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.     

Dispersing of a charged drop in an electrostatic field

The characteristics of the breakup of a charged drop in a uniform electrostatic field are calculated on the basis of Onsager’s principle of minimum dissipation of energy in nonequilibrium processes. The ranges of the physical parameters where daughter droplets are emitted from two tips and from one tip of an unstable parent drop and when emission is completely absent are found. The dimensionless radii, charges, and specific charges of the daughter droplets are determined. Zh. Tekh. Fiz. 69, 26–30 (December 1999)     

Nonlinear oscillations of an uncharged conducting drop in an external uniform electrostatic field

The problem of nonlinear oscillations of the finite amplitude of an uncharged drop of an ideal incompressible conducting liquid in an external uniform electrostatic field is solved for the first time by analytical asymptotic methods. The problem is solved in an approximation quadratic in amplitude of the initial deformation of the equilibrium shape of the drop and in eccentricity of its equilibrium spheroidal deformation. Compared with the case of nonlinear oscillations of charged drops in the absence of the field, the curvature of the vertices of uncharged drops nonlinearly oscillating in the field is noticeably higher, whereas the number of resonant situations (in the sense of internal resonant interaction of modes) is much smaller.     

Nonlinear vibrations of a charged drop in an electrostatic suspension

The problem of nonlinear vibrations of a charged drop of an ideal incompressible conducting fluid in an electrostatic suspension is analytically solved in an approximation quadratic in two small parameters: vibration amplitude and equilibrium deformation of the shape of the drop in an electrostatic field. To solve the problem analytically, the desired quantities are expanded in semiinteger powers of the small parameters. It is shown that the charge of the drop and the gravitational field influence the shape of the drop, nonlinear corrections to the vibration frequencies, and critical conditions for instability of the drop against the surface charge. At near-critical values of the charge, the shape of the nonlinearly vibrating drop falls far short of being a sphere or a spheroid, which should be taken into account in treating experimental data.     

Nonlinear oscillations of a drop moving at a constant velocity in an insulating medium subjected to an electrostatic field

Nonlinear asymptotic calculations of the second order of smallness in the amplitude of the initial deformation of an ideally conducting liquid drop show that the laminar flow of an ideal conducting incompressible dielectric liquid flowing about the drop in an external electrostatic field parallel to the flow causes oscillation mode’s interaction in the first and second orders of smallness. Both linear and nonlinear interactions between the oscillation modes of the drop excite modes that are absent in the spectrum of modes governing the initial deformation of the drop’s equilibrium shape. In the second order of smallness, the mode interaction decreases the electrostatic field strength, as well as the velocity and density of the environment, that are critical for development of instability of the drop against the polarization charge.     

Mechanisms behind spheroidal oscillations and electrostatic instability of a charged drop

Mechanisms behind the oscillations of a charged spheroidal drop deformed at the zero time and the sequence of oscillation modes are investigated. It is shown that two modes adjacent to those governing the initial deformation are also excited on either side due to interaction between the spheroidal deformation and oscillation modes. If the charge of the drop is so close to a value critical for electrostatic instability that the finite-amplitude virtual initial deformation makes the fundamental mode unstable, its amplitude, as well as the amplitude of the nearest neighbor coupled to the fundamental mode through deformation, starts to exponentially grow with time. If the charge is equal to, or slightly exceeds the critical value, the amplitudes of the fundamental mode and all modes deformation-coupled with it lose stability almost simultaneously. This qualitatively changes the conditions under which the charged drop becomes unstable against the self-charge. The superposition of higher oscillation modes at the vertices of the spheroidal drop generates dynamic (i.e., time-oscillating) hillocks emitting an excessive charge.     

Instability of a charged spherical drop moving parallel to an external electrostatic field

It is shown that the pressure of electrostatic fields induced by the self-charge of a drop and by the polarization charge and aerodynamic pressure of a laminar gas flow around a moving charged drop acting simultaneously reduce the critical instability conditions for the surface of the drop. For these conditions, the spectrum of capillary oscillations of the drop is calculated. It is found that, at various values of the charge, field strength, and velocity of the drop, the vibrational instability of the drop surface develops through the interaction of different oscillation modes, namely, second and third, second and fourth, and third and fifth.     

Nonlinear analysis of the equilibrium shape of a charged conducting drop in an electrostatic suspension

An analytical asymptotic expression is derived that describes the equilibrium shape of a charged drop of an ideal incompressible conducting liquid suspended in superposed collinear uniform electrostatic and gravitational fields. The expression is obtained in an approximation quadratic in the small amplitude of deviation of the equilibrium drop from a sphere, with the electrostatic field dimensionless strength taken as a measure of the deviation amplitude. With allowance for the gravitational and electrostatic fields and interaction between the drop self-charge and external electrostatic field, the equilibrium shape of the drop is found to be very close to a spheroid when the charge and the electrostatic field strength are far from their critical values. The analysis is carried out with a refined procedure of calculation of the equilibrium shape of drops placed in external force fields.     

