Skip to main content Accessibility help

A variational principle for three-dimensional interactions between water waves and a floating rigid body with interior fluid motion

  • Hamid Alemi Ardakani (a1)


A variational principle is given for the motion of a rigid body dynamically coupled to its interior fluid sloshing in three-dimensional rotating and translating coordinates. The fluid is assumed to be inviscid and incompressible. The Euler–Poincaré reduction framework of rigid body dynamics is adapted to derive the coupled partial differential equations for the angular momentum and linear momentum of the rigid body and for the motion of the interior fluid relative to the body coordinate system attached to the moving rigid body. The variational principle is extended to the problem of interactions between gravity-driven potential flow water waves and a freely floating rigid body dynamically coupled to its interior fluid motion in three dimensions.


Corresponding author

Email address for correspondence:


Hide All
Alemi Ardakani, H. 2016 A symplectic integrator for dynamic coupling between nonlinear vessel motion with variable cross-section and bottom topography and interior shallow-water sloshing. J. Fluids Struct. 65, 3043.
Alemi Ardakani, H. 2017 A coupled variational principle for 2D interactions between water waves and a rigid body containing fluid. J. Fluid Mech. 827, R2 1–12.
Alemi Ardakani, H. & Bridges, T. J. 2011 Shallow-water sloshing in vessels undergoing prescribed rigid-body motion in three dimensions. J. Fluid Mech. 667, 474519.
Bateman, H. 1932 Partial Differential Equations of Mathematical Physics. Cambridge University Press.
Bokhove, O. & Oliver, M. 2006 Parcel Eulerian–Lagrangian fluid dynamics of rotating geophysical flows. Proc. R. Soc. A 462, 25752592.
Bretherton, F. P. 1970 A note on Hamilton’s principle for perfect fluids. J. Fluid Mech. 44, 1931.
Broer, L. J. F. 1974 On the Hamiltonian theory for surface waves. Appl. Sci. Res. 29, 430446.
Calderer, A., Guo, X., Shen, L. & Sotiropoulos, F. 2018 Fluid–structure interaction simulation of floating structures interacting with complex, large-scale ocean waves and atmospheric turbulence with application to floating offshore wind turbines. J. Comput. Phys. 355, 144175.
Chernousko, F. L. 1965 Motion of a rigid body with cavities filled with viscous fluid at small Reynolds numbers. USSR Comput. Maths Math. Phys. 5, 99127.
Chernousko, F. L.1972 Motion of a Rigid Body with Cavities Containing a Viscous Fluid. NASA Technical Translations.
Cotter, C. & Bokhove, O. 2010 Variational water-wave model with accurate dispersion and vertical vorticity. J. Engng Maths 67, 3354.
Van Daalen, E. F. G., Van Groesen, E. & Zandbergen, P. J. 1993 A Hamiltonian formulation for nonlinear wave-body interactions. In Proceedings of the Eight International Workshop on Water Waves and Floating Bodies, Canada, pp. 159163. IWWWFB.
Daniliuk, I. I. 1976 On integral functionals with a variable domain of integration. In Proceedings of the Steklov Institute of Mathematics, vol. 118, pp. 144. American Mathematical Society.
Desbrun, M., Gawlik, E. S., Gay-Balmaz, F. & Zeitlin, V. 2014 Variational discretization for rotating stratified fluids. J. Discrete Continuous Dyn. Syst. 34, 477509.
Disser, K., Galdi, G. P., Mazzone, G. & Zunino, P. 2016 Intertial motions of a rigid body with a cavity filled with a viscous liquid. Arch. Rat. Mech. Anal. 221, 487526.
