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High-resolving photo and videorecording was used to visualize the fine structure of the flow pattern upon the coalescence of a droplet, $$0.39 < D < 0.43$$ cm in diameter, falling at a velocity $$3.1 < U < 3.9$$ m/s with water in a reservoir. The spreading of droplets of water, water solution of ink, and copper and iron sulphates is studied. For all pairs of substances, in addition to cavities, crowns, and sheets with shaped outer edges, capillary waves radiated by the annular region of fluid coalescence and thin jetlets at the cavity bottom and the crown walls were observable in the splash formation regime. The jetlets penetrate the sheet and emerge from the tooth tips in the form of spikes. Sequences of small droplets fly out from the spike tips, whose positions change with time. The fluid acceleration is favored by the conversion of the available potential energy upon the annihilation of the free surfaces of the coalescing fluids.

The dispersion relation for capillary waves of an arbitrary azimuthal symmetry on an elliptic cross-section jet in a uniform electrostatic field which is perpendicular to the axis of symmetry of the jet is derived. The stability of the first three azimuthal modes is investigated. It is shown that the stability of capillary waves on the jet reduces with increase in the external electrostatic field strength and the related eccentricity of transverse jet cross-section, as well as with increase in the polarization charge on the jet surface. Comparison with a jet in a radial electrostatic field is carried out. In the case of the transverse electrostatic field the instability growth rates of the capillary waves of any symmetry on the jet surface are turned out to be significantly higher than those in the case of the radial electrostatic field.

Combined high-resolution optic and acoustic methods are used for the first time to synchronously detect the fine structure of processes initiated by submersion of a freely falling water drop in a tank with degassed liquid. The emphasis is on analyzing the evolution of the fine structure of flow pattern, the accompanying capillary waves, and the amplitude–frequency characteristics of sound packets. The sequence of acoustic signals includes a shock pulse, which is caused by the initial contact of the drop with the surface of the liquid at rest, and a series of consecutive sound packets. The shock pulse being reliably reproduced in experiments has a complex structure, which includes short segments of high-frequency oscillations and longer segments of low-frequency oscillations. The number of emitted sound packets and their parameters vary in an arbitrary way from one experiment to another while the size of drop and the distance of fall remain invariable. A series of experiments reveals cases of single and secondary packet emission with different spectral compositions of packets, as well as cases of their total absence. Analysis of data is performed with allowance for the effect of the “double layer”, which replaces the coalescence surface of liquids and retains the disturbances caused by transformation of available potential surface energy to other forms, such as thermal energy, pressure disturbance, and energy of fine currents actively affecting the gas cavities.

Experimental data are used to consider the behavior of gravity–capillary waves in the regions of a sharp change in the surface tension (phase transitions in films). The classical theory of surface waves is applied to systems with a model dependence of the surface tension on the surfactant concentration. The frequency shift in the regions of a sharp change in the surface tension is shown to remain small at all wavelengths and surface tensions used in experiments. However, the attenuation in these regions increases noticeably as compared to constant surface tension, which can be used to detect such transitions experimentally.

International audience; A new type of instability of standing gravity—capillary waves has been detected under the irradiation of the liquid surface by a continuous plane ultrasonic beam with a constant amplitude. At an intensity of 1.03-MHz ultrasound above a certain threshold, the excitation of one of the eigenmodes of gravity—capillary waves at a frequency of 6.56 Hz has been observed. The time of establishment of the stationary amplitude of oscillations of the surface significantly exceeds the intrinsic time of relaxation of the mode and decreases with an increase in the threshold excess. The scenario of development of instability is similar to an inverse energy cascade.

The theoretically calculated dependence of the sizes of the main and satellite droplets on the dimensionless wave number of the perturbation of the surface of the Newtonian fluid jet is proved experimentally. It is also shown that an increase in the fluid viscosity significantly changes the dynamics of development of capillary waves in the jet and the mechanism of formation of satellite droplets.