Disintegration of a drop in an external electrostatic field

Regular features of the disintegration of both a drop of a perfectly conducting liquid and a drop of a dielectric liquid into two or three parts in an external uniform electric field are studied using the principle of minimizing the potential energy of the final state of a closed system with spontaneous processes.     

Capillary oscillations of a drop in an electric field

The capillary oscillations of a conducting spheroidal drop with small eccentricity in an electric field are studied. The effect of the electric field is to lower the natural frequencies of oscillations and the damping coefficient.     

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 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 capillary oscillation of a charged drop

In the quadratic approximation with respect to the amplitudes of capillary oscillation and velocity field of the liquid moving inside a charged drop of a perfectly conducting fluid, it is shown that the liquid drop oscillates about a weakly prolate form. This refines the result obtained in the linear theory developed by Lord Rayleigh, who predicted oscillation about a spherical form. The extent of elongation is proportional to the initial amplitude of the principal mode and increases with the intrinsic charge carried by the drop. An estimate is obtained for the characteristic time of instability development for a critically charged drop.     

Modification of the boundary layer theory for calculating viscous drop oscillations in a uniform electrostatic field

Capillary oscillations on the free surface of a viscous conductive liquid drop placed in an electrostatic field are calculated. In an approximation linear in stationary deformation amplitude, the drop in this field has the shape of a spheroid extended along the field. The initial problem is modified and simplified in terms of the boundary layer theory by applying an approximation that is linear in the oscillation amplitude and quadratic in the eccentricity of the drop. The accuracy of the approximate solution relative to an exact one is estimated. It is shown that, with a rise in the electrostatic field strength (with an increase in the eccentricity of the drop) and in the viscosity of the liquid, the boundary layer at the free surface of the drop becomes thicker.     

Capillary oscillations of an emitting charged viscous drop of finite conductivity

Capillary oscillations of a charged drop of a viscous incompressible liquid with finite conductivity emitting electromagnetic waves are considered. A dispersion relation for the capillary oscillations has been derived and analyzed using a linear approximation to oscillation amplitudes.     

Linear (in oscillation dimensionless amplitude) interaction between the modes of a nonspherical charged drop in an external electrostatic field

The stability of a heavily charged drop in a weak uniform electrostatic field (in which the equilibrium shape of the drop can be represented by a prolate spheroid) is calculated in the fourth order of smallness in the eccentricity of the spheroidal drop and in the first order of smallness in the drop oscillation dimensionless amplitude. It is found that as the order of approximation in eccentricity grows, so does the number of modes interacting with the initially excited mode. In the given order of smallness, the preferred (initially excited) mode is shown to interact with the nearest eight modes. The drop becomes unstable if such is the second mode.     

Electromagnetic radiation produced by linear oscillations of a charged drop

A dispersion equation is derived for the capillary oscillations of a charged drop of an ideal incompressible liquid travelling in an ideal incompressible ambient allowing for the energy lost via emission of electromagnetic radiation. The intensity of electromagnetic radiation emitted by a single drop and a storm cloud has been estimated.     

Electromagnetic radiation due to nonlinear oscillations of a charged drop

The nonlinear oscillations of a spherical charged drop are asymptotically analyzed under the conditions of a multimode initial deformation of its equilibrium shape. It is found that if the spectrum of initially excited modes contains two adjacent modes, the translation mode of oscillations is excited among others. In this case, the center of the drop’s charge oscillates about the equilibrium position, generating a dipole electromagnetic radiation. It is shown that the intensity of this radiation is many orders of magnitude higher than the intensity of the drop’s radiation, which arises in calculations of the first order of smallness and is related to the drop’s charged surface oscillations.     

On nonlinear corrections to oscillations frequencies of a charged drop in an incompressible ambient medium

An analytic expression in the third order of smallness in the amplitude of the initial deformation of an equilibrium, spherical, charged, ideally conducting drop in an incompressible dielectric medium is derived for its generatrix and for nonlinear corrections to oscillation frequencies. It is shown that the presence of the ambient liquid reduces the absolute values of the corrections to frequency and of the self-charge critical for the realization of drop instability.     

Some laws of polarization and dispersion of a drop in an electrostatic field

The laws of distribution among contributions in various interactions to the total polarization energy of a conductor in a uniform electrostatic field was analyzed. It is shown that in a closed system, spontaneous shape variations of a liquid conductor with a free surface in an external magnetic field are possible only if they are accompanied by an increase in the conductor dipole moment. Variations of the intrinsic energy of a conductor are studied by the example of a conductive liquid drop in the case where a drop affected by a polarization charge becomes unstable. Analytical expressions defining the sizes and charges of the droplets ejected out of the initial drop under the conditions of instability are derived.