Mathematical Study of the Small Oscillations of a Spherical Layer of Viscous Fluid about a Rigid Spherical Core in the Gravitational Field

The problem of the small oscillations of a spherical layer of an inviscid fluid about a rigid spherical body in the gravitational field has been studied by Laplace in the case of a fluid layer of small depth; his results have been rediscovered by R.Wavre using his method of the uniform process. In this paper, the authors consider the case of a layer of viscous fluid. Using the methods of functional analysis, they obtain from the variational form of the equations of motions, an operatorial equation in a Hilbert space. They reduce the problem of the small oscillations to the study of an operator pencil and, so, they can determine the spectrum of the problem and specify the set of eigenvalues according to the coefficient of viscosity. They also give results concerning the solutions of the evolution problem by means of the theory of the semi-group. © 2014 Springer Basel.