Small oscillations of a liquid in a partitioned cylindrical vessel

The equations of motion of a rigid body whose cavity is partially filled with an ideal fluid have been obtained in works of Moiseev [1, 2, 3], Okhotsimskii [4], Narimanov [5], and Rabinovich [6]. All the equation coefficients have been calculated for a cavity in the form of a circular cylinder or two concentric cylinders.The problem of fluid motion in a partitioned cylindrical cavity was considered by Rabinovich [7]. It was also considered by Bauer [8], who analyzed the particular case of vessel motion in the plane of one of the partitions.In the following we consider the two-dimensional motion of a cylinder with radial and annular baffles, and a definition is given of the velocity potential in the case of arbitrary positioning of the radial baffles with respect to the motion plane. Formulas are obtained for determining the parameters of a mechanical analog of the wave oscillations, which consists of two mathematical pendulum subsystems.     

Damping of the nonlinear oscillations of a viscous liquid partly filling a vessel

The results of a numerical experiment to determine the damping of the free axisymmetric oscillations of a viscous incompressible liquid partly filling a right circular cylinder are presented. The experiment was carried out for Reynolds numbers at the lower limit of applicability of the laminar boundary layer approximation and for the case of finite oscillation amplitude. The influence of nonlinear effects on estimates of the logarithmic decrement is discussed at the quantitative level.Tomsk. Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 5, pp. 158–162, September–October, 1994.     

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 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.     

Flow of a viscous fluid in a cylindrical vessel with a rotating cover

The problem considered arises in solving various technical problems associated with flows of a viscous fluid in a closed space near rotating plane surfaces, turbomachine disks, thrust bearings, rotational viscosimeters, etc. The approximate solution of the problem on the basis of a simplified flow scheme was first obtained by Schultz-Grunow [1], The most complete investigation has been made recently by Grohne [2], who outlined a program for solving the problem by joining several partial solutions on the basis of definite hypotheses concerning the flow core.With the development of electronic digital computers and the necessary numerical methods, the most effective means of solving the considered problem is the use of the grid methods for solving partial differential equations. The present paper is devoted to presenting the results of the solution of the problem using the grid method on a digital computer.     

Example of motion of a cylindrical solid in a viscous liquid

The problem of motion of a pulsating solid (an infinitely long circular cylinder) in an oscillating viscous liquid in the presence (or absence) of an external stationary force is considered. The perturbation method is applied. It is found that the solution of the time-average motion of a body exists if and only if body pulsations, liquid vibrations, and external forces satisfy a certain relation. The presence of a plane analog of the phenomenon of predominantly unidirectional motion of a compressible solid in an oscillating liquid is established.     

Motion of a free axisymmetric viscous liquid film

Axisymmetric free-film flows are encountered in connection with the atomization of liquids and the collision of jets [1, 2]. In [3] steady motion with transverse symmetry is examined and its inviscid instability is studied. Here, steady flow with an arbitrary velocity profile is investigated numerically by the collocation method. The study of the stability of the steady flow under the assumption of local plane-parallelism leads to the formulation of a sixth-order eigenvalue problem which is solved numerically. The existence of unstable disturbances of two types is demonstrated.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 4, pp. 23–29, July–August, 1990.     

Active transient elasto-acoustic response damping of a thick-walled liquid-coupled piezolaminated cylindrical vessel

The linear 3D piezoelasticity theory in conjunction with the versatile transfer matrix approach and the wave equation for the internal acoustic domain are employed for active non-stationary vibroacoustic response control of an arbitrarily thick, tri-laminate, fluid-filled, simply supported, piezocomposite cylindrical tank, excited by arbitrary (non-axisymmetric) time-dependent on-surface mechanical loads. The smart structure is composed of a supporting core layer of functionally graded orthotropic material perfectly bonded to inner and outer spatially distributed radially polarized functionally graded piezoceramic sensor and uniform force actuator (FGPM) layers. Active vibration damping is implemented by transferring the accumulated voltage on the sensor layer to the piezoelectric actuator layer in context of proportional and derivative control laws. Durbin's numerical inverse Laplace transform scheme is utilized to calculate the time response histories of the relevant interface displacement/stress components, center-point acoustic pressure, and actuator voltage, for selected loading configurations (i.e., concentrated step, impulse, and moving external loads). Numerical simulations demonstrate the effectiveness of the adopted distributed sensing/actuation configuration together with the active damping control strategy in suppressing the vibroacoustic response of a three-layered (Ba₂NaNb₅O₁₅/Al/PZT4) water-filled piezoelastic cylindrical tank. Limiting cases are considered and the validity of results is established by comparison with the available data as well as with the aid of a commercial finite element package.     

Axisymmetric motion of a viscous fluid in the finite annular cavity of a rotating cylindrical vessel

The method of nets is used to investigate unsteady axisymmetric viscous flow in a cylindrical gap of finite height. This situation is characterized by vortex motion in a plane passing through the axis of the coaxial cylinders. These flows have previously been studied in relation to the case of stepwise variation of the angular velocity of the cylinders [1]. In the present case the angular velocity is varied linearly in the acceleration stage and the acceleration interval is a parameter of the problem. After acceleration the rotation rate is determined from the ordinary differential equation describing the process of deceleration of the system as a whole.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 2, pp. 37–42, March–April, 1988.The authors are grateful to E. M. Zhukhovitskii for his interest and valuable comments.     

Influence of rotation on the damping of surface waves in a cylindrical layer of viscous fluid

A study is made of the oscillations of the free surface of a cylindrical layer of viscous incompressible fluid attracted by gravity to a solid, uniformly rotating cylinder. Logarithmic decay rates are found for the damping of the surface gravitational waves when large Reynolds numbers are assumed. It is shown that rotation introduces an asymmetry into the damping of the waves traveling in and against the direction of rotation of the fluid.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 6. pp. 171–173, November–December, 1984.     

Mathematical study of the small oscillations of a pendulum containing a viscous liquid

The authors study the small oscillations of a pendulum containing a viscous liquid.Using the principle of virtual work, they determine the equations of motions and the variational formulation of the problem. Renouncing a little long method of the Russian scientists, they obtain directly from the variational equation, an operator equation in a Hilbert space.The corresponding bundle of operators belongs to a known class, as that the authors can study the spectrum of the problem and specify the set of eigenvalues according to the coefficient of viscosity.     

Calculation of the decay rate of oscillations of a low-viscosity fluid rotating in a cylindrical vessel

A study is made of the problem of small oscillations of a low-viscosity fluid rotating under conditions of weightlessness in a cylindrical vessel. For fixed volume of the fluid and different values of the angular velocity and the wetting angle the decay rate and frequency of the characteristic oscillations are calculated. For an ideal fluid, the shapes of the characteristic oscillations of the free surface of the fluid are also calculated.