Capillary Waves Research Articles

Evaluating the forces involved in bubble management in DMEK surgery: mathematical and computational model with clinical implications.

To model postoperative forces involved in Descemet membrane endothelial keratoplasty (DMEK) tissue adherence and bubble management, including the impact of surface tension on graft support, with a view towards clinical applications. Tennent Institute of Ophthalmology, Glasgow, and James Watt School of Engineering, University of Glasgow, Glasgow, United Kingdom. Mathematical modelling and computer simulation. Theoretical modelling of biphasic flow and interaction of gas, liquid and tissue within the anterior chamber for static horizontal scenario A (adherent DMEK with mobile bubble) and dynamic vertical scenario B (release of bubble due to pupil block following DMEK). The model assumed incompressibility for both fluids within realistically achievable pressure ranges. Cahn-Hilliard Navier-Stokes equations were discretised through the application of the Finite Element Method. Mathematical modelling and computer simulation showed bubble size, corneal curvature and force intensity influences surface tension support for DMEK tissue in scenario A. Scenario B demonstrated complex, uneven distribution of surface pressure on the DMEK graft during uncontrolled bubble release. Uneven pressure concentration can cause local tissue warping, with air/fluid displacement via capillary waves generated on the fluid-air interface adversely impacting DMEK support. We have quantitatively and qualitatively modelled the forces involved in DMEK adherence in normal circumstances. We have shown releasing air/gas can abruptly reduce DMEK tissue support via generation of large pressure gradients at the liquid/bubble/graft interfaces, creating negative local forces. Surgeons should consider these principles to reduce DMEK graft dislocation rates via optimised bubble size to graft size, longer acting bubble support and avoiding rapid decompression where possible.

Traveling waves at the surface of active liquid crystals.

Active liquid crystals exert nonequilibrium stresses on their surroundings through constant consumption of energy, giving rise to dynamical steady states not present in equilibrium. The paradigmatic example of an active liquid crystal is a suspension of microtubule bundles powered by kinesin motor proteins, which exhibits self-sustained spatiotemporal chaotic flows. This system has been modelled using continuum theories that couple the microtubule orientation to active flows. Recently the focus has shifted to the interfacial properties of mixtures of active liquid crystals and passive fluids. Active/passive interfaces have been shown to support propagating capillary waves in the absence of inertia and offer a promising route for relating experimental parameters to those of the continuum theory. In this paper we report the derivation of a minimal model that captures the linear dynamics of the interface between an active liquid crystal and a passive fluid. We show that the dynamics of the interface, although powered by active flows throughout the bulk, is qualitatively captured by equations that couple non-reciprocally interface height and nematic director at the interface. This minimal model reproduces the dynamical structure factor evaluated from numerical simulations and the qualitative form of the wave dispersion relation seen in experiments.

A NEW PERSPECTIVE ON THE NONLINEAR DATE–JIMBO–KASHIWARA–MIWA EQUATION IN FRACTAL MEDIA

In this paper, we first created a fractal Date–Jimbo–Kashiwara–Miwa (FDJKM) long ripple wave model in a non-smooth boundary or microgravity space recorded. Using fractal semi-inverse skill (FSIS) and fractal traveling wave transformation (FTWT), the fractal variational principle (FVP) was derived, and the strong minimum necessary circumstance was attested with the He Wierstrass function. We have discovered two distinct solitary wave solutions, the square form of the hyperbolic secant function and the hyperbolic secant function form. Then, soliton solutions are cultivated through FVP and the minimum steady state condition. Finally, the influences of non-smooth boundaries on solitons were tackled, and the properties of the solution were demonstrated through three-dimensional contour lines. Fractal dimension can impact waveforms, but cannot affect their vertex values. The presentation of soliton solutions (SWS) using techniques is not only laudable but also noteworthy. The technique employed can also be used to investigate solitary wave solutions of other local fractional calculus partial differential equations.

Simulation of the Wave Turbulence of a Liquid Surface Using the Dynamic Conformal Transformation Method

The dynamic conformal transformation method has been generalized for the first time to numerically simulate the capillary wave turbulence of a liquid surface in the plane symmetric anisotropic geometry. The model is strongly nonlinear and involves effects of surface tension, as well as energy dissipation and pumping. Simulation results have shown that the system of nonlinear capillary waves can pass to the quasistationary chaotic motion regime (wave turbulence). The calculated exponents of spectra do not coincide with those for the classical Zakharov–Filonenko spectrum for isotropic capillary turbulence but are in good agreement with the estimate obtained under the assumption of the dominant effect of five-wave resonant interactions.

Dynamics of 2D fluid in bounded domain via conformal variables

AbstractIn the present work, we compute numerical solutions of an integro‐differential equation for traveling waves on the boundary of a 2D blob of an ideal fluid in the presence of surface tension. We find that solutions with multiple lobes tend to approach Crapper capillary waves in the limit of many lobes. Solutions with a few lobes become elongated as they become more nonlinear. It is unclear whether there is a limiting solution for small number of lobes, and what are its properties. Solutions are found from solving a nonlinear pseudodifferential equation by means of the Newton conjugate‐residual method. We use Fourier basis to approximate the solution with the number of Fourier modes up to .

Current modified higher-order Schrödinger equation of broader bandwidth capillary-gravity waves

A higher-order nonlinear Schrödinger equation of capillary-gravity waves for broader bandwidth on infinite depth of water including the effect of depth uniform current is established. The derivation is made from Zakharov's integral equation by extending the narrow bandwidth restriction to make it more suited for application to problems involving actual sea waves. On the basis of this equation, a stability analysis is made for uniform Stokes waves. After obtaining an instability condition, instability regions in the perturbed wave number space are displayed that are in good agreement with the exact numerical findings. It is found that the modifications in the stability characteristics at the fourth-order term are due to the interaction between the frequency-dispersion term and the mean flow. It is seen that the growth rate of sideband instability decreases due to the effects of both surface tension and depth uniform following currents. Significant deviations of the instability regions are observed between narrow-banded and broader-banded results. In addition, we have depicted the instability growth rate for the case of pure capillary waves.

Generation of Surface Liquid Flow in Channels by Capillary Vibrations and Waves

Generation of Surface Liquid Flow in Channels by Capillary Vibrations and Waves

The partial-rogue ripple solutions of nonlocal Kadomtsev–Petviashvili equation

The derivation of new exact solutions for integrable systems has always been an open problem in mathematical physics and engineering. In this paper, a large class of partial-rogue ripple solutions of a nonlocal Kadomtsev–Petviashvili [KP] equation are derived, which means that these ripple waves are semi-localized in time. Firstly, the Wronskian and Grammian determinant expression of the nonlocal KP equation are constructed. Then, three kinds of partial-rogue ripplon solutions of nonlocal KP equations, including (i) partial-rogue ripple soliton solutions, (ii) partial-rogue ripple lump chain solutions, and (iii) partial-rogue ripple lump solutions, are proposed. These solutions approach to the zero background plane as t→∞, which is the characteristic of partial-rogue ripple waves. These solutions are rarely reported, and their dynamics have been analytically and numerically discussed. The results of this paper are more practical and closely match the attenuated ripple waves stirred up in water waves. Additionally, the research method in this paper provides new references for existing results.

