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 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 secondary Raman resonances of oscillations of the fundamental mode of a charged drop moving in an environment

Second-order calculations show that, when a gas flows about a charged drop, the fundamental mode of the multimode initial deformation of its equilibrium shape builds up through nonlinear secondary Raman resonant interaction with higher modes if this mode is present in the mode spectrum specifying the initial deformation. This circumstance accounts for large-amplitude spheroidal oscillations of drops in natural liquid-drop systems and provides an insight into corona initiation in the vicinity of drops in thunderstorm clouds and into lightning initiation.     

Stability of a charged drop having the form of a triaxial ellipsoid

The stability of a highly charged, isolated conductive drop is analyzed within the principle of minimum potential energy of a closed system. A treatment of the stability of drops of ellipsoidal shape shows that both spherical drops and drops having an oblate spheroidal shape experience instability at sufficiently large charges according to a single scheme, i.e., they deform to a prolate spheroid. Zh. Tekh. Fiz. 68, 48–51 (November 1998)     

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.     

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.     

Rain induced depolarization from 1 GHz to 300 GHz in a tropical environment

The rain induced depolarization in a tropical environment has been studied using a tropical raindrop size distribution developed by Ajayi and Olsen (A-O). The differential attenuation, differential phase shift and cross polarization discrimination, XPD, were computed over a frequency range of 1GHz to 300GHz for spheroidal drops and up to 33GHz for Pruppacher-Pitter drops. The variations of XPD with frequency, rainfall rate and copolar attenuation, CPA, were investigated. A mathematical relationship was established between the XPD and the CPA, canting angle and frequency of propagation from 5GHz to 300 GHz for spheroidal drops and up to 33 GHz for Pruppacher-Pitter drops. The results obtained using the A-O drop size distribution have been compared with those assuming the Laws and Parsons (L-P) distribution. The Pruppacher-Pitter drop shape has been found to give rise to higher XPD, especially at low CPA and high frequencies.     

油水两相分散流的相界面浓度输运方程研究

通过对连续相湍流导致的油滴聚合和破裂现象的合理模化,本文建立了适用于垂直上升管油水两相分散流的相界面浓度输运方程.其中对连续相湍流驱动油滴随机碰撞引起的聚合现象模化时,考虑了两油滴间连续相液膜排出过程中的凹面膜效应.而在描述油滴与湍流涡随机碰撞导致的破裂现象时,考虑了破裂过程所产生油滴的大小不等的因素.利用所建立的相界面浓度输运方程对沿实验段轴向的相界面浓度变化进行了预测,结果与实验数据比较表明吻合很好.     

Critical conditions for instability of a highly charged oblate spheroidal drop

The spectrum of capillary oscillations of a charged oblate spheroidal drop is calculated in neglect of the interaction between modes by means of a perturbation expansion in the small deviation of the equilibrium shape of the drop from spherical. The critical conditions for instability of its nth mode with respect to the self-charge are calculated in the form of an analytical function describing how the dimensionless Rayleigh parameter characterizing the stability of the drop depends on the value of the spheroidal deformation. Zh. Tekh. Fiz. 69, 10–14 (July 1999)     

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)     

Estimating surfactant surface coverage and decomposing its effect on drop deformation

A novel method is introduced to estimate surface coverage and the equation of state of insoluble surfactant on droplets, involving measurement of interfacial tension on a single parent drop and progressively subdivided generations of daughter drops. This has enabled quantitative decomposition of the dilution, tip-stretching, and Marangoni effects of surfactants on drop deformation. For a small viscosity ratio of 0.09, the Marangoni effect dominates, increasing first and then decreasing with surface coverage, the dilution effect is significant at high, and tip-stretching only at low surface coverage. For a viscosity ratio of 2.3, the dilution effect dominates, and neither Marangoni nor tip-stretching effects play an important role.     

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)     

Breakup of charged drops into droplets of comparable dimensions under strong spheroidal virtual deformations

On the basis of the principle of minimization of energy of a closed system in which spontaneous processes are occurring, we investigate the breakup of a highly charged drop into two and three droplets of comparable dimensions under conditions of virtual spheroidal deformations. Zh. Tekh. Fiz. 68, 31–38 (August 1998)     

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.     

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 analysis of the equilibrium shape of a charged drop rotating around its symmetry axis

From the condition of pressure balance on the free surface of a charged rotating conducting-liquid drop, an analytical expression for the equilibrium shape of the drop is derived in the second-order approximation in a small parameter, the ratio of the deformation amplitude to the radius of the initial spherical shape. It is found that, in the linear approximation in the small parameter, the drop takes the form of an oblate spheroid, while in the quadratic approximation, the equilibrium shape of the drop differs from the spheroidal one.     

Nonlinear analysis of the equilibrium shape of a charged drop in the tornado funnel wall

From the pressure balance condition on the free surface of a conducting liquid charged drop, an expression is derived for the equilibrium shape of the drop placed in the field of centrifugal forces acting in the tornado wall. The analysis is carried out in an approximation quadratic in small parameter (the ratio of the deformation amplitude to the radius of the initially spherical drop). In the linear approximation, the drop is a spheroid oblate in the direction normal to the tornado axis. The eccentricity of the spheroid squared is proportional to the angular velocity squared and the radius of the drop cubed. In the quadratic approximation, the equilibrium shape of the drop is other than spheroidal.     

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.     

On the absence of charge symmetry in electrostatic interaction between charged drops and hard particles

It is shown that the analytical expressions for the energy and force of electrostatic interaction between charged conducting particles (drops), a point charge, and a finite-size particle, as well as between a particle (a drop or a point charge) and a conducting plane, are asymmetric with respect to the sign of one of the charges. This is because the polarization interaction is always attractive irrespective of the signs of interacting particles. The absence of this symmetry leads to the self-constriction of charged aerodispersed systems containing a condensed phase, for example, plasma or liquid-droplet systems.     

Electrical properties of water drops inside the dropwise cluster

The Letter shows that inside a dropwise cluster formed over the heated water surface, water drops are electrically charged. The charge of a separate drop reaches 10³ units of an elementary charge. The drops are positioned from each other at the distance of double Debye radius length. It is fixed up that drops levitate over the water surface in consequence of the Stokes force acting from the side of gas-vapor flow rising from water surface. The Stokes force thousand times exceeds the Coulomb drop repulsion force from the water surface.