Electric dispersion of fluid under the oscillatory instability of its free surface

A semiphenomenological analysis is performed of possible modes of electric dispersion of drops and menisci at the end of the capillary used to deliver the liquid into the discharge system under an oscillatory instability of the charged liquid surface. The instability is assumed to be induced by a time-dependent external force acting on the liquid surface, a finite rate of charge redistribution over the surface under virtual deformations, and tangential discontinuity of the velocity field across the interface.     

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.     

Electrostatic instability of a heavily charged conducting liquid jet

The effect of electric charge on the jet surface on the capillary instability of the jet and its disintegration into drops is analyzed. A theoretical explanation is given for the electrostatic mechanism of instability development and jet disintegration that is akin to the mechanisms behind the instability of a heavily charged drop (Rayleigh instability) and flat uniformly charged liquid surface (Tonks-Frenkel instability) but differs qualitatively from the conventional capillary mechanism of instability and disintegration.     

Exact solutions of the equations of motion of liquid helium with a charged free surface

The dynamics of the development of instability of the free surface of liquid helium, which is charged by electrons localized above it, is studied. It is shown that, if the charge completely screens the electric field above the surface and its magnitude is much larger than the instability threshold, the asymptotic behavior of the system can be described by the well-known 3D Laplacian growth equations. The integrability of these equations in 2D geometry makes it possible to describe the evolution of the surface up to the formation of singularities, viz., cuspidal point at which the electric field strength, the velocity of the liquid, and the curvature of its surface assume infinitely large values. The exact solutions obtained for the problem of the electrocapillary wave profile at the boundary of liquid helium indicate the tendency to a change in the surface topology as a result of formation of charged bubbles.     

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.     

Nonlinear analysis of the equilibrium shape of a charged conducting drop in an electrostatic suspension

An analytical asymptotic expression is derived that describes the equilibrium shape of a charged drop of an ideal incompressible conducting liquid suspended in superposed collinear uniform electrostatic and gravitational fields. The expression is obtained in an approximation quadratic in the small amplitude of deviation of the equilibrium drop from a sphere, with the electrostatic field dimensionless strength taken as a measure of the deviation amplitude. With allowance for the gravitational and electrostatic fields and interaction between the drop self-charge and external electrostatic field, the equilibrium shape of the drop is found to be very close to a spheroid when the charge and the electrostatic field strength are far from their critical values. The analysis is carried out with a refined procedure of calculation of the equilibrium shape of drops placed in external force fields.     

Evaporation of Charged Drops

A model of evaporation of a multiply charged liquid drop is developed. The model self-consistently takes into account the main factors influencing the charged drop evaporation, including effects of the drop surface curvature and charge on the saturated vapor pressure, repeated fragmentation of drops during evaporation, and the capability of drops having a unit charge and a certain stabilization radius not to evaporate even in an unsaturated vapor medium. Analytical dependences are derived that can be used to calculate an integral lifetime of a charged drop with allowance for its fragmentation into smaller drops. Our estimates demonstrate that the evaporation time of charged drops is much smaller than that of uncharged drops.     

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.     

Mass transfer due to nonlinear capillary-gravitational waves on the surface of a viscous liquid

An analytical expression of the second order of smallness in wave amplitude-to-wavelength ratio is derived for a horizontal flow arising in a finite-depth layer of a viscous liquid under the action of a periodic nonlinear capillary wave. It is found that the liquid flow is determined by the nonlinear component of the velocity field vortex part and the flow rate increases with increasing viscosity and decreasing wavelength irrespective of the layer thickness. In thin layers, the flow rate rapidly drops from its maximal value with increasing viscosity, wavelength, and surface charge density. If the liquid surface is charged, the horizontal liquid flow decreases rapidly as the surface charge density approaches the threshold of the Tonks-Frenkel instability.     

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.     

Influence of charge relaxation on the capillary disintegration of a jet of a viscous dielectric liquid placed in an electrostatic field collinear with the axis of the jet

A dispersion relation is derived for capillary waves with an arbitrary symmetry on the surface of a charged jet of a finite-conductivity viscous liquid placed in an electric field collinear with the axis of the jet. Analytical calculations are carried out in an approximation that is linear in dimensionless wave amplitude. In the case of axisymmetric waves, the instability of which causes disintegration of the jet into drops, the finiteness of the potential equalization rate has a noticeable effect only for jets of poorly conducting liquids. The charge relaxation shows up in that “purely relaxation” periodic and aperiodic liquid flows arise. When the conductivity of the liquid declines, the instability growth rates for unstable waves increase and their spectrum extends toward short waves. A charge present on the surface of the jet enhances its instability. An increase in the charge surface diffusion coefficient variously influences the capillary and relaxation branches of the solution: the damping ratio increases in the former case and decreases in the latter. As the diffusion coefficient rises, relaxation flows may become unstable.     