We have studied the instability increments of capillary waves with an arbitrary symmetry (arbitrary azimuthal numbers m) on the surface of a space-charged cylindrical jet of an ideal incompressible dielectric liquid moving relative to an ideal incompressible material dielectric medium. It is shown that for not very high jet velocities, the axisymmetric mode (m = 0) is the first to become unstable upon an increase in the space charge density, and then the kink mode (m = 1) and flexural-strain mode (m = 2) exhibit instability. It is this sequence of realization of instability of azimuthal modes that determines the regularities of charged jet dispersion in experiments. For jet velocities comparable with those critical for aerodynamic instability realization, the mode with m = 1 is the first to lose its stability. The dependences of maximal increments on wavenumbers have been determined for all azimuthal modes.

The surface flows of a liquid in the presence of a partially submerged vibrating plate are investigated. In the case of low-frequency plate vibrations, two vortices are formed on the surface of the liquid on each side of the plate in which liquid particles move away from plate surfaces. Under parametric excitation of transverse capillary waves near the boundary of the liquid layer wetting the plate surface, the flow direction in vortices becomes opposite. High-frequency vibrations of the plate create additional secondary vortices on the liquid surface.

The processes of the formation and decay of vortex motion on the surface of water are studied experimentally. The vortex motion is formed by capillary waves, which are generated as a result of harmonic oscillations of a cell with water in the vertical direction. It is shown that, at oscillation amplitudes smaller than the critical value, a vortex lattice with a period that is equal to the excitation wavelength is formed. The characteristic time of the formation and decay of vortex motion is close to the time of viscous damping of the pumping wave. At pumping amplitudes that are greater than the critical value, a vortex lattice is initially formed on the surface; then it is destroyed gradually by large-scale movements. The energy distribution over the wave vector can be described by a power function that is close to k–5/3. It is suggested that large-scale vortex motion arises as a result of the interaction of low-frequency waves appearing on the surface as a consequence of the nonlinear interaction of waves.

The problem of the stability of capillary waves on the surface of a charged jet of an ideal incompressible electroconducting liquid, which moves with respect to a material dielectric medium, is considered. There is a tangential discontinuity of the velocity field on the interface between the media. Solutions to the problem in two idealized models have been compared, i.e., when the jet has a finite and infinite length. It has been shown that the instability increments and the wave numbers of the most unstable waves, computed in both models, are linearly related, and velocity of motion of the jet acts as a coefficient of proportionality.

Results of laboratory studies of the damping of gravity–capillary waves on a water surface covered with a film of petroleum product (diesel fuel) in a wide range of change in a film thickness are presented. A nonmonotonic dependence (with a local maximum) of a damping coefficient on film thickness is discovered. Numerical calculations of the dispersion equation for gravity–capillary waves in the presence of a viscoelastic film of arbitrary thickness, which confirmed the existence of the maximum of the damping coefficient as a function of film thickness, are performed. Based on a comparison of the calculation results and the data of laboratory measurements of wave characteristics, the values of parameters in the diesel fuel films are estimated in a wide range of a change in their thickness.

The laws of implementation of electrostatic instability of the surface of a cylindrical volume charged jet of an ideal incompressible dielectric liquid moving relative to the ideal incompressible dielectric medium and the stability of bending-deformation capillary waves developed on the surface are investigated analytically. It is found that there are thresholds for the critical conditions of implementation of the instability with respect to the jet velocity relative to the medium (Weber number) and with respect to the electric space charge (relative to the ratio of the electrostatic pressure on the jet surface to the Laplace pressure). The critical analytic dependence between these dimensionless parameters is found.

