# American Institute of Mathematical Sciences

March  2014, 6(1): 39-66. doi: 10.3934/jgm.2014.6.39

## Fluid-structure interaction in the Lagrange-Poincaré formalism: The Navier-Stokes and inviscid regimes

 1 Department of Mathematics, Imperial College London, London SW7 2AZ, United Kingdom

Received  December 2012 Revised  February 2014 Published  April 2014

In this paper, we derive the equations of motion for an elastic body interacting with a perfect fluid via the framework of Lagrange-Poincaré reduction. We model the combined fluid-structure system as a geodesic curve on the total space of a principal bundle on which a diffeomorphism group acts. After reduction by the diffeomorphism group we obtain the fluid-structure interactions where the fluid evolves by the inviscid fluid equations. Along the way, we describe various geometric structures appearing in fluid-structure interactions: principal connections, Lie groupoids, Lie algebroids, etc. We finish by introducing viscosity in our framework as an external force and adding the no-slip boundary condition. The result is a description of an elastic body immersed in a Navier-Stokes fluid as an externally forced Lagrange-Poincaré equation. Expressing fluid-structure interactions with Lagrange-Poincaré theory provides an alternative to the traditional description of the Navier-Stokes equations on an evolving domain.
Citation: Henry Jacobs, Joris Vankerschaver. Fluid-structure interaction in the Lagrange-Poincaré formalism: The Navier-Stokes and inviscid regimes. Journal of Geometric Mechanics, 2014, 6 (1) : 39-66. doi: 10.3934/jgm.2014.6.39
##### References:
 [1] R. Abraham and J. E. Marsden, Foundations of Mechanics, 2nd edition, American Mathematical Society, 2000. [2] R. Abraham, J. E. Marsden and T. S. Ratiu, Manifolds, Tensor Analysis, and Applications, vol. 75 of Applied Mathematical Sciences, 3rd edition, Spinger, 2009. [3] V. I. Arnold, Sur la géométrie différentielle des groupes de Lie de dimension infinie et ses applications à l'hydrodynamique des fluides parfaits, Annales de l'Institut Fourier, 16 (1966), 316-361. doi: 10.5802/aif.233. [4] V. I. Arnold and B. A. Khesin, Topological Methods in Hydrodynamics, vol. 24 of Applied Mathematical Sciences, 125. Springer-Verlag, New York, 1998. [5] G. K. Batchelor, An Introduction to Fluid Dynamics, Cambridge University Press, Cambridge, 1999. [6] H. Cendra, J. E. Marsden and T. S. Ratiu, Lagrangian Reduction by Stages, Mem. Amer. Math. Soc., 152 (2001), x+108 pp. doi: 10.1090/memo/0722. [7] R. L. Fernandes and I. Struchiner, Lie algebroids and classification problems in geometry, São Paulo J. Math. Sci., 2 (2008), 263-283. [8] E. S. Gawlik, P. Mullen, D. Pavlov, J. E. Marsden and M. Desbrun, Geometric, variational discretization of continuum theories, Physica D: Nonlinear Phenomena, 240 (2011), 1724-1760. doi: 10.1016/j.physd.2011.07.011. [9] F. Gay-Balmaz and T. S. Ratiu, The geometric structure of complex fluids, Advances in Applied Mathematics, 42 (2009), 176-275. doi: 10.1016/j.aam.2008.06.002. [10] E. Hairer, C. Lubich and G. Wanner, Geometric Numerical Integration, Structure Preserving Algorithms for Ordinary Differential Equations, vol. 31 of Series in Computational