Jet from a very large, axisymmetric, surface-gravity wave

We demonstrate that gravity acting alone at large length scales (compared to the capillary length) can produce a jet from a sufficiently steep, axisymmetric surface deformation imposed on a quiescent, deep pool of liquid. Mechanistically, the jet owes it origin to the focusing of a concentric, surface wave towards the axis of symmetry, quite analogous to such focusing of capillary waves and resultant jet formation observed during bubble collapse at small scales. A weakly nonlinear theory based on the method of multiple scales in the potential flow limit is presented for a modal (single-mode) initial condition representing the solution to the primary Cauchy–Poisson problem. A pair of novel, coupled, amplitude equations are derived governing the modulation of the primary mode. For moderate values of the perturbation parameter $\epsilon$ (a measure of the initial perturbation steepness), our second-order theory captures the overshoot (incipient jet) at the axis of symmetry quite well, demonstrating good agreement with numerical simulation of the incompressible, Euler equation with gravity (Popinet 2014, Basilisk. http://basilisk.fr) and no surface tension. We demonstrate that the underlying wave focusing mechanism may be understood in terms of radially inward motion of nodal points of a linearised, axisymmetric, standing wave. This explanation rationalises the ubiquitous observation of such jets accompanying cavity collapse phenomena, spanning length scales from microns to several metres. Expectedly, our theory becomes inaccurate as $\epsilon$ approaches unity. In this strongly nonlinear regime, slender jets form with surface accelerations exceeding gravity by more than an order of magnitude. In this inertial regime, we compare the jets in our simulations with the inertial, self-similar, analytical solution by Longuet-Higgins (J. Fluid Mech., 1983, vol. 127, pp. 103–121) and find qualitative agreement with the same. This analysis demonstrates, from first principles, an example of a jet created purely under gravity from a smooth initial perturbation and provides support to the analytical model of Longuet-Higgins (J. Fluid Mech., 1983, vol. 127, pp. 103–121).

Dimensionless Groups by Entropic Similarity: II-Wave Phenomena and Information-Theoretic Flow Regimes.

The aim of this study is to explore the insights of the information-theoretic definition of similarity for a multitude of flow systems with wave propagation. This provides dimensionless groups of the form Πinfo=U/c, where U is a characteristic flow velocity and c is a signal velocity or wave celerity, to distinguish different information-theoretic flow regimes. Traditionally, dimensionless groups in science and engineering are defined by geometric similarity, based on ratios of length scales; kinematic similarity, based on ratios of velocities or accelerations; and dynamic similarity, based on ratios of forces. In Part I, an additional category of entropic similarity was proposed based on ratios of (i) entropy production terms; (ii) entropy flow rates or fluxes; or (iii) information flow rates or fluxes. In this Part II, the information-theoretic definition is applied to a number of flow systems with wave phenomena, including acoustic waves, blast waves, pressure waves, surface or internal gravity waves, capillary waves, inertial waves and electromagnetic waves. These are used to define the appropriate Mach, Euler, Froude, Rossby or other dimensionless number(s)-including new groups for internal gravity, inertial and electromagnetic waves-to classify their flow regimes. For flows with wave dispersion, the coexistence of different celerities for individual waves and wave groups-each with a distinct information-theoretic group-is shown to imply the existence of more than two information-theoretic flow regimes, including for some acoustic wave systems (subsonic/mesosonic/supersonic flow) and most systems with gravity, capillary or inertial waves (subcritical/mesocritical/supercritical flow). For electromagnetic wave systems, the additional vacuum celerity implies the existence of four regimes (subluminal/mesoluminal/transluminal/superluminal flow). In addition, entropic analyses are shown to provide a more complete understanding of frictional behavior and sharp transitions in compressible and open channel flows, as well as the transport of entropy by electromagnetic radiation. The analyses significantly extend the applications of entropic similarity for the analysis of flow systems with wave propagation.

Effect of Surface Tension Relaxation on the Stability of the Charged Jet

In the asymptotic calculations of the first order of smallness by the dimensionless amplitude of capillary waves on the surface of charged jets of polar liquid, the effect of the relaxation effect of surface tension on the regularities of their implementation is investigated. Calculations are carried out on the model of an ideal non-compressible electrically conductive fluid. It has been shown that taking into account the effect of dynamic surface tension leads to an increase in the order of the dispersion equation, which has another damping root, which is obliged to destroy the near-surface double electric layer (destruction of the ordering of polar molecules in the near-surface layer), which undergoes electrostatic instability at sufficiently large charges (pre-breakdown in the sense of ignition of corona discharge in air). In the ideal fluid mathematical model used, the relaxation motion of the jet surface disturbances that occurs when the surface tension relaxation effect is turned on and the attenuation decrements of capillary wave motions are purely of a relaxation nature.

