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 calculating the oscillations of a viscous liquid layer over a solid spherical core in terms of the boundary layer theory

The theory of a boundary layer near the periodically oscillating free surface of a spherical viscous liquid layer over a solid core (bottom) is modified. Two boundary layers are considered to adequately describe a liquid viscous flow in the system: one at the free surface of the liquid and the other at the solid bottom. The thicknesses of the boundary layers are estimated, which provide any given discrepancy between an exact solution to the model problem and a solution obtained in the small viscosity approximation. Taking into account the boundary layer near the solid bottom is shown to be significant only for lower oscillation modes. For higher modes, the flow near the core can be considered potential. In the case of lower modes and shallow liquid, the surface and bottom boundary layers overlap and an eddy flow occupies the entire volume of the liquid.     

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.     

Instability development on the charged free surface of a horizontal liquid layer with thermal convection

The instability of the charged free surface of a horizontal liquid layer heated from the solid bottom against excess electric charge is studied theoretically for the case in which this type of instability is combined with thermal-convective instability. The structure of the total spectrum of unstable wave flows and physical parameters influencing the structure of the spectrum are determined.     

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.     

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.     

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.     

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 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.     

Boundary layer theory modified for analyzing finite-amplitude oscillations of a viscous liquid charged drop

The existing concepts of the boundary layer arising near the free surface of a viscous liquid, which is related to its periodic motion, are revised with the aim to calculate finite-amplitude linear oscillations of a viscous liquid charged drop. Equations complementing the boundary layer theory are derived for the vicinity of the oscillating free spherical surface of the drop. An analytical solution to these equations is found, comparison with an exact solution is made, and an estimate of the boundary layer thickness is obtained. The domain of applicability of the modified theory is defined.     

On the possibility of initiating a corona discharge at the crests of nonlinear waves on the surface of a charged thin layer of a viscous conducting liquid

An analytic expression for the electrostatic field strength at the free surface of a thin layer of a uniformly charged viscous incompressible liquid is obtained in second-order asymptotic calculations in the amplitude of a periodic capillary-gravity wave propagating over the liquid surface. It is shown that a corona discharge at the crests of the waves can be initiated at subcritical values of the field strength (in the sense of possible realization of the Tonks-Frenkel instability). The electrostatic field strength at the crests of nonlinear waves increases with the wavenumber and the wave amplitude.     

Instability of a charged spherical drop moving relative to a medium

It is shown that the critical self-charge for the onset of instability of a charged drop in a flow of an ideal fluid decreases as the flow velocity of the fluid past the drop increases, i.e., a complex instability arises which is a superposition of the instabilities of the free surface of the drop with respect to the tangential discontinuity of the velocity field at the free surface of the drop and with respect to the self-charge. Zh. Tekh. Fiz. 69, 7–14 (May 1999)     

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.     

Resonance effect of the bottom topography on the surface of an inclined layer of a viscous liquid

The reaction of the film interface to low-amplitude waviness of the wall was studied. A linearized version of the problem described by the Orr — Sommerfeld equation was considered; the solution was sought by asymptotic expansion in small parameter 1/Re, and usual spectral problem concerning stability to perturbations of exp[iα(x-ct)] type was solved. According to calculations, for some specially chosen wave numbers α the drift and dispersion effects balance each other, providing zero resulting velocity c _R = 0. If we assume that a rigid wall is corrugated with the same α, we can say that stationary waves caused by the wavy wall are in resonance with intrinsic perturbations of the second kind. The work was financially supported by the Russian Foundation for Basic Research (Grants Nos. 05-08-33585a and 06-08-96637-r-yug-a.)     

Modification of the boundary layer theory for analysis of finite-amplitude oscillations of a charged bubble in a viscous liquid

The prevailing concepts concerning the boundary layer near the free surface of a viscous liquid associated with oscillatory motion are modified for calculating finite-amplitude linear oscillations of a charged bubble in this liquid. Equations of the boundary layer theory for the neighbourhood of the oscillating free spherical surface of a charged bubble in a dielectric liquid are derived, their analytic solution is obtained and compared with the exact solution, and the thickness of the boundary layer is assessed. The range of applicability of the modified theory is determined.     

Long-wavelength structure on a charged liquid surface

The problem of the equilibrium form of a charged surface of a dielectric liquid in a strong electric field, such that a flat surface becomes unstable, is studied. A periodic long-wavelength structure with a small amplitude can arise when the gap between the surface and a charged electrode is small compared with the capillary length and the charge completely screens the electric field. The equilibrium form of the surface is calculated assuming that the resulting wave is one-dimensional. The effect of the boundary conditions at the vessel walls on the dependence of the amplitude of the standing wave on the applied voltage is estimated. It is shown that this dependence is very sensitive to the conditions of contact between the vessel walls and the liquid. The possibility is discussed of using the theory developed in this paper to explain the experimental results obtained with a charged liquid-hydrogen surface. Zh. éksp. Teor. Fiz. 115, 43–49 (January 1999)     

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.     

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.