We have derived and analyzed the dispersion equation for capillary waves with an arbitrary symmetry (with arbitrary azimuthal numbers) on the surface of a space-charged cylindrical jet of an ideal incompressible dielectric liquid moving relative to an ideal incompressible dielectric medium. It has been proved that the existence of a tangential jump of the velocity field on the jet surface leads to a periodic Kelvin–Helmholtz- type instability at the interface between the media and plays a destabilizing role. The wavenumber ranges of unstable waves and the instability increments depend on the squared velocity of the relative motion and increase with the velocity. With increasing volume charge density, the critical value of the velocity for the emergence of instability decreases. The reduction of the permittivity of the liquid in the jet or an increase in the permittivity of the medium narrows the regions of instability and leads to an increase in the increments. The wavenumber of the most unstable wave increases in accordance with a power law upon an increase in the volume charge density and velocity of the jet. The variations in the permittivities of the jet and the medium produce opposite effects on the wavenumber of the most unstable wave.

The experimental results of studying wave turbulence in a system of capillary waves on the surface of water in a cylindrical vessel are presented. Waves on the surface are excited by vertical vibrations of the water-filled vessel with accelerations below the parametric-instability threshold at a fixed frequency f p or in the frequency band from 30 to 50 Hz. The deviation of the surface from equilibrium is recorded by a laser beam reflecting from water. It is shown experimentally that the position of the high-frequency boundary of the inertial range f b of the turbulent cascade is shifted towards high frequencies with an increase in the pump amplitude А and described by the power function f b ~ A β with index β equal to 1.2 ± 0.1 for monochromatic pumping and 1.10 ± 0.15 for broadband pumping. The value of the index β for monochromatic pumping is close to the theoretical estimate; for broadband pumping, it is less by a factor of 2.5. In the dissipation region f > f b , the turbulent distribution is damped exponentially with the characteristic frequency f d depending on the amplitude and spectral characteristic of the excitation force. It follows from the amplitude dependences that the frequency f d is proportional to the frequency of the boundary of the inertial range f b .

We calculate the radiation forces in a spherical drop lying on a solid substrate. The forces form as a result of the action of a capillary wave on a fluid as it propagates along the free spherical surface. We study the structure of acoustic currents excited by the radiation forces.

In the domain of capillary waves, a bicubic dispersion relation is derived and analyzed for surface and internal capillary–gravitational waves in a three-layer liquid with a free surface. It is shown that the ratio of the internal wave amplitudes to the surface wave amplitudes is fairly large if the trivial condition of a “homogeneous liquid” is discarded. The amplitude ratio between the internal waves themselves (generated at different interfaces) may be both greater and smaller than unity depending on the physical parameters of the system. Specifically, it strongly depends on the densities of the layers and their thicknesses.

The dependences of the frequency and damping rate of a potential surface wave on the wavenumber and the degree of anisotropy of a bi-Maxwellian electron distribution characterized by different temperatures along and across the plasma surface are established. It is demonstrated that the influence of electron thermal motion along the plasma surface on the surface wave properties is similar to the influence of thermal motion on the properties of a bulk Langmuir wave. On the contrary, thermal motion across the surface qualitatively affects the dispersion relation and substantially increases the damping rate.

A fine structure of the flows developing during primary contact of freely falling drops with a deep quiescent fluid is studied using the macrophotography and high-speed video filming methods. Water drops falling in water, alcohol, and oil, as well as drops of oil, petroleum, and aqueous solutions of salt or alcohol falling in water are investigated. The work is focused on the visualization of the finespray scattering from the primary contact area. The collisions of small droplets with the surface of the submerging drop are first recorded. The direction of the spray and streamer scattering is determined by the surface tension coefficients of the coalescing liquids. The conditions under which the spray droplets collide with the drop surface are determined.

The wave motion in a cylindrical layer of an ideal conducting liquid on a hard rod kept at a constant electrical potential is calculated accurate to the first order of smallness in dimensional perturbation of the free surface. The instability of the free surface is also considered. A dispersion relation is derived. It is shown that the range of instability waves depends on only the electric field strength near the free surface and the instability increments of capillary waves decrease as the layer gets thinner. The influence of the hard rod becomes tangible only when its radius becomes comparable to the thickness of the liquid layer.

