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)     

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.     

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.     

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.     

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.     

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.     

Parametric instabilities of a viscous liquid layer covered by a thin elastic plate under vertical periodic motion

Parametric instabilities of a horizontal liquid layer with a finite depth covered by a thin elastic plate under a vertical periodic motion are investigated with account taken of the viscosity of the liquid layer. The primary regions and the secondary ones of dynamic instabilities are determined by using the equation of a thin elastic plate including the normal component of the viscous stress, but not the tangential component of it. The critical amplitude of the imposed oscillation, beyond which a parametric instability occurs (that is, the neutral stability curves) is found in the space of the frequency and amplitude of the imposed vertical oscillation. These results are confirmed by experimental ones for a liquid layer of glycerine covered with a thin rubber plate.     

On the theory of a boundary layer associated with wave motion on the free surface of a liquid

A modified theory of a boundary layer associated with a periodic capillary-gravitational motion on the free surface of an infinitely deep viscous liquid is proposed. The flow in the boundary layer is described in terms of a simplified (compared with the complete statement) model problem a solution to which correctly reflects the main features of an exact asymptotic solution: the rapid decay of the flow eddy part with depth of the liquid and insignificance of some terms appearing in the complete statement. The boundary layer thickness at which the discrepancy between the exact asymptotic solution and model solution is within a given margin is estimated.     

On parametric buildup of waves on the charged surface of polar liquids under periodic decay of the double electric layer

Analytical calculations show that the time-periodic decay of the double electric layer near the free surface of a viscous polar liquid causes periodic variation of the surface tension coefficient of the liquid and may be a reason for parametric buildup of capillary-gravitational finite-amplitude wave motion.     

Cavitation in a lubrication flow between a moving sphere and a boundary

A heavy sphere is free to move inside a rotating horizontal cylinder filled with viscous liquid. The steady motion is essentially Stokesian, and the sphere rotates at a fixed location with a lubrication layer between the ball and the wall. The symmetry of the flow field suggests there will be no force to balance the normal component of the ball's weight. However, we show that a normal force can arise when a cavitation bubble is present. The bubble size was measured as a function of the cylinder rotation rate and agrees well with a model which uses the force and torque balances on the sphere.     

Boundary conditions in the vicinity of a dynamic contact line: experimental investigation of viscous drops sliding down an inclined plane

To probe the microscopic balance of forces close to a moving contact line, the boundary conditions around viscous drops sliding down an inclined plane are investigated. At first, the variation of the contact angle as a function of the scale of analysis is discussed. The dynamic contact angle is measured at a scale of 6 mum all around sliding drops for different volumes and speeds. We show that it depends only on the capillary number based on the local liquid velocity, measured by particle tracking. This velocity turns out to be normal to the contact line everywhere. It indirectly proves that, in comparison with the divergence involved in the normal direction, the viscous stress is not balanced by intermolecular forces in the direction tangential to the contact line, so that any motion in this last direction gets damped.     

Hydroelastic oscillations of a viscous infinitely long liquid column

The coupled frequencies of hydroelastic systems in zero gravity consisting of an elastic shell and a viscous liquid layer with a free surface have been treated. The system exhibits either an annular liquid layer around an elastic cylindrical center shell or a liquid layer inside an elastic infinitely long container. The first case has been evaluated numerically, where the influence of the liquid surface tension parameter, the elasticity parameter of the shell, its axial deflection wave length l and the thickness of the layer have been determined. In contrast to the hydroelastic system with an ideal liquid, the system with viscous liquid exhibits instability of the liquid surface as well as the shell.     

Rotation of a droplet in a viscous fluid

Unique capabilities for modeling the bulk motion of one liquid in another arise from the use of droplets made of a magnetic liquid. In this paper the low-frequency rotational motion of a magnetic droplet suspended in a viscous liquid is investigated. In this frequency range, the shape of the droplet does not depend on the field frequency and is determined only by its amplitude. An analytic solution has been found in the Stokes approximation to the problem, which generalizes the classic problem of Jeffrey to the case of a liquid ellipsoidal particle. This solution makes it possible to determine the velocity field inside and outside the liquid particle, the moments of the viscous forces acting on the droplet, its coefficient of rotational mobility. Zh. éksp. Teor. Fiz. 112, 1340–1350 (October 1997)     

Instability of a charged layer of a viscous liquid on the surface of a solid spherical core

The dispersion relation for the spectrum of capillary waves of a spherical layer of a viscous liquid coating a solid spherical core with a layer of finite thickness is introduced and analyzed. It is shown that the existence of two mechanisms for the viscous dissipation of the energy of the capillary-wave motions of the liquid, viz., damping in the bulk of the layer and on the solid core, leads to restriction of the spectrum of the realizable capillary waves of the liquid on both the high-and low-mode sides. At a fixed value of the system charge which is supercritical for the first several capillary modes, the maximum growth rates in the case of a small solid core are possessed by modes from the middle of the band of unstable modes, while in thin liquid layers the highest of the unstable modes have the largest growth rates. This points out differences in the realization of the instability of the charged surface of the spherical layer for small and large relative sizes of the solid core. Zh. Tekh. Fiz. 67, 8–13 (September 1997)     

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.     

A Problem on a Viscous Layer Deformation

Doklady Physics - The axisymmetric motion of a viscous incompressible fluid in a layer bounded by a solid plane and the free surface parallel to it is considered. There are three regimes of motion...     

Acoustic wave dispersion in a cylindrical elastic tube filled with a viscous liquid

This paper deals with the study of the velocity and the attenuation of an acoustic wave propagating inside a cylindrical elastic tube filled with a viscous liquid. A theory describing the propagation of the axisymmetrical modes in such waveguides is presented, with special attention given to the absorption produced by the viscous mechanisms in the liquid. One of these mechanisms is related to the momentum transfer between the compression and rarefaction regions of a propagating wave. The other viscous mechanism is due to the momentum transport inside the viscous boundary layer, close to the tube wall. Numerical calculations were carried out to investigate the influence of different parameters (frequency, tube radii, viscosity coefficient) on the propagation of acoustic waves.     

Differential criterion of a bubble collapse in viscous liquids

The present work is devoted to a model of bubble collapse in a Newtonian viscous liquid caused by an initial bubble wall motion. The obtained bubble dynamics described by an analytic solution significantly depends on the liquid and bubble parameters. The theory gives two types of bubble behavior: collapse and viscous damping. This results in a general collapse condition proposed as the sufficient differential criterion. The suggested criterion is discussed and successfully applied to the analysis of the void and gas bubble collapse.     

Three-dimensional perturbations of the radial-rotational spread–sink of a viscous cylindrical layer

The temporal evolution of the three-dimensional pattern of perturbations superposed on a radialrotational spread or sink of a viscous layer the parameters of which depend on the radial coordinate and time has been studied. The motion of the layer boundaries is specified both in the main motion and in the perturbed motion. Based on the integral relations method applied to the linearized problem in perturbations, sufficient estimates of exponential stability (in particular, on a finite time interval) of the main motion have been derived.     

有粒滞液层负载时薄板中类Lamb波的传播

本文从弹性波传播理论出发,结合边界条件,导出了薄板一面有粘滞液层负载时板中类Lamb波传播的色散方程;研究了液层粘滞引起的类Lamb波衰减.数值计算表明,板和液层的驻波共振振动引起极中类Lamb波对称和反对称模式的模式转换以及衰减系数的变化。文中对k_td ≤ 1时,A₀模式类Lamb波的微质量和粘滞传感特性也作了讨论。