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Journal of Applied Nonlinear Dynamics 2(3) (2013) 261--283 | DOI:10.5890/JAND.2013.08.003

Bachir Meziani; Ouerdia Ourrad

Laboratoire de Physique Théorique, Université A. Mira of Béjaia, Campus de Targua Ouzemour, 06000 Béjaia, Algeria

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Abstract

The nonlinear sloshing of two superposed fluids in a rectangular tank is studied. The equations governing the nonlinear motion of the interface and the free surface have been resolved at first and second order. The shape of these surfaces depends on the perturbation parameter ε and density ratio ρ of the fluids. The analysis of nonlinear effects shows new aspects that are not pre-viously observed. Linear theory is still valid for very small sloshing amplitude. When the amplitude increases, nonlinear effects can not be neglected. Superposed fluids with neighbors densities are more sensitive to nonlinear effects. they occurs in this case, by sub harmonic instabilities in space and beats phenomena in time. The first non-linear effects are observed at the interface. when ε increases, the influence of terms related to the first order eigenfrequency ω11 decreases and the influence of terms related to the eigenfrequency 2ω11 increases faster than those relating to the second order eigenfrequency ω21. At the free surface, the influence of terms related to the second order eigenfrequency ω21 is more pronounced when they appear as the influence of those relating to the eigenfrequency 2ω11.

Acknowledgments

B. MEZIANI and O. OURRAD would like to thank Xavier LEONCINI, Theoretical Physics Centre, UMR 6207, Université d ’Aix-Marseille, Luminy, Case 907 F-13288 Marseille Cedex 9, France, for all discussions and suggestions.

References

1. [1]	Ibrahim, R.A. (2005), Liquid sloshing dynamics, Theory and Applications, Cambridge University Press, Cambridge.

2. [2]	Choi, W. and Camassa R. (1999), Fully nonlinear internal waves in a two-fluid system, Journal of Fluid Mechanics, 396, 1-36.

3. [3]	La Rocca, M., Sciortino, G., and Boniforti, M.A. (2002), Interfacial gravity waves in a two-fluid system, Fluid Dynamics Research, 30, 31-66.

4. [4]	La Rocca, M., Sciortino, G., Adduce, C., and Boniforti, M.A. (2005), Experimental and theoretical investigation on the sloshing of a two-liquid system with free surface, Pysics of Fluids, 17, 062101.

5. [5]	Mackey, D. and Cox, E.A. (2003), Dynamics of a two-layer fluid sloshing problem, IMA Journal of Applied Mathematics, 68, 665-686.

6. [6]	Generalis, S.C. and Nagata, M. (1995), Faraday resonance in a two-liquid layer system, Fluid Dynamics Research, 15,145-165.

7. [7]	Lu, D., Dai, S. and Zhang, B. (1999), Hamiltonian formulation of nonlinear water waves in a two fluid system, Applied Mathematics and Mechanics, 20(4), 343-349.

8. [8]	Hara, K. and Takahara, H. (2008), Hamiltonian formulation for nonlinear sloshing in layered two immiscible fluids, Journal of System Design and Dynamics,2(5), 1183-1193.

9. [9]	Faltinsen, O.M., Rognebakke, O.F., Lukovsky, I. A. and Timokha, A. N. (2000), Multidimensional modal analysis of nonlinear sloshing in a rectangular tank with finite water depth, Journal of Fluid Mechanics, 407, 201-234.

10. [10]	Amaouche, M. and Meziani, B. (2008), Oscillations of two superposed fluids in an open and flexible container, Comptes Rendus de Mecanique,336, 329-335.

11. [11]	Frandsen, J.B. (2004), Sloshing motions in excited tanks, Journal of Computational Physics, 196, 53-87.