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.     

On the stability of capillary waves on the surface of a space-charged dielectric liquid jet moving in a material medium

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.     

On the stability of capillary waves on the surface of a charged jet moving relative to the environment

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.     

On the capillary stability of a cylindrical dielectric liquid jet in a longitudinal electrostatic field

A dispersion relation is derived for capillary waves with arbitrary symmetry (with arbitrary azimuthal numbers) on the surface of a cylindrical jet of an ideal incompressible dielectric liquid subjected to an electrostatic field aligned with the symmetry axis of the jet. It is shown that only long axisymmetric waves can experience capillary instability in such a system. The wavenumber range into which unstable waves fall begins with a zero value, and its width depends on the permittivities of the liquid and ambient and on the electrostatic field strength squared. As the field strength grows, the wavenumber range for unstable waves rapidly narrows and the capillary instability growth rate, as well as the wavenumber of the wave with the greatest growth rate, decreases.     

Electrostatic instability of a heavily charged conducting liquid jet

The effect of electric charge on the jet surface on the capillary instability of the jet and its disintegration into drops is analyzed. A theoretical explanation is given for the electrostatic mechanism of instability development and jet disintegration that is akin to the mechanisms behind the instability of a heavily charged drop (Rayleigh instability) and flat uniformly charged liquid surface (Tonks-Frenkel instability) but differs qualitatively from the conventional capillary mechanism of instability and disintegration.     

Capillary oscillations of a flat charged surface of liquid with finite conductivity

A dispersion relation is proposed and analyzed for the spectrum of capillary motion at a charged flat liquid surface with allowance made for the finite rate of charge redistribution accompanying equalization of the potential as a result of the wave deformation of the free surface. It is shown that when the conductivity of the liquid is low, a highly charged surface becomes unstable as a result of an increase in the amplitude of the aperiodic chargerelaxation motion of the liquid and not of the wave motion, as is observed for highly conducting media. The finite rate of charge redistribution strongly influences the structure of the capillary motion spectrum of the liquid and the conditions for the establishment of instability of its charged surface when the characteristic charge relaxation time is comparable with the characteristic time for equalization of the wave deformations of the free surface of the liquid. Zh. Tekh. Fiz. 67, 34–41 (August 1997)     

On bending instability of a volumetrically charged capillary jet of dielectric liquids

It is shown that jets of volumetrically charged dielectric liquids in the vicinity of point k = 0 are characterized by a finite range of wavenumbers, in which a jet exhibits bending instability. The interval of wavenumbers corresponding to unstable waves with azimuth number m = 1, as well as the increment of the most unstable wave and its wavenumber, increase in proportional to the electric charge per unit length of the jet and the permittivity of the liquid.     

On the stability of a cylindrical jet moving in a material medium along an external electrostatic field

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.     

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)     

On the stability of a nonaxisymmetric charged jet of a viscous conducting liquid

A dispersion equation is derived for axisymmetric and nonaxisymmetric capillary oscillations in a jet of viscous conducting liquid subjected to a constant potential. It is shown that conditions arising when the surface charge density in the jet is high cause the instability of nonaxisymmetric, rather than axisymmetric, modes with the resulting disintegration of the jet into drops of various sizes. This theoretical finding allows one to correctly interpret of experimental data for the spontaneous disintegration of charged jets.     

Spontaneous disintegration of a noncylindrical jet emitted from the liquid surface unstable against self-charge

A dispersion relation for waves on the surface of a charged viscous incompressible conducting liquid jet with an arbitrary azimuthal number is derived. It is shown that the influence of deformation on the growth rate and wavenumber of the most unstable mode varies according to the sign of local deformation relative to the cylindrical jet (the locality is specified by the wavelength), azimuthal number, and electric charge per unit length of the jet. This circumstance should be taken into account to comprehend conditions of liquid spontaneous electrodispersion.     

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.     

On the time evolution of the liquid meniscus at the end of a capillary placed in an external electric field

Using a linearized set of equations of electrodynamics, the stability of the uniformly charged meniscus of a viscous conducting incompressible liquid at the end of a capillary is investigated and analytical expressions are derived for the electric field outside the meniscus, velocity fields in the liquid flow and meniscus, and generatrix of the meniscus shape. It is found that, if an external electric field near the meniscus exceeds that at which the free liquid surface becomes unstable against the surface charge, a finite number of longest waves become unstable with their instability growth rates nonmonotonically depending on the wavenumber. Analysis of the time evolution of the meniscus shape under various initial conditions shows that cylindrical waves with the highest instability growth rates play a decisive role in this process, while the influence of the initial deformation amplitude is insignificant.     

Influence of charge carrier diffusion on the stability of a charged drop of a finite-conductivity liquid

A dispersion relation for the capillary oscillations of a spherical drop of a viscous incompressible liquid with a charge transfer finite rate is derived and analyzed with emphasis on the role of diffusion. It is shown that diffusion has the strongest influence on the stability of rapidly damped quasi-periodic motions of a low-conductivity liquid. The instability growth rate of capillary oscillations grows with the charge diffusion coefficient and decreases with rising conductivity of the liquid.     

Sufficient conditions for linear long-wave instability of steady-state axisymmetric flows of an ideal liquid with a free boundary in an azimuthal magnetic field

The problem of linear stability of steady-state axisymmetric shear jet flows of an inviscid ideally conducting incompressible liquid with a free surface and “frozen-in” azimuthal magnetic field is analyzed. The sufficient conditions for theoretical (on semi-infinite time intervals) and practical (on finite time intervals) instability of these flows relative to small axisymmetric long-wave perturbations are obtained by the direct Lyapunov method. An a priori lower estimate indicating (at least) an exponential increase with time of small perturbations under investigation is constructed in the case when these conditions are valid for theoretical as well as practical instability. In addition, an illustrative analytic example of steady-state flows under investigation and small axisymmetric long-wave perturbations superimposed on them is constructed (according to our estimate, these perturbations increase with time).     

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.     

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.     

Nonlinear analysis of a wave motion on the surface of a jet moving in a dielectric medium placed in a longitudinal electric field

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.     

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)     

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)