Mathematical study of the small oscillations of a finite cylindrical column liquid-gas under zero gravity
DOI:
https://doi.org/10.14419/gjma.v7i1.26540الكلمات المفتاحية:
Gas Dynamics، Free Boundaries، Capillarity (Surface Tension)، Small Oscillations، Variational Method.الملخص
This paper deals with the mathematical study of the small motions of a system formed by a cylindrical liquid column bounded by two parallel circular rings and an internal cylindrical column constituted by a barotropic gas under zero gravity. From the equations of motion, the authors deduce a variational equation. Then, the study of the small oscillations depends on the coerciveness of a hermitian form that appears in this equation. It is proved that this last problem is reduced to an auxiliary eigenvalues problem. The discussion shows that, under a simple geometric condition, the problem is a classical vibration problem.
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المراجع
[1] Bauer, H.F. Oscillations of immiscible liquids in free space or in spherical containers in zero gravity environnement. Ingenieur. Archiv. 363-381(1982).https://doi.org/10.1007/BF00536679.
[2] Bauer, H.F. Natural frequencies and stability circular cylindrical immiscible systems, Appl. Micrograv. Technol. 27-44 (1989).
[3] Bauer, H.F. Natural frequencies and response of spinning liquid column with apparently sliding contact line, Acta. Mech. 115-122 (1993).https://doi.org/10.1007/BF01173146.
[4] Capodanno, P. The stability of a configuration of a liquid layer in an axially symmetric container uniformly rotating under conditions of weightlesness, Prikladnaya Matematikai Mekhanika ( .M.M). 65(4). 619-630 (2001) (in Russian). English translation ; J. Appl. Math. Mech. 65(4). 605-616 (2001). https://doi.org/10.1016/S0021-8928(01)00065-X.
[5] Capodanno, P. Etude mathématique des petites oscillations d’un pont liquide sphérique entre deux membranes circulaires en apesanteur (in french). Rev. Roum. Math. Pure. Appl, 46. 207-224 (2001).
[6] Capodanno, P. The stability and the small oscillations of an onduloÏŠdal fluid bridge between two coaxial disks under conditions of weightlessness. Prikladnaya Matematikai Mekhanika ( .M.M). 67(2). 283-299 (2003). (In Russian). English translation; J. Appl. Math. Mech. 67(2). 253-268 (2003).https://doi.org/10.1016/S0021-8928(03)90012-8.
[7] Kopachevskii, N.D., Krein, S.G. Operator approach to linear problems of hydrodynamics. Vol. 1, Birkhauser, Basel (2001).https://doi.org/10.1007/978-3-0348-8342-9_1.
[8] Lamb, H.: Hydrodynamics. Cambridge at the University Press, Cambridge (1932).
[9] Langbein, D. Capillary Surfaces-Shape- Stability- Dynamics, in particular under weightlessness. In: Springer Tracts in modern Physics. Springer- Verlag, Berlin (2002).https://doi.org/10.1007/3-540-45267-2.
[10] Moiseyev, N.N., Rumyantsev, V.V. Dynamic stability of bodies containing fluid. Springer- Verlag, Berlin (1968).https://doi.org/10.1007/978-3-642-86452-0.
[11] Morand, H.J-P., Ohayon, R. Interactions fluides-structures. Masson, Paris (1992).
[12] Roseau, M. Vibrations des systèmes mécaniques. Masson, Paris (1984).
[13] Sanchez H.J., Sanchez P.E. Vibration and coupling of continuous systems. Asymptotic methods. Springer-Verlag, Berlin (1989).
[14] Schwartz, L. Théorie des distributions. Hermann, Paris (1966).
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