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.     

Flow stability of a liquid dielectric cylinder in a transverse electrostatic field

A perturbation method is used to study the stability of the interface between a cylinder formed by the flow of an inviscid liquid dielectric and air in the presence of an external electrostatic field produced by coaxial electrodes. The field is perpendicular to the interface. The outer electrode is at a potential V₀ and the inner electrode is grounded. An equation is found for the neutral-equilibrium curve, and the threshhold values of V₀, i.e., those at which the surface becomes unstable, are calculated for the case in which the outer radius of the stream is twice the inner radius.Translated from Izvestiya VUZ. Fizika, No. 12, pp. 34–38, December, 1969.     

Stability of a liquid-dielectric jet in a longitudinal electrostatic field

Conclusions A longitudinal electrostatic field tends to stabilize a jet of liquid dielectric.     

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.     

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.     

Asymptotic analysis of nonlinear nonaxisymmetric waves on the surface of a neutral dielectric jet in a longitudinal electrostatic field

The problem of calculation of finite-amplitude waves on the cylindrical surface of an ideal incompressible dielectric liquid jet in a uniform electrostatic field collinear to the unperturbed jet axis is solved using a second-order asymptotic analytic procedure in ratio of the wave amplitude to the jet radius. Nonlinear corrections to the jet profile, velocity field potential, and electrostatic potentials inside and outside the jet are of resonant nature. The degenerate resonant interaction between the wave determining the initial strain and the waves excited due to nonlinearity of the hydrodynamic equations can take place for waves with different symmetries (different azimuth numbers).     

On the capillary vibration and stability of a charged bubble in a dielectric liquid

The capillary vibration and stability of a charged bubble against infinitesimal volume and shape perturbations when the bubble is immersed in a viscous incompressible dielectric liquid are studied. The range of physical parameters where the noncentrosymmetric radial and axisymmetric surface motions of the bubble are unstable is found. Asymptotic analytical expressions for the damping constant of the axisymmetric capillary vibration in the low-and high-viscosity approximations are derived.     

Effect of longitudinal electric field on capillary instability of a thin axisymmetric layer of liquid dielectric coating a dielectric fiber

The flow of a viscous dielectric liquid surrounded with a gas is investigated in the process of capillary disintegration of a thin axisymmetric liquid layer on an undeformable cylindrical dielectric fiber in a uniform electric field is investigated. An asymptotic analysis of the system of equations and hydrodynamic boundary conditions written with allowance for surface ponderomotive forces is carried out for the case when the average thickness of the layer is much smaller than the radius of the fiber cross section. The problem of the transition of the liquid configuration from the state of a stationary cylindrical layer to the hydrodynamic state in the form of a regular sequence of drops is formulated. In this formulation, a nonlinear parabolic equation that describes the evolution of the local thickness of the layer on the time interval to the instant of drop formation is derived. The effect of the key parameters on the capillary instability is analyzed based on the linearized version of the resultant equation and the linearized electrostatic problem of calculating the field perturbations.     

On the stability of nonaxisymmetric modes of a spatially charged jet of a viscous dielectric liquid

A dispersion equation is derived for nonaxisymmetric capillary vibrations of a spatially charged jet of a viscous dielectric liquid. It is shown that the critical conditions with respect to the instability of nonaxisymmetric jet modes are less stringent than those for axisymmetric modes, this effect being more pronounced for liquids characterized by lower values of the dielectric permittivity.     

Stability of a bubble in a dielectric liquid in an external electrostatic field

Critical instability conditions are found for a gas bubble in a liquid dielectric in a uniform external electrostatic field E ₀. It is shown that they depend both on the magnitude of E ₀ and on the properties of the liquid, as well as on the gas pressure in the bubble. In a linear approximation with respect to the square of the eccentricity of an equilibrium spheroidal form, the equilibrium eccentricity of the bubble exceeds the equilibrium eccentricity of a drop in the field E ₀. The gas pressure in the bubble lowers the critical electric field E ₀ for development of an instability in the bubble. Zh. Tekh. Fiz. 69, 43–48 (August 1999)     

Instability of a bubble in a liquid dielectric in an external electrostatic field

The sizes, charges, and number of daughter bubbles emitted during the development of instability with respect to the polarization charge in a uniform electrostatic field of a gas bubble in a liquid dielectric are found on the basis of the Onsager principle of minimum energy dissipation for nonequilibrium processes. Zh. Tekh. Fiz. 69, 10–13 (November 1999)     

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.     

Critical equilibrium spheroidal deformation of a drop of dielectric liquid in a uniform electrostatic field

The stability of a dielectric drop, which in an external electrostatic field takes on the equilibrium shape of a prolate spheroid, is analyzed using the principle of minimum total potential energy of an isolated system. The values of the Taylor parameter and degree of spheroidal deformation at which the drop loses stability are determined for a wide range of dielectric constants of the substance of the drop. Zh. Tekh. Fiz. 69, 23–28 (July 1999)     

Effect of a strong longitudinal magnetic field on capillary disintegration of a magnetic liquid film covering the inner surface of a narrow cylindrical tube

The effect of magnetic forces on the viscous regime of disintegration of a thin cylindrical ferroliquid film magnetized to saturation is studied on the basis of an asymptotic model describing the evolution of small perturbations during capillary disintegration of this film. A formula describing the dependence of the wavelength of the most rapidly increasing mode on the physical parameters characterizing this phenomenon is derived analytically.     

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.     

Amplification of a convective wave in a longitudinal electrostatic field

Radio Engineering Institute, Taganrog. Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Radiofizika, Vol. 32, No. 8, pp. 1026–1033, August, 1989.     

Effect of vacuum magnetic field on the gravitodynamic stability of liquid jet

Summary  The MHD stability of a liquid jet (radiusR ₀ and density ϱ) under the influence of self-gravitating force has traditionally been studied using a normal-mode type of analysis. The dispersion relation is obtained and studied analytically and numerically. The axial magnetic fields inside and outside the jet have always stabilizing effects. The transverse magnetic field has a destabilizing effect. It is found that the largest doman of instability is fastly shrinking (as α> ?) with increasingH ₀ /H _G values,H _G 4(4πGϱ² R ² ₀ μ)^1/2, whereG is the gravitational constant. Some reported works are recovered as limiting cases from the present work.     

A statistical model of the electrostatic field at a dielectric surface

A method is proposed for determination of electric-field intensity distribution on a plane over which charges of random magnitude and sign are distributed. The distribution functions for normal and tangent components of electric field intensity in this plane are obtained.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 6, pp. 95–99, June, 1975.     

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.     

Nonlinear development of capillary waves in a viscous liquid jet

The one-dimensional approximate equations describing the dynamics of a Newtonian viscous fluid are used to analyze the nonlinear development of capillary waves in a jet. It is shown that the size of satellite droplets resulting from a nonuniform jet breakup decreases with the Reynolds number at a constant wavenumber. The satellite-droplet formation ceases at a certain value of the Reynolds number, which depends on the wavenumber and initial perturbation amplitude.