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 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.     

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.     

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 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.     

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.     

Modification of the boundary layer theory for calculating viscous drop oscillations in a uniform electrostatic field

Capillary oscillations on the free surface of a viscous conductive liquid drop placed in an electrostatic field are calculated. In an approximation linear in stationary deformation amplitude, the drop in this field has the shape of a spheroid extended along the field. The initial problem is modified and simplified in terms of the boundary layer theory by applying an approximation that is linear in the oscillation amplitude and quadratic in the eccentricity of the drop. The accuracy of the approximate solution relative to an exact one is estimated. It is shown that, with a rise in the electrostatic field strength (with an increase in the eccentricity of the drop) and in the viscosity of the liquid, the boundary layer at the free surface of the drop becomes thicker.     

On the thickness of a boundary layer related to the oscillating free surface of a charged drop of a viscous liquid

The capillary oscillations of a charged drop of a viscous liquid are calculated in terms of the boundary layer theory in an approximation linear in oscillation amplitude. Calculation is accompanied with the estimation of a relative error that arises when the exact solution is replaced by an approximate one. It is shown that, for the calculation accuracy in the framework of the boundary layer theory to be about several percent, the thickness of the boundary layer near the free surface of the drop must be several times larger than that at which the intensity of the eddy flow caused by the oscillating surface decreases by e times. As the viscosity of the liquid grows, so does the thickness of the boundary layer.     

Field of a point source in a semi-infinite elastic medium

An exact solution for the tensor Green's function of a harmonic field for a semi-infinite elastic medium is presented in an easy-to-use form in the theory of wave scattering. The solution is derived in the form of a sum of the Green's functions for an infinite medium and the term satisfying the homogeneous wave equation for a semi-infinite elastic medium. The results reproduce the known far-field asymptotics containing longitudinal, transversal and surface Rayleigh-type wave modes. The near-field asymptotic is essentially different for the regions far and near the boundary.     

Resonance oscillation damping of a scanning microscope probe by a near-surface viscous liquid layer

Viscous liquid layer motion between a probe with a tip shaped as a paraboloid of revolution and a surface is considered for semicontact-mode operation of a scanning probe microscope. The presence of a viscous liquid layer leads to energy dissipation and is one of the factors responsible for the decrease in the probe oscillation amplitude. The Reynolds equation for viscous liquid motion is used to obtain an analytic solution to the problem. The formula derived for the loss is compared with experimental data obtained for probes and layers with various curvature radii and viscosities.     

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)     

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.     

Diffraction at an elliptical cylinder with a strongly prolate cross section

The problem of diffraction of a high-frequency plane wave by an infinite cylinder, whose cross section is a strongly prolate ellipse, was considered. The field asymptotics in the boundary layer near the surface were obtained. These asymptotics contain a parameter that characterizes the degree of ellipse elongation, which is equal to the ratio of the transverse wave size squared to the longitudinal size. The approximation accuracy and the applicability domain of the asymptotic formulas were investigated by comparing them to the numerical data obtained in the pdetool environment of the MatLab package.     

Density of states of a one-dimensional disordered photonic crystal

An analytic theory of the density of states in one-dimensional disordered photonic crystals is proposed. It is shown that the problem of the density of optical modes can be reduced in the small dielectric contrast approximation to solving a generalized Fokker-Planck equation for the distribution function of the logarithmic derivative of the electric field (the wave phase). The exact analytic solution and density-of-states asymptotics deep in the band gap of the photonic crystal and close to the band gap edge are derived. The results obtained agree well with the empirical relations derived earlier from numerical experiments.     

A unified compatibility approach to solve certain classical engineering models involving Newtonian fluid flow due to stretching or shrinking surface: A comprehensive study

A unified compatibility method of differential equations is employed to solve some nonlinear two-point boundary value problems arising in the study of the classical model of viscous (Newtonian) fluid flow due to impermeable shrinking and stretching sheets. The solution procedure allows us to find the exact solution of the nonlinear models in the form of a closed-form exponential function. The solution methodology is easy as well as systematic and provides a unified treatment to already known ad hoc solutions of these models found in the literature before. Moreover, some new exact solutions of the various extended versions of this classical engineering boundary layer problem under different physical considerations are discussed. Hence, several misrepresented solutions related to this boundary layer model which are discussed before in the literature are identified, corrected, and clarified in this paper.     

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.     

Stability of travelling wave solutions for coupled surface and grain boundary motion

We investigate the spectral stability of the travelling wave solution for the coupled motion of a free surface and grain boundary that arises in materials science. In this problem a grain boundary, which separates two materials that are identical except for their crystalline orientation, evolves according to mean curvature. At a triple junction, this boundary meets the free surfaces of the two crystals, which move according to surface diffusion. The model is known to possess a unique travelling wave solution. We study the linearization about the wave, which necessarily includes a free boundary at the location of the triple junction. This makes the analysis more complex than that of standard travelling waves, and we discuss how existing theory applies in this context. Furthermore, we compute numerically the associated point spectrum by restricting the problem to a finite computational domain with appropriate physical boundary conditions. Numerical results strongly suggest that the two-dimensional wave is stable with respect to both two- and three-dimensional perturbations.     

The factorization method in a rough-surface scattering problem: I

In the present paper the wave scattering problem on rough surface is considered for the Helmholtz equation with the Dirichlet boundary condition. An approximate solution is derived with using a factorization approach to the original Helmholtz equation. As a result, the system of two equations of parabolic type appears. The first system equation has an exact analytical solution whereas for the second one, an approximate solution, is considered in terms of perturbation series. It is shown that the obtained approximate solution is the modified classical small perturbation series with respect to small Rayleigh parameter. In Appendix A it is demonstrated that, when the derived perturbation series is converged, it is possible to summarize it and to represent the exact solution of original boundary problem in an analytical symbolical form.     

The factorization method in a rough-surface scattering problem: I

In the present paper the wave scattering problem on rough surface is considered for the Helmholtz equation with the Dirichlet boundary condition. An approximate solution is derived with using a factorization approach to the original Helmholtz equation. As a result, the system of two equations of parabolic type appears. The first system equation has an exact analytical solution whereas for the second one, an approximate solution, is considered in terms of perturbation series. It is shown that the obtained approximate solution is the modified classical small perturbation series with respect to small Rayleigh parameter. In Appendix A it is demonstrated that, when the derived perturbation series is converged, it is possible to summarize it and to represent the exact solution of original boundary problem in an analytical symbolical form.     

除垢超声波传播影响因素的理论研究

从一维平面波理论入手分析了超声波声压分布特性。依据多普勒频移原理,在声场的运动方程,连续性方程,波动方程的基础上,建立一个超声波在流动的液体中传播的控制方程。根据轴对称模型的实际特点,简化了所得方程,并求出解析解。结果表明流动液体可以产生声波的衰减。液体的黏滞性是产生超声波衰减的重要原因。超声波的频率较高,液体的黏滞性对超声波衰减影响明显。依据黏滞力与速度梯度的关系,建立一个超声波在黏滞液体传播的控制方程,并依据边界条件求出解析解,反映了媒质黏滞性对超声波传播尤其是衰减特性的影响。