Stability of a velocity field tangential discontinuity in a three-layer density-stratified liquid with a moving middle layer

A dispersion relation is analytically derived for gravitational waves in an ideal incompressible threelayer liquid with a free surface in the presence of a velocity field tangential discontinuity between the layers. The discontinuity results from the motion of the middle layer. The instability of the tangential discontinuity is shown to depend on the relative velocity of contacting layers, which, in turn, depends on the ratio of their densities. The closer the density ratio to unity, the lower the moving layer velocity causing instability. In the given case, instability involves internal waves arising at the second and third interfaces in accordance with the Kelvin–Helmholtz concept of instability development. Internal waves with wavelengths far exceeding the thickness of the middle layer are found to interact with each other. Surface waves only change their frequencies.     

On the time evolution of the liquid meniscus at the end of a capillary placed in an external electric field

Using a linearized set of equations of electrodynamics, the stability of the uniformly charged meniscus of a viscous conducting incompressible liquid at the end of a capillary is investigated and analytical expressions are derived for the electric field outside the meniscus, velocity fields in the liquid flow and meniscus, and generatrix of the meniscus shape. It is found that, if an external electric field near the meniscus exceeds that at which the free liquid surface becomes unstable against the surface charge, a finite number of longest waves become unstable with their instability growth rates nonmonotonically depending on the wavenumber. Analysis of the time evolution of the meniscus shape under various initial conditions shows that cylindrical waves with the highest instability growth rates play a decisive role in this process, while the influence of the initial deformation amplitude is insignificant.     

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.     

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.     

Linear wave interaction at the charged fluid-fluid interface under tangential discontinuity of the velocity field

Capillary wave flow in a two-layer fluid with the upper layer moving parallel to the charged interface at a constant velocity is treated within a linear mathematical model. Interaction between waves excited on the free surface of the upper layer and at the interface results not only in classical Kelvin-Helmholtz instability (at low velocities of the upper layer) but also in oscillatory instability of the interface. The instability increment depends on the fluid density ratio, translational velocity, and charge density at the interface.     

Capillary vibrations of a viscoelastic medium under the action of a constant external force

A layer of a viscoelastic liquid was found to exhibit two types of instabilities, aperiodic and vibrational, when its free surface was subjected to an external force. For the aperiodic instability, the critical condition and increment value were derived analytically. If the angle between the force direction and external normal to the free surface of the liquid is smaller than 45 degrees, only the vibrational instability sets up in the system; if the angle is larger, the aperiodic one alone is observed.     

Internal transverse fluctuation waves in a viscous liquid at a hard boundary

A liquid layer resting on a hard bottom and bounded by hard walls is considered. Analysis of the dispersion relation shows that both slowly decaying solitary waves and rapidly decaying periodic waves may arise inside the layer at the interface between stratified ～100-nm-thick near-surface regions of low-viscous liquids. Stratification is due to the orienting influence of the hard bottom and to fluctuation forces. The dispersion laws for the waves are characteristic of purely capillary waves in both a deep and shallow liquid.     

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.     

Dynamics of the free surface of a conducting liquid in a near-critical electric field

Near-critical behavior of the free surface of a perfectly conducting liquid in an external electric field is considered. Based on an analysis of three-wave processes using the method of integral estimates, sufficient criteria for hard instability of a planar surface are formulated. It is shown that the higher-order nonlinearities do not saturate the instability, for which reason the growth of disturbances has an explosive character.     

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.     

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.     

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.     

On nonlinear periodic capillary-fluctuation wave flow in a thin liquid film on a solid substrate

Analytical calculation of a nonlinear periodic wave flow on the free surface of a charged layer of an ideal incompressible conducting liquid resting on a solid substrate is carried out for the case when fluctuation-induced forces (the dispersion component of the wedging pressure) have a decisive effect on the system. It is shown that wave flows emerge in the liquid in calculations of the second order of smallness in the wave amplitude, which is assumed to be small compared with the thickness of the liquid layer. These flows result from nonlinear interaction as nonlinear corrections to the waves set at the zero time. The field of fluctuation-induced forces displaces these flows toward the periphery of the area of influence of these forces. This effect takes place both in the presence of an external electric field near the free surface and in its absence. The sign and value of the nonlinear corrections depend on whether an electric field is present near the free surface of the liquid. In the presence of an electric field, the curvature of the crest of the nonlinear waves increases; in its absence, the curvature decreases.     

