Free vibrations of a self-gravitating liquid sphere with viscous and capillary forces

Abstract

KREIN has shown [1] that the spectrum of the problem of free oscillations of a heavy viscous liquid in a partially filled vessel, where the surface tension forces are not considered, has two limit points λ = 0 and λ = ∞; all the eigen values λ, except perhaps a finite number, are real. The surface forces, as has been shown in a number of works [2, 3] must substantially change the structure of the spectrum, moving the limit point of the spectrum to zero. This fact is also confirmed by the model of the problem considered in [4]. At the International Mathematical Congress in Moscow (1966) Krein raised the question of investigating the structure of the spectrum of this problem with surface forces. In this paper we consider the classical problem of small oscillations of a viscous incompressible liquid sphere under the action of forces of self-gravitation and surface tension, and also a similar plane problem. Unlike [3], where the case of low viscosity is considered by the “boundary layer” method, we here investigate the exact characteristic equation for an arbitrary value for the viscosity. In Section 1 the characteristic equations of both problems are deduced. They appear to be very similar and therefore in this paper, for definiteness, the characteristic equation for the plane problem is considered in more detail (Sections 2–5), and in Section 6 the necessary changes relating to the space problem are briefly described. In this paper it is shown that the spectrum of both problems is discrete with a unique limit point at infinity, while all the eigenvalues, except perhaps for a finite number, are real. With a fairly high viscosity there are no unreal eigenvalues. On the other hand for fairly low viscosity the number of unreal eigenvalues can be made as large as we please. If we leave out of account the surface tension, the limit point λ = 0 is added to the limit point λ = ∞ of the spectrum. Consideration of the model problem of [4] and the two physical problems of this paper gives us confidence that these conclusions are true for the case of free oscillations of a viscous liquid in a partially filled vessel in an arbitrary field of force.