Abstract
The authors study the small oscillations of a system of two nonmixing fluids, the lower inviscid, the upper viscoelastic, in an open container, restricting themselves for the second to the more simple Oldroyd model. From the equations of motion, they obtain a variational formulation of the problem, from which they deduce a variational equation for the viscoelastic fluid only, then a system of operatorial equations in suitable Hilbert space. They show the existence and the symmetry of the spectrum, prove the stability of the system and specify the location of the eigenvalues. They prove the existence of two sets of positive real eigenvalues having, the first l’infinity, the second a point of the real axis, as point of accumulation. Finally, after a suitable transformation of the operatorial equations of motion, they obtain an existence and unicity theorem of the solution of the associated evolution problem by means of the semigroups theory.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Capodanno, P.: Planar small oscillations of two nonmixing heavy liquids in a fixed vessel. Mechanics Research Communications 15(4), 243–247 (1988) (in french)
Capodanno, P.: Mathematical study of the small oscillations of two nonmixing liquids in a container. Mechanics Research Communications 23(1), 75–80 (1996) (in french)
Dautray, R., Lions, J.L.: Analyse mathématique et calcul numérique, vol. 4. Masson, Paris (1988)
Friedman, A., Shinbrot, M.: The initial value problem for the linearized equation of water waves. J. Math. Mech 2, 107–180 (1967)
Kopachevsky, N.D., Krein, S.G.: Operator approch to linear problems of Hydrodynamics, vol. 1. Birkhauser, Basel (2001)
Kopachevsky, N.D., Krein, S.G.: Operator approch to linear problems of Hydrodynamics, vol. 2. Birkhauser, Basel (2003)
Lions, J.L.: Equations différentielles opérationnelles et problèmes aux limites. Springer Verlag, Berlin (1961)
Moiseyev, N.N.: About the oscillations of an ideal incompressible liquid in a container, Dokl. AN. SSSR, \(\text{N}^{o}\)85(5), (1952)
Moiseyev, N.N., Rumyantsev, V.V.: Dynamic stability of bodies containing fluid. Springer, Berlin (1968)
Morand, H.J.-P., Ohayon, R.: Interactions fluides-structures. Masson, Paris (1992)
Riesz, F., Nagy, B.S.Z.: Leçons d’analyse fonctionnelle. Gauthier villars, Paris (1968)
Sanchez Hubert, J., Sanchez Palencia, E.: Vibration and coupling of continuous systems. Asymptotic methods. Springer Verlag, Berlin (1989)
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by A.V. Fursikov
Rights and permissions
About this article
Cite this article
Essaouini, H., El Bakkali, L. & Capodanno, P. Mathematical Study of the Small Oscillations of Two Nonmixing Fluids, the Lower Inviscid, the Upper Viscoelastic, in an Open Container. J. Math. Fluid Mech. 19, 645–657 (2017). https://doi.org/10.1007/s00021-016-0300-7
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00021-016-0300-7