A. N. Zharov

Stability of a bubble in a dielectric liquid in an external electrostatic field

Critical instability conditions are found for a gas bubble in a liquid dielectric in a uniform external electrostatic field E0. It is shown that they depend both on the magnitude of E0 and on the properties of the liquid, as well as on the gas pressure in the bubble. In a linear approximation with respect to the square of the eccentricity of an equilibrium spheroidal form, the equilibrium eccentricity of the bubble exceeds the equilibrium eccentricity of a drop in the field E0. The gas pressure in the bubble lowers the critical electric field E0 for development of an instability in the bubble.

Effect of the initial deformation of a charged drop on nonlinear corrections to critical conditions for instability

A nonlinear (proportional to the vibration amplitude squared) decrease in the critical (in terms of instability) charge of a vibrating drop is found to be limited, as follows from third-order asymptotic calculations. This effect occurs when the spectrum of modes specifying the initial deformation of the drop contains, along with the fundamental mode, higher modes. The influence of the environment density on nonlinear corrections to the critical conditions for instability is analyzed.

On nonlinear resonant four-mode interaction between capillary vibrations of a charged drop

Energy transfer from higher modes of capillary vibrations of an incompressible liquid charged drop to the lowest fundamental mode under four-mode resonance is studied. The resonance appears when the problem of nonlinear axisymmetric capillary vibration of a drop is solved in the third-order approximation in amplitude of the multimode initial deformation of the equilibrium shape of the drop. Although the resonant interaction mentioned above builds up the fundamental mode even in the first order of smallness, its amplitude turns out to be comparable to a quadratic (in small parameter) correction arising from nonresonant nonlinear interaction, since the associated numerical coefficients are small.

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

Analytical expressions are derived for the shape generatrix of an ideally conducting drop immersed in an incompressible dielectric medium as well as for nonlinear corrections to the frequencies of the oscillations of the drop. The solutions are obtained in an approximation of the third order of smallness with respect to the amplitude of the initial deformation of the equilibrium spherical shape of the drop. It is shown that the presence of the ambient liquid results in a reduction of the absolute magnitudes of corrections both to the oscillation frequencies and the self-charge critical for the development of instability of the drop.

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 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.

Analytical study of nonlinear vibrations of a charged viscous liquid drop

The generatrix of a nonlinearly vibrating charged drop of a viscous incompressible conducting liquid is found by directly expanding the equilibrium spherical shape of the drop in the amplitude of initial multimode deformation up to second-order terms. A fact previously unknown in the theory of nonlinear interaction is discovered: the energy of an initially excited vibration mode of a low-viscosity liquid drop is gradually (within several vibrations periods) transferred to the mode excited by only nonlinear interaction. Irrespectively of the form of the initial deformation, an unstable viscous drop bearing a charge slightly exceeding the critical Rayleigh value takes the shape of a prolate spheroid because of viscous damping of all the modes (except for the fundamental one) for a characteristic time depending on the damping rates of the initially excited modes and the further evolution of the drop is governed by the fundamental mode. In a high-viscosity drop, the rate of rise of the unstable fundamental mode amplitude does not increase continuously with time, contrary to the predictions of nonlinear analysis in terms of the ideal liquid model: it first decreases to a value slightly differing from zero (which depends on the extent of supercriticality of the charge and viscosity of the liquid), remains small for a while (the unstable mode amplitude remains virtually time-independent), and then starts growing.

Properties of expansions in Legendre polynomial derivatives that show up in analysis of nonlinear vibrations of a viscous liquid drop

Analytical expressions for the coefficients of expansions of Legendre polynomial products in the first and second derivatives of the polynomials with respect to polar angles, as well as for the coefficients of expansions of derivative products in Legendre polynomials and their first derivatives, are derived. An interrelation between these expansion coefficients and a relationship between these coefficients and the Clebsch-Gordan coefficients are found. When axisymmetric nonlinear vibrations of a viscous liquid drop are investigated analytically, the toroidal component of the velocity field can be ignored, which significantly cuts the body of computation.

Internal nonlinear four-mode interactions of capillary oscillations of a charged drop

Analytical expressions for the shape of a charged drop of an incompressible liquid nonlinearly oscillating upon multimode initial deformation have been obtained for the first time to within the third order of smallness. The second-order corrections to oscillation frequencies are calculated depending on the spectrum of modes determining the initial deformation. The third-order calculations show that the fundamental mode amplitude may increase due to the energy exchange with higher modes in a large number of possible four-mode resonance interactions.

On the calculation of the translational mode amplitude for a drop nonlinearly vibrating in an environment

The second-order amplitudes of the capillary vibration modes of a drop of an ideal incompressible liquid placed in an incompressible ideal medium are calculated. The approximation is quadratic in initial multimode deformation of the equilibrium spherical shape caused by nonlinear interaction. The mathematical statement of the problem is such that the immobility condition for the center-of-mass of the drop is met automatically. When the translational mode amplitude is calculated, a set of hydrodynamic boundary conditions at the interface, rather than the condition of center-of-mass immobility (which is usually applied for simplicity in the problems of drops vibration in a vacuum), should be used.