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.     

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.     

微液滴振荡过程的光滑粒子动力学方法数值模拟

本文对传统的光滑粒子动力学方法进行了改进, 改进的光滑粒子动力学方法对传统粒子方法中的核梯度进行了修正, 采用了一种新型的核函数和离散格式, 应用改进的光滑粒子动力学方法对微液滴振荡过程进行了数值研究. 研究了不同纵横比和雷诺数(Re)下振荡阻尼与振荡的周期、振幅与Re数的关系. 研究表明: 对于纵横比λ≤ 4时的微液滴振荡过程, 其他参数恒定不变的前提下, Re数越大, 液滴形状变化越剧烈, 波的阻尼作用越弱, 液滴振荡周期变长; 在Re数一定的前提下, 随着液滴初始的纵横比的增大, 液滴振动的振幅增大, 液滴振荡的周期变长.     

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.     

Anisotropic surface tension and the liquid drop model

The liquid drop model assumes that the surface tension of all nuclei is isotropic. The consequences of relaxing this assumption are studied. First a macroscopic theory of anisotropic surface tension is presented. Then with a particular choice for the functional form of the surface tension tensor, this theory is applied to a charged liquid drop. A small anisotropy in the surface tension can significantly distort a highly charged drop and lower its fission barrier.     

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.     

On nonlinear corrections to oscillations frequencies of a charged drop in an incompressible ambient medium

An analytic expression in the third order of smallness in the amplitude of the initial deformation of an equilibrium, spherical, charged, ideally conducting drop in an incompressible dielectric medium is derived for its generatrix and for nonlinear corrections to oscillation frequencies. It is shown that the presence of the ambient liquid reduces the absolute values of the corrections to frequency and of the self-charge critical for the realization of drop instability.     

Nonlinear oscillations and stability of a charged drop moving relative to a dielectric medium

Nonlinear calculations to within the second order of smallness with respect to the initial deformation of a liquid drop show that a stream of an ideal incompressible dielectric liquid streamlining the charged ideally conducting drop causes interaction between modes both in the first and second orders of smallness. Both the linear and nonlinear interactions of the oscillation modes result in the excitation of modes absent in the spectrum of the initial drop deformation. The relative motion of the drop and the medium leads to broadening of the spectrum of modes excited in the second order of smallness. The presence of the flow streamlining the drop and the intermode interaction result in decreasing the critical magnitudes of the drop charge and the velocity and density of the medium determining drop instability development.     

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.     

液滴振荡模型及与数值模拟的对比

由于碰撞壁面后液滴内部流动的复杂性, 以及气-液-固三相间的相互作用, 对液滴碰撞壁面形态变化的数学理论研究有较大的难度, 因此所见者多为实验和数值模拟. 本文通过对液滴受力状态的分析, 得到了惯性力、黏性力和表面张力带经验系数的表达式, 并进一步建立了液滴碰撞壁面振荡模型, 得到了液滴铺展半径的振荡表达式, 以及表面张力、黏性系数等参数对液滴铺展的影响. 最后通过与液滴衰减振荡数值模拟结果的对比, 确定了液滴振荡模型中的修正系数, 验证了模型的可行性. 关键词： 液滴碰撞 振荡 铺展半径/高度 数值模拟     

Nonlinear resonance interaction between the oscillation modes of a spherical liquid layer on the surface of a melting hailstone

Evolutionary equations are derived and solved that describe the time dependence of the oscillation mode amplitudes on the surface of a charged conducting liquid layer resting on a solid core. It is assumed that the layer experiences a multimode initial deformation. The equations are solved asymptotically in the second order of smallness in the small dimensionless amplitude of capillary oscillations on the surface of the layer. Mechanisms behind internal nonlinear resonance interaction between the modes of the liquid layer oscillations and behind energy transfer between the modes both in degenerate and in secondary combination resonances are investigated. It is found that in the degenerate resonance interaction between oscillation modes, the energy may be transferred not only from lower to higher modes but also vice versa if the higher mode is excited at the zero time. This conclusion is valid not only for a liquid layer on the surface of a solid core but also for a drop.     

倾斜均质表面上等径水滴聚合过程的可视化实验

对倾斜均匀表面上等直径水滴的聚合过程及特性进行了可视化实验研究,获得了水滴直径和表面倾角等参数对液滴聚合过程中液滴液桥半径、接触角和接触线变化特性的影响,分析了水滴聚合对其运动的影响.实验结果表明:表面倾角越大,下滑的临界半径越小;液滴的直径越大,液滴聚合后越容易下滑;液滴聚合可以加快液滴的运动,使下滑临界半径减小.     

