On capillary motion of a viscoelastic liquid with a charged free surface

The structure of the capillary-relaxation motion spectrum in a liquid with a charged free surface has been investigated taking into account the viscosity relaxation effect. On the basis of numerical analysis of the dispersion equation for the wave motion in a viscoelastic incompressible liquid, it is shown that for a given wave number the range of characteristic relaxation times in which relaxation-type wave motion exists is limited and expands with increasing wave number. The growth rate of instability of the charged liquid surface markedly depends on the characteristic relaxation time and increases with its growth; in liquids with elastic properties, the energy dissipation rate of capillary motion is enhanced. At a surface charge density that is supercritical for the onset of Tonks-Frenkel instability, both purely gravitational waves and waves of a relaxational nature exist.     

Capillary oscillations and Tonks-Frenkel instability of a liquid layer of finite thickness

A dispersion relation is derived and analyzed for the spectrum of capillary motion at a charged flat surface of viscous liquid covering a solid substrate with a layer of finite thickness. It is shown that for waves whose wavelengths are comparable with the layer thickness, viscous damping at the solid bottom begins to play an important role. The spectrum of capillary liquid motion established in this system has high and low wave number limits. The damping rates of the capillary liquid motion with wave lengths comparable with the layer thickness are increased considerably and the Tonks-Frenkel instability growth rates are reduced compared with those for a liquid of infinite depth. Zh. Tekh. Fiz. 67, 27–33 (August 1997)     

Influence of charge relaxation on the capillary oscillations of a charged viscous spheroidal drop

A dispersion relation is derived for the spectrum of capillary modes of a charged spheroidal drop of a viscous liquid with allowance for charge relaxation. It is shown that the finite charge transport rate leads to lowering of the instability growth rates for various capillary modes of a spheroidal drop of a low-viscosity liquid. As the degree of deformation of the drop increases, the magnitude of the absolute change in the growth rate caused by the finite rate of charge redistribution decreases. Zh. Tekh. Fiz. 69, 28–36 (August 1999)     

Influence of charge relaxation on the capillary disintegration of a jet of a viscous dielectric liquid placed in an electrostatic field collinear with the axis of the jet

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.     

Capillary oscillations of a charged viscous spheroidal droplet

Dispersion relations are derived for the capillary oscillations of a charged viscous spheroidal droplet by scalarization within perturbation theory using an expansion in two small parameters, viz., the magnitude of the perturbation of the spheroidal surface as a result of thermal fluctuations and the magnitude of the deviation of the equilibrium spheroidal droplet shape from a spherical shape. It is shown analytically that the motion spectrum of the liquid consists of two components that interact in the linear theory: toroidal vortex motion and poloidal potential motions. A numerical analysis reveals that the instability growth rates of the higher modes of a highly charged droplet increase with enhancement of the degree of spheroidal strain and decrease rapidly as the viscosity of the liquid increases. Zh. Tekh. Fiz. 68, 20–27 (April 1998)     

Characteristics of the capillary motions of surfactant solutions with a charged free surface

Abastract A dispersion relation is derived and analyzed for the spectrum of capillary motions on the charged plane surface of a liquid in which a surfactant is dissolved. It is shown that two additional wave motions are generated in this kind of system by bulk diffusion and surface diffusion of the surfactant and are sensitive to the diffusion coefficients and elastic properties of the surfactant films and to the viscosity of the solution and the presence of a surface charge. In solutions of inactive surfactants the growth rate of Tonks-Frenkel instability increases as the surfactant concentration increases. Zh. Tekh. Fiz. 68, 22–29 (February 1998)     

Electric dispersion of fluid under the oscillatory instability of its free surface

A semiphenomenological analysis is performed of possible modes of electric dispersion of drops and menisci at the end of the capillary used to deliver the liquid into the discharge system under an oscillatory instability of the charged liquid surface. The instability is assumed to be induced by a time-dependent external force acting on the liquid surface, a finite rate of charge redistribution over the surface under virtual deformations, and tangential discontinuity of the velocity field across the interface.     

Thickness of a boundary layer attributed to the wave motion on the charged free surface of a viscous liquid

It is shown that the analytical estimator for the boundary layer thickness that contains the wave frequency in the denominator and is proposed for approximate calculation of the wave motion on the free surface of a viscous liquid cannot be formally applied to the wave motion on the uniformly charged liquid surface. The fact is that, when the surface charge density attains a value critical in terms for the Tonks-Frenkel instability, the wave frequency tends to zero. From the analysis of liquid motions near the electric charge critical density, a technique is proposed for calculating the thickness of a boundary layer attributed to flows of various kinds. It is found that the thickness of the boundary layer due to aperiodic flows with amplitudes exponentially growing with time (such flows take place at the stage of instability against the surface charge) does not exceed a few tenths of the wavelength, whereas the thickness of the boundary layer due to exponentially decaying liquid flows is roughly equal to the wavelength.     

Nonlinear periodic waves on the charged surface of a finite-thickness ideal liquid layer

In the fourth order of smallness in the amplitude of a periodic capillary-gravitational wave travelling over the uniformly charged free surface of an ideal incompressible conducting liquid of a finite depth, analytical expressions for the evolution of the nonlinear wave, velocity field potential of the liquid, electrostatic field potential above the liquid, and nonlinear frequency correction that is quadratic in a small parameter are derived. It is found that the dependence of the amplitude of the nonlinear correction to the frequency on the charge density on the free liquid surface and on the thickness of the liquid layer changes qualitatively when the layer gets thinner. In thin liquid layers, the resonant wavenumber depends on the surface charge density, while in thick layers, this dependence is absent.     