Droplet entrainment characteristics of HTPB/Paraffin blended fuels for hybrid rocket motors

Droplet entrainment contributes to excellent regression rates of HTPB/paraffin blended fuels in hybrid rocket motors, but also leads to more complicated mass transfer mechanisms in application. The liquid film dynamic behaviors and droplet entrainment mechanisms of the fuels in a windowed two-dimensional slab facility under hot gas stream were investigated using the backlight method, image processing technique and fast Fourier transform. The results showed that the liquid film with a thickness of less than 3 mm could be formed on the fuel surface, and the gas-liquid interface experienced rolling and ripple waves before evolving into singlet disturbance-crests, bags and ligaments, the sources of entrained droplets. Moreover, interfacial waves propagated periodically at frequencies within 100 Hz, which was stimulated by gas shear stress and inhibited by liquid film viscosity damping, thus the regression rate was positively correlated with the temperature and mass flux of hot gas stream, while negatively correlated with the chamber pressure and the HTPB concentration. Additionally, the burst of the main part and the rim of bag provided polarized droplet diameters among the three droplet entrainment mechanisms involving the ligament break-up and the shearing-off of crest. Therefore, the averaged droplet diameter and its coefficient of variation even reduced to 47.98 μm and 0.48 with the disappearance of bag break-up, when the HTPB content increased to 40 wt%.

On the role of unsteadiness in impulsive-flow-driven shear instabilities: precursors of fragmentation

Shear instabilities at the interface of two fluids, such as classical Kelvin–Helmholtz instability (KHI), is the precursor of interface destabilization, leading to fluid fragmentation critical in a wide range of applications. While many insights into such instabilities are derived for steady background forcing flow, unsteady impulse flows are ubiquitous in environmental and physiological processes. Yet, little is understood on how unsteadiness shapes the initial interface amplification necessary for the onset of its topological change enabling subsequent fragmentation. In this combined theoretical, numerical and experimental study, we focus on an air-on-liquid interface exposed to canonical unsteady shear flow profiles. Evolution of the perturbed interface is formulated theoretically as an impulse-driven initial value problem using both linearized potential flow and nonlinear boundary integral methods. We show that the unsteady airflow forcing can amplify the interface's inherent gravity–capillary wave, up to wave-breaking transition, even if the configuration is classically KH stable. For impulses much shorter than the gravity–capillary wave period, it is the cumulative action, akin to total energy, that determines amplification, independent of the details of the impulse profile. However, for longer impulses, the details of the impulse profile become important. In this limit, akin to a resonance, it is the entangled history of the interaction of the forcing, i.e. the impulse, that changes rapidly in amplitude, and the response of the oscillating interface that matters. The insights gained are discussed and experimentally illustrated in the context of interface distortion and destabilization relevant for upper respiratory mucosalivary fluid fragmentation in violent exhalations.

A Comparative Analysis of Conductance Probes and High-Speed Camera Measurements for Interfacial Behavior in Annular Air-Water Flow.

Different techniques are used to analyze annular flow, but the more interesting ones are those techniques that do not perturb the flow and provide enough resolution to clearly distinguish the interfacial phenomena that take place at the interface, especially the disturbance waves (DW) and the ripple waves (DW). The understanding of these events is important because it influences the heat and mass transfer taking place through the thin film formed near the walls in this flow regime. The laser-induced fluorescence (LIF) and the three-electrode conductance probe are two commonly used techniques to study experimentally annular flow phenomena. In this paper, a set of experiments at different temperatures of 20 °C, 30 °C and 40 °C and different liquid Reynolds numbers have been performed in the annular flow regime, the characteristic of the DW and RW as average height and frequency of these waves has been measured by both techniques LIF and conductance probes. In addition, we also measured the mean film thickness. It was found that the mean film thickness and the DW height are practically the same when measured by both techniques; however, the height of the RW is smaller when measured by the conductance probe and this difference diminishes when the temperature increases.

