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.     

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 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.     

Lasing characteristics of a pendant drop deformed by an applied electric field

The lasing properties of an oval-shaped resonant cavity (ORC) with a continuously variable aspect ratio have been studied. The ORC was formed with a dye-doped pendant drop placed inside a variable static electric field. When the drop ORC was pumped by a nitrogen laser, lasing from the ORC was found to have strong directional emission characteristics and an intensity enhancement factor as great as 19.5. Calculated results of light rays escaping from ORC's by refraction are in good agreement with the experimental data.     

The shape of a drop in a constant electric field

The variation of the shape of a drop immersed in an immiscible liquid under the action of an electric field is calculated. The charge is transferred both by ohmic current through the interface and by the convective component over the interface. A solution quadratic in the parameter that is the ratio of the electric pressure to the capillary pressure is analyzed. Conditions where the drop transforms into a spheroid that is prolate or oblate along the electric field vector are found. An experimental study of the drop deformation by electric forces is carried out.     

Instability of a spherical drop in a nonuniform electric field

Analytical calculation in the first order of smallness shows that the equilibrium shape of a drop in the field of a point charge is axisymmetric about the plane passing through the center of mass of the drop normally to the axis connecting the center of mass with the point charge. Whether the equilibrium shape of the drop is stable or not depends on the value of the field parameter, which, in turn, depends on the point charge and the distance to it. There is an asymptotic value of the critical parameter above which all modes become unstable. In the field of the point charge, the mode coupling grows; that is, a mode excited at the zero time generates oscillations of the six nearest modes with amplitudes proportional to that of the initially excited mode. If the initially excited mode loses stability, all the modes coupled with it also become unstable. The surface instability of the drop also develops when the initially excited mode is stable but at least one of the modes coupled with it is unstable.     

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)     

Equilibrium shape and stability of a charged drop in an air flow under an electric field

The pressure balance on the surface of a charged liquid drop moving along a uniform electrostatic field is analyzed. The liquid is assumed to be nonviscous and incompressible. In the approximation linear in deformation amplitude, the equilibrium shape of the drop as a function of the charge, field strength, and velocity of travel can be both a prolate and an oblate spheroid. Critical conditions for the surface instability of such a drop are obtained analytically in the form of a relationship between the charge, field strength, and velocity of travel. An instability criterion is found by extrapolating to large Reynolds numbers. This makes it possible to fit the earlier model of a corona-initiated lightning in the vicinity of large charged water drops or hailstones to the charges of the drops, field strengths, and velocities of travel (relative to the medium) typical of thunderclouds.     

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.     

Capillary oscillations and stability of a charged drop rotating about the axis of symmetry

The stability of a charged conductive liquid drop rotating about the axis of symmetry against the pressure of the self-charge electric field and inertial force pressure is investigated in an approximation linear in oscillation amplitude and square of the spheroidal drop deformation eccentricity. It is found that the axisymmetric modes of the rotating drop are stable. Only nonaxisymmetric modes with azimuthal numbers maximal for a given mode may be unstable. The Coriolis force plays a stabilizing role.     

Equation of solvated ion oscillations in solution under an external periodic electric field

An equation describing ion oscillations with respect to solvent molecules is derived as a result of the consideration of the interaction of an external electric field with free and bound charges in the volume of a salt solution in a polar dielectric liquid. An analysis of the equation shows that the ion motion represents a drift toward one of the electrodes forming the field in the electroinsulated solution volume if amplitudes and durations of sinusoidal field half-periods are different.     

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.     

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.     

Capillary disintegration of a suspended filamentary drop of a viscous magnetic fluid in a longitudinal magnetic field

A differential equation that describes the axisymmetric motion of two immiscible magnetic fluids of the same density and viscosity is derived. It includes in explicit form the contribution of capillary forces localized at the interface between the fluids, which has the form of a weakly distorted cylindrical surface. With this equation, a dispersion relation for the problem of capillary instability of an extended axisymmetric drop placed in a uniform longitudinal magnetic field is obtained. The effect of magnetic forces on the capillary disintegration of the drop for the extreme cases (large and small Ohnesorge numbers) is analyzed.     

On the equilibrium shapes of a conducting drop in a uniform and nonuniform electric field

The equilibrium shape of a drop in the electrostatic field of a point charge and a point dipole is asymptotically calculated in terms of the dimensionless deformation of the shape and a ratio between the drop’s radius and the distance to the point charge (dipole). Irrespective of the degree of nonuniformity of the field, the prolate spheroidal deformation (typical of the uniform field) is shown to be the main reason for the change in the equilibrium shape of the spherical drop. When the nonuniformity of the field grows, the equilibrium shape becomes more and more asymmetric and different from the spheroidal one. This, all other things being equal, may influence the critical conditions for the instability of the drop’s surface against an induced charge. It follows from the aforesaid that the drop in the field of the dipole will be the first to undergo instability with the electrostatic pressure on the drop being the same.     

Dielectrophoresis and deformation of a liquid drop in a non-uniform, axisymmetric AC electric field

Analytical theory for the dielectrophoresis and deformation of a leaky dielectric drop, suspended in a leaky dielectric medium, subjected to non-uniform, axisymmetric Alternating Current (AC) fields is presented in the small deformation limit. The applied field is assumed to be a combination of a uniform part and a quadrupole component. The analysis shows that the magnitude and the sign of the steady and time-periodic dielectrophoretic velocity depend upon the frequency of the applied voltage. The frequency of oscillatory motion is twice that of the applied frequency and the phase lag is a consequence of charge dynamics. A deformed drop under non-uniform axisymmetric AC fields admits Legendre modes l = 2, 3, 4 . The deformation has a frequency-dependent steady and time-periodic parts due to charge and interface dynamics. The steady deformation can be zero at a certain critical frequency in leaky dielectric systems. The time-periodic deformation also has a frequency which is twice the frequency of the applied voltage. In perfect dielectric systems, unlike the steady state deformation which is a balance of Maxwell and curvature stresses, the time-periodic deformation additionally includes viscous stresses associated with the oscillatory shape changes of the drop. A consequence of this effect is a phase lag that is dependent on the charge and interface hydrodynamics and a lag of π/2 at high frequencies. It also results in vanishing amplitude of the oscillatory deformation at high frequencies. The study should lead to a better understanding of dielectrophoresis under non-uniform axisymmetric AC fields and better electrode design to affect drop breakup.     

Forced oscillations of a magnetic liquid drop

Oscillations of a magnetic liquid drop suspended in a nonmagnetic viscous medium were studied analytically and experimentally. Oscillations occurred under the action of a linearly polarized alternating magnetic field. The frequency dependence of the amplitude and phase of oscillations was determined in the weak field and laminar flow approximation. The influence of the viscosity of the liquid on the natural drop oscillation frequency was studied. The obtained solution was used in experiments on determining interphase surface tension.