Faltinsen, O. M & Timokha, A. N. 2009 Sloshing. Cambridge University Press.
Flanders, H. 1973 Differentiation under the integral sign. Am. Math. Mon. 80, 615627.
Gagarina, E., Ambati, V. R., Nurijanyan, S., van der Vegt, J. J. W. & Bokhove, O. 2016 On variational and symplectic time integrators for Hamiltonian systems. J. Comput. Phys. 306, 370389.
Gagarina, E., Ambati, V. R., van der Vegt, J. J. W. & Bokhove, O. 2014 Variational space–time (dis)continuous Galerkin method for nonlinear free surface water waves. J. Comput. Phys. 275, 459483.
Gagarina, E., van der Vegt, J. & Bokhove, O. 2013 Horizontal circulation and jumps in Hamiltonian wave models. Nonlinear Process. Geophys. 20, 483500.
Gawlik, E. S., Mullen, P., Pavlov, D., Marsden, J. E. & Desbrun, M. 2011 Geometric, variational discretization of continuum theories. Physica D 240, 17241760.
Gay-Balmaz, F., Marsden, J. E. & Ratiu, T. S. 2012 Reduced variational formulations in free boundary continuum mechanics. J. Nonlinear Sci. 22, 463497.
Gerrits, J. & Veldman, A. E. P. 2003 Dynamics of liquid-filled spacecraft. J. Engng Maths 45, 2138.
Greenhill, A. G. 1880 On the general motion of a liquid ellipsoid. Proc. Camb. Phil. Soc. 4, 414.
Van Groesen, E. & Andonowati 2017 Hamiltonian Boussinesq formulation of wave–ship interactions. Appl. Math. Model. 42, 133144.
Holm, D. D., Marsden, J. E. & Ratiu, T. S. 1998a The Euler–Poincaré equations and semidirect products with applications to continuum theories. Adv. Maths 137, 181.
Holm, D. D., Marsden, J. E. & Ratiu, T. S. 1998b The Euler–Poincaré models of ideal fluids with nonlinear dispersion. Phys. Rev. Lett. 80, 41734176.
Holm, D. D., Marsden, J. E. & Ratiu, T. S.1999 The Euler–Poincaré equations in geophysical fluid dynamics. arXiv:chao-dyn/9903035.
Holm, D. D., Schmah, T. & Stoica, C. 2009 Geometric Mechanics and Symmetry: From Finite to Infinite Dimensions. Oxford University Press.
Hough, S. S. 1895 The oscillations of a rotating ellipsoidal shell containing fluid. Phil. Trans. R. Soc. Lond. A 186, 469506.
Ibrahim, R. A. 2005 Liquid Sloshing Dynamics. Cambridge University Press.
Kalogirou, A. & Bokhove, O. 2016 Mathematical and numerical modelling of wave impact on wave-energy buoys. In Proceedings of the International Conference on Offshore Mechanics and Arctic Engineering, p. 8. The American Society of Mechanical Engineers.
Kostyuchenko, A. G., Shkalikov, A. A. & Yurkin, M. Y. 1998 On the stability of a top with a cavity filled with a viscous fluid. Funct. Anal. Appl. 32, 100113.
Lewis, D., Marsden, J. E., Montgomery, R. & Ratiu, T. S. 1986 The Hamiltonian structure for dynamic free boundary problems. Physica D 18, 391404.
Leybourne, M., Batten, W. M. J., Bahaj, A. S., Minns, N. & O’Nians, J. 2014 Preliminary design of the OWEL wave energy converter pre-commercial demonstrator. Renewable Energy 61, 5156.
Luke, J. C. 1967 A variational principle for a fluid with a free surface. J. Fluid Mech. 27, 395397.
Lukovsky, I. A. 1976 Variational method in the nonlinear problems of the dynamics of a limited liquid volume with free surface. In Oscillations of Elastic Constructions with Liquid, pp. 260264. Volna (in Russian).
Lukovsky, I. A. 2015 Nonlinear Dynamics: Mathematical Models for Rigid Bodies with a Liquid. De Gruyter.
Marsden, J. E. & Ratiu, T. S. 1999 Introduction to Mechanics and Symmetry. Springer.
