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.     

Boundary layer theory modified for analysis of wave flow on the surface of a viscous liquid finite-thickness layer resting on a hard bottom

The theory of a boundary layer that is adjacent to the surface of an indefinitely deep viscous liquid and caused by its periodic motion is modified for analysis of finite-amplitude flow motion on the charged surface of a viscous conductive finite-thickness liquid layer resting on a hard bottom (the thickness of the layer is comparable to the wavelength). With the aim of adequately describing the viscous liquid flow, two boundary layers are considered: one at the free surface and the other at the hard bottom. The thicknesses of the boundary layers are estimated for which the difference between an exact solution and a solution to a model problem (stated in terms of the modified theory) may be set with a desired accuracy in the low-viscosity approximation. It is shown that the presence of the lower (bottom) boundary layer should be taken into account (with a relative computational error no more than 0.001) only if the thickness of the viscous layer does not exceed two wavelengths. For thicker layers, the bottom flow may be considered potential. In shallow liquids (with a thickness of two tenths of the wavelength or less), the upper (near-surface) and bottom layers overlap and the eddy flow entirely occupies the liquid volume. As the surface charge approaches a value that is critical for the onset of instability against the electric field negative pressure, the thicknesses of both layers sharply grow.     

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.     

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.     

Effect of the marangoni convection on the injection mechanism of instability

Electroconvective instability of a nonisothermal layer of a weakly conductive liquid with a free boundary whose surface tension depends linearly on temperature is considered for the case where charge injection is performed through this surface. When calculating the unperturbed stationary distribution of the charge and field, we supposed that the injector is separated from the liquid by an air gap of finite thickness. It was, however, assumed when analyzing the stability of the system that the motion in the air gap has no effect on the motion in the liquid phase and the disturbances of the electric field and charge in the air gap decay rapidly because of its high conductivity.     

Capillary oscillations and Tonks-Frenkel instability of a liquid layer of finite thickness

A dispersion relation is derived and analyzed for the spectrum of capillary motion at a charged flat surface of viscous liquid covering a solid substrate with a layer of finite thickness. It is shown that for waves whose wavelengths are comparable with the layer thickness, viscous damping at the solid bottom begins to play an important role. The spectrum of capillary liquid motion established in this system has high and low wave number limits. The damping rates of the capillary liquid motion with wave lengths comparable with the layer thickness are increased considerably and the Tonks-Frenkel instability growth rates are reduced compared with those for a liquid of infinite depth. Zh. Tekh. Fiz. 67, 27–33 (August 1997)     

On the stability of capillary waves on the surface of a space-charged dielectric liquid jet moving in a material medium

We have derived and analyzed the dispersion equation for capillary waves with an arbitrary symmetry (with arbitrary azimuthal numbers) on the surface of a space-charged cylindrical jet of an ideal incompressible dielectric liquid moving relative to an ideal incompressible dielectric medium. It has been proved that the existence of a tangential jump of the velocity field on the jet surface leads to a periodic Kelvin–Helmholtz- type instability at the interface between the media and plays a destabilizing role. The wavenumber ranges of unstable waves and the instability increments depend on the squared velocity of the relative motion and increase with the velocity. With increasing volume charge density, the critical value of the velocity for the emergence of instability decreases. The reduction of the permittivity of the liquid in the jet or an increase in the permittivity of the medium narrows the regions of instability and leads to an increase in the increments. The wavenumber of the most unstable wave increases in accordance with a power law upon an increase in the volume charge density and velocity of the jet. The variations in the permittivities of the jet and the medium produce opposite effects on the wavenumber of the most unstable wave.     

Modification of the boundary layer theory for analysis of wave motions in a cylindrical jet of a viscous liquid

The existent concepts of the boundary layer near the free surface of a viscous liquid, which is related to its periodic motion, are modified with the aim of analyzing the finite-amplitude wave motion on the surface of a thick charged jet of a viscous conducting liquid. To describe the flow in the boundary layer, a model problem is proposed that is simpler in statement compared with the complete problem and the solution of which uses the governing properties of the exact solution obtained in the low-viscosity asymptotics: the form of the dispersion relation, wave profile, and rate of velocity field viscous damping with time. An estimate is made of the boundary layer thickness at which the discrepancy between the exact solution and solution to the model problem (stated in terms of the theory proposed) falls into a given interval in the low-viscosity asymptotics. The domain of applicability of the modified theory is determined.     

Electrostatic stability of the liquid layer surface on a hard wettable cylindrical substrate

The wave motion in a cylindrical layer of an ideal conducting liquid on a hard rod kept at a constant electrical potential is calculated accurate to the first order of smallness in dimensional perturbation of the free surface. The instability of the free surface is also considered. A dispersion relation is derived. It is shown that the range of instability waves depends on only the electric field strength near the free surface and the instability increments of capillary waves decrease as the layer gets thinner. The influence of the hard rod becomes tangible only when its radius becomes comparable to the thickness of the liquid layer.     

On capillary motion of a viscoelastic liquid with a charged free surface

The structure of the capillary-relaxation motion spectrum in a liquid with a charged free surface has been investigated taking into account the viscosity relaxation effect. On the basis of numerical analysis of the dispersion equation for the wave motion in a viscoelastic incompressible liquid, it is shown that for a given wave number the range of characteristic relaxation times in which relaxation-type wave motion exists is limited and expands with increasing wave number. The growth rate of instability of the charged liquid surface markedly depends on the characteristic relaxation time and increases with its growth; in liquids with elastic properties, the energy dissipation rate of capillary motion is enhanced. At a surface charge density that is supercritical for the onset of Tonks-Frenkel instability, both purely gravitational waves and waves of a relaxational nature exist.     