Mathematics, Springer Verlag, 2002. [11] H. O. Jacobs, T. S. Ratiu and M. Desbrun, On the coupling between an ideal fluid and immersed particles, Phys. D, 265 (2013), 40-56. arXiv:1208.6561v1 doi: 10.1016/j.physd.2013.09.004. [12] E. Kanso, J. E. Marsden, C. W. Rowley and J. B. Melli-Huber, Locomotion of articulated bodies in a perfect fluid, Journal of Nonlinear Science, 15 (2005), 255-289. doi: 10.1007/s00332-004-0650-9. [13] S. D. Kelly, The Mechanics and Control of Robotic Locomotion with Applications to Aquatic Vehicles, PhD thesis, California Institute of Technology, 1998. [14] S. Kobayashi and K. Nomizu, Foundations of differential geometry. Vol I, Interscience Publishers, John Wiley & Sons, 1963. [15] I. Kolar, P. W. Michor and J. Slovak, Natural Operations in Differential Geometry, Springer-Verlag, Berlin, 1993. [16] H. Lamb, Hydrodynamics, Reprint of the 1932 Cambridge University Press edition, Cambridge Mathematical Library. Cambridge University Press, Cambridge, 1993. [17] T. Lee, M. Leok and N. H. McClamroch, Computational geometric optimal control of rigid bodies, Communications in Information and Systems, 8 (2008), 445-472. doi: 10.4310/CIS.2008.v8.n4.a5. [18] D. Lewis, J. E. Marsden, R. Montgomery and T. S. Ratiu, The Hamiltonian structure for dynamic free boundary problems, Phys. D, 18 (1986), 391-404. doi: 10.1016/0167-2789(86)90207-1. [19] J. C. Marrero, D. Martín de Diego and E. Martínez, Discrete Lagrangian and Hamiltonian mechanics on Lie groupoids, Nonlinearity, 19 (2006), 1313-1348. doi: 10.1088/0951-7715/19/6/006. [20] J. E. Marsden and T. J. R. Hughes, Mathematical Foundations of Elasticity, Corrected reprint of the 1983 original. Dover Publications, Inc., New York, 1994. [21] J. E. Marsden and J. Scheurle, Lagrangian reduction and the double spherical pendulum, ZAMP, 44 (1993), 17-43. doi: 10.1007/BF00914351. [22] J. E. Marsden and J. Scheurle, The reduced Euler-Lagrange equations, Fields Institute Communications, 1 (1993), 139-164. [23] J. E. Marsden and M. West, Discrete mechanics and variational integrators, Acta Numerica, 10 (2001), 357-514. doi: 10.1017/S096249290100006X. [24] J. E. Radford, Symmetry, Reduction and Swimming in a Perfect Fluid, PhD thesis, California Institute of Technology, 2003. [25] G. Schwarz, Hodge Decomposition-A Method for Solving Boundary Value Problems, vol. 1607 of Lecture Notes in Mathematics, Springer-Verlag, Berlin, 1995. [26] A. Shapere and F. Wilczek, Geometry of self-propulsion at low Reynolds number, Journal of Fluid Mechanics, 198 (1989), 557-585. doi: 10.1017/S002211208900025X. [27] M. Troyanov, On the Hodge decomposition in $\mathbb{R}^{N}$, Mosc. Math. J., 9 (2009), 899-926, 936. [28] J. Vankerschaver, E. Kanso and J. E. Marsden, The geometry and dynamics of interacting rigid bodies and point vortices, Journal of Geometric Mechanics, 1 (2009), 223-266. doi: 10.3934/jgm.2009.1.223. [29] J. Vankerschaver, E. Kanso and J. E. Marsden, The dynamics of a rigid body in potential flow with circulation, Reg. Chaot. Dyn., 15 (2010), 606-629. doi: 10.1134/S1560354710040143. [30] A. Weinstein, Lagrangian mechanics and groupoids, in Mechanics Day, 207-231, Fields Inst. Commun., 7, Amer. Math. Soc., Providence, RI, 1996.

