Nonlinear nonaxisymmetric oscillations of a space-charged jet

The problem of nonlinear nonaxisymmetric oscillations of a jet of a space-charged dielectric (zeroconductivity) liquid is solved asymptotically. The solution is accurate up to second-order terms in initial deformation amplitude. It is shown that the difference between the amplitude values of second-order corrections to the solutions for perfectly conducting and perfectly dielectric liquids depends on the permittivity of the latter and may amount to several tens of percent. The permittivity of the liquid also governs the positions of internal resonances, in the vicinity of which nonlinear energy transfer between modes takes place.     

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 corrections to the oscillation mode frequencies of an ideal liquid charged jet

An analytical asymptotic expression is derived for the time evolution of an ideal incompressible conducting liquid jet with a uniformly charged surface that moves with a constant velocity along the symmetry axis of an undisturbed cylindrical surface. The expression is obtained in the third order of smallness in jet oscillation amplitude under the conditions when the initial deformation of the equilibrium jet surface is specified by one (axisymmetric or nonaxisymmetric) mode. Analytical expressions are also found for the positions of internal nonlinear degenerate three-and four-mode resonances, which are typical of nonlinear corrections to the equilibrium shape of the jet, liquid velocity field potential in the jet, and electrostatic field near the jet, and also for a nonlinear frequency correction. It is established that the nonlinear oscillations of the jet take place about a surface depending on the type of initial (generally asymmetric) deformation rather than about the equilibrium cylindrical shape.     

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.     

Nonlinear capillary vibration of a charged bubble in an ideal dielectric liquid

The problem of nonlinear radial pulsations and surface vibrations of a charged bubble placed in an ideal incompressible dielectric liquid is asymptotically solved up to the second order of smallness by the method of many scales. It is shown that, in the case of nonlinear vibrations, resonance energy exchange may take place not only between surface modes but also between the radial mode and a surface mode. A new type of instability (other than Rayleigh instability against the self-charge), instability against the excess vapor pressure in the bubble, is discovered. The new type of instability shows up as energy transfer from the centrosymmetric pulsation mode to all initially excited surface vibration modes simultaneously.     

Nonaxisymmetric oscillations of a charged jet of a finite-conductivity viscous liquid

A dispersion relation for nonaxisymmetric oscillations of a charged jet of a viscous incompressible finite-conductivity liquid is derived. In the low-viscosity limit, when the dispersion relation can be reduced to a power algebraic equation, analytical expressions for its roots are found and their dependences on the conductivity of the liquid and carrier mobility are analyzed. Taking into account a finite rate of electric charge transfer introduces an additional damping and somewhat decreases the instability growth rates.     

Nonlinear oscillations of a charged jet of a conducting liquid under multimode initial deformation of its surface

The subject of consideration is a uniformly charged jet of an ideal incompressible conducting liquid moving with a constant velocity along the symmetry axis of an undisturbed cylindrical surface. An evolutionary expression for the jet shape is derived accurate to the second order of smallness in oscillation amplitude for the case when the initial deformation of the equilibrium surface is a superposition of a finite number of both axisymmetric and nonaxisymmetric modes. The flow velocity field in the jet and the electric field distribution near it are determined. The positions of internal nonlinear secondary combined three-mode resonances are found, which are typical of nonlinear corrections to the analytical expressions for the jet shape, flow velocity field potentials, and electrostatic field in the vicinity of the jet.     

Nonlinear corrections to the frequencies of nonaxisymmetric Modes of a space-charged dielectric liquid jet

An expression for the shape of a uniformly charged (over volume) jet of an ideal incompressible dielectric liquid as a function of time is derived with analytical asymptotic calculations of the third order of smallness in jet oscillation amplitude for the case when the initial deformation of the equilibrium surface is governed by one (in general nonaxisymmetric) mode. Analytical expressions for nonlinear corrections to the frequencies and positions of inner nonlinear degenerate three-and four-mode resonances are found.     

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.     

Nonlinear oscillations of a charged drop accelerated in an electrostatic field

An analytical asymptotic solution to the problem of nonlinear oscillations of a charged drop moving with acceleration through a vacuum in a uniform electrostatic field is found. The solution is based on a quadratic approximation in two small parameters: the eccentricity of the equilibrium spheroidal shape of the drop and the amplitude of the initial deformation of the equilibrium shape. In the calculations carried out in an inertial frame of reference with the origin at the center of mass of the drop, expansions in fractional powers of the small parameter are used. Corrections to the vibration frequencies are always negative and appear even in the second order of smallness. They depend on the stationary deformation of the drop in the electric field and nonlinearly reduce the surface charge critical for development of the drops’s instability. It is found that the evolutions of the shapes of nonlinearly vibrating unlike-charged drops differ slightly owing to inertial forces.     

