Experimental study of surface waves with Faraday resonance excitation

The results of an investigation of standing two-dimensional gravity waves on the free surface of a homogeneous liquid, induced by the vertical oscillations of a rectangular vessel under Faraday resonance conditions, are presented. The frequency ranges of excitation are determined and resonance relationships for the second and third modes are obtained and analyzed. Nonlinearities of the waves generated, such as wave profile asymmetry and node oscillations, are evaluated. Wave breakdown and the onset of unstable oscillation modes are considered. Experimental results are compared with the theoretical data.The experimental studies [1–4], devoted to Faraday resonance, deal mainly with the conditions under which resonance arises and the frequency response.Moscow. Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 1, pp. 122–129, January–February, 1995.     

Observation of standing waves of large amplitude in harmonic Faraday resonance

Stationary two-dimensional gravity waves on the free surface of a homogeneous fluid in a rectangular vessel in the presence of vertical oscillations excited in harmonic Faraday resonance with identical frequencies are investigated. For the second mode, the dependence of the amplitude on the excitation frequency is found and the wave profiles are obtained. The experimental results are compared with the theoretical model. Moscow. Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 1, pp. 185–188, January–February, 2000. The work was supported by the Russian Foundation for Basic Research (project No. 99-01-01080) and CONACYT (Mexico, Catedra Patrimonial, S. Ya. Sekerzh-Zen’kovich, Ref. 489100-2, Exp. 950060).     

Experimental investigation of secondary steady flows in Faraday surface waves

It is shown that in the two-dimensional Faraday surface waves excited in a vertically oscillating rectangular water-filled vessel there is a system of secondary circulatory flows that occupies the entire fluid volume between the vessel bottom and the free surface. In parallel with the oscillations at the wave frequency, the fluid particles are slowly displaced in accordance with these circulatory flows. The secondary flow velocity field is measured and the trajectories of individual fluid particles in the standing wave are determined. The experimental data are compared with the Longuet-Higgins model. It is shown that the initial stage of formation of regular structures on the surface of a sediment layer on the vessel bottom may be related with the presence of secondary circulatory flows.     

Coupled magnetoacoustic waves in a conducting paramagnetic fluid

We consider the propagation of small disturbances in a paramagnetic conducting fluid in a uniform constant magnetic field. Because of coupling of the mechanical and magnetic effects, coupled magnetoacoustic oscillations of a wave nature develop in a certain (resonant) frequency region. The usual MHD waves and uniform magnetization oscillations occur far from resonance. Dissipative processes are accounted for.The equations of motion for a conducting paramagnetic fluid in which interaction of the hydrodynamic velocity with the magnetization and the magnetic field was taken into account phenomenologically were obtained in [1], One of the consequences of this interaction is the propagation of coupled magnetoelastic waves in the fluid; this phenomenon is examined in the present paper.     

Rock vibration and finite oil recovery

Basic information concerning the possibility of mechanical stimulation of an oil reservoir is presented. The positive effect of vibration on the oil fraction in the output of flooded wells is demonstrated. The effect is attributed to the restoration of permeability for dispersed oil as a result of drop clusterization or breakdown. A mathematical model illustrating the special role of dominant vibration frequencies is proposed. This model is based on the nonlinear effects associated with internal viscoelastic resonance. The corresponding evolution equation of the seismic waves emitted by the vibrator is a generalization of the Burgers-Korteweg-de Vries equation. For this equation the existence of an asymptotic regular wave structure is proved. Taking the microparticle rotation effect into account leads to bimodal wave vibrations, and under conditions of long-short-wave resonance the nonlinear generation of high ultrasonic frequencies by seismic waves is possible. The ultrasonic vibrations created enable the oil drops to recover their mobility.Based on paper presented to the fluid mechanics section of the Seventh Congress on Theoretical and Applied Mechanics, Moscow, August 1991.Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No.5, pp. 110–119, September–October, 1992.     