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##### References:
 [1] R. Abraham and J. E. Marsden, Foundations of Mechanics, 2nd edition, American Mathematical Society, 2000. [2] R. Abraham, J. E. Marsden and T. S. Ratiu, Manifolds, Tensor Analysis, and Applications, vol. 75 of Applied Mathematical Sciences, 3rd edition, Spinger, 2009. [3] V. I. Arnold, Sur la géométrie différentielle des groupes de Lie de dimension infinie et ses applications à l'hydrodynamique des fluides parfaits, Annales de l'Institut Fourier, 16 (1966), 316-361. doi: 10.5802/aif.233. [4] V. I. Arnold and B. A. Khesin, Topological Methods in Hydrodynamics, vol. 24 of Applied Mathematical Sciences, 125. Springer-Verlag, New York, 1998. [5] G. K. Batchelor, An Introduction to Fluid Dynamics, Cambridge University Press, Cambridge, 1999. [6] H. Cendra, J. E. Marsden and T. S. Ratiu, Lagrangian Reduction by Stages, Mem. Amer. Math. Soc., 152 (2001), x+108 pp. doi: 10.1090/memo/0722. [7] R. L. Fernandes and I. Struchiner, Lie algebroids and classification problems in geometry, São Paulo J. Math. Sci., 2 (2008), 263-283. [8] E. S. Gawlik, P. Mullen, D. Pavlov, J. E. Marsden and M. Desbrun, Geometric, variational discretization of continuum theories, Physica D: Nonlinear Phenomena, 240 (2011), 1724-1760. doi: 10.1016/j.physd.2011.07.011. [9] F. Gay-Balmaz and T. S. Ratiu, The geometric structure of complex fluids, Advances in Applied Mathematics, 42 (2009), 176-275. doi: 10.1016/j.aam.2008.06.002. [10] E. Hairer, C. Lubich and G. Wanner, Geometric Numerical Integration, Structure Preserving Algorithms for Ordinary Differential Equations, vol. 31 of Series in Computational Mathematics, Springer Verlag, 2002. [11] H. O. Jacobs, T. S. Ratiu and M. Desbrun, On the coupling between an ideal fluid and immersed particles, Phys. D, 265 (2013), 40-56. arXiv:1208.6561v1 doi: 10.1016/j.physd.2013.09.004. [12] E. Kanso, J. E. Marsden, C. W. Rowley and J. B. Melli-Huber, Locomotion of articulated bodies in a perfect fluid, Journal of Nonlinear Science, 15 (2005), 255-289. doi: 10.1007/s00332-004-0650-9. [13] S. D. Kelly, The Mechanics and Control of Robotic Locomotion with Applications to Aquatic Vehicles, PhD thesis, California Institute of Technology, 1998. [14] S. Kobayashi and K. Nomizu, Foundations of differential geometry. Vol I, Interscience Publishers, John Wiley & Sons, 1963. [15] I. Kolar, P. W. Michor and J. Slovak, Natural Operations in Differential Geometry, Springer-Verlag, Berlin, 1993. [16] H. Lamb, Hydrodynamics, Reprint of the 1932 Cambridge University Press edition, Cambridge Mathematical Library. Cambridge University Press, Cambridge, 1993. [17] T. Lee, M. Leok and N. H. McClamroch, Computational geometric optimal control of rigid bodies, Communications in Information and Systems, 8 (2008), 445-472. doi: 10.4310/CIS.2008.v8.n4.a5. [18] D. Lewis, J. E. Marsden, R. Montgomery and T. S. Ratiu, The Hamiltonian structure for dynamic free boundary problems, Phys. D, 18 (1986), 391-404. doi: 10.1016/0167-2789(86)90207-1. [19] J. C. Marrero, D. Martín de Diego and E. Martínez, Discrete Lagrangian and Hamiltonian mechanics on Lie groupoids, Nonlinearity, 19 (2006), 1313-1348. doi: 10.1088/0951-7715/19/6/006. [20] J. E. Marsden and T. J. R. Hughes, Mathematical Foundations of Elasticity, Corrected reprint of the 1983 original. Dover Publications, Inc., New York, 1994. [21] J. E. Marsden and J. Scheurle, Lagrangian reduction and the double spherical pendulum, ZAMP, 44 (1993), 17-43. doi: 10.1007/BF00914351. [22] J. E. Marsden and J. Scheurle, The reduced Euler-Lagrange equations, Fields Institute Communications, 1 (1993), 139-164. [23] J. E. Marsden and M. West, Discrete mechanics and variational integrators, Acta Numerica, 10 (2001), 357-514. doi: 10.1017/S096249290100006X. [24] J. E. Radford, Symmetry, Reduction and Swimming in a Perfect Fluid, PhD thesis, California Institute of Technology, 2003. [25] G. Schwarz, Hodge Decomposition-A Method for Solving Boundary Value Problems, vol. 1607 of Lecture Notes in Mathematics, Springer-Verlag, Berlin, 1995. [26] A. Shapere and F. Wilczek, Geometry of self-propulsion at low Reynolds number, Journal of Fluid Mechanics, 198 (1989), 557-585. doi: 10.1017/S002211208900025X. [27] M. Troyanov, On the Hodge decomposition in $\mathbb{R}^{N}$, Mosc. Math. J., 9 (2009), 899-926, 936. [28] J. Vankerschaver, E. Kanso and J. E. Marsden, The geometry and dynamics of interacting rigid bodies and point vortices, Journal of Geometric Mechanics, 1 (2009), 223-266. doi: 10.3934/jgm.2009.1.223. [29] J. Vankerschaver, E. Kanso and J. E. Marsden, The dynamics of a rigid body in potential flow with circulation, Reg. Chaot. Dyn., 15 (2010), 606-629. doi: 10.1134/S1560354710040143. [30] A. Weinstein, Lagrangian mechanics and groupoids, in Mechanics Day, 207-231, Fields Inst. Commun., 7, Amer. Math. Soc., Providence, RI, 1996.
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