Nonlinear periodic waves at the charged surface of an electrically conducting viscous liquid

A correct solution to the problem of periodic-wave propagation along the charged surface of a deep viscous liquid in the second-order approximation in the wave amplitude is given for the first time. It is shown that the second-order correction in the amplitude to the profile of the wave being considered plays a decisive role in the realization of the instability of a liquid with respect to its intrinsic charge.     

On the initiation of a corona discharge near the surface of a nonlinearly oscillating liquid layer on the surface of a charged hailstone

An expression is derived for the electric field strength near a wet hailstone in an approximation quadratic in the oscillation amplitude of a charged liquid layer on its surface. It is found that the electric field strength in a small neighborhood of the capillary wave crests grows with the number of a mode governing the initial deformation of the equilibrium (spherical) shape of the liquid layer. Even if the charge is small (when the Rayleigh parameter of the hailstone equals one-hundredth of the value critical for stability against the self-charge), the electric field near the hailstone is high enough for initiating a corona discharge in its vicinity.     

On the possibility of corona initiation at the surface of a liquid layer nonlinearly oscillating on the surface of a charged hailstone placed in a uniform electrostatic field

An expression for the electric field strength near a watered hailstone is derived in an approximation quadratic in the amplitude of capillary oscillations of a charged conducting liquid layer covering the hailstone. As the number of the mode governing the initial deformation of the equilibrium spherical free surface of the liquid layer increases and its thickness decreases, the electric field strength in the neighborhood of the capillary wave crests rises. Even in the case of small charges and low electric fields, the electric field near the hailstone is high enough to initiate a corona.     

Flow pattern due to capillary-gravitational waves in a charged layer of a viscous conducting liquid on a solid support

Analytical solutions for the time evolution of a capillary-gravitational wave in a charged layer of a viscous conducting liquid on a solid support are found. It is shown that the velocity field eddy component of the wave-induced liquid flow arises not only near the free surface of the liquid, but also at the solid bottom. The ratio between the amplitudes of these eddy components depends on the relationship between the thickness of the layer and the wavelength. If the wavelength far exceeds the thickness, the eddy flow amplitude near the bottom exceeds that near the free surface and the eddy flow occupies the whole volume of the liquid.     

Nonlinear capillary-gravitational periodic waves on the charged surface of a finite-thickness viscous liquid

An analytical expression for the time evolution of the profile of a nonlinear periodic capillary-gravitational wave traveling over the charged surface of a viscous incompressible finite-thickness liquid is found. The calculation is carried out in the second order of smallness in wave amplitude. It is shown that the dependence of a nonlinear correction to a linear solution on the liquid viscosity and liquid layer thickness changes qualitatively in going from thick to thin liquid layers.     

Nonlinear periodic waves on the charged surface of a deep low-viscosity conducting liquid

An analytical expression for the profile of a finite-amplitude wave on the free charged surface of a deep low-viscosity conducting liquid is derived in an approximation quadratic in wave amplitude-to-wavelength ratio. It is shown that viscosity causes the wave amplitude to decay with time and makes the wave profile asymmetric at surface charge densities subcritical in terms of Tonks-Frenkel instability. At supercritical values of the surface charge density, taking account of viscosity decreases the growth rate of emissive protrusions on the unstable free surface, slightly broadens them for short waves, and narrows for long ones. Analytical expressions for the wave frequencies, damping rates, and instability growth rates with regard to viscosity are found.     

On the stability of a nonaxisymmetric charged jet of a viscous conducting liquid

A dispersion equation is derived for axisymmetric and nonaxisymmetric capillary oscillations in a jet of viscous conducting liquid subjected to a constant potential. It is shown that conditions arising when the surface charge density in the jet is high cause the instability of nonaxisymmetric, rather than axisymmetric, modes with the resulting disintegration of the jet into drops of various sizes. This theoretical finding allows one to correctly interpret of experimental data for the spontaneous disintegration of charged jets.     

On internal nonlinear resonant interaction of capillary-gravitational waves on the flat surface of a viscous liquid

Mechanisms behind internal nonlinear resonant interaction of periodic capillary-gravitational waves on the uniformly charged flat surface of an infinitely deep viscous conducting liquid are considered. A mathematical procedure modifying the well-known method of many scales is proposed for constructing an asymptotically valid solution near the resonance. It is shown that the internal nonlinear resonant interaction results in effective energy transfer from long waves to shorter ones. An increase in the viscosity of the liquid diminishes the rate of energy transfer between resonantly interacting waves.     

Nonlinear periodic waves on the charged surface of a finite-thickness ideal liquid layer

In the fourth order of smallness in the amplitude of a periodic capillary-gravitational wave travelling over the uniformly charged free surface of an ideal incompressible conducting liquid of a finite depth, analytical expressions for the evolution of the nonlinear wave, velocity field potential of the liquid, electrostatic field potential above the liquid, and nonlinear frequency correction that is quadratic in a small parameter are derived. It is found that the dependence of the amplitude of the nonlinear correction to the frequency on the charge density on the free liquid surface and on the thickness of the liquid layer changes qualitatively when the layer gets thinner. In thin liquid layers, the resonant wavenumber depends on the surface charge density, while in thick layers, this dependence is absent.     

