the cylinder expands the unstable region. The flow profiles for different power-law coefficients are consistent with the result of the linear stability analysis, proving the unstable role of the power-law coefficient.

Scaling of turbulence-forced capillary waves

We simulate the propagation of capillary waves at the interface between two liquid layers of equal density, viscosity and thickness, which flow in stratified conditions inside a turbulent Poiseuille channel. This setup gives us the possibility to single out the effect of inertia and surface tension – in isolation from gravity – on the propagation of interfacial capillary waves. Numerical simulations, which are based on a combined pseudo spectral/phase field method, are run keeping the shear Reynolds number constant, Reτ=300, and considering four different values of the Weber number: We=0.5, We=1.0, We=1.5 and We=2.0. In the present case, waves are naturally forced by turbulence over a broad range of scales, from the larger ones, whose size is of order of the channel height h, down to the smaller dissipative scales. We show that, by increasing the Weber number, the interface is deformed more vigorously, and is prone to sustain the propagation of waves over a broader range of scales. Upon computation of the power spectra of wave elevation, we show that the behavior of waves agrees with the theoretical predictions given by the weak wave turbulence theory, which predicts a decay Sη(k)∼k−4 in the inertial regime, followed by a steeper decay at large k, in the surface-tension-dominated regime. We clearly show that the transition between the inertial and the surface-tension-dominated regime is pushed towards larger k by increasing the Weber number. In addition, we show that the two-dimensional frequency–wavenumber spectra of the wave elevation – the dispersion relation – is in good agreement with well established theoretical prediction, ω(k)∼k3/2, and that the range of validity of such prediction broadens for increasing Weber number.

A new mechanistic model to predict liquid loading in inclined natural gas wells

Accurately predicting the critical gas velocity and inclination angle is crucial for gas field development. However, the mechanism of liquid loading remains elusive due to insufficient attention given to the multivaluedness of flow structures and phase non-uniformity. This paper theoretically studied the flow pattern and gas-liquid configuration in wellbore based on the minimum energy principle. The Kelvin-Helmholtz instability of liquid film was analyzed, and a new correlation for the interfacial friction factor was proposed based on wave properties. The results indicated that the flow pattern transitioned from stratified to annular flow as the inclination angle increased. The gas-liquid interface presented a curved shape, with the interface curvature being influenced by various factors such as fluid properties, tube diameter, inclination angle, and gas and liquid velocities. As the inclination angle increases, the critical gas velocity first increases and then decreases, reaching a peak between 44° and 59°, aligning well with experimental and conventional results. Larger tube diameter, higher liquid density, and viscosity result in a higher critical gas velocity. Waves are more prone to develop in inclined pipes, and capillary waves can break up liquid film due to instability, leading to a discontinuous flow in inclined tubes. Film instability is exacerbated by a smaller diameter, higher liquid holdup, or greater inclination angle. Finally, a new model was devised to predict liquid loading in inclined natural gas wells, demonstrating better performance than existing models. In practice, the present model can achieve real-time prediction of critical gas velocity and continuously monitor the location of liquid accumulation in gas wells over the lifespan.

Surface light scattering: the capability of precisely measuring the interfacial properties of Newtonian fluids within the capillary wave transition region

This work employs a theoretical power spectrum of the dynamics of surface fluctuations in the frequency domain to elucidate the dissipation of capillary waves of fluids in the near-critical oscillation or transition region, seeking to resolve the problem of large discrepancies in viscosity and surface tension measurements in the region. The frequency-domain analysis approach was suggested, involving the discrete Fourier transform to transform the time domain data to the frequency domain and the fitting algorithm for the power spectrum. A total of three refrigerants, four alkanes and six fatty esters were chosen to examine the approach, with a range of the dimensionless number Y between 1.3 and 101.9. The findings demonstrate that frequency-domain approach is an effective way of dealing with dissipation issues for surface light scattering method when approaching the critical oscillation region, and thus could extend the technique for determining the viscosity and surface tension of Newtonian fluids to a sufficient wide viscosity range.

Surface properties of the seas of Titan as revealed by Cassini mission bistatic radar experiments

Saturn’s moon Titan was explored by the Cassini spacecraft from 2004 to 2017. While Cassini revealed a lot about this Earth-like world, its radar observations could only provide limited information about Titan’s liquid hydrocarbons seas Kraken, Ligeia and Punga Mare. Here, we show the results of the analysis of the Cassini mission bistatic radar experiments data of Titan’s polar seas. The dual-polarized nature of bistatic radar observations allow independent estimates of effective relative dielectric constant and small-scale roughness of sea surface, which were not possible via monostatic radar data. We find statistically significant variations in effective dielectric constant (i.e., liquid composition), consistent with a latitudinal dependence in the methane-ethane mixing-ratio. The results on estuaries suggest lower values than the open seas, compatible with methane-rich rivers entering seas with higher ethane content. We estimate small-scale roughness of a few millimeters from the almost purely coherent scattering from the sea surface, hinting at the presence of capillary waves. This roughness is concentrated near estuaries and inter-basin straits, perhaps indicating active tidal currents.