Instability of a charged interface of two immiscible viscous liquids with charge relaxation taken into account

On the basis of an analysis of a derived dispersion relation, it is demonstrated that there can be two different types of instability relative to the free charge of a charged, planar interface between two viscous immiscible liquids with finite electrical conductivity in a gravitational field. For large values of the surface charge density, depending on the viscosities and ratio of conductivities of the media, one can observe either an aperiodic (of the Tonks-Frenkel type) or oscillatory instability of the interface. Increasing the viscosity of the lower liquid leads to a substantial drop in the increments of the mentioned instability types and alters the critical conditions for manifestation of the oscillatory instability, whereas varying the viscosity of the upper surface has only a very weak effect on these characteristics. Zh. Tekh. Fiz. 68, 13–19 (September 1998)     

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.     

Simulations of coulombic fission of charged inviscid drops

We present boundary-integral simulations of the evolution of critically charged droplets. For such droplets, small perturbations are unstable and eventually lead to the formation of a lemon-shaped drop with very sharp tips. For perfectly conducting drops, the tip forms a self-similar cone shape with a subtended angle identical to that of a Taylor cone, and quantities such as pressure and velocity diverge in time with power-law scaling. In contrast, when charge transport is described by a finite conductivity, we find that small progeny drops are formed at the tips, whose size decreases as the conductivity is increased. These small progeny drops are of nearly critical charge, and are precursors to the emission of a sustained flow of liquid from the tips as observed in experiments of isolated charged drops.     

Some effects of the electrostatic interaction of water drops in the atmosphere

The electric field at the surface of two conducting spherical charged particles and their interaction force are calculated. It is shown that as particles carrying like charge approach each other, the force changes sign and becomes attractive. The case where the charge on each particle varies as the square of its radius is an exception (repulsion at any distance between the particles). Self-similar asymptotic solutions for the interaction force and energy are found for particles of identical size. For a pair of charged water drops falling simultaneously in the atmosphere, a numerical simulation shows that a drop formed by coalescence of the pair may be subject to the Rayleigh instability. Zh. Tekh. Fiz. 69, 12–17 (December 1999)     

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.     

On the time evolution of the liquid meniscus at the end of a capillary placed in an external electric field

Using a linearized set of equations of electrodynamics, the stability of the uniformly charged meniscus of a viscous conducting incompressible liquid at the end of a capillary is investigated and analytical expressions are derived for the electric field outside the meniscus, velocity fields in the liquid flow and meniscus, and generatrix of the meniscus shape. It is found that, if an external electric field near the meniscus exceeds that at which the free liquid surface becomes unstable against the surface charge, a finite number of longest waves become unstable with their instability growth rates nonmonotonically depending on the wavenumber. Analysis of the time evolution of the meniscus shape under various initial conditions shows that cylindrical waves with the highest instability growth rates play a decisive role in this process, while the influence of the initial deformation amplitude is insignificant.     

Nonlinear capillary oscillation of a charged drop

In the quadratic approximation with respect to the amplitudes of capillary oscillation and velocity field of the liquid moving inside a charged drop of a perfectly conducting fluid, it is shown that the liquid drop oscillates about a weakly prolate form. This refines the result obtained in the linear theory developed by Lord Rayleigh, who predicted oscillation about a spherical form. The extent of elongation is proportional to the initial amplitude of the principal mode and increases with the intrinsic charge carried by the drop. An estimate is obtained for the characteristic time of instability development for a critically charged drop.     

Instability of a charged plane interface between two liquids with respect to the tangential discontinuity of a time-dependent velocity field

The differential equation that describes the evolution of perturbations of a charged plane boundary between immiscible liquids when the upper liquid moves relative to the lower one with a time-dependent velocity parallel to the boundary is the Hill equation. In this system, the interface can exhibit instabilities of three types at various values of physical parameters: the Kelvin-Helmholtz, Tonks-Frenkel, and parametric instability. When physical parameters have certain values, the interface that is unstable with respect to surface charge and the tangential discontinuity of the velocity field across the interface can be parametrically stabilized.