倾斜均质表面上非等径液滴的聚合特性

对倾斜均匀表面上非等径液滴的聚合过程及特性进行了可视化实验研究,获得了液滴半径和表面倾角等参数对液滴聚合过程中液滴液桥半径、接触角和接触线变化特性的影响规律,进一步说明了倾斜表面上液滴聚合可以加快液滴的运动.     

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.     

Upper limit on the viscosity of a nuclear liquid

Small oscillations of the atomic nucleus in the model of a viscous charged incompressible liquid drop under the action of capillary and Coulomb forces about an equilibrium spherical form are considered. As a result of investigating the frequency spectrum of intrinsic surface oscillations of the nucleus, the existence of an upper limit on the viscosity of the nuclear liquid is proven, and a precise formula for this limit is obtained.The present work is based on a paper presented at the Second All-Union Working Conference on Gravitation and the Unification of Fundamental Fields, Kiev, October 25–28, 1082.Translated from Izvestiva Vysshikh Uchebnykh Zavedenii, Fizika, No. 4, pp. 31–35, April, 1985.It remains to thank Prof. K. P. Stanyukovich for his interest in the work and numerous useful discussions and also Prof. D. D. Ivanenko for discussions of the result obtained.     

水平均质表面上液滴聚合过程的可视化实验研究

对水平均匀表面上液滴的聚合过程及特性进行了可视化实验研究,获得了液滴半径和液滴物性等参数对液滴聚合过程中液滴液桥半径和接触角变化特性的影响规律。实验结果表明:液滴聚合中液桥半径和接触角都呈衰减振荡变化; 聚合前液滴半径越小,液桥半径振荡频率越大,振幅越小,振荡时间越短;液滴的粘度越大,液桥半径的振荡频率越小, 振幅越小,振荡时间越短;液滴聚合前的接触角明显大于聚合液滴静止后的接触角,其差值与固体界面状况和气、固、液物性相关。     

黏性液滴变形过程的核梯度修正光滑粒子动力学模拟

本文提出了一种核梯度改进光滑粒子动力学(KGC-SPH)方法,模拟了黏性液滴形变自由表面问题.首先,通过模拟等温黏性液滴拉伸和旋转变形,验证了KGC-SPH法较SPH法具有较高精度和更好稳定性,且能很好地保持总角动量守恒.其次,基于非等温van der Waals模型对平衡态圆形液滴的形成过程进行数值研究,观察到小幅度振荡现象,并给出了一种新的克服张力不稳定性的方法和一种适合KGC-SPH方法的新的表面张力处理技术.最后,研究了van der Waals液滴的周期性振荡现象,讨论了初始椭圆形液滴长短半轴比 关键词： 光滑粒子动力学 黏性液滴 van der Waals模型 表面张力     

Nonlinear corrections to the oscillation mode frequencies of an ideal liquid charged jet

An analytical asymptotic expression is derived for the time evolution of an ideal incompressible conducting liquid jet with a uniformly charged surface that moves with a constant velocity along the symmetry axis of an undisturbed cylindrical surface. The expression is obtained in the third order of smallness in jet oscillation amplitude under the conditions when the initial deformation of the equilibrium jet surface is specified by one (axisymmetric or nonaxisymmetric) mode. Analytical expressions are also found for the positions of internal nonlinear degenerate three-and four-mode resonances, which are typical of nonlinear corrections to the equilibrium shape of the jet, liquid velocity field potential in the jet, and electrostatic field near the jet, and also for a nonlinear frequency correction. It is established that the nonlinear oscillations of the jet take place about a surface depending on the type of initial (generally asymmetric) deformation rather than about the equilibrium cylindrical shape.     

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.     

Deformation and nonlinear corrections to critical conditions of charged drop instability in an electrostatic suspension

Nonlinear asymptotic analysis of a charged drop placed in electrostatic and gravitational fields reveals a correction to the oscillation frequency and, accordingly, to the critical Rayleigh parameter. The analysis uses approximations quadratic in oscillation amplitude and linear in dimensionless equilibrium deformation of the drop. The correction is found to be proportional to the product of the oscillation amplitude and deformation. It is natural to name this correction deformational. In computations of the third order of smallness in oscillation dimensionless amplitude, a correction to the frequency and Rayleigh parameter appears, which is due to a nonlinear interaction between oscillation modes. This correction is larger than the deformational one in magnitude. Deformational corrections can be eliminated by experimenting under no-gravity conditions, but corrections due to the nonlinearity of hydrodynamic equations cannot be eliminated in this way. It is these corrections that are responsible for a critical Rayleigh parameter measurement inaccuracy.