Exact solution for capillary waves on the surface of a liquid of finite depth

Using the Schwartz function method, we have obtained a new exact solution for the problem of stationary capillary waves of finite amplitude on the surface of a liquid that has a finite depth. The reliability of the solution was confirmed by the results of numerical verification of the main boundary equation. The obtained solution of the problem is general in the sense that for any Weber number one can find the corresponding wave configuration. Parametric analysis showed a nonmonotonic dependence of the wave-length and its amplitude on the Weber number. The fact that the problem has one more branch of the solution (the trivial solution) indicates the possibility of the existence of other branches. The Schwartz function method cannot guarantee finding all solutions of the problem even from the specified class of functions. Therefore, the question of reproducing the known exact solution of W. Kimmersley for this problem and its reliability remains open. Note that for the parameter ß included in the main boundary equation, W. Kimmersley preliminarily laid down assumption ß = 1. The found exact solution has the property that ß > 1 and cannot coincide with W. Kimmersley’s solution.

Gas entrapment and pore formation in metal droplet-based 3D printing

Gas entrapment behavior during metal droplet deposition tends to form pore defects and discourages interfacial bonding between droplets, resulting in lower densities and poorer performance of printed components. Moreover, the gas entrapment process is influenced by the interaction of gas-liquid two-phase flow and non-isothermal solidification, which makes the interfacial behavior of droplets challenging to identify. Therefore, gaining insight into the gas entrapment behavior and pore formation mechanism at the bottom of molten droplets is essential. This paper established a model of two-dimensional axisymmetric numerical by combining the fluid volume tracking method and heat transfer solidification model. Single metal molten droplet deposition experiments verified the accuracy of the model. The evolution of solid phase fraction, velocity field, pressure field, and contact surface heat flux during the deposition of the molten droplet at low Weber numbers (0.1-10) was investigated comprehensively. The mechanisms of gas entrapment and pore formation were revealed. The effects of molten droplet impact velocity, contact angle, and surface tension on the gas film evolution process and pore morphology were further investigated. The results show that decreasing the droplet impact velocity causes capillary waves to contact the substrate surface and form local solidification, allowing the gas disk to evolve into central and two peripheral pores. Moreover, the pore breaks away from the substrate and remains inside the droplet when decreasing the droplet contact angle to 50°. Furthermore, the rate of gas disk retraction decreases as surface tension reduces, which may accentuate the resistance effect of the contact surface solidification layer on the gas disk retraction, thus changing the final morphology of the pores. These findings elucidate the mechanism of pore formation at the bottom of molten metal droplets and further understand the interfacial behavior of droplets in 3D-printed metal structures.

Investigation of capillary waves and thermal diffusion at the air–liquid interface of imidazolium-cation-based ionic liquids using transient grating spectroscopy

The capillary wave and thermal diffusion at the gas–liquid interface of 1-alkyl-3-methylimidazolium bis(trifluoromethylsulfonyl)amide [Cnmim][NTf2] (n = 4, 8, 12; where n indicates the number of carbons in the alkyl chain) were investigated using interface-selective transient grating (TG) spectroscopy. At the interface, the TG signal exhibited capillary wave damping (with no capillary wave oscillation). Moreover, the capillary wave damping constant was greater than that predicted by the hydrodynamic theory; the deviation increased with the cation alkyl-chain length. The thermal diffusivities at the gas–liquid interface of [C8mim][NTf2] and [C12mim][NTf2] were lower than those in the bulk, whereas both the values at the interface and in the bulk were similar for [C4mim][NTf2]. The difference is possibly owing to the formation of a layered structure at the interface.

