Hydrodynamics in weak gravitational fields small oscillations of an ideal liquid in a cylindrical vessel

The problem of determining the frequencies and forms of small natural oscillations of an ideal liquid in a cylindrical vessel under conditions close to weightlessness is examined. It is assumed that a weak homogeneous gravitational field acts parallel to the vertical generatrix forming the cylinder. In contrast to [1], where only the first antisymmetric oscillation frequency is found for a semiinfinite cylindrical vessel, the frequencies of several axiosymmetric, antisymmetric, etc. oscillations are obtained as functions of the gravitational-field intensity and other parameters of the problem. The Ritz method is employed for two different variations of the problem, equivalent to that of oscillations of an ideal liquid under conditions of weightlessness [1–5].Translated from Izvestiya Akademii Nauk SSSR. Mekhanika Zhidkosti i Gaza, No. 2, pp. 3–13, March–April, 1973.     

Example of parametric resonance in a rotating flow

Parametric resonance is one of the common types of instability of mechanical systems [1]. A standard example of the equations describing parametric oscillations is the Mathieu equation and its generalizations. In hydrodynamics these oscillations have been closely studied in connection with the problem of the vertical oscillations of a vessel containing an incompressible fluid in a uniform gravity field [1–5]. In this paper a new example of a flow whose stability problem reduces to the Mathieu equation is given. This is a flow of special type in a rotating cylindrical channel. The direction of the angular velocity is perpendicular to the channel axis, and its magnitude varies periodically with time. Flows with this geometry are of potential interest in technical applications [6, 7].Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 2, pp. 175–177, March–April, 1987.     

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.     

Influence of surface tension on damping of the oscillations of a viscous incompressible fluid in a cylindrical channel

A numerical calculation is made of small oscillations of a viscous incompressible fluid that fills half of a horizontal cylindrical channel. The calculation is made with and without allowance for surface tension. The results of the calculation show that allowance for surface tension increases the damping of the oscillations. The general properties of problems of the normal oscillations of a heavy and capillary viscous incompressible fluid were studied in [1–3], in which the possibility of applying the Bubnov-Galerkin method to these problems was pointed out. A method for calculating the oscillations of a viscous incompressible fluid that partly fills an arbitrary vessel at large Reynolds numbers was developed in [3–5].Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 3, pp. 128–132, May–June, 1980.     

Free oscillations of a liquid rotating in a cylindrical vessel under conditions of weightlessness

The plane problem of determination of the natural frequencies of small oscillations of a viscous liquid rotating in a partially filled cylindrical vessel under conditions of weight-lessness is examined. If the angular velocity of vessel rotation is sufficiently low, surface forces acting on the liquid-gas boundary prove to be of the same order as the centrifugal forces and significantly affect the oscillatory frequencies. Asymptotic formulas expressing the dependence of the oscillatory frequencies on the parameters of the problem are obtained by the boundary layer method, with the assumption that the ratio of viscous to centrifugal forces is low.Khar'kov. Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkostii Gaza, No. 4, pp. 3–9, July–August, 1972.     

Hydrodynamics in weak gravitational fields two-dimensional oscillations of an ideal fluid in a rectangular channel

The two-dimensional problem of determining the frequencies and modes of small natural osciliations of an ideal fluid in a rectangular channel under near-weightless conditions is considered. It is assumed that a weak gravitational field acts parallel to the vertical walls of the channel. The Ritz method is employed for the variational problem, which is equivalent to the problem of oscillations of a fluid under weightless conditions [1, 2].Khar'kov. Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 3–13, September–October, 1972.     

Parametric Excitation of a Secondary Flow in a Vertical Layer of a Fluid in the Presence of Small Solid Particles

Thermal convection is studied in an inhomogeneous medium consisting of a fluid and a solid admixture under conditions of finite–frequency vibrations. Convection equations are derived within the framework of the generalized Boussinesq approximation, and the problem of flow stability in a vertical layer of a viscous fluid with horizontal oscillations along the layer to infinitely small perturbations is considered. A comparison with experimental data is made.     

Oscillations of a vessel containing a viscous fluid

The oscillations of a rigid body having a cavity partially filled with an ideal fluid have been studied in numerous reports, for example, [1–6]. Certain analogous problems in the case of a viscous fluid for particular shapes of the cavity were considered in [6, 7]. The general equations of motion of a rigid body having a cavity partially filled with a viscous liquid were derived in [8]. These equations were obtained for a cavity of arbitrary form under the following assumptions: 1) the body and the liquid perform small oscillations (linear approximation applicable); 2) the Reynolds number is large (viscosity is small). In the case of an ideal liquid the equations of [8] become the previously known equations of [2–6]. In the present paper, on the basis of the equations of [8], we study the free and the forced oscillations of a body with a cavity (vessel) which is partially filled with a viscous liquid. For simplicity we consider translational oscillations of a body with a liquid, since even in this case the characteristic mechanical properties of the system resulting from the viscosity of the liquid and the presence of a free surface manifest themselves.The solutions are obtained for a cavity of arbitrary shape. We then consider some specific cavity shapes.     

