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.     

Influence of the viscosity of a nonlinearly vibrating charged liquid drop on the positions of internal resonances

An asymptotic analytical expression for the generatrix of a viscous charged liquid drop is for the first time derived in the second order of smallness in the axisymmetric initial deformation of the drop. The expression is represented as an infinite series in the roots of the dispersion relation and a finite sum of the numbers of modes specifying the initial deformation. In some of the terms of the analytical expression, the denominators involve the differences between the mode frequencies. These differences may become small under certain values of the charge, causing internal nonlinear resonant mode interaction. Analytical and numerical investigations of the effect of viscosity on the vibrating frequency show that the resonant values of the self-charge of the drop tend to increase with increasing viscosity. The viscosity of the liquid does not affect the spectrum of modes excited via nonlinear mode interaction.     

Nonlinear capillary vibrations of a charged drop placed in a dielectric medium: Single-mode initial deformation of the drop shape

The nonlinear vibrations of the equilibrium spherical shape of a charged drop placed in a perfect incompressible dielectric medium are asymptotically calculated in the second-order approximation in single-mode initial deformation of the drop surface. The drop is assumed to be a perfect incompressible liquid. It is shown that the nonlinear vibration amplitudes, as well as the energy distribution between nonlinearly excited modes, depend significantly on the parameter ρ, where ρ is the ratio of the environmental density to that of the drop. It is also demonstrated that an increase in ρ raises the amplitude of the highest of the vibration modes excited due to second-order nonlinear interaction. In the second order of smallness, the amplitude of the zeroth mode is independent of the density ratio. As ρ grows, the effect of the self-charge of the drop, the interfacial tension, and the permittivity of the environment on the nonlinear oscillations increases.     

Nonlinear resonance interaction between the oscillation modes of a spherical liquid layer on the surface of a melting hailstone

Evolutionary equations are derived and solved that describe the time dependence of the oscillation mode amplitudes on the surface of a charged conducting liquid layer resting on a solid core. It is assumed that the layer experiences a multimode initial deformation. The equations are solved asymptotically in the second order of smallness in the small dimensionless amplitude of capillary oscillations on the surface of the layer. Mechanisms behind internal nonlinear resonance interaction between the modes of the liquid layer oscillations and behind energy transfer between the modes both in degenerate and in secondary combination resonances are investigated. It is found that in the degenerate resonance interaction between oscillation modes, the energy may be transferred not only from lower to higher modes but also vice versa if the higher mode is excited at the zero time. This conclusion is valid not only for a liquid layer on the surface of a solid core but also for a drop.     

On nonlinear resonant four-mode interaction between capillary vibrations of a charged drop

Energy transfer from higher modes of capillary vibrations of an incompressible liquid charged drop to the lowest fundamental mode under four-mode resonance is studied. The resonance appears when the problem of nonlinear axisymmetric capillary vibration of a drop is solved in the third-order approximation in amplitude of the multimode initial deformation of the equilibrium shape of the drop. Although the resonant interaction mentioned above builds up the fundamental mode even in the first order of smallness, its amplitude turns out to be comparable to a quadratic (in small parameter) correction arising from nonresonant nonlinear interaction, since the associated numerical coefficients are small.     

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.     

Nonlinear capillary vibrations and stability of a highly charged drop under single-mode initial deformation of large amplitude

Nonlinear vibrations of a charged drop that are caused by a virtual initial perturbation of the equilibrium spherical shape of the drop were considered. The perturbation can be proportional to any of the free vibration modes. The spectrum and stability of the vibrations were studied correct to second order of smallness.     

On acoustic radiation from a nonlinearly vibrating charged drop

An expression for the time-varying shape of an incompressible liquid drop immersed in a compressible dielectric medium is derived to the second-order approximation in drop vibration amplitude. It is shown that the acoustic radiation spectrum of the drop has a monopole component, which makes a considerable contribution to the integral radiation intensity. Its appearance is associated with the time variation of the zeroth-mode vibration amplitude, showing up in the second order of smallness.     

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.     

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

Analytical expressions are derived for the shape generatrix of an ideally conducting drop immersed in an incompressible dielectric medium as well as for nonlinear corrections to the frequencies of the oscillations of the drop. The solutions are obtained in an approximation of the third order of smallness with respect to the amplitude of the initial deformation of the equilibrium spherical shape of the drop. It is shown that the presence of the ambient liquid results in a reduction of the absolute magnitudes of corrections both to the oscillation frequencies and the self-charge critical for the development of instability of the drop.     

Nonlinear vibrations of a charged drop in a third-order approximation in the amplitude of multimode initial deformation

A solution to the problem of nonlinear surface vibration of a charged ideal liquid drop is found in a third-order approximation in initial multimode deformation of the equilibrium spherical shape by the method of many scales. It is shown that the spectrum of modes that are responsible for the shape of the drop at an arbitrary time instant depends considerably on the spectrum of modes governing the initial deformation of the drop. The latter spectrum also has an effect on nonlinear corrections to the vibration frequencies and, consequently, on a nonlinear correction to the critical Rayleigh parameter, which specifies the stability of the drop against self-charge.     

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.     

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.     

On the calculation of the translational mode amplitude for a drop nonlinearly vibrating in an environment

The second-order amplitudes of the capillary vibration modes of a drop of an ideal incompressible liquid placed in an incompressible ideal medium are calculated. The approximation is quadratic in initial multimode deformation of the equilibrium spherical shape caused by nonlinear interaction. The mathematical statement of the problem is such that the immobility condition for the center-of-mass of the drop is met automatically. When the translational mode amplitude is calculated, a set of hydrodynamic boundary conditions at the interface, rather than the condition of center-of-mass immobility (which is usually applied for simplicity in the problems of drops vibration in a vacuum), should be used.     

Analytical study of nonlinear vibrations of a charged viscous liquid drop

The generatrix of a nonlinearly vibrating charged drop of a viscous incompressible conducting liquid is found by directly expanding the equilibrium spherical shape of the drop in the amplitude of initial multimode deformation up to second-order terms. A fact previously unknown in the theory of nonlinear interaction is discovered: the energy of an initially excited vibration mode of a low-viscosity liquid drop is gradually (within several vibrations periods) transferred to the mode excited by only nonlinear interaction. Irrespectively of the form of the initial deformation, an unstable viscous drop bearing a charge slightly exceeding the critical Rayleigh value takes the shape of a prolate spheroid because of viscous damping of all the modes (except for the fundamental one) for a characteristic time depending on the damping rates of the initially excited modes and the further evolution of the drop is governed by the fundamental mode. In a high-viscosity drop, the rate of rise of the unstable fundamental mode amplitude does not increase continuously with time, contrary to the predictions of nonlinear analysis in terms of the ideal liquid model: it first decreases to a value slightly differing from zero (which depends on the extent of supercriticality of the charge and viscosity of the liquid), remains small for a while (the unstable mode amplitude remains virtually time-independent), and then starts growing.     

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.     

On internal mode resonance in a nonlinearly vibrating volumetrically charged dielectric drop

In the approximation quadratic in the amplitude of an arbitrary initial deformation of an equilibrium spherical uniformly (volumetrically) charged drop of a dielectric liquid, an analytical expression for the drop surface generatrix as a function of time is derived in the case when the drop shape executes axisymmetric vibrations. A condition that must be imposed on mode frequencies in order for resonant interaction between modes to take place in the quadratic approximation is found. It is shown that many resonances, rather than one known previously, are realized when the self-charge is insufficient (subcritical) for drop surface instability against self-charge to arise. Nonlinear two-and three-mode resonant interactions are studied.     

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.     

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.