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 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.     

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.     

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.     

On the structure of the velocity field under the free spheroidal surface of a viscous liquid drop oscillating in an electrostatic field

An asymptotic analytical solution to an initial boundary-value problem considering (i) the time evolution of the capillary oscillation amplitude as applied to a viscous spheroidal liquid drop placed in a uniform electrostatic field and (ii) the liquid flow velocity field inside the drop is found. The problem is solved in an approximation that is linear in two small parameters: the dimensionless oscillation amplitude and the dimensionless field-induced constant deformation of the equilibrium (spherical) shape of the drop. Terms proportional to the product of the small parameters are retained. In this approximation, interaction between oscillation modes is revealed. It is shown that the intensity of the eddy component of the oscillation-related velocity field depends on the liquid viscosity and the external uniform electrostatic field strength. The intensity of the eddy component decays rapidly with distance from the free surface. The depth to which the eddy flow (which is caused by periodical flows on the free surface) penetrates into the drop is a nonmonotonic function of the polar angle and increases with dimensionless viscosity and field strength.     

Instability of a spherical drop in the field of an electrical dipole

Analytical calculations show that, as a field in which an initially spherical charged conducting incompressible drop is placed becomes more and more nonuniform, coupling between the drop’s oscillation modes grows and the threshold of stability against the electrical field pressure declines. When an electrostatic parameter characterizing the electric field pressure exceeds a value that is critical for a certain mode to be unstable, the amplitude of this mode exponentially grows in an aperiodic manner and the amplitudes of modes coupled with this mode build up in an oscillatory manner, each mode having its own instability growth rate. In all cases, there exists a threshold value of the dimensionless electric parameter above which all oscillation modes are unstable.     

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 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 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.     

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 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 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.     

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.     

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 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 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.     

Deformation and nonlinear corrections to critical conditions of charged drop instability in an electrostatic suspension

Nonlinear asymptotic analysis of a charged drop placed in electrostatic and gravitational fields reveals a correction to the oscillation frequency and, accordingly, to the critical Rayleigh parameter. The analysis uses approximations quadratic in oscillation amplitude and linear in dimensionless equilibrium deformation of the drop. The correction is found to be proportional to the product of the oscillation amplitude and deformation. It is natural to name this correction deformational. In computations of the third order of smallness in oscillation dimensionless amplitude, a correction to the frequency and Rayleigh parameter appears, which is due to a nonlinear interaction between oscillation modes. This correction is larger than the deformational one in magnitude. Deformational corrections can be eliminated by experimenting under no-gravity conditions, but corrections due to the nonlinearity of hydrodynamic equations cannot be eliminated in this way. It is these corrections that are responsible for a critical Rayleigh parameter measurement inaccuracy.     

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.     

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.     

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.