In a laboratory pool, the fine structure of flows arising from the primary contact of freely falling droplet with a liquid at rest is investigated by the methods of macrophotography and high-speed videotaping. Primary attention is paid to visualization of short capillary waves on the droplet surface formed from the impact of small splashes. The angular positions of the trajectories of splashes determine the values of the surface-tension coefficients of the liquids of the droplet and the accepting environment. The conditions under which the splashes hit the droplet surface are determined.

We calculate the field of radiation forces in a cylindrical fluid layer on a solid substrate formed as a result of the action on a fluid of a capillary wave propagating from the axis along a free surface. We study the structure of acoustic flows excited by the radiation forces. We discuss the action of flows on small-sized particles and the possibilities of these particles to form ordered structures.

Two-dimensional wave turbulence on the charged surface of liquid hydrogen in a rectangular cell has been studied. It has been found that, after intense monochromatic pumping at frequencies of 14–30 Hz in an electric field partially compensating the gravitational field of the Earth, incommensurate subharmonics at low frequencies until 0.8 Hz can be observed on the surface of the liquid with a certain time delay in addition to the direct turbulent cascade of Kolmogorov-Zakharov capillary waves. The nonlinear interaction of these subharmonics with each other and with a wave excited at the pump frequency results in the appearance of additional combination waves in the direct turbulence spectrum.

The spectra of capillary waves were studied for bidistilled water and a NADH aqueous solution, and their dynamics under electromagnetic irradiation in the millimeter range was determined at different space frequencies of the surface wave k.

We consider nonlinear gravity-capillary waves with the nonlinearity parameter ɛ ∼ 0.1–0.25. For this nonlinearity, time scale separation does not occur and the kinetic wave equation does not hold. An energy cascade in this case is built at the dynamic time scale (D-cascade) and is computed by the increment chain equation method first introduced in [15]. We for the first time compute an analytic expression for the energy spectrum of nonlinear gravity-capillary waves as an explicit function of the ratio of surface tension to the gravity acceleration. We show that its two limits—pure capillary and pure gravity waves on a fluid surface—coincide with the previously obtained results. We also discuss relations of the D-cascade model with a few known models used in the theory of nonlinear waves such as Zakharov’s equation, resonance of modes with nonlinear Stokes-corrected frequencies, and the Benjamin-Feir index. These connections are crucial in understanding and forecasting specifics of the energy transport in a variety of multicomponent wave dynamics, from oceanography to optics, from plasma physics to acoustics.

The processes of formation of alcohol and water drops, as well as formation of small fragments—satellites, are traced using the high-speed filming. The trajectory of a water drop satellite is nonmonotonic, at first the satellite moves upward against the gravity force, reaches the oscillating residual fluid at the nozzle exit, and then starts to move down. From the satellite, a microdroplet is ejected, which bounces off the residual fluid at the nozzle, returns back to the satellite and merges. In the case of an alcohol drop, no accompanying microdroplet is formed, and the satellite follows a nearballistic trajectory.

Packets of capillary waves on the surface of a horizontal water layer generated at coalescence with a layer of microdroplets (with a characteristic diameter of 50–100 μm) of the dissipative structure “droplet cluster” have been detected by high-speed video recording and the schlieren method. The shortest experimentally observed waves have a length of about 130 μm and their phase velocity exceeds 1.8 m/s. It has been found that the coalescence of a single drop initiates a self-sustained wave process, which induces the coalescence of hundreds of droplets in a time shorter than 1 ms, which form a cluster.

Investigation of damping of gravity-capillary waves (GCWs) in the presence of turbulence is a classical hydrodynamic problem which has important geophysical applications, one of which is related with the problem of forming a radar and optical image of a ship wake on wavy water surface. In this work a new method for the laboratory study of surface wave damping in turbulized liquid is described and the results are presented. The damping of standing GCWs by turbulence on the water surface in a tank mounted on a vibration table is studied. GCWs and turbulence are excited using a two-frequency mode of vibration table oscillations. A high-frequency small amplitude signal is used for parametric GCW excitation; a low-frequency large amplitude signal is used for generating turbulence due to water flowing through a fixed perforated grid submerged into the tank. The coefficient of GCW damping is determined by measured threshold of parametric excitation of the waves; turbulence characteristics are determined by the PIV and PTV techniques. Dependences of GCW damping coefficients on their frequency at different turbulence intensities are obtained, estimates for turbulent viscosity are presented, and a comparison with empirical models proposed earlier is performed.

