On free oscillations of a viscous fluid in a vessel

Abstract

A number of papers is devoted to problems of small oscillations of viscous fluids. Waves on the surface of a viscous fluid of infinite depth were examined for example in [1]. In [3] a boundary layer method is developed which is applied to problems of oscillations of a fluid in vessels in the case of small viscosity, [3–6] and others use this method to solve a series of problems on oscillations of a low viscosity fluid in certain regions. Certain general theorems on properties of characteristic oscillations of a heavy viscous fluid in a vessel are established in [7]. In [8] the approximate expression for the decrement of damping of free oscillations of a heavy viscous fluid in a cylindrical vessel of infinite depth is obtained. Results of experimental investigation of oscillations of a fluid in vessels are given in [9]. In this paper free small oscillations of a viscous incompressible fluid are studied in a stationary vessel of arbitrary shape in presence of gravity. In the main part of this paper the Reynolds' number is assumed to be large (viscosity small) which makes it possible, as in [2–6], to apply the boundary layer method. The investigation is carried ont by a method which is analogous to the one which was used in [10] in the study of motion of a body with a cavity completely filled with viscous fluid. Asymptotic relationships are obtained for eigenvalues and eigen functions of the problem on free oscillations of a viscous fluid in an arbitrary vessel. Decrements of damping and corrections to eigenfrequencies due to viscosity are expressed through equations which depend only on the corresponding eigenfrequencies and eigenfunctions of the problem on oscillations of an ideal fluid. Computations are carried out for some specific forms of vessels. In the last part of the paper the special character of motion of a viscous fluid near the line of contact of the free surface with the wall of the vessel is elucidated. Here free oscillations are examined for arbitrary Reynolds' number.