Hamiltonian structure for a neutrally buoyant rigid body interacting with N vortex rings of arbitrary shape: The case of arbitrary smooth body shape

Theoretical and Computational Fluid Dynamics

Volumn 22, Issue 1, 2008, Pages 37-64

  Shashikanth, Banavara N ^a   Sheshmani, Artan ^b   Kelly, Scott David ^c   Marsden, Jerrold E ^d  

^a Horeshoe Cir Jett Hall   (United States)

^b UNIVERSITY OF ILLINOIS AT URBANA CHAMPAIGN   (United States)

^c University of North Carolina at Charlotte   (United States)

^d CALIFORNIA INSTITUTE OF TECHNOLOGY   (United States)

Author keywords

Hamiltonian structure; Hydrodynamical interaction; Ideal hydrodynamics; Lie Poisson bracket; Poisson bracket; Vortex ring; Vorticity

Indexed keywords

HAMILTONIAN STRUCTURE; HYDRODYNAMICAL INTERACTION; IDEAL HYDRODYNAMICS; LIE-POISSON BRACKET; POISSON BRACKET; VORTEX RING;

BUOYANCY; EQUATIONS OF MOTION; HAMILTONIANS; MATHEMATICAL MODELS;

VORTICITY;

BUOYANCY; EQUATIONS OF MOTION; HAMILTONIANS; MATHEMATICAL MODELS; VORTICITY;

BIOT THEORY; FLOW VELOCITY; HYDRODYNAMICS; KINETIC ENERGY; RIGIDITY; VORTICITY;

EID: 37249023735     PISSN: 09354964     EISSN: 14322250     Source Type: Journal    
DOI: 10.1007/s00162-007-0065-y     Document Type: Article

References (40)

