Comptes Rendus
On the fundamental sloshing frequency in the ice-fishing problem
Comptes Rendus. Mécanique, Volume 330 (2002) no. 11, pp. 723-728.

The sloshing problem is considered in a half-space covered by a rigid dock with apertures. The dependence of the fundamental sloshing frequency on the shape of the free surface region is studied. It is proved that the inequality holds between the fundamental eigenvalues corresponding to two different regions if some conditions are fulfilled. These conditions are verified for particular classes of regions of a fixed area in order to demonstrate that the disk yields the maximum of the fundamental eigenvalue for each of these classes. On the other hand, examples of regions are constructed for which the fundamental eigenfrequency is larger than that for the circular aperture of the same area and even as large as one wishes.

On considère le problème du ballottement dans un demi-espace couvert part un couvercle rigide avec des ouvertures. On étudie la dépendance de la valeur propre fondamentale par rapport à la forme de la région de surface libre, sous l'hypothèse que l'aire de cette région est fixée.

Received:
Revised:
Published online:
DOI: 10.1016/S1631-0721(02)01530-9
Keywords: fluid mechanics, sloshing problem, fundamental eigenvalue, integral operator, variational principle
Mot clés : mécanique des fluides, oscillations libres d'une fluide, valeur propre fondamenale, opérateur intégral, principe variationnelle

Vladimir Kozlov 1; Nikolay Kuznetsov 2

1 Department of Mathematics, Linköping University, 58183 Linköping, Sweden
2 Laboratory for Mathematical Modelling of Wave Phenomena, Institute for Problems in Mechanical Engineering, Russian Academy of Sciences, V.O., Bol'shoy pr. 61, St. Petersburg 199178, Russia
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Vladimir Kozlov; Nikolay Kuznetsov. On the fundamental sloshing frequency in the ice-fishing problem. Comptes Rendus. Mécanique, Volume 330 (2002) no. 11, pp. 723-728. doi : 10.1016/S1631-0721(02)01530-9. https://comptes-rendus.academie-sciences.fr/mecanique/articles/10.1016/S1631-0721(02)01530-9/

[1] C.M. Linton; P. McIver Handbook of Mathematical Techniques for Wave/Structure Interactions, Chapman & Hall, CRC, 2001

[2] C. Bandle Isoperimetric Inequalities and Applications, Pitman, 1980

[3] N. Kuznetsov; O. Motygin Sloshing problem in a half-plane covered by a dock with two gaps: monotonicity and asymptotics of eigenvalues, C. R. Acad. Sci. Paris, Série IIb, Volume 329 (2001), pp. 791-796

[4] P. Henrici; B.A. Troesch; L. Wuytack Sloshing frequencies for a half-space with circular or strip-like aperture, Z. Angew. Math. Phys, Volume 21 (1970), pp. 285-318

[5] J.W. Miles On the eigenvalue problem for fluid sloshing in a half-space, Z. Angew. Math. Phys, Volume 23 (1972), pp. 861-868

[6] H.F. Weinberger An isoperimetric inequality for the N-dimensional free membrane problem, J. Rational Mech. Anal, Volume 5 (1956), pp. 633-636

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