Curvilinear motion of an ellipsoidal bubble

The Lagrange equations are used to investigate the curvilinear motion of an ellipsoidal bubble. At a small inclination of the minor axis of the ellipsoid to the vertical, the ellipsoid begins to oscillate about the equilibrium position and its trajectory begins to swing. Analytic expressions are presented for the oscillation frequency and for the ratio of the swing amplitude to the amplitude of the oscillations of the ellipsoid. It is assumed that the bubble has the form of an axially symmetrical ellipsoid [1].Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 3, pp. 90–93, May–June, 1971.     

Parametric oscillations of liquid in communicating vessels

It is well known that a periodic change in the equilibrium or flow parameters of an incompressible liquid exerts a material influence on the hydrodynamic stability. As an example we may quote the parametric excitation of surface waves (gravitational-capillary [1], electrohydrodynamic [2], magnetohydrodynamic [3]) and the oscillations of liquid in communicating vessels [4, 5]. The chief object of the foregoing experimental investigations was that of determining the boundaries of the regions of unstable equilibrium with respect to small perturbations. In the present investigation we made an experimental study of the parametric resonance and finite-amplitude parametric oscillations arising in a liquid-filled U-tube subject to alternating vertical overloadings. We shall describe two forms of oscillations in the liquid, and we shall determine the corresponding ranges of unstable equilibrium with respect to small random perturbations (self-excitation) and also to finite-amplitude perturbations. We shall study nonlinear modes of excitation and mutual transitions between the two forms of oscillations. We shall find the ranges of existence of steady-state oscillations.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 2, pp. 36–42, March–April, 1976.The authors wish to thank G. I. Petrova and the participants in his seminar for useful discussions, and S. S. Grigoryan for valuable advice.     

Stability of the Interface of a Liquid-Suspension System under High-Frequency Nonlinearly Polarized Vibration

The stability of the state of a two-layer system consisting of a homogeneous liquid and a solid particle suspension in the same liquid with a plane interface between them is investigated. The system performs both horizontal and vertical high-frequency vibration with an arbitrary phase shift. It is shown that a mean flow develops under the simultaneous action of both horizontal and vertical vibration; quantitative flow stability characteristics are determined numerically using the differential sweep method. It is shown that a traveling wave relief exists on the liquid-suspension interface. Transverse oscillations in phase with longitudinal oscillations destabilize the entire system. The presence of an oscillation phase shift can lead to an increase in the stability limit; the direction of motion of the wave relief can differ depending on the value of the phase shift. Instability of the system in the presence of strictly vertical vibration is observed, the crisis being associated with longwave monotonic perturbations.__________Translated from Izvestiya Rossiiskoi Academii Nauk, Mekhanika Zhidkosti i Gaza, No. 3, 2005, pp. 3–13.Original Russian Text Copyright © 2005 by Lobov, Lyubimov, and Lyubimova.     

Five problems on the effect of a solid wall on the added mass of floating cylindrical bodies subject to vertical impact

Very few studies have been made of the effect of the channel wall on the added mass of floating cylindrical bodies subject to vertical impact. These include [1–4].In the present paper we use the direct approach in curvilinear coordinates (under the usual assumptions adopted in hydrodynamic impact theory) to solve the following vertical impact problems: ellipses with semimajor axes horizontal and vertical in confocal vessels; and floating cylinders which are positioned eccentrically in a semicylindrical vessel, near a vertical wall, and near a stationary cylinder. The solutions are valid for any distance between the impacted contour and the channel wall.     

Vertical displacements of a Wave-Riding buoy

The influence of water waves on the free vertical oscillations of a spar buoy with an attached line of measuring instruments is investigated. The equations of motion of such a system are derived on the basis of the Lagrangian approach. Local parametric splines are used to reduce the problem to a system of second-order nonlinear ordinary differential equations, which is solved numerically by Geer's method. The period of free oscillations of the buoy is plotted as a function of its waterline area and the elasticity of the dropline, and the amplitude of the buoy oscillations is plotted as a function of the period and amplitude of the water waves and the elasticity of the dropline.S. P. Timoshenko Institute of Mechanics, National Academy of Sciences of Ukraine, Kiev. Translated from Prikladnaya Mekhanika, Vol. 31, No. 7, pp. 83–88, July, 1995.     

