On calculating the oscillations of a viscous liquid layer over a solid spherical core in terms of the boundary layer theory

The theory of a boundary layer near the periodically oscillating free surface of a spherical viscous liquid layer over a solid core (bottom) is modified. Two boundary layers are considered to adequately describe a liquid viscous flow in the system: one at the free surface of the liquid and the other at the solid bottom. The thicknesses of the boundary layers are estimated, which provide any given discrepancy between an exact solution to the model problem and a solution obtained in the small viscosity approximation. Taking into account the boundary layer near the solid bottom is shown to be significant only for lower oscillation modes. For higher modes, the flow near the core can be considered potential. In the case of lower modes and shallow liquid, the surface and bottom boundary layers overlap and an eddy flow occupies the entire volume of the liquid.     

Modification of the boundary layer theory for analysis of finite-amplitude oscillations of a charged bubble in a viscous liquid

The prevailing concepts concerning the boundary layer near the free surface of a viscous liquid associated with oscillatory motion are modified for calculating finite-amplitude linear oscillations of a charged bubble in this liquid. Equations of the boundary layer theory for the neighbourhood of the oscillating free spherical surface of a charged bubble in a dielectric liquid are derived, their analytic solution is obtained and compared with the exact solution, and the thickness of the boundary layer is assessed. The range of applicability of the modified theory is determined.     

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 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 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.     

Modification of the boundary layer theory for analysis of wave motions in a cylindrical jet of a viscous liquid

The existent concepts of the boundary layer near the free surface of a viscous liquid, which is related to its periodic motion, are modified with the aim of analyzing the finite-amplitude wave motion on the surface of a thick charged jet of a viscous conducting liquid. To describe the flow in the boundary layer, a model problem is proposed that is simpler in statement compared with the complete problem and the solution of which uses the governing properties of the exact solution obtained in the low-viscosity asymptotics: the form of the dispersion relation, wave profile, and rate of velocity field viscous damping with time. An estimate is made of the boundary layer thickness at which the discrepancy between the exact solution and solution to the model problem (stated in terms of the theory proposed) falls into a given interval in the low-viscosity asymptotics. The domain of applicability of the modified theory is determined.     

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)     

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)     

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)     

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 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.     

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.     

Initial-value problem for plasma oscillations in a uniform magnetic field

The solution is given of the initial-value problem for the nonrelativistic linearised Vlasov-Maxwell equations describing longitudinal and transverse plasma oscillations in an external uniform magnetic field. The problem is solved for all directions of propagation except normal to the external magnetic field, and the equilibrium distribution is not assumed isotropic. The method of solution is an extension of Van Kampen's eigenfunction expansion technique, already developed considerably by Zelazny and McCure, in which the problem is reduced to the solution of a system of singular integral equations.     

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.     

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.     

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.     

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.