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.     

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.     

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 interaction between a charged drop and an external acoustic field

An interaction between capillary oscillations of a charged drop and an external acoustic field is investigated under conditions in which nonlinear components of the acoustic pressure on the drop surface may be neglected. It is shown that equations describing the temporal evolution of modes of the capillary waves in this case may be either the Mathieu-Hill equations or ordinary inhomogeneous equations of the second order describing forced oscillations. In both cases, the drop instability (of a parametric or resonance type) may result in its disintegration due to deformation caused by the acoustic field at its own drop charge, subcritical in the sense of the Rayleigh criterion.     

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.     

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.     

Nonlinear electrohydrodynamic phenomena and droplet generation in charged jets of conducting liquid

Phenomena occurring at the tip of a charged conducting jet are analyzed in detail using numerical methods developed for axially symmetric flows. Universal mechanisms (independent of the method for producing the jet) for droplet formation with different ratios of the Laplace and electrical pressures on the lateral surface are identified. An explanatory analysis is given for all of the nonlinear stages of the classical Rayleigh instability of a charged conducting drop, beginning with the formation of a jet at the surface of the drop and culminating in the generation of a developed jet of secondary droplets. Zh. Tekh. Fiz. 69, 1–9 (November 1999)     

Capillary disintegration of a ferrofluid cylindrical film magnetized to saturation by an axial magnetic field

The subject of consideration is a film of a magnetic fluid applied on the surface of a thin magnetically soft cylinder. Quantities figuring in equations and boundary conditions describing the axisymmetric flow due to capillary and magnetic forces at capillary disintegration of the film are estimated in order of magnitude. The effect of magnetization on the capillary disintegration of the film is studied using simplified equations of ferrohydrodynamics. It is shown that magnetization shifts the range of Rayleigh capillary instability toward longer wave modes. As a result, drops arising at the final stage are larger than in the case of the nonmagnetic liquid.     

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.     

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 Increments for Waves with Different Symmetries on a Space-Charged Jet Moving Relative to a Material Medium

Technical Physics - 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...     

Experimental evidence of a Rayleigh-plateau instability in free falling granular jets

A granular jet falling out of a funnel shaped container, subjected to small vertical vibrations, develops an instability farther downstream as may happen for ordinary liquid jets. Our results show that this instability is reminiscent of the Rayleigh-Plateau capillary instability leading to breakup of the jet at large scales. The first stages of this instability are captured in detail allowing a determination of the dispersion relation. Surface tensions measured in this unstable regime (of the order of mN/m) are in agreement with previously reported measurements carried out at much smaller scales. This instability and the breakup of the jet can be inhibited when the effect of the surrounding medium (air) is reduced by enclosing the jet in an evacuated chamber, showing that the effective surface tension measured is the result of a strong interaction with the surrounding air.     

Capillary disintegration of a suspended filamentary drop of a viscous magnetic fluid in a longitudinal magnetic field

A differential equation that describes the axisymmetric motion of two immiscible magnetic fluids of the same density and viscosity is derived. It includes in explicit form the contribution of capillary forces localized at the interface between the fluids, which has the form of a weakly distorted cylindrical surface. With this equation, a dispersion relation for the problem of capillary instability of an extended axisymmetric drop placed in a uniform longitudinal magnetic field is obtained. The effect of magnetic forces on the capillary disintegration of the drop for the extreme cases (large and small Ohnesorge numbers) is analyzed.     

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.     

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.     

Instabilities and decay rates of charged viscous liquid jets

The eigenmodes and characteristic frequencies of surface deformations of real fluids in the geometry of a cylindrical jet, which is homogeneously charged, are calculated using the linearized Navier-Stokes equation with free surface boundary conditions. A stability phase diagram in the wavenumber-fissility plane is given. Two types of instabilities are described, surface tension and Coulomb force induced. The known modes for ideal potential flow are extended to arbitrary viscosity. In addition new types of modes are described (eigenfunctions and decay rates) which represent rotational flow and have no correspondence to Rayleigh's jet instability.     

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.     

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.     

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)     

毛细喷孔超高速水射流的脉动及破碎

水切割射流的动力学特性的诸多方面尚未得到认识和理解。本文对毛细喷孔产生的超高压水射流展开可视化研究,分析了常规压力及超高压条件下毛细水射流的液体破碎机制并对超高压毛细射流的脉动现象进行了讨论。常规条件下的毛细射流遵从经典的破碎模式;在超高压条件下,射流完整段呈瑞利模式,完整段以下呈雾化破碎模式,射流集束性呈现周期性变化。结果表明,传统理论不能够表达小孔径时超高速毛细水射流的破碎特性;喷孔内部流动情况如流动分离及空化成为该条件下射流破碎和脉动的重要原因。