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

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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

Henry Jacobs ^1, and Joris Vankerschaver ^1,

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

Received December 2012 Revised February 2014 Published April 2014

Abstract
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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.

Keywords: Fluid mechanics, fluid-structure interaction, solid mechanics, Lie groups..

Mathematics Subject Classification: 76T99, 53Z05, 22A22, 22E7.

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.

show all references

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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