Stability of vertical oscillations in an electrodynamic suspension system with a discrete guideway structure

Conclusions 1. The coupling of the vehicle oscillation equation and the equations for the currents in the guideway structures must be taken into account in the stability analysis of vertical oscillations in an electrodynamic suspension system with a discrete guideway structure.2. The reported calculations indicate that vertical oscillations are unstable over a wide range of travel velocities. Consequently, the execution of steady-state vehicle motion requires vertical stabilization of the system.3. Effective stabilization can be achieved by means of a dynamic vibration damper. Even for a small relative mass =0.025 of the optimally tuned damper the perturbed motion decays rapidly.Physicotechnical Problems Branch of the Institute of Geotechnical Mechanics, National Academy of Sciences of Ukraine, Dnepropetrovsk. Translated from Prikladnaya Mekhanika, Vol. 31, No. 7, pp. 88–93, July, 1995.     

Experimental and theoretical investigation of thermal convection in a terrestrial model of a convection detector

The effect of small inclinations and low-amplitude rocking on thermal convection in an air-filled cube with sides 3 cm long has been investigated experimentally and numerically for steady and periodic heating from above. Expressions have been obtained for the dependence of the temperature difference between two points inside the cube on the angle of inclination, the applied vertical temperature difference, the rocking frequencies and the oscillations of the heating power. The possibility of estimating the magnitude and direction of the microacceleration vector on the basis of temperature field measurements made under orbital flight conditions is discussed.Moscow. Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 5, pp. 67–75, September–October, 1994.     

Two exact solutions of the Navier-stokes equations for the concentration of eddies in a viscous liquid

In the final analysis, vorticity in a liquid or gas is broken down by viscosity [1]; however, there are known cases of the appearance and long-term existence of three-dimensional eddies in water, air, and other media. Therefore, the conditions under which vorticity can even rise with viscosity are of interest. For example, with the flow of a liquid out of an opening in the bottom of a rotating cylindrical vessel, the total momentum with respect to the vertical axis of the vessel increases with the time [2, 3]. For some flows, there exist contradictory opinions: In [4, 5] it is asserted that an eddy around a flat sink in a viscous liquid is damped, while, in [6, 7], it is argued that, with determined Reynolds numbers, there is an increase in the vorticity around a sink. The present article gives exact solutions of the Navier—Stokes equations, demonstrating the development of eddies in a viscous liquid.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 6, pp. 77–81, November –December, 1977.     

Stable nonlinear waves in a poiseuille flow

Nonlinear Tolmin-Schlichting waves are studied [1–8]. The investigation is carried out by means of a modified Stuart-Watson method [1–3]. In the case of a rigid regime of excitation terms to the fifth order are taken into account in expansions with respect to the amplitude of self-excited oscillations. The stability of self-excited oscillations with respect to two- and three-dimensional disturbances is examined.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 40–45, September–October, 1978.The author thanks S. Ya. Gertsenshtein for attention to the work and discussion of the results.     

Liquid flow in the cylindrical vessel filling process

The axisymmetric flow of a highly viscous incompressible liquid as it fills the space between semi-infinite vertical coaxial cylinders is investigated. The liquid flows into the vessel at a constant rate along the surface of the inner cylinder. The flow is characterized by the presence of a free surface. The problem is formulated in the creep flow approximation and is solved numerically. The numerical solution leads to an investigation of the hydrodynamics of the filling process, including the stage in which the liquid spreads over the horizontal surface and the filling stage which follows the arrival of the free surface front at the wall of the outer cylinder.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 1, pp. 65–70, January–February, 1990.     

On the determination of nonlinear oscillations of a liquid in a circular cylinder sector

The main hydrodynamic coefficients of equations, describing large oscillations of an ideal incompressible and homogeneous liquid in tanks having the form of a cylindrical sector are calculated. Nonlinear oscillations of a liquid in cylindrical containers have been investigated in [1–3]. Here we use the method of solving some nonlinear problems of the oscillations of an ideal liquid in arbitrary containers, proposed in [4]. The dependence of the calculated coefficients on the geometrical parameters of the tank, which is important in practical applications, is analyzed.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 124–131, September–October, 1970.The authors thank G. S. Narimanov for attention and advice.     

Parametric excitation of waves on the surface of an inviscid liquid

A study is made of the stability of the equilibrium of the free surface of an infinite layer of inviscid incompressible liquid executing oscillations along the vertical axis. The problem is solved in the nonlinear formulation by series expansion with respect to the amplitude of the excitation. Soft and hard excitation regimes of the surface waves are obtained. The stability of the regimes is investigated. It is shown that the plane wave formed on the surface of the liquid is unstable.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 68–75, September–October, 1982.I thank V. A. Briskman for suggesting the problem and for constant interest in the work and also A. A. Nepomnyashchii for discussing the results.