On the structure of parasitic gravity-capillary standing waves in the small surface tension limit

We present new numerical solutions for nonlinear standing water waves when the effects of both gravity and surface tension are considered. For small values of the surface tension parameter, solutions are shown to exhibit highly oscillatory capillary waves (parasitic ripples), which are both time- and space-periodic, and which lie on the surface of an underlying gravity-driven standing wave. Our numerical scheme combines a time-dependent conformal mapping together with a shooting method, for which the residual is minimised by Newton iteration. Previous numerical investigations typically clustered gridpoints near the wave crest, and thus lacked the fine detail across the domain required to capture this phenomenon of small-scale parasitic ripples. The amplitude of these ripples is shown to be exponentially small in the zero surface tension limit, and their behaviour is linked to (or explains) the generation of an elaborate bifurcation structure.

СОВЕРШЕНСТВОВАНИЕ МЕТОДИКИ ПРОЕКТИРОВАНИЯ ПОЧВОЗАЩИТНЫХ ТЕХНОЛОГИЙ НА СКЛОНОВЫХ АГРОЛАНДШАФТАХ

The main purpose of the work is to clarify the values of soil erosion resistance. In the existing methods for assessing the erosion resistance of soils, the kinetic and potential energy of the water flow is studied. It is believed that the decrease in total energy in the process of flowing along the channel is associated with the work of the water flow in destroying and washing away the soil. However, this does not take into account the energy of capillary waves existing on the water surface during laminar flow. The frequency of the capillary waves is usually such that the waves are not visible to the naked eye and are therefore not taken into account, although their energy is, in order of magnitude, comparable to the energy of the flow. Refinement of the value of erosion resistance is associated with the ability to evaluate and take into account the energy of capillary waves. The work is related to the development of a methodology for accounting and evaluating the contribution of the energy of capillary waves to the total energy of a water microflow, which has a destructive and eroding effect on the soil of a sloping agricultural landscape. The paper developed a mathematical model for estimating the energy of capillary waves, on the basis of which an expression was obtained to determine the ratio of the energy of capillary waves formed on the surface of a water flow to the kinetic energy of the water flow. To experimentally study the process of development of capillary waves on smooth and rough surfaces and determine the operating parameters, an installation was created in the form of a rectangular-section tray with an adjustable inclination angle, adjustable intensity of water supply and replaceable working surfaces. The geometry of the microflow and the shape of its surface were determined from the results of video filming and the readings of the laser rangefinder. In the range of regime parameters studied in this work, the numerical estimation of the dependence of the energy of capillary waves on the kinetic energy of the water flow for the first time revealed a significant (up to 40–60%) contribution of capillary waves formed on the surface of microflows to the total energy of the water flow. The results obtained in this work indicate the need to take into account the contribution of capillary waves in the energy analysis of the initial stage of water erosion on slopes.

Experimental investigation of three-dimensional free-surface and interfacial sloshing in a vertical cylindrical tank

In the present study, hundreds of experiments have been conducted on the three-dimensional free-surface and interfacial sloshing in a vertical cylindrical tank containing two immiscible liquids. The bounds of different free-surface and interfacial wave regimes are determined by maintaining fixed excitation amplitude and slowly increasing excitation frequency until another type of wave regime began to appear. In general, three types of the free-surface wave regimes are observed when the excitation frequency is in the neighborhood of the lowest natural frequency of the free surface, i.e., planar gravity wave, chaotic gravity wave, and swirling gravity wave. Similarly, when the excitation frequency is near the lowest natural frequency of the internal interface, three types of interfacial wave regimes, i.e., planar gravity wave, chaotic gravity-capillary wave, and swirling gravity-capillary wave, are generated. Besides, it is worth pointing out that when the excitation frequency is near the lowest natural frequency of the internal interface as well as very close to a third of the lowest natural frequency of the free surface, large-amplitude rotating wave motion occurs at both the free surface and the internal interface. This is due to even though the excitation frequency is far away from the natural frequency of the free surface, the secondary resonance can still become dominant and lead to large-amplitude motion of the free-surface rotating wave and subsequently influences the internal interface. This paper reveals that the sloshing behaviors of two-layer liquid in the vertical cylindrical tank are much more complicated than those of single-layer liquid.