垂直参数激励表面波研究进展

受外激励的充液刚性容器中流体的波动问题有实际的工程应用背景.竖直方向的受周期性外激励的充液容器的自由表面波问题--Faraday波问题是流体力学三大不稳定性难题之一(另外两个不稳定性问题是Rayleigh-B\'enard对流和Taylor-Couette流).本文综述了在理想流体中和弱粘性流体中Faraday波的研究成果;介绍了作者在底部垂直激励的圆柱形容器中流体表面波图谱的实验研究和理论分析的结果.最后提出有待进一步研究的问题.图13,参74      

Transversal oscillations of a fluid in a long cylindrical container with membrane or elastic plate on the free surface

We propose approximate solutions of two-dimensional hydroelastic problems that describe free oscillations of an ideal fluid in a horizontal long cylindrical container with arbitrary symmetric cross section. The free surface of the fluid is covered by a plane membrane or an elastic plate. Using specific examples, we analyze the obtained solutions and the results of computation of frequencies and forms of oscillations of the mechanical system under consideration.     

Solitary waves in media with dispersion and dissipation (a review)

The latest results relating to the theory of nonlinear waves in dispersive and dissipative media are reviewed. Attention is concentrated on small-amplitude solitary waves and, in particular, on the classification of types of solitary waves, their conditions of existence, the evolution of local perturbations associated with the presence of solitary waves of various types, and problems of the existence of nonlinear waves localized with respect to a particular direction as the space dimension increases (spontaneous dimension breaking). As examples of dispersive and dissipative media admitting plane solitary waves of various types, we consider a cold collisionless plasma, an ideal incompressible fluid of finite depth beneath an elastic plate and with surface tension, and a fluid in a rapidly oscillating rectangular vessel (Faraday resonance). Examples of spontaneous dimension breaking are considered for the generalized Kadomtsev-Petviashvili equation. Moscow. Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 2, pp. 3–27. March–April, 2000. The work was carried out with financial support from the Russian Foundation for Basic Research (project No. 99-0101150).     

Pressure induced by the interaction of water waves with nearly equal frequencies and nearly opposite directions

We present second-order expressions for the free-surface elevation, velocity potential and pressure resulting from the interaction of surface waves in water of arbitrary depth. When the surface waves have nearly equal frequencies and nearly opposite directions, a second-order pressure can be felt all the way to the sea bottom. There are at least two areas of applications: reflective structures and microseisms.Microseisms generated by water waves in the ocean are small vibrations of the ground resulting from pressure oscillations associated with the coupling of ocean surface gravity waves and the sea floor. They are recorded on land-based seismic stations throughout the world and they are divided into primary and secondary types, as a function of spectral content. Secondary microseisms are generated by the interaction of surface waves with nearly equal frequencies and nearly opposite directions. The efficiency of microseism generation thus depends in part on ocean wave frequency and direction. Based on the second-order expressions for the dynamic pressure, a simple theoretical analysis that quantifies the degree of nearness in amplitude, frequency, and incidence angle, which must be reached to observe the phenomenon, is presented.     

Experimental investigation of Faraday waves of maximum height

The profiles of standing gravity waves of maximum height, parametrically excited on the free surface of a deep fluid in a vertically oscillating rectangular vessel (Faraday waves), are investigated experimentally. For a small modulation index of the excitation parameter, waves of three types are distinguished: regular, temporally periodic and symmetric about the vertical line passing through their crest; irregular but retaining the connectivity of the liquid volume; and breaking waves with drops separating from the free surface of the fluid. It is established that the profile of the maximum-height regular waves is smooth with a steepness of 0.255 and a limiting angle at the crest of less than 80°. Certain realizations of irregular and breaking waves, with profiles similar to those of regular waves but with much smaller steepnesses, 0288 and 0.429, respectively, are detected.     

Second-harmonic resonance with Faraday excitation

Capillary-gravity waves in an inviscid liquid exhibit second- or sub-harmonic resonance at precise frequencies. When the container performs small periodic vertical vibrations, either wave may also experience Faraday (‘parametric’) excitation. Equations describing this situation are derived, incorporating slight detuning from two-wave and Faraday resonances. Similar equations arise in other physical contexts.With Faraday forcing of the wave with lower frequency, the evolution equations (without detuning) are transformable to the corresponding unforced equations, the general solution of which is known. With Faraday forcing of the wave with higher frequency no such simplification is possible. Here, various transformed equations are considered and numerical results elucidate their solutions. For some initial data, solutions remain bounded; but other initial values give unbounded solutions. We establish the form of the boundaries that separate these two classes.     