1. Allen J.J., Jouanne, Y., Shashikanth, B.N.: Vortex interaction with a moving sphere. J. Fluid Mech. 587, 337-346
2. Amrouche C., Bernardi C., Dauge M. and Girault V. (1998). Vector potentials in three dimensional non-smooth domains. Math. Methods Appl. Sci. 21: 823-864
3. Arnold V.I. (1966). Sur la géométrie differentielle des groupes de Lie de dimenson infinie et ses applications à l'hydrodynamique des fluids parfaits. Ann. Inst. Fourier Grenoble 16: 319-361
4. Arnold, V.I., Khesin, B.: Topological Methods in Hydrodynamics. Appl. Math. Sciences, vol. 125 Springer, Heidelberg (1998)
5. Arnold V.I. and Khesin B. (1992). Topological methods in hydrodynamics. Ann. Rev. Fluid Mech. 24: 145-166
6. Batchelor G.K. (1967). An Introduction to Fluid Dynamics. Cambridge University Press, London
7. Bendali A., Dominguez J.M. and Gallic S. (1985). A variational approache for the vector potential formulation of the Stokes and Navier-Stokes problems in three-dimensional domains. J. Math. Anal. Appl. 107: 537-560
8. Blackmore D. and Knio O. (2001). Hamiltonian structure for vortex filament flows. ZAMM 81: 45-48
9. Blackmore, D., Knio, O.: Hamiltonian formulation of the dynamics of interaction between a vortex ring and a rigid sphere. In: Proceedings of the European Congress on Computational Methods in Applied Sciences and Engineering. ECCOMAS 2000, Barcelona, Spain, 11-14 September (2000)
10. n with asymptotic conditions. J. Funct. Anal. 18: 73-84
11. Coutand D. and Shkoller S. (2005). Motion of an elastic solid inside an incompressible visoucs fluid. Arch. Ration. Mech. Anal. 176(1): 25-102
12. Desjardins B. and Esteban M.J. (1999). Existence of weak solutions for the motion of a rigid body in a visocus fluid. Arch. Ration. Mech. Anal. 146(1): 59-71
13. Ebin D.G. and Marsden J.E. (1970). Groups of diffeomorphisms and the motion of an incompressible fluid. Ann. Math. 92: 102-163
14. Hirasaki G.J. and Hellums J.D. (1967). A general formulation of the boundary conditions on the vector potential in three-dimensional hydrodynamics. Quart. Appl. Math. 26(3): 331-342
15. Kanso E., Marsden J.E., Rowley C.W. and Melli-Huber J.B. (2005). Locomotion of articulated bodies in a perfect fluid. J. Nonlinear Sci. 15: 255-289
16. Kelly, S.D.: The mechanics and control of robotic locomotion with applications to aquatic vehicles. Ph.D. Thesis, Institute of Technology, California (1998)
17. Klein R., Majda A.J. and Damodaran K. (1995). Simplified equations for the interaction of nearly parallel vortex filaments. J. Fluid Mech. 288: 201-248
18. Knio O.M. and Ting L. (1997). Vortical flow outside a body and sound generation. SIAM J. Appl. Math. 57(4): 972-981
19. Lamb, H.: Hydrodynamics. Dover, NY, sixth edition (1932)
20. Langer J. and Perline R. (1991). Poisson geometry of the filament equation. J. Nonlinear Sci. 1: 71-94
21. Lewis D., Marsden J.E., Montgomery R. and Ratiu T.S. (1986). The Hamiltonian structure for dynamic free boundary problems. Phys. D 18: 391-404
22. Lighthill M.J. (1956). The image system of a vortex element in a rigid body. Proc. Camb. Phil. Soc. 52: 317-321
23. Ma, Z., Shashikanth, B.N.: Dynamics and control of the system of a 2D rigid circular cylinder and point vortices. In: Proceedings of the American Control Conference. Minneapolis, Minnesota, USA, 14-16 June (2006)
24. Marsden, J.E.: Lectures on Mechanics, vol. 174. London Math. Soc. Lecture Note Ser., vol. 174. Cambridge University Press, London (1992)
25. Marsden J.E., Montgomery R. and Ratiu T. (1984). Gauged Lie-Poisson structures. Contempl. Math. 28: 101-114
26. Marsden, J.E., Montgomery, R., Ratiu, T.S.: Reduction, symmetry and phases in mechanics, vol. 436. Memoirs of the AMS, vol. 436. Am. Math. Soc. Providence (1990)
27. Marsden, J.E., Ratiu, T.S.: Introduction to Mechanics and Symmetry, vol. 17. Texts in Applied Mathematics, vol. 17; 1994, 2nd edn. Springer, Heidelberg (1999)
28. Marsden J.E. and Weinstein A. (1983). Coadjoint orbits, vortices and Clebsch variables for incompressible fluids. Phys. D 7: 305-323
29. Milne-Thomson L.M. (1996). Theoretical Hydrodynamics, 5th edn. Dover, New York
30. n J. Math. Anal. Appl. 42: 271-301
31. Radford, J.: Symmetry, reduction and swimming in a perfect fluid. Ph.D. Thesis, California Institute of Technology (2003)
32. Saffman P.G. (1992). Vortex Dynamics. Cambridge Monographs on Mechanics and Applied Mathematics. Cambridge Univ. Press., London
33. Shashikanth B.N. (2005). Poisson brackets for the dynamically interacting system of a 2D rigid boundary and N point vortices: The case of arbitrary smooth cylinder shapes. Reg. Chaotic Dyn. 10(1): 1-14
34. Shashikanth B.N. (2006). Symmetric pairs of point vortices interacting with a neutrally buoyant 2D circular cylinder. Phys. Fluids 18: 127103
35. Shashikanth B.N. and Marsden J.E. (2003). Leapfrogging vortexrings: Hamiltonian structure, geometricphases and disete reduction. Fluid Dyn. Res. 33: 333-356
36. Shashikanth, B.N., Marsden, J.E., Burdick, J.W., Kelly, S.D.: The Hamiltonian structure of a 2D rigid circular cylinder interacting dynamically with N point vortices. Phys. Fluids 14(3), 1214-1227 (2002) (see also Erratum. Phys. of Fluids 14(11), 4099)
37. Ting, L., Klein, R.: Viscous Vortical Flows. Lecture Notes in Physics, vol. 374. Springer, Berlin (1991)
38. Truesdell C. (1954). The Kinematics of Vorticity. Indiana University Press, Bloomington
39. Vladimirov, V.A.: Vortical momentum of bounded ideal incompressible fluid flows. J. Appl. Mech. Tech. Phys. 20, 157-163 (1979)
40. (Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No.2, 53-61, (1979))