Marsden, J. E. & West, M. 2001 Discrete mechanics and variational integrators. Acta Numerica 10, 357514.
Mazer, A. & Ratiu, T. S. 1989 Hamiltonian formulation of adiabatic free-boundary Euler flows. J. Geom. Phys. 6, 271291.
Miles, J. W. 1976 Nonlinear surface waves in closed basins. J. Fluid Mech. 75, 419448.
Miles, J. W. 1977 On Hamilton’s principle for surface waves. J. Fluid Mech. 83, 153158.
Miloh, T. 1984 Hamilton’s principle, Lagrange’s method, and ship motion theory. J. Ship Res. 28, 229237.
Moiseyev, N. N. & Rumyantsev, V. V. 1968 Dynamic Stability of Bodies Containing Fluid. Springer.
Morrison, P. J. 1998 Hamiltonian description of the ideal fluid. Rev. Mod. Phys. 70, 467521.
Murray, R. M., Lin, Z. X. & Sastry, S. S. 1994 A Mathematical Introduction to Robotic Manipulation. CRC Press.
Oliver, M. 2014 A variational derivation of the geostrophic momentum approximation. J. Fluid Mech. 751, R2 1–10.
Oliver, M. 2006 Variational asymptotics for rotating shallow water near geostrophy: a transformational approach. J. Fluid Mech. 551, 197234.
O’Reilly, O. M. 2008 Intermediate Dynamics for Engineers: a Unified Treatment of Newton–Euler and Lagrangian Mechanics. Cambridge University Press.
Pavlov, D., Mullen, P., Tong, Y, Kanso, E., Marsden, J. E. & Desbrun, M. 2011 Sructure-preserving discretization of incompressible fluids. Physica D 240, 443458.
Ramodanov, S. M. & Sidorenko, V. V. 2017 Dynamics of a rigid body with an ellipsoidal cavity filled with viscous fluid. Intl J. Non-Linear Mech. 95, 4246.
Rumiantsev, V. V. 1966 On the theory of motion of rigid bodies with fluid-filled cavities. J. Appl. Math. Mech. 30, 5777.
Rumyantsev, V. V. 1963 Lyapunov’s method in the study of the stability of rigid bodies with fluid-filled cavities. Izv. Akad. Nauk SSSR (Series Mekh. Mashinostr.) 6, 119140.
Salmon, R. 1983 Practical use of Hamilton’s principle. J. Fluid Mech. 132, 431444.
Salmon, R. 1988 Hamiltonian fluid mechanics. Annu. Rev. Fluid Mech. 20, 225256.
Shepherd, T. G. 1990 Symmetries, conservation laws, and Hamiltonian structure in geophysical fluid dynamics. Adv. Geophys. 32, 287338.
Stokes, G. G. 1880 Mathematical and Physical Papers, vol. 1. Cambridge University Press.
Strygin, V. V. & Sobolev, V. A. 1988 Separation of Motions By the Integral Manifolds Method. Nauka (in Russian).
Timokha, A. N. 2016 The Bateman-Luke variational formalism in a sloshing with rotational flows. Dopov. Nac. Akad. Nauk Ukr. 4, 3034.
Veldman, A. E. P., Gerrits, J., Luppes, R., Helder, J. A. & Vreeburg, J. P. B. 2007 The numerical simulation of liquid sloshing on board spacecraft. J. Comput. Phys. 224, 8299.
Zakharov, V. E. 1968 Stability of periodic waves of finite-amplitude on the surface of a deep fluid. J. Appl. Mech. Tech. Phys. 9, 190194.
Zhukovskii, N. Y. 1885 On the motion of a rigid body with cavities filled with a homogeneous liquid drop. Zh. Fiz.-Khim. Obs. Phys. 17, 81113.
Zhukovskii, N. E. 1948 Motion of a rigid body having a cavity filled with fluid. Collected Works, vol. 1, pp. 31152. Gostekhizdat.
MathJax is a JavaScript display engine for mathematics. For more information see

JFM classification

A variational principle for three-dimensional interactions between water waves and a floating rigid body with interior fluid motion

  • Hamid Alemi Ardakani (a1)


Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed