Vibrations of a body in a bounded volume of viscous fluid

The problem of the vibrations of a body in a bounded volume of viscous fluid has been studied on a number of occasions [1–4]. The main attention has been devoted to determining the hydrodynamic characteristics of elements in the form of rods. Analytic solution of the problem is possible only in the simplest cases [2]. In the present paper, in which large Reynolds numbers are considered, the asymptotic method of Vishik and Lyusternik [5] and Chernous' ko [6] is used to consider the general problem of translational vibrations of an axisymmetric body in an axisymmetric volume of fluid. Equations of motion of the body and expressions for the coefficients due to the viscosity of the fluid are obtained. It is shown that in the first approximation these coefficients differ only by a constant factor and are completely determined if the solution to the problem for an ideal fluid is known. Examples are given of the determination of the “viscous” added mass and the damping coefficient for some bodies and cavities. In the case of an ideal fluid, general estimates are obtained for the added mass and also for the influence of nonlinearity. Ritz's method is used to solve the problem of longitudinal vibrations of an ellipsoid of revolution in a circular cylinder. The hydrodynamic coefficients have been determined numerically on a computer. The theoretical results agree well with the results of experimental investigations.     

Oscillations of liquid in a vessel in the form of a rectangular parallelepiped

We consider the forced and the free oscillations of a liquid partially filling a cavity in the form of a rectangular parallelepiped. The characteristics of these oscillations are studied for small deformations of the free surface. It is shown that for definite frequencies and amplitudes of two-dimensional translational motions of the parallelepiped the fundamental of the liquid oscillations is excited in the plane perpendicular to the plane of motion of the vessel. The effect of small linear damping of the liquid oscillations on the shape of the boundaries of the principal region of instability of the liquid oscillations is evaluated. Fairly large oscillations of a liquid in a cylinder were considered in [1]. The same problem for a cavity of arbitrary configuration was studied in [2]. We note also that the conclusions of the study presented here are in qualitative agreement with the basic results obtained by a somewhat different method in [3] for a cavity in the form of a right circular cylinder.     

A mechanical model of a liquid performing small oscillations in a spherical cavity

We construct a system of approximate nonlinear equations describing the small oscillations of an ideal incompressible liquid which partiallyfills a spherical cavity. These equations are obtained for the case when the cavity undergoes small harmonic translational displacements with a frequency close to the fundamental frequency of the liquid oscillations in the direction perpendicular to the gradient of the mass force field acting on the liquid.     

Nonsmall liquid oscillations in moving vessels

The system of approximate nonlinear equations describing liquid oscillations in axisymmetric vessels is constructed. The equations are obtained for the case in which two coordinates belonging to the family of generalized coordinates characterizing the liquid motion are not small. This family is selected so that from the resulting nonlinear equations we can obtain as a particular case the nonlinear equations of [1–3], which are valid for the class of cylindrical vessels, and the requirements are satisfied that the resulting nonlinear equations correspond to the widely adopted linearized equations of liquid oscillations [4–6], Nonlinear equations are obtained which describe liquid oscillations in arbitrary vessels of rotation with radial baffles.     

Angular vibrations of an ellipsoid of revolution in a confined viscous fluid

Vibrations of bodies in confined viscous fluids have been studied on a number of occasions, transverse vibrations of rods being the main subject of investigation [1–3]. The present authors [4] have considered the general problem of translational vibrations of an axisymmetric body in an axisymmetric region containing a low-viscosity fluid. The present paper follows the same approach and considers the problem of small angular vibrations of an ellipsoid of revolution in a circular cylinder with flat ends. In the general case, the hydrodynamic coefficients in the equation of motion of the ellipsoid are determined numerically for different values of the dimensionless geometrical parameters using Ritz's method. In the case of an unconfined fluid, analytic dependences in terms of elementary functions are obtained for the hydrodynamic coefficients. The theoretical results agree well with experimental investigations.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 2, pp. 34–39, March–April, 1984.     

Small vibrations of a sphere in a spherical volume of viscous fluid

Problems of the vibration of bodies in confined viscous fluids have been solved to determine the added masses and damping coefficients of rods [1–3] and floats [4–5]. The solutions of these problems, based on the use of simplifications of the boundary-layer method [4–6], are obtained analytically in general form and are in good agreement with the experimental data. However, in each specific case the possibility of using such solutions for given values of the fluid viscosity and vibration frequency must be justified either experimentally [2, 4, 5] or theoretically as, for example, in [1], where an analytic solution was obtained for concentric cylinders. The present paper offers a general solution of the problem of the small vibrations of a sphere in a spherical volume of fluid valid over a broad range of variation of the dimensionless kinematic viscosity. The limiting cases of this solution for both high and low viscosity are considered. The asymptotic expressions obtained are compared with calculations based on the analytic solution.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 2, pp. 29–34, March–April, 1986.     

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.     

Nonsmall liquid oscillations in a right circular cylinder

In this study we consider certain nonlinear effects which occur during oscillations of a liquid partially filling a right circular cylinder. The problem of nonlinear oscillations of a liquid in a circular cylinder has been considered in [1, 2]. The same problem has been solved in [3, 4] for arbitrary cavities by a somewhat different method.In the present paper we investigate the stability of forced oscillations of a liquid in a cylinder when the latter performs small harmonic oscillations in a plane passing through its axis.