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)     

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.     

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 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.     

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 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.     

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 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.     

On the possibility of corona initiation near a conducting drop nonlinearly vibrating in an external electrostatic field

An analytical asymptotic expression for the field strength near an ideal incompressible electrically conducting liquid drop nonlinearly vibrating in external electrostatic field E ₀ is found in an order of 5/2 in a small parameter. The small parameter here is the amplitude of deformation of the spherical shape of the drop. It is found that the strength of the electric field resulting at the tops of the drop exceeds the corona-initiating field even if E ₀ is one order of magnitude lower than the value at which the drop becomes unstable against the induced charge (that is, at such values of E ₀ as are observed in storm clouds in full-scale experiments).     

The laws of electromagnetism and an electrostatic analogue

Starting from the idea of a current-carrying coil as a magnetic ‘dipotential’, i.e. a dipole of constant potential difference, the relationships between the electromagnetic laws of Ampere, Biot-Savart, Grassmann and Faraday are discussed.

Different text-books on electricity and magnetism have used various methods of relating the laws of electromagnetism, but there remain difficulties whichever method is adopted. The present work is an attempt to put elementary work on a more logical basis without elaborate mathematics, by a greater use of the concept of magnetic potential. While it is not claimed that anything fundamentally new will be found here, it is hoped that the ideas will be useful for teaching.     

Critical equilibrium spheroidal deformation of a drop of dielectric liquid in a uniform electrostatic field

The stability of a dielectric drop, which in an external electrostatic field takes on the equilibrium shape of a prolate spheroid, is analyzed using the principle of minimum total potential energy of an isolated system. The values of the Taylor parameter and degree of spheroidal deformation at which the drop loses stability are determined for a wide range of dielectric constants of the substance of the drop. Zh. Tekh. Fiz. 69, 23–28 (July 1999)     

Local enhancement of a uniform electrostatic field near the tip of a spheroidal drop

The enhancement in a uniform electrostatic field at the tip of a spheroidal drop is shown to depend on the dielectric constant of the drop material, its initial radius, and the external electric field and to become greater as these increase. The loss of stability of a drop in an external electrostatic field that is accompanied by a very rapid growth in the magnitude of the spheroidal deformation causes a rapid, transient enhancement of the field at its tip. Zh. Tekh. Fiz. 69, 49–54 (August 1999)     

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.     

Analytical properties of the polarization operator and dispersion relations for of a photon in an external non-abelian field

Thresholds of photoproduction of SU(2) color quarks in an external field are determined. Analyticity of the polarization operator in a chromomagnetic field is studied. The dispersion relation for the trace of the polarization operator of a photon with an internal quark loop is derived. M. E. Rasulzade Baku State University; M. V. Lomonosov Moscow State University. Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 6, pp. 11–16, June, 2000.     

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.     

Internal circulation in a drop in an acoustic field

An investigation of the internal flow field for a drop at the antinode of a standing wave has been carried out. The main difference from the solid sphere case is the inclusion of the shear stress and velocity continuity conditions at the liquid-gas interface. To the leading order of calculation, the internal flow field was found to be quite weak. Also, this order being fully time dependent has a zero mean flow. At the next higher order, steady internal flows are predicted and, as in the case of a solid sphere, there is a recirculating layer consisting of closed streamlines near the surface. In the case of a liquid drop, however, the behavior of this recirculating Stokes layer is quite interesting. It is predicted that the layer ceases to have recirculation when [formula: see text], where [symbol: see text] is the liquid viscosity, mu is the exterior gas-phase viscosity, and M is the dimensionless frequency parameter for the gas phase, defined by M = i omega a2 rho/mu, with a being the drop radius. Thorough experimental confirmation of this interesting new development needs to be conducted. Although it seems to agree with many experiments with levitated drops where no recirculating layer has been clearly observed, a new set of experiments for specifically testing this interesting development need to be carried out.     

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.     

Motion and radiation of an electron in a homogeneous magnetic field and a weak electrostatic field

The spectral and angular distributions characterizing the intensity of the spontaneous radiation emitted by an electron moving in crossed electrostatic and homogeneous magnetic fields is determined in both the relativistic and the nonrelativistic case.     

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.     

Polarization fluctuations in a dipolar medium in an electrostatic field

Radio Engineering and Electronics Institute, Academy of Sciences of the USSR. Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Radiofizika, Vol. 32, No. 9, pp. 1113–1124, September, 1989.