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.     

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.     

Stability of a charged drop of a viscous electrically conducting liquid in a viscous electrically conducting medium

A dispersion relation is derived for capillary oscillations of a charged electrically conducting viscous drop in an electrically conducting viscous medium. It is shown that aperiodic instability of the charged interface between the two media can arise in this system, with a growth rate that depends qualitatively differently on the ratio of their conductivities in different ranges of values of this ratio. In a certain range of conductivity ratios the drop undergoes oscillatory instability. Zh. Tekh. Fiz. 69, 34–42 (October 1999)     

Numerical analysis of decaying nonlinear oscillations of a viscous liquid drop

An adaptive grid numerical model is developed for simulating the dynamics of a viscous liquid drop whose initial shape is strongly disturbed by an external field. Simulated oscillations of a drop in microgravity and on a horizontal surface are compared with available numerical and experimental results.     

Properties of expansions in Legendre polynomial derivatives that show up in analysis of nonlinear vibrations of a viscous liquid drop

Analytical expressions for the coefficients of expansions of Legendre polynomial products in the first and second derivatives of the polynomials with respect to polar angles, as well as for the coefficients of expansions of derivative products in Legendre polynomials and their first derivatives, are derived. An interrelation between these expansion coefficients and a relationship between these coefficients and the Clebsch-Gordan coefficients are found. When axisymmetric nonlinear vibrations of a viscous liquid drop are investigated analytically, the toroidal component of the velocity field can be ignored, which significantly cuts the body of computation.     

Boundary layer theory modified for analyzing finite-amplitude oscillations of a viscous liquid charged drop

The existing concepts of the boundary layer arising near the free surface of a viscous liquid, which is related to its periodic motion, are revised with the aim to calculate finite-amplitude linear oscillations of a viscous liquid charged drop. Equations complementing the boundary layer theory are derived for the vicinity of the oscillating free spherical surface of the drop. An analytical solution to these equations is found, comparison with an exact solution is made, and an estimate of the boundary layer thickness is obtained. The domain of applicability of the modified theory is defined.     

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.     

On the possibility of corona-discharge ignition in the vicinity of a weakly charged drop executing nonlinear vibrations

An analytic expression for the electric-field strength in the vicinity of a charged drop of an electrically conducting liquid is obtained for the case where the initial shape of the drop executing nonlinear vibrations is specified by a virtual excitation of an arbitrary single mode of capillary vibrations. It turns out that, even at small charges (such that the Rayleigh parameter for the drop is equal to one-tenth of the critical value associated with stability against the intrinsic charge), the electric-field strength at the drop surface in the case of an initial excitation of one of high modes is sufficient for the ignition of a corona discharge.     

Capillary oscillations and stability of a charged, viscous drop in a viscous dielectric medium

The scalarization method is used to obtain a dispersion relation for capillary oscillations of a charged, conducting drop in a viscous, dielectric medium. It is found that the instability growth rate of the charged interface depends substantially on the viscosity and density of the surrounding medium, dropping rapidly as they are increased. In the subcritical regime the influence of the viscosity and density of both media leads to a nonmonotonic dependence of the damping rate of the capillary motions of the liquid on the viscosity or density of the external medium for a fixed value of the viscosity or density of the internal medium. The falloff of the frequencies of the capillary motions with growth of the viscosity or density of the external medium is monotonic in this case. Zh. Tekh. Fiz. 68, 1–8 (September 1998)     

Instability of a charged spherical viscous drop moving relative to a medium

It is shown that, as the velocity of the flow around a charged drop of viscous liquid increases the drop charge value critical for the occurrence of drop instability rapidly decreases. It is found that, for some domains of values of the charge, the ratio of densities of the media, and the ambient velocity, the even and odd modes of the drop capillary oscillations pairwise couple with each other, which represents drop vibrational instability against the tangential discontinuity of the velocity field at the drop surface. At medium velocities larger than those associated with such domains, the instability growth rates for odd modes exceed the increments of even modes with smaller orders, which corresponds to the parachute-like deformation of the drop in the flow.     