Stability of a charged drop of a viscous electrically conducting liquid in a viscous electrically conducting medium

A dispersion relation is derived for capillary oscillations of a charged electrically conducting viscous drop in an electrically conducting viscous medium. It is shown that aperiodic instability of the charged interface between the two media can arise in this system, with a growth rate that depends qualitatively differently on the ratio of their conductivities in different ranges of values of this ratio. In a certain range of conductivity ratios the drop undergoes oscillatory instability. Zh. Tekh. Fiz. 69, 34–42 (October 1999)     

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.     

Convective flows in a layer of a viscous liquid with the uniformly charged free surface

It is found theoretically that the critical conditions under which a charged liquid surface becomes unstable against the electric charge relax as a result of interaction between capillary-gravitational and convective flows in the liquid. As the surface charge density approaches a value critical in terms of development of Tonks-Frenkel instability, convection in the liquid arises at a temperature gradient however small, this effect depending on the liquid layer thickness.     

Instability of a charged layer of a viscous liquid on the surface of a solid spherical core

The dispersion relation for the spectrum of capillary waves of a spherical layer of a viscous liquid coating a solid spherical core with a layer of finite thickness is introduced and analyzed. It is shown that the existence of two mechanisms for the viscous dissipation of the energy of the capillary-wave motions of the liquid, viz., damping in the bulk of the layer and on the solid core, leads to restriction of the spectrum of the realizable capillary waves of the liquid on both the high-and low-mode sides. At a fixed value of the system charge which is supercritical for the first several capillary modes, the maximum growth rates in the case of a small solid core are possessed by modes from the middle of the band of unstable modes, while in thin liquid layers the highest of the unstable modes have the largest growth rates. This points out differences in the realization of the instability of the charged surface of the spherical layer for small and large relative sizes of the solid core. Zh. Tekh. Fiz. 67, 8–13 (September 1997)     

Nonlinear oscillations of a charged layer of a conducting liquid on the surface of a solid spherical core

Nonlinear oscillations of a layer of an ideal incompressible perfectly conducting liquid on the surface of a charged melting hailstone (solid core) are studied using analytical asymptotic calculations of the second order of smallness in initial deformation amplitude. Specifically, it is shown that, when the thickness of the layer is much less than the characteristic linear size (radius) of the solid core, the size of the core considerably influences the amplitudes of capillary oscillation modes arising on the surface of the charged layer via nonlinear interaction. It is found that, as the liquid layer on the surface of the solid core gets thinner, the energy in the spectrum of nonlinearly excited modes is redistributed with its maximum shifting toward higher (larger number) modes.     

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.     

Nonlinear periodic waves on the charged surfactant-covered surface of a viscous fluid

The profile of a periodic capillary-gravitational wave propagating over the surfactant-covered surface of a fluid is found in the second-order approximation in initial deformation amplitude. It is shown that the surfactant film appreciably affects the intensity of nonlinear interaction between harmonics constituting the nonlinear wave.     

Nonlinear periodic waves on the charged surface of a viscous finite-conductivity fluid

The profile of a periodic capillary-gravitational wave propagating over the surface of a viscous finite-conductivity fluid is found in a second-order approximation in initial deformation amplitude. When the finiteness of the rate with which the potential of the fluid smoothes out as capillary-gravitational waves travel over its free surface is taken into account, the intensity of nonlinear interaction between the waves changes. This intensity is found to depend on the electric charge surface density, conductivity of the fluid, and wavenumbers. The finiteness of the potential smoothing rate influences the nonlinear interaction between the waves nonmonotonically.     

Finite-amplitude waves on the surface of a viscous deep liquid

For the first time a rigorous solution to the problem on time evolution of the periodic wave shape on the surface of a viscous infinitely deep liquid is found in the quadratic approximation with respect to the wave amplitude. It is found, in particular, that the damping rate of the quadratic component with respect to the wave amplitude is twice as high as the damping rate of the linear term. It is shown that inclusion of viscosity leads to asymmetry of the wave profile.     

Thickness of a boundary layer attributed to the wave motion on the charged free surface of a viscous liquid

It is shown that the analytical estimator for the boundary layer thickness that contains the wave frequency in the denominator and is proposed for approximate calculation of the wave motion on the free surface of a viscous liquid cannot be formally applied to the wave motion on the uniformly charged liquid surface. The fact is that, when the surface charge density attains a value critical in terms for the Tonks-Frenkel instability, the wave frequency tends to zero. From the analysis of liquid motions near the electric charge critical density, a technique is proposed for calculating the thickness of a boundary layer attributed to flows of various kinds. It is found that the thickness of the boundary layer due to aperiodic flows with amplitudes exponentially growing with time (such flows take place at the stage of instability against the surface charge) does not exceed a few tenths of the wavelength, whereas the thickness of the boundary layer due to exponentially decaying liquid flows is roughly equal to the wavelength.     

On the time evolution of the surface shape of a charged viscous liquid drop deformed at zero time

The problem of capillary oscillations of the equilibrium spherical shape of a charged viscous incompressible liquid drop is solved in an approximation linear in amplitude of the initial deformation that is represented by a finite sum of axisymmetric modes. In this approximation, the shape of the drop as a function of time, as well as the velocity and pressure fields of the liquid in it, may be represented by infinite series in roots of the dispersion relation and by finite sums in numbers of the initially excited modes. In the cases of low, moderate, and high viscosity, the infinite series in roots of the dispersion relation can be asymptotically correctly replaced by a finite number of terms to find compact analytical expressions that are convenient for further analysis. These expressions can be used for finding higher order approximations in amplitude of the initial deformation.