Influence of the thermocapillary effect on the dynamics and stability of motion of a thin evaporating film

We consider the dynamics of evaporation of a thin layer of a polar liquid (e.g., water) with the free surface, which is located on a solid substrate. Thermocapillary instability takes place at the liquid-vapor free boundary. The surface energy of the substrate-liquid contact region is a nonmonotonic function and is the sum of the interactions of the Van der Waals force and the force of the double electric layer. The influence of the Marangoni effect on the velocity, profile, and stability of the liquid evaporation front is analyzed.     

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 capillary stability of a cylindrical dielectric liquid jet in a longitudinal electrostatic field

A dispersion relation is derived for capillary waves with arbitrary symmetry (with arbitrary azimuthal numbers) on the surface of a cylindrical jet of an ideal incompressible dielectric liquid subjected to an electrostatic field aligned with the symmetry axis of the jet. It is shown that only long axisymmetric waves can experience capillary instability in such a system. The wavenumber range into which unstable waves fall begins with a zero value, and its width depends on the permittivities of the liquid and ambient and on the electrostatic field strength squared. As the field strength grows, the wavenumber range for unstable waves rapidly narrows and the capillary instability growth rate, as well as the wavenumber of the wave with the greatest growth rate, decreases.     

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.     

Morphological stability analysis of partial wetting

We develop a macroscopic static theory of the morphological stability of partial wetting. The system we studied consist of a smooth horizontal solid surface and some non-volatile liquid on it. A necessary condition for the stable equilibrium of such systems is known as the Young condition on the contact angle made at the contact line where the free surface of liquid meets the solid surface. But this condition is local and is not sufficient for the stability. We present a formulation for studying the stability of systems which satisfy the Young condition. Then we apply this to several morphologies of wetting. We find that there are at least two fundamental morphologies that we call a hole and a ridge, which are thermodynamically unstable against certain infinitesimal deformations of the contact lines. The hole type instability has also been found recently [D. J. Srolovitz and S. A. Safran, J. Appl. Phyys., 60 (1986), 1]. We also derived a reduced expression for the wetting energy as a functional of the contact line positions under the assumption of almost flat free surface of the liquid. This serves us to understand the characteristic length scale which appears in the ridge type instability. Besides these instabilities there is another category of morphological instability in which the system becomes unstable against an infinitesimal deformation of the free surface of liquid. We show this by an illustrating example in which the instability is described as the so-called tangent bifureation in nonlinear systems.     

Formation of singularities on the charged surface of a liquid-helium layer with a finite depth

The dynamics of the development of an instability of a charged surface of a liquid-helium layer with a finite depth is investigated. The equations describing the evolution of the free surface are derived with the use of conformal variables for the case in which the charge completely screens the electric field above the liquid. A model of the evolution of a spatially localized perturbation of a liquid-helium surface is proposed for the strong-field limit where the dynamics of the liquid is predominantly determined by the effect of electrostatic forces. This model describes the development of an instability of the initially planar boundary to the point of the formation of cuspidal dimples. The limit of an infinitely deep liquid is considered. The stability of the previously revealed liquid flow regime described by the Laplacian growth equations is proved without significant constraints on the surface geometry.     

Electrostatic instability of the lateral surface of a viscous liquid finite-conductivity jet in a collinear electrical field

The influence of the finiteness of the charge transfer rate on the electrostatic instability of the lateral surface of a viscous liquid jet is studied. The study is based on the analysis of a dispersion relation for flexural-deformation capillary waves on the surface of the jet with allowance for charge relaxation. The jet is subjected to a superposition of two electrostatic fields one of which is collinear with the jet’s axis and the other is directed radially to the former. It is found that the finiteness of the potential equalization rate influences jets of a poorly conducting liquid most strongly. The charge relaxation shows up in the appearance of both periodic and aperiodic “purely relaxation” flows. Relaxation flows give rise to electrostatic instability in low-permittivity liquids. When the conductivity of the liquid drops, the instability growth rate of relaxation waves grows and their spectrum expands toward shorter waves. An increase in the charge surface diffusion coefficient introduces destabilization into the relaxation flows of the liquid, which may eventually become unstable.