On nonlinear vibrations of a charged drop in the third-order approximation in the amplitude of initial single-mode excitation

Nonlinear axisymmetric motions of the free surface of a charged drop of an ideal liquid under the single-mode initial deformation of its equilibrium shape is investigated in the third-order approximation in the initial perturbation amplitude. An analytical expression for the drop shape generatrix is derived. Nonlinear corrections to the vibration frequencies for the initial perturbation of an arbitrary mode are found for the first time. The effect of vibration nonlinearity on the instability of the drop against its self-charge is studied.     

On the time evolution of the surface shape of a charged viscous liquid drop deformed at zero time

The problem of capillary oscillations of the equilibrium spherical shape of a charged viscous incompressible liquid drop is solved in an approximation linear in amplitude of the initial deformation that is represented by a finite sum of axisymmetric modes. In this approximation, the shape of the drop as a function of time, as well as the velocity and pressure fields of the liquid in it, may be represented by infinite series in roots of the dispersion relation and by finite sums in numbers of the initially excited modes. In the cases of low, moderate, and high viscosity, the infinite series in roots of the dispersion relation can be asymptotically correctly replaced by a finite number of terms to find compact analytical expressions that are convenient for further analysis. These expressions can be used for finding higher order approximations in amplitude of the initial deformation.     

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.     

Acoustic radiation from a drop due to its nonlinear vibrations

It is shown that the intensity of acoustic radiation from a vibrating drop depends mainly on the monopole and dipole components appearing only in the second order of smallness in vibration amplitude. The intensity of the quadrupole acoustic radiation generated by the vibration fundamental mode in the first order of smallness in amplitude turns out to be much weaker. This is associated with the fact that, if the acoustic wavelength is much larger than the drop characteristic size, their ratio becomes a governing small parameter, being lesser than the ratio of the drop vibration amplitude to the drop linear size. Analytical estimates of the amplitudes of monopole, dipole, and quadrupole components of the velocity field associated with the acoustic field of the drop.     

On the thickness of a boundary layer related to the oscillating free surface of a charged drop of a viscous liquid

The capillary oscillations of a charged drop of a viscous liquid are calculated in terms of the boundary layer theory in an approximation linear in oscillation amplitude. Calculation is accompanied with the estimation of a relative error that arises when the exact solution is replaced by an approximate one. It is shown that, for the calculation accuracy in the framework of the boundary layer theory to be about several percent, the thickness of the boundary layer near the free surface of the drop must be several times larger than that at which the intensity of the eddy flow caused by the oscillating surface decreases by e times. As the viscosity of the liquid grows, so does the thickness of the boundary layer.     

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.     

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.     

Nonlinear vibrations of a charged drop due to the initial excitation of neighboring modes

The asymptotic analysis of the nonlinear vibrations of a charged drop that are induced by a multi-mode initial deformation of its equilibrium shape is performed. It is shown that when two, three, or several neighboring modes are present in the initial deformation spectrum, the mode with the number one (translational mode) appears in the second-order mode spectrum. The excitation of the translational mode follows from the requirement of center-of-mass immobility and causes the dipole components (which are absent in the linear analysis) to appear in the spectra of the acoustic and electromagnetic radiation of the charged drop.     

Influence of charge relaxation on the capillary oscillations of a charged viscous spheroidal drop

A dispersion relation is derived for the spectrum of capillary modes of a charged spheroidal drop of a viscous liquid with allowance for charge relaxation. It is shown that the finite charge transport rate leads to lowering of the instability growth rates for various capillary modes of a spheroidal drop of a low-viscosity liquid. As the degree of deformation of the drop increases, the magnitude of the absolute change in the growth rate caused by the finite rate of charge redistribution decreases. Zh. Tekh. Fiz. 69, 28–36 (August 1999)     

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.