Locomotion of articulated bodies in an ideal fluid: 2d model with buoyancy, circulation and collisions

Articulated solid bodies (ASB) is a basic model for the study of shape-changing underwater vehicles made of rigid parts linked together by pivoting joints. In this paper we study the locomotion of such swimming mechanisms in an ideal fluid. Our study ranges over a wide class of problems: several ASBs can be involved (without being hydrodynamically decoupled), the fluid-bodies system can be partially or totally confined and fluid circulation, buoyancy force and possible collisions between bodies are taken into account. We derive the Euler–Lagrange equation governing the dynamics of the system, study its well-posedness and describe a numerical scheme implemented in a Matlab toolbox (Biohydrodynamics Toolbox).

[1]  J. A. Sparenberg.  Survey of the mathematical theory of fish locomotion , 2002 .

[2]  Vivette Girault,et al.  Weighted Sobolev spaces for Laplace's equation in Rn , 1994 .

[3]  S. Mikhlin,et al.  Mathematical Physics, An Advanced Course , 1973 .

[4]  Keiji Kawachi,et al.  Regular Article: A Numerical Study of Undulatory Swimming , 1999 .

[5]  G. Taylor.  Analysis of the swimming of long and narrow animals , 1952, Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences.

[6]  S. K. Gupta,et al.  Fluid Mechanics and its Applications , 2011 .

[7]  Antoine Henrot,et al.  Variation et optimisation de formes , 2005 .

[8]  T. Y. Wu,et al.  Mathematical biofluiddynamics and mechanophysiology of fish locomotion , 2001 .

[9]  Joel W. Burdick,et al.  Propulsion and control of deformable bodies in an ideal fluid , 1999, Proceedings 1999 IEEE International Conference on Robotics and Automation (Cat. No.99CH36288C).

[10]  Giovanni P. Galdi,et al.  On the Steady Self‐Propelled Motion of a Body in a Viscous Incompressible Fluid , 1999 .

[11]  JUAN B. MELLI,et al.  MOTION PLANNING FOR AN ARTICULATED BODY IN A PERFECT FLUID , 2006 .

[12]  J. L. Nayler.  Mathematical Biofluiddynamics. Sir James Lighthill. Regional Conference Series in Applied Mathematics. Society for Industrial and Applied Mathematics, Philadelphia. 281 pp. Illustrated. , 1976, The Aeronautical Journal (1968).

[13]  Clarence W. Rowley,et al.  Motion Planning for an Articulated Body in a Perfect Planar Fluid , 2006, SIAM J. Appl. Dyn. Syst..

[14]  Marsden,et al.  ON THE SELF-DISPLACEMENT OF DEFORMABLE BODIES IN A POTENTIAL FLUID FLOW , 2008 .

[15]  Marius Tucsnak,et al.  An Initial and Boundary Value Problem Modeling of Fish-like Swimming , 2008 .

[16]  Jerrold E. Marsden,et al.  Locomotion of Articulated Bodies in a Perfect Fluid , 2005, J. Nonlinear Sci..

[17]  J.E. Marsden,et al.  Optimal Motion of an Articulated Body in a Perfect Fluid , 2005, Proceedings of the 44th IEEE Conference on Decision and Control.

[18]  K. Atkinson.  The Numerical Solution of Integral Equations of the Second Kind , 1997 .

[19]  M. Triantafyllou,et al.  Hydrodynamics of Fishlike Swimming , 2000 .

[20]  G. Taylor.  Analysis of the swimming of microscopic organisms , 1951, Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences.

[21]  T. J. Rivlin.  The Chebyshev polynomials , 1974 .

[22]  J. Lions,et al.  Non-homogeneous boundary value problems and applications , 1972 .

[23]  Alexandre Munnier,et al.  Locomotion of Deformable Bodies in an Ideal Fluid: Newtonian versus Lagrangian Formalisms , 2009, J. Nonlinear Sci..

[24]  Dmitry A. Onishchenko,et al.  Motion of a Body with Undeformable Shell and Variable Mass Geometry in an Unbounded Perfect Fluid , 2003 .

[25]  Michael S. Triantafyllou,et al.  Three-dimensional flow structures and vorticity control in fish-like swimming , 2002, Journal of Fluid Mechanics.

[26]  S. Childress.  Mechanics of swimming and flying: Frontmatter , 1977 .

[27]  P. Heywood.  Trigonometric Series , 1968, Nature.

[28]  R. Kress,et al.  Inverse Acoustic and Electromagnetic Scattering Theory , 1992 .

[29]  K. Atkinson,et al.  Theoretical Numerical Analysis: A Functional Analysis Framework , 2001 .

[30]  Antoine Henrot,et al.  Variation et optimisation de formes : une analyse géométrique , 2005 .

[31]  J. A. Sparenberg.  Hydrodynamic Propulsion and Its Optimization , 1995 .

[32]  E. Kanso.  Swimming due to transverse shape deformations , 2009, Journal of Fluid Mechanics.

[33]  R. Kress.  Numerical Analysis , 1998 .

[34]  Alexandre Munnier,et al.  On the motion and collisions of rigid bodies in an ideal fluid , 2008, Asymptot. Anal..

[35]  Vivette Girault,et al.  DIRICHLET AND NEUMANN EXTERIOR PROBLEMS FOR THE n-DIMENSIONAL LAPLACE OPERATOR AN APPROACH IN WEIGHTED SOBOLEV SPACES , 1997 .

[36]  Williams,et al.  Self-propelled anguilliform swimming: simultaneous solution of the two-dimensional navier-stokes equations and Newton's laws of motion , 1998, The Journal of experimental biology.

[37]  Richard M. Murray,et al.  Modelling efficient pisciform swimming for control , 2000 .