A small lying drop is investigated considering the linear tension at the three-phase boundary under thermodynamic and mechanical equilibrium. Formulas for the surface tension at the liquid-vapor interface, the linear tension, and the contact angle are found by the Gibbs dividing surface method. It is shown that the surface tension and the contact angle decrease while the absolute value of the linear tension increases with decreasing radius of the surface curvature. Comparison of theoretical predictions with experimental data obtained by atomic-force microscopy shows a satisfactory agreement and evidences that the linear tension must be negative.

The field of radiation forces in a fluid layer on a solid substrate is calculated. This field is formed during propagation of surface capillary wave along a free surface. The wave is excited by sub ...

Analytically, in the second order approximation, the nonlinear internal resonance energy exchange between waves on the surface of a cylindrical incompressible dielectric fluid jet that moves relative to a material medium in the longitudinal electrostatic field is investigated.

The damping ratios of waves and oscillations in nonlinear dispersion equations are found for planar, cylindrical, and spherical geometries as applied to finite-volume liquids. For a cylindrical jet and a plane interface between viscous liquids, the damping ratios are determined for the first time. When the radius of curvature of the liquid jet surface decreases, so does the damping ratio of capillary waves. In a system of immiscible liquids, the damping ratio may be both larger and smaller than that for the pure liquid depending on the viscosity of the liquids and the ratio of their densities. This is because the damping ratio depends on the kinematic viscosities of pure liquids. The damping ratio is also estimated for waves arising at the liquidgas interface due to a tangential discontinuity of the velocity field.

The influence of a solid bottom on the “dead water” effect for gravitational and fluctuation waves (having the same dispersion law), as well as capillary waves, is calculated in a linear statement. It is found that the presence of a solid bottom aggravates the dead water effect.

Laboratory study results on the damping of gravity-capillary waves (GCWs) on water covered with oil slicks are presented. The nonmonotonic behavior (the presence of a local maximum) of the dependence of the damp coefficient on the slick thickness has been ascertained. The dispersion relation for GCWs in the presence of a viscoelastic slick of arbitrary thickness has been numerically calculated; this confirmed the presence of a damp coefficient maximum as a function of slick thickness. The parameters of oil slicks (interfacial elasticity and viscosity) have been estimated in a wide range of their thickness from a comparison of the calculation results with the data from laboratory measurements of wave parameters on water covered with an oil slick.

The possibility of degenerate internal nonlinear resonance interaction between capillary waves with arbitrary symmetry (arbitrary azimuthal numbers) on the surface of a charged cylindrical jet of an ideal incompressible conducting liquid is demonstrated. The jet moves in an ideal incompressible dielectric medium collinearly with an external uniform electrostatic field. It is shown, in particular, that six different resonance situations take place for axisymmetric waves in which primary waves and waves due to the nonlinearity of the equations of hydrodynamics exchange energy.

In the dissipative range at frequencies above the inertial frequency range, the turbulent cascade of capillary waves on the surface of liquid helium and hydrogen decays according to an exponential law. The characteristic frequency of the quasi-Planck distribution is determined by the spectral characteristic of an exciting force. In the case of harmonic pumping on the surface of superfluid helium in the discrete turbulence regime, energy condensation is observed near the high-frequency edge of the inertial range. The effect is due to the influence of discreteness in the spectrum of the eigenfrequencies of surface excitations and in the turbulence distribution on the energy transfer through the cascade.