Capillary oscillations of a flat charged surface of liquid with finite conductivity

A dispersion relation is proposed and analyzed for the spectrum of capillary motion at a charged flat liquid surface with allowance made for the finite rate of charge redistribution accompanying equalization of the potential as a result of the wave deformation of the free surface. It is shown that when the conductivity of the liquid is low, a highly charged surface becomes unstable as a result of an increase in the amplitude of the aperiodic chargerelaxation motion of the liquid and not of the wave motion, as is observed for highly conducting media. The finite rate of charge redistribution strongly influences the structure of the capillary motion spectrum of the liquid and the conditions for the establishment of instability of its charged surface when the characteristic charge relaxation time is comparable with the characteristic time for equalization of the wave deformations of the free surface of the liquid. Zh. Tekh. Fiz. 67, 34–41 (August 1997)     

Instability of a flat horizontal interface between a thin layer of a ferrofluid and a thin layer of a nonmagnetic liquid in the presence of a vertical magnetic field

An asymptotic analysis of the equations and boundary conditions of fluid dynamics is performed, and a nonlinear model is constructed for the onset of the development of Rosensweig instability in a thin horizontal ferrofluid layer at rest covered with a thin layer of a lighter nonmagnetic liquid. The surface of a nonmagnetized slab is the lower boundary of the ferrofluid, and the interface with a gas is the upper boundary of the nonmagnetic liquid. The pressure in the gas is constant. The instability being considered arises upon the application of a rather strong uniform vertical magnetic field. The proposed model involves five dimensionless parameters. The critical magnetization of the initial ferrofluid layer with a flat upper boundary and the threshold wave number are found. The effect of the governing parameters on the instability region and on the wavelength of the fastest growing mode is studied in the linear formulation of the problem.     

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.     

Nonlinear periodic waves on the charged free surface of a perfect fluid

Analytical expressions for the profile of a nonlinear wave and for a nonlinear correction to its frequency are derived in the fourth-order approximation in amplitude of a periodic traveling wave on a uniformly charged free surface of an infinitely deep perfect incompressible fluid. It is found that corrections to the amplitude and frequency of the nonlinear wave are absent if the problem is solved under the initial condition that provides the constancy of the first-order amplitude and wavelength in time. Nonlinear analysis of conditions for instability of the fluid free surface against the surface charge shows that the critical charge density and wave-number of the least stable wave are not constant (as in the linear theory) and decrease with growing amplitude of the wave.     

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.     

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.     

Nonlinear analysis of the Rayleigh-Taylor instability at the charged interface

An asymptotic solution to the problem of analyzing the nonlinear stage of the Rayleigh-Taylor instability at the uniformly charged interface between two (conducting and insulating) immiscible ideal incompressible liquids is derived in the third order of smallness. It is found that the charge expands the range of waves experiencing instability toward shorter waves and decreases the length of the wave with a maximum growth rate. It turns out that the characteristic linear scale of interface deformation, which arises when the heavy liquid flows into the light one, decreases as the charge surface density increases in proportion to the square root of the Tonks-Frenkel parameter characterizing the stability of the interface against the distributed self-charge.     

Stability of a thermal boundary layer in the presence of vibration in weightlessness environment

Several weightless experiment with supercritical fluids have shown that thermal boundary layers can be destabilized when submitted to a harmonic vibration. A study of the phenomenon is given here in a regular fluid during a sudden change of wall temperature in the presence of harmonic tangential vibrations and under weightlessness. A semi-infinite space is filled with a fluid and bounded by a flat wall oscillating in its plane. For this configuration, a state with the fluid velocity parallel to the wall is possible but this fluid motion does not influence the heat transfer. Then the propagation of thermal waves can be described by classical relations. The stability of this state is studied under the assumption of a “frozen” temperature profile. The vibration frequency is assumed to be high such that the viscous boundary layer thickness is small in comparison with the thermal boundary layer thickness. The calculations show that the instability develops when the thickness of the thermal boundary layer attains a critical value. The wavelength of the most dangerous perturbations is found to be about twice the critical thermal boundary layer thickness.     

Influence of charge relaxation on the capillary disintegration of a jet of a viscous dielectric liquid placed in an electrostatic field collinear with the axis of the jet

A dispersion relation is derived for capillary waves with an arbitrary symmetry on the surface of a charged jet of a finite-conductivity viscous liquid placed in an electric field collinear with the axis of the jet. Analytical calculations are carried out in an approximation that is linear in dimensionless wave amplitude. In the case of axisymmetric waves, the instability of which causes disintegration of the jet into drops, the finiteness of the potential equalization rate has a noticeable effect only for jets of poorly conducting liquids. The charge relaxation shows up in that “purely relaxation” periodic and aperiodic liquid flows arise. When the conductivity of the liquid declines, the instability growth rates for unstable waves increase and their spectrum extends toward short waves. A charge present on the surface of the jet enhances its instability. An increase in the charge surface diffusion coefficient variously influences the capillary and relaxation branches of the solution: the damping ratio increases in the former case and decreases in the latter. As the diffusion coefficient rises, relaxation flows may become unstable.     

Exact partial solutions for the surface dynamics of a dielectric liquid with a charged surface in the gravitational field

We study the evolution of instability in the boundary of a perfect dielectric liquid with a free surface charge in an external electric field. Conformal variables are used to find exact partial solutions to the equations of motion for the case when the charge completely shields the field above the liquid, the electrostatic and gravitational forces being taken into account. The solutions describe the development of instability of the initially planar boundary until sharp dimples are formed on it.