Abstract
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 by using his method of the uniform process. The second author and his collaborators have studied the case of a layer of viscous fluid by means of the methods of the functional analysis. In this paper, we consider the case of a layer of viscoelastic fluid that obeys to the simpler Oldroyd’s law. Using the classical methods for the calculation of the potential and the methods of the functional analysis, we obtain from the variational form of the equations of the motion, an operatorial equation in a suitable Hilbert space. We reduce the problem of the small oscillations to the study of an operator pencil and so, we can precise the location of the spectrum and prove the existence of three sets of real eigenvalues. We give a theorem of existence and unicity of the solution of the associated evolution problem by means of the semi - groups theory.
Similar content being viewed by others
References
Kopachevsky, N.D.: On small motions of a system of two viscoelastic fluids contained in a fixed vessel. Dyn. Syst. 7(35): 1–2, 109–145 (2017)
Kopachevsky, N.D.: To the problem on small motions of the system of two viscoelastic fluids in a fixed vessel. Contemp. Math. Fund. Direct. 64(3), 547–572 (2018)
Kopachevsky, N.D., Sëmkina, E.V.: Small movements of partially dissipative hydrosystem in stationary containers. Din. Sist. Simferopol’ 8(36), 103–126 (2018)
Zakora, D.A., Kopachevsky, N.D.: To the problem on small oscillations of a system of two viscoelastic fluids filling immovable vessel: model problem. Contemp. Math. Fund. Direct. 66(2), 182–208 (2020)
Kopachevsky, N.D., Krein, S.G.: Operator Approach to Linear Problems of Hydrodynamics, vol. 2. Birkhäuser, Basel (2003)
Dautray, R., Lions, J.L.: Analyse mathématique et calcul numérique, vol. 5. Masson, Paris (1988)
Mikhlin, S.G.: Mathematical Physics. An Advanced Course. North Holland Publishing Company, Amsterdam (1970)
Riesz, F., Nagy, B.S.Z.: Leçons d’analyse fonctionnelle. Gauthier villars, Paris (1968)
Lions, J.L.: Equations différentielles opérationnelles et problèmes aux limites. Springer, Berlin (1961)
Kopachevsky, N.D., Krein, S.G.: Operator Approach to Linear Problems of Hydrodynamics, vol. 1. Birkhäuser, Basel (2001)
Sanchez Hubert, J., Sanchez Palencia, E.: Vibration and Coupling of Continuous Systems. Asymptotic Methods. Springer, Berlin (1989)
Vivona, D., Capodanno, P.: Mathematical study of the small oscillations of a spherical layer of viscous fluid about a rigid spherical core in the gravitational field. Mediterr. J. Math. 12, 245–262 (2015)
Dautray, R., Lions, J.L.: Analyse mathématique et calcul numérique, vol. 3. Masson, Paris (1988)
Hobson, E.W.: The Theory of Spherical and Ellipsoidal Harmonics. Chelsea Publishing Company, New York (1965)
Wavre, R.: Essai sur les petites vibrations des astres fluides. Commentarii Mathematici Helvetici 4, 74–96 (1932)
Wavre, R.: Figures planétaires et géodésie. Gauthier villars, Paris (1932)
Acknowledgements
The authors are grateful to the referee and the editorial board for some useful comments that improved the presentation of the manuscript.
Author information
Authors and Affiliations
Corresponding author
Ethics declarations
Conflict of interest
The authors declare that they have no conflict of interest.
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
Essaouini, H., Capodanno, P. Mathematical study of the small oscillations of a spherical layer of viscoelastic fluid about a rigid spherical core in the gravitational field. Z. Angew. Math. Phys. 72, 109 (2021). https://doi.org/10.1007/s00033-021-01545-3
Received:
Revised:
Accepted:
Published:
DOI: https://doi.org/10.1007/s00033-021-01545-3
Keywords
- Viscoelastic fluid
- Small oscillations
- Gravitational field
- Variational and spectral methods
- Semi-groups theory