Among fast natural phenomena, exchange pro� cesses involving the atmosphere and the ocean are of particular interest. These processes include the inter� action of rainfall (first of all, rain) with the oceanic surface. The kinetic energy of a rain droplet and the surface energy that becomes free once the droplet has been absorbed result in a series of hydrodynamic phe� nomena. These are formation of a cavern, a cumula� tive jet, streamers, regular capillary waves, and irregu� lar waves. The formation and fillingin of the caverns are accompanied by the detachment of air cavities of both regular and irregular shape, with a part of them emitting sound. The acoustic rain noise generated by falling drop� lets forms the total sonic background. This is an important source of information on the intensity and localization of rain, which is a significant fraction of the total ocean acoustic noise (1). In records of under� water signals obtained within the rain period, the spec� tral peak between 10 and 15 kHz is surely localized (2). Modernization of experimental methods has opened the possibility to measure fluctuations of sound pressure in water, which arise as a result of the fall of both the set of (rain) droplets and an individual droplet with the simultaneous registration of the flow pattern (2). Among competing mechanisms of sound emission, the impact pulse, hydrodynamic impact, and aircavity resonances were analyzed. In experi� ments, along with the sound registration, air bubbles were always observed. Therefore, basic attention was paid to the resonance mechanism for which the soundemission frequency was connected to the bub� ble size by a simple calculation relationship (3). The available experimental data made it possible to obtain a diagram of flow modes in the plane of the dropletfall velocity as a function of the droplet diam� eter. In this plane, a region of stable registration is iso� lated for gas bubbles of the size corresponding to the spectralpeak frequency (4). However, for the reso� nance mechanism, the question remained open on the transformation of the fallingdroplet energy to that of bubble resonance vibrations accompanied by sound emission. In further experiments, it was noted that gas cavities emit underwater sound in the course of recon� structing their irregular shape (with tapers and folds) to the equilibrium (spherical) one. This process is accompanied by the liberation of the potential surface energy (5). The analysis of the processes allows for the fact that a falling droplet leads to accepting the substance liq� uid, the momentum, and the mechanical energy, which results in generating visible and latent flows of different scales. Caverns, coronas, cumulative jets, and streamers correspond to visible flows and manifest themselves in the changing the shape of the free sur� face (6). Largescale latent flows cause chaotic displace� ments of gas bubbles in the bulk of the liquid. The vir� tually unstudied finestructure latent flows manifest themselves in peculiarities of both the freesurface shape and the substancetransport pattern near the interface. They also determine the properties of emit� ted sound packets. In the present paper, for the first time, we thoroughly study the frequencytime param� eters of sound signals initiated in air and water by freely falling droplets.

A dispersion relation is derived for capillary waves with arbitrary symmetry (with arbitrary azimuthal numbers) on the surface of a jet of an ideal incompressible dielectric liquid moving in an ideal incompressible dielectric medium along an external uniform electrostatic field. A tangential discontinuity in the velocity field on the jet surface is shown to cause Kelvin-Helmholtz periodical instability at the interface and destabilize axisymmetric, flexural, and flexural-deformational waves. Both the flexural and flexural-deformational instabilities have a threshold and are observed not at an arbitrarily small velocity of the jet but starting from a certain finite value. It is shown that the instability of waves generated by the tangential discontinuity of the velocity field is periodic only formally (from the pure mathematical point of view). Actually, the jet disintegrates within the time of instability development, which is shorter than the half-cycle of the wave.

The nonlinear capillary wave motion in a two-layer liquid with a free surface is analytically investigated accurate to the second order of smallness in ratio of the wave amplitude to the layer thickness. The layers differ in physicochemical properties. A capillary analogue to the “dead water” effect is observed in the system in both linear and quadratic approximations. In the absence of an electric charge at the interfaces, internal nonlinear resonance interaction between capillary waves is also absent regardless of the place of their origination. When there is a charge at the interlayer boundary, capillary waves resonantly interact with each other.