Instability of a charged layer of a viscous liquid on the surface of a solid spherical core

The dispersion relation for the spectrum of capillary waves of a spherical layer of a viscous liquid coating a solid spherical core with a layer of finite thickness is introduced and analyzed. It is shown that the existence of two mechanisms for the viscous dissipation of the energy of the capillary-wave motions of the liquid, viz., damping in the bulk of the layer and on the solid core, leads to restriction of the spectrum of the realizable capillary waves of the liquid on both the high-and low-mode sides. At a fixed value of the system charge which is supercritical for the first several capillary modes, the maximum growth rates in the case of a small solid core are possessed by modes from the middle of the band of unstable modes, while in thin liquid layers the highest of the unstable modes have the largest growth rates. This points out differences in the realization of the instability of the charged surface of the spherical layer for small and large relative sizes of the solid core. Zh. Tekh. Fiz. 67, 8–13 (September 1997)     

Long-wavelength structure on a charged liquid surface

The problem of the equilibrium form of a charged surface of a dielectric liquid in a strong electric field, such that a flat surface becomes unstable, is studied. A periodic long-wavelength structure with a small amplitude can arise when the gap between the surface and a charged electrode is small compared with the capillary length and the charge completely screens the electric field. The equilibrium form of the surface is calculated assuming that the resulting wave is one-dimensional. The effect of the boundary conditions at the vessel walls on the dependence of the amplitude of the standing wave on the applied voltage is estimated. It is shown that this dependence is very sensitive to the conditions of contact between the vessel walls and the liquid. The possibility is discussed of using the theory developed in this paper to explain the experimental results obtained with a charged liquid-hydrogen surface. Zh. éksp. Teor. Fiz. 115, 43–49 (January 1999)     

Nonlinear analysis of interaction between finite-amplitude capillary waves in a layered inhomogeneous liquid

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.     

Instability development on the charged free surface of a horizontal liquid layer with thermal convection

The instability of the charged free surface of a horizontal liquid layer heated from the solid bottom against excess electric charge is studied theoretically for the case in which this type of instability is combined with thermal-convective instability. The structure of the total spectrum of unstable wave flows and physical parameters influencing the structure of the spectrum are determined.     

On the Stability of the Surface of a Short Charged Jet Moving with Respect to an External Material Medium

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.     

Capillary oscillations of a thin film of viscous liquid on a solid substrate and its instability against surface charge

In a numerical analysis of the dispersion relation describing capillary motions in a thin film of a viscous, surface-charged liquid with fluctuation forces taken into account, it is found that the critical conditions of instability of the free surface of the liquid for a fixed thickness d of the liquid film in the region where the influence of the fluctuation forces is large (d<100 nm) depend strongly on the wave number and do not depend on the viscosity of the liquid, that the fluctuation forces strongly affect the wave number of the most unstable wavelength and decrease the instability growth rate, and that the capillary motions of the liquid admit an analogy with gravity-capillary motion and can be interpreted as fluctuation-capillary motions. Zh. Tekh. Fiz. 68, 27–31 (October 1998)     

Mass transfer due to nonlinear capillary-gravitational waves on the surface of a viscous liquid

An analytical expression of the second order of smallness in wave amplitude-to-wavelength ratio is derived for a horizontal flow arising in a finite-depth layer of a viscous liquid under the action of a periodic nonlinear capillary wave. It is found that the liquid flow is determined by the nonlinear component of the velocity field vortex part and the flow rate increases with increasing viscosity and decreasing wavelength irrespective of the layer thickness. In thin layers, the flow rate rapidly drops from its maximal value with increasing viscosity, wavelength, and surface charge density. If the liquid surface is charged, the horizontal liquid flow decreases rapidly as the surface charge density approaches the threshold of the Tonks-Frenkel instability.     

Radiation of a charge moving over the diffraction grating

The states of a charged particle with a finite free path are determined in the field of a resonant electromagnetic wave. The exact resonance conditions, the modulation and beam instability mechanisms, the charge and current densities (Ohm's law) are obtained for the collisionless beam of resonance particles. Quantum theory of radiation is developed for the resonant adiabatic interaction between a particle and a wave taking into account the interaction with a constant magnetic field induced at the grating surface by the charge and nonresonant waves. The radiation power, the spectrum, and the range of generated frequencies are determined. The results obtained can be used in the plasma and solid-state theories and in electronics.     

Quasiadiabatic decay of capillary turbulence on the charged surface of liquid hydrogen

We study the free decay of capillary turbulence on the charged surface of liquid hydrogen. We find that decay begins from the high frequency end of the spectral range, while most of the energy remains localized at low frequencies. The apparent discrepancy with the self-similar theory of nonstationary wave turbulent processes is accounted for in terms of a quasiadiabatic decay wherein fast nonlinear wave interactions redistribute energy between frequency scales in the presence of finite damping at all frequencies. Numerical calculations based on this idea agree well with experimental data.     

Effect of the marangoni convection on the injection mechanism of instability

Electroconvective instability of a nonisothermal layer of a weakly conductive liquid with a free boundary whose surface tension depends linearly on temperature is considered for the case where charge injection is performed through this surface. When calculating the unperturbed stationary distribution of the charge and field, we supposed that the injector is separated from the liquid by an air gap of finite thickness. It was, however, assumed when analyzing the stability of the system that the motion in the air gap has no effect on the motion in the liquid phase and the disturbances of the electric field and charge in the air gap decay rapidly because of its high conductivity.     

Electrostatic instability of the lateral surface of a viscous liquid finite-conductivity jet in a collinear electrical field

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.