S. O. Shiryaeva

Capillary analogue of the “dead water” effect in a stratified liquid with a charged interface

In terms of a linear mathematical model of a capillary-gravitational flow in a two-layer liquid with a finite-thickness upper layer, it is shown analytically that an analogue of the dead water phenomenon exists in the domain of capillary waves, which was previously observed only in gravitational waves. This phenomenon shows up as an exponential increase in the capillary wave amplitude at the interface with the surface tension coefficient at the interface tending to zero. It is found that an external electric field displaces this phenomenon toward the range of finite surface tension coefficients.

Nonlinear analysis of interaction between finite-amplitude capillary waves in a layered inhomogeneous liquid

The nonlinear capillary wave motion in a two-layer liquid with a free surface is analytically investigated accurate to the second order of smallness in ratio of the wave amplitude to the layer thickness. The layers differ in physicochemical properties. A capillary analogue to the “dead water” effect is observed in the system in both linear and quadratic approximations. In the absence of an electric charge at the interfaces, internal nonlinear resonance interaction between capillary waves is also absent regardless of the place of their origination. When there is a charge at the interlayer boundary, capillary waves resonantly interact with each other.

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.

On the Equilibrium Shape of a Heavily Charged Drop Suspended in a Weak Electrostatic Field

The object of investigation is the shape of a heavily charged drop placed in a weak electrostatic field. Calculations were carried out in the fourth order of smallness in the eccentricity, through which the steady-state deformation of the drop is measured. It is shown that the equilibrium shape of such a drop can be approximated by a prolate spheroid. This shape is due to the self-charge of the drop rather than to the external field, which is very weak and merely specifies a preferred direction. In an electrostatic suspension, where such a situation may take place, the deformation-related measurement inaccuracy of the critical charge can be totally eliminated if fine droplets are used.

Internal nonlinear resonance on a charged jet

Nonlinear analytical asymptotic calculations of the second order of smallness show that the motion of a charged jet in a material medium generates periodic wave motions of the jet-medium interface (Kelvin-Helmholtz instabilities), which grow in time. In addition, the motion of the jet gives rise to nonlinear internal resonance interaction of waves. The parameters of this interaction (intensity and characteristic time) depend on the physical parameters of the system: electric charge density of the jet, its velocity in the medium, mass density, wavenumbers of interacting waves, and the interfacial tension coefficient.

On the stability of capillary waves on the surface of a charged jet moving relative to the environment

A dispersion relation is derived for capillary waves with arbitrary symmetry (arbitrary azimuthal numbers) on the surface of a charged cylindrical jet of an ideal incompressible conducting liquid moving relative to an ideal incompressible dielectric medium. It is shown that a tangential discontinuity in the velocity field on the surface of the jet leads to periodic instability of waves (similar to the Kelvin-Helmholtz instability) at the interface and destabilizes both axisymmetric and flexural waves. The wavenumber range for unstable waves and the instability growth rate increase with the field strength and relative speed of motion, varying as the square of these parameters. In the case of the neutral jet, the flexural instability is of the threshold character and sets in starting from a certain finite value of the speed rather than at an arbitrary small speed.

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.

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.

Nonlinear asymptotic calculation of the Kelvin-Helmholtz instability

The problem of periodic capillary-gravitational wave motion on the uniformly charged interface between two ideal immiscible incompressible liquids is solved in the third order of smallness. The lower liquid is assumed to be ideally conducting, while the upper one is a dielectric executing translational motion parallel to the interface with a constant velocity. A nonlinear frequency correction in the resonance form is found. It is shown that the positions of internal nonlinear resonances depend on the sum of the field and Weber parameters, the density ratio of the liquids, and the wave number. When the upper liquid is denser than the lower one, resonances are absent.

Electromagnetic Radiation of a Nonlinearly Oscillating Charged Drop

Nonlinear oscillations of a charged drop caused by a multimode initial deformation of the equilibrium spherical shape are studied using an asymptotic analytical technique. It is shown that, when the spectrum of modes describing the initial deformation contains two adjacent modes, the center of the drop charge oscillates near the center of mass, which causes the generation of an electromagnetic radiation of the dipole type.