The Mathieu differential equation for the evolution of the amplitudes of arbitrarily symmetric capillary waves (with arbitrary azimuthal numbers) propagating over the surface of a incompressible dielectric cylindrical liquid jet is analyzed. The jet is placed in a time-periodic uniform electric field that is parallel to the symmetry axis of the jet unperturbed by the wave flow. It is found that the time-varying electric field pressure parametrically builds up both axisymmetric waves on the jet surface, flexural waves, and flexural deformation waves. At a fixed frequency of the external field, waves with different wavelengths and symmetries (different azimuthal numbers) may build up simultaneously in the main demultiplication resonance, as well as in secondary and tertiary resonances. The parametric buildup of flexural deformation waves has a threshold relative to the external field frequency: it takes place at the field frequency exceeding a certain value depending on the jet radius and physicochemical properties of the liquid.

A liquid layer resting on a hard bottom and bounded by hard walls is considered. Analysis of the dispersion relation shows that both slowly decaying solitary waves and rapidly decaying periodic waves may arise inside the layer at the interface between stratified ∼100-nm-thick near-surface regions of low-viscous liquids. Stratification is due to the orienting influence of the hard bottom and to fluctuation forces. The dispersion laws for the waves are characteristic of purely capillary waves in both a deep and shallow liquid.

The influence of the finiteness of the charge transfer rate on the electrostatic instability of the lateral surface of a viscous liquid jet is studied. The study is based on the analysis of a dispersion relation for flexural-deformation capillary waves on the surface of the jet with allowance for charge relaxation. The jet is subjected to a superposition of two electrostatic fields one of which is collinear with the jet’s axis and the other is directed radially to the former. It is found that the finiteness of the potential equalization rate influences jets of a poorly conducting liquid most strongly. The charge relaxation shows up in the appearance of both periodic and aperiodic “purely relaxation” flows. Relaxation flows give rise to electrostatic instability in low-permittivity liquids. When the conductivity of the liquid drops, the instability growth rate of relaxation waves grows and their spectrum expands toward shorter waves. An increase in the charge surface diffusion coefficient introduces destabilization into the relaxation flows of the liquid, which may eventually become unstable.

We report the first observation of “quasi-Planck” spectra of capillary turbulence on the surface of liquid hydrogen in the dissipation domain. Capillary waves have been driven by low-frequency random force. We have observed that the frequency spectrum of surface elevation changes its dependence from power-like 〈|ηω2|〉 ∼ ω−2,8 at middle-frequency domain to “quasi-Planck” distribution ∼eω/ωd at higher frequencies. The frequency ωd is proportional to the boundary frequency between inertial interval and dissipation domain and it is scaled up with the increase of driving force.

A dispersion relation is derived for capillary waves with an arbitrary symmetry on the surface of a charged jet of a finite-conductivity viscous liquid placed in an electric field collinear with the axis of the jet. Analytical calculations are carried out in an approximation that is linear in dimensionless wave amplitude. In the case of axisymmetric waves, the instability of which causes disintegration of the jet into drops, the finiteness of the potential equalization rate has a noticeable effect only for jets of poorly conducting liquids. The charge relaxation shows up in that “purely relaxation” periodic and aperiodic liquid flows arise. When the conductivity of the liquid declines, the instability growth rates for unstable waves increase and their spectrum extends toward short waves. A charge present on the surface of the jet enhances its instability. An increase in the charge surface diffusion coefficient variously influences the capillary and relaxation branches of the solution: the damping ratio increases in the former case and decreases in the latter. As the diffusion coefficient rises, relaxation flows may become unstable.