Nonlinear oscillations of a charged layer of a conducting liquid on the surface of a solid spherical core

Nonlinear oscillations of a layer of an ideal incompressible perfectly conducting liquid on the surface of a charged melting hailstone (solid core) are studied using analytical asymptotic calculations of the second order of smallness in initial deformation amplitude. Specifically, it is shown that, when the thickness of the layer is much less than the characteristic linear size (radius) of the solid core, the size of the core considerably influences the amplitudes of capillary oscillation modes arising on the surface of the charged layer via nonlinear interaction. It is found that, as the liquid layer on the surface of the solid core gets thinner, the energy in the spectrum of nonlinearly excited modes is redistributed with its maximum shifting toward higher (larger number) modes.     

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.     

On the oscillations of a charged drop of a finite-conductivity viscous liquid

A dispersion relation for the capillary oscillations of a charged spherical drop of a viscous incompressible finite-conductivity liquid is derived and analyzed. It is found that electric currents inside the charged drop equalize its potential and produce liquid flows interacting with both potential and eddy poloidal liquid flows inside the drop that are due to drop oscillations. Taking into account the finiteness of the rate of potential equalization over the drop surface leads to an additional damping of the capillary oscillations that arises because of the increased role of energy dissipation.     

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)     

Nonlinear oscillations and stability of a charged drop moving relative to a dielectric medium

Nonlinear calculations to within the second order of smallness with respect to the initial deformation of a liquid drop show that a stream of an ideal incompressible dielectric liquid streamlining the charged ideally conducting drop causes interaction between modes both in the first and second orders of smallness. Both the linear and nonlinear interactions of the oscillation modes result in the excitation of modes absent in the spectrum of the initial drop deformation. The relative motion of the drop and the medium leads to broadening of the spectrum of modes excited in the second order of smallness. The presence of the flow streamlining the drop and the intermode interaction result in decreasing the critical magnitudes of the drop charge and the velocity and density of the medium determining drop instability development.     

Nonlinear analytical calculation of the equilibrium shape of a charged drop uniformly accelerated in an electrostatic field

An analytical asymptotic expression for the equilibrium shape of a charged drop of an ideal incompressible conducting liquid uniformly accelerated in collinear electrostatic and gravitational fields is derived in an approximation quadratic with respect to the deviation of the equilibrium shape of the drop from a sphere. It is found that the equilibrium shape of the drop is close to a prolate spheroid when its self-charge and the external electric field strength are far from their values critical in terms of instability against the self-charge and induced charge. This spheroid experiences an insignificant pear-shaped distortion even when the charge of the drop and the electrostatic field strength are high.     

Nonlinear secondary Raman resonances of oscillations of the fundamental mode of a charged drop moving in an environment

Second-order calculations show that, when a gas flows about a charged drop, the fundamental mode of the multimode initial deformation of its equilibrium shape builds up through nonlinear secondary Raman resonant interaction with higher modes if this mode is present in the mode spectrum specifying the initial deformation. This circumstance accounts for large-amplitude spheroidal oscillations of drops in natural liquid-drop systems and provides an insight into corona initiation in the vicinity of drops in thunderstorm clouds and into lightning initiation.     

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.     

Nonlinear degenerate resonance interaction of waves on a charged liquid surface

The physics of nonlinear degenerate resonance energy exchange between waves on the flat free charged surface of a conducting liquid is analytically (asymptotically) studied up to the second order of smallness. A set of differential equations for the evolution of the amplitudes of nonlinearly resonantly interacting waves is derived. It turns out that nonlinear computations (taking into account the dependence of the wave frequency on the finite amplitude) yield an infinite number of degenerate resonances, although computations based on frequencies found in the linear theory give a finite number of resonances. In nonlinear computations, the positions of the degenerate resonances depend on the surface charge density (or on the external electric field normal to the free surface of the liquid) in contrast to the results of linear computations (based on frequencies found in the linear theory). It is found that as the wavenumber of an exact degenerate resonance is approached (that is, in the vicinity of this number), the direction of energy transfer changes sign: now the energy is transferred from a shorter wave to a longer one and not the reverse.     

Nonlinear periodic waves at the charged surface of an electrically conducting viscous liquid

A correct solution to the problem of periodic-wave propagation along the charged surface of a deep viscous liquid in the second-order approximation in the wave amplitude is given for the first time. It is shown that the second-order correction in the amplitude to the profile of the wave being considered plays a decisive role in the realization of the instability of a liquid with respect to its intrinsic charge.