Free oscillations of a fluid in a cylindrical container with arbitrary axisymmetric bottom and elastic elements on the free surface of the fluid

We consider the problem of free oscillations of an ideal fluid in a container that has the form of a right circular cylinder with arbitrary axisymmetric bottom in the case where the unperturbed free surface of the fluid is covered by an elastic membrane or plate. Using the expansion in eigenfunctions of an auxiliary spectral problem with a parameter in boundary conditions and the method of decomposition of the domain of meridional cross-section of a container, we obtain an analytic solution of the problem. Individual examples of mechanical systems are considered, for which we construct solutions by using the proposed algorithm, analyze these solutions, and compute the frequencies and forms of oscillations.     

Energy pumping for mechanical systems involving non-smooth Saint-Venant terms

This paper deals with energy transfer from initial one degree of freedom system including non-smooth terms of friction (Saint-Venant) type to an auxiliary mass via an adapted non-linear coupling. We describe design of auxiliary degree of freedom and non-linear smooth coupling to the initial one degree of freedom system. First the auxiliary system is designed as if the initial one degree of freedom system was simply elastic. Then we study numerically the transfer for the non-smooth system in the cases of free transient oscillations or forced oscillations under periodic external forcing. Efficiency of energy pumping is discussed.     

Wave attenuation in a vessel equipped with damper plates

The effect of vertical damper plates mounted at the center of a rectangular vessel normal or at an angle to a wave flow is experimentally investigated for the first mode of the standing surface waves excited at the parametric resonance. The variation of the resonance curves and the wave attenuation degree are discussed. The fluid depth effect on the wave motion damping is evaluated.     

Oscillatory Motion of a Viscous Fluid in Contact with a Flat Layer of a Porous Medium

Analytical solutions are obtained for two problems of transverse internal waves in a viscous fluid contacting with a flat layer of a fixed porous medium. In the first problem, the waves are considered which are caused by the motion of an infinite flat plate located on the fluid surface and performing harmonic oscillations in its plane. In the second problem, the waves are caused by periodic shear stresses applied to the free surface of the fluid. To describe the fluid motion in the porous medium, the unsteady Brinkman equation is used, and the motion of the fluid outside the porous medium is described by the Navier–Stokes equation. Examples of numerical calculations of the fluid velocity and filtration velocity profiles are presented. The existence of fluid layers with counter-directed velocities is revealed.     

Stability of a Liquid Film Flowing Down an Oscillating Inclined Surface

The stability of flow of a liquid film along an inclined plate subject to periodic oscillations under the action of the gravity force is investigated with allowance for the surface tension. An equation of the Orr-Sommerfeld type with time-periodic coefficients is used. A method for determining the eigenvalues of the linear stability problem is developed on the basis of Floquet theory, spectral representation of the variables, and multistep methods of integration of ordinary differential equations. The bifurcation spectrum of the resonance modes is investigated, and the amplification coefficients and phase velocities are calculated for the surface waves, Tollmien-Schlichting waves, and resonance waves. The influence of external parameters, namely, the inclination, the surface tension, and the layer thickness, on the resonance modes and the steady-state flow modes is studied.     

密频近线性系统2阶超谐共振Hopf分叉

研究了一个两自由度密频近线性系统２阶超谐共振Ｈｏｐｆ分叉，揭示出由于密频内共振的作用，系统将存在较复杂的动力学行为。     

Dynamic Model of a Periodic Medium with Double Porosity

The paper presents a unified mathematical approach for describing the dynamic stressstrain state of mechanical structures from heterogeneous materials possessing a double coupled system of pore channels filled with fluid. New dynamic equations describing the oscillations of poroelastic systems based on the developed model of a continuous medium with additional degrees of freedom in the form of various pressures of the components constituting the liquid phase of the material are obtained. The equations and the method of obtaining them have a greater degree of generalization than those encountered in the literature. Theoretical results can be used to study the propagation of vibrations in fractured geological rocks saturated with liquid, to develop technical systems of new structural materials with a porous structure, for the analysis of micro streams of fluid in the hierarchical system of microporous bone tissue.     

Occurrence of quasiperiodic flows between two rotating permeable cylinders

This paper considers viscous incompressible fluid flows between two infinite rotating permeable concentric cylinders near the bifurcation point, which result in secondary steady flow and oscillations with azimuthal waves. Steady periodic and quasiperiodic fluid flow regimes with two, three, and four independent frequencies are obtained by methods of the theory of codimension-two bifurcations of hydrodynamic flows having cylindrical symmetry.