The hydrodynamic, acoustical, and optical phenomena arising in the collision of a freely falling droplet with a liquid surface at rest have been investigatedexperimentally for more than 100 years since the pioneer studies of Thomson, Newall, and Worthington [1,2]. Nowadays, attention is given to studying the practically important finestructure components of arisingflows. The interest is related to the development ofmethods of remote sounding [3] and caused by thedependence of the scatteredsignal properties on thesizes of the watersurface inhomogeneities. For thefrequencies of about a few hundred megahertz, theprobing electromagnetic wavelength becomes comparable to the diameters of the cavity formed from theimpact of a rain droplet. With a further increase in frequency, it becomes comparable to the length of capillary waves diverging from the splash region, with thecumulativejet diameters, and with the secondarydroplets ejected from the liquid layer [4]. Pronouncedsymmetric splash patterns also are widely used in theforming design [5].The characteristics of the flow arising as a result ofthe droplet fall depend on the properties of collidingliquids (density, viscosity, and surfacetension coefficient), the droplet sizes and velocity (or the height ofits free fall), and the conditions of the experiment (theliquidlayer depth, the bottom slope, the liquid anddroplet temperature, and the atmospheric pressure).The theory of the processes proceeding in the dropletimpact is developed mainly in the perfectliquidapproximation when the Froude number Fr =

A dispersion relation is derived for capillary waves with arbitrary symmetry (arbitrary azimuthal numbers) on the surface of a charged cylindrical jet of an ideal incompressible conducting liquid moving relative to an ideal incompressible dielectric medium. It is shown that a tangential discontinuity in the velocity field on the surface of the jet leads to periodic instability of waves (similar to the Kelvin-Helmholtz instability) at the interface and destabilizes both axisymmetric and flexural waves. The wavenumber range for unstable waves and the instability growth rate increase with the field strength and relative speed of motion, varying as the square of these parameters. In the case of the neutral jet, the flexural instability is of the threshold character and sets in starting from a certain finite value of the speed rather than at an arbitrary small speed.

Regularities of the nonlinear gravitational wave motion in a two-layer density-stratified fluid are investigated for a finite thickness of the upper, lighter, layer. The characteristics of the nonlinear internal resonant interaction of the gravity waves generated by the free surface of the upper layer and the medium interface are considered. It is shown that in second-order calculations both degenerate (two-wave) and secondary combined (three-wave) resonant interactions may be realized.

Waves propagating along the interface between a thin vapor film and a liquid layer in the presence of a heat flux are investigated. The boundary conditions on the vapor-liquid phase surface take into account the temperature dependence of the pressure and the possibilities of formation of the metastable state of the superheated liquid and mass flow. Variations in the saturation pressure as functions of the temperature and mass flux lead to generation of weakly damped periodic waves of low amplitude whose velocity can be much higher than the velocity of the gravity waves. The waves ensure stability of the vapor film beneath the liquid layer in the gravity field. The finite-amplitude waves on the surface of the vapor film differ from the Stokes surface waves on the free surface of isothermal fluid. Instability regimes related with superheating of the liquid ant its explosive boiling when the amplitude of an initially small wave increases to infinity in a finite time can develop in a certain working-parameter regime.

An expression for the electric field strength near a watered hailstone is derived in an approximation quadratic in the amplitude of capillary oscillations of a charged conducting liquid layer covering the hailstone. As the number of the mode governing the initial deformation of the equilibrium spherical free surface of the liquid layer increases and its thickness decreases, the electric field strength in the neighborhood of the capillary wave crests rises. Even in the case of small charges and low electric fields, the electric field near the hailstone is high enough to initiate a corona.

In terms of a linear mathematical model of a capillary-gravitational flow in a two-layer liquid with a finite-thickness upper layer, it is shown analytically that an analogue of the dead water phenomenon exists in the domain of capillary waves, which was previously observed only in gravitational waves. This phenomenon shows up as an exponential increase in the capillary wave amplitude at the interface with the surface tension coefficient at the interface tending to zero. It is found that an external electric field displaces this phenomenon toward the range of finite surface tension coefficients.

A mathematical model of regular wavy microrelief formation on a semiconductor surface subjected to an obliquely incident medium-energy ion be am is stated and analyzed. Taking into account the influence of fluctuation forces (the molecular component of the disjoining pressure) and of the ion-beam-transferred electric charge is shown to bring together model predictions and observations.