Eulerian and Lagrangian Descriptions

This chapter derives the relations between Eulerian and Lagrangian descriptions of displacement and velocity fields, relations between the time derivatives of system properties, variations, and introduces Jourdain’s variational principle. Jourdain’s principle is then applied to viscous incompressible fluids, and the derivation of the energy rate equation. These equations will be utilized in the subsequent chapter for the derivation of the flow-oscillator model for vortex-induced vibration.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Subscribe and save

Springer+ Basic €32.70 /Month

Buy Now

Price includes VAT (France)

eBook EUR 85.59 Price includes VAT (France)

Hardcover Book EUR 105.49 Price includes VAT (France)

Tax calculation will be finalised at checkout

Purchases are for personal use only

References

  1. Baruh H (1999) Analytical dynamics. WCB/McGraw-Hill, Boston Google Scholar
  2. Bateman H (1931) On dissipative systems and related variational principles. Phys Rev 38:815–819 ArticleGoogle Scholar
  3. Cengel YA, Cimbala JM (2006) Fluid mechanics: fundamentals and applications. McGraw-Hill series in mechanical engineering. McGraw-Hill, New York City Google Scholar
  4. Dill EH (2006) Continuum mechanics: elasticity, plasticity, viscoelasticity. CRC Press, Boca Raton Google Scholar
  5. Gelfand IM, Fomin SV (1963) Calculus of variations. Prentice-Hall Inc, Englewood Cliffs MATHGoogle Scholar
  6. Goldstein H, Poole C, Safko J (2002) Classical mechanics, 3rd edn. Addison-Wesley, Boston MATHGoogle Scholar
  7. Jourdain PEB (1909) Note on an analogue of gauss principle of least constraint. Q J Pure Appl Math 40:153–157 MATHGoogle Scholar
  8. Kövecses J, Cleghorn WL (2003) Finite and impulsive motion of constrained mechanical systems via Jourdain’s principle: discrete and hybrid parameter models. Int J Non-Linear Mech 38(6):935–956 ArticleGoogle Scholar
  9. Kundu PK, Cohen IM (2008) Fluid mechanics, 4th edn. Elsevier Inc., Oxford Google Scholar
  10. Lamb H (1963) Hydrodynamics. Cambridge University Press, Cambridge Google Scholar
  11. Lanczos C (1970) The variational principles of mechanics, 4th edn. Courier Dover Publications, Mineola MATHGoogle Scholar
  12. Leech CM (1977) Hamilton’s principle applied to fluid mechanics. Q J Mech Appl Math 30:107–130 ArticleMathSciNetGoogle Scholar
  13. McIver DB (1973) Hamilton’s principle for systems of changing mass. J Eng Mech 7(3):249–261 MATHGoogle Scholar
  14. Millikan CB (1929) On the steady motion of viscous, incompressible fluids; with particular reference to a variation principle. Philos Mag Ser 7, 7(44):641–662 Google Scholar
  15. Moon FC (1998) Applied dynamics: with applications to multibody and mechatronic systems. Wiley, New York Google Scholar
  16. Mottaghi S (2018) Modeling vortex-induced fluid-structure interaction using an extension of Jourdain’s principle. PhD dissertation, Rutgers, the State University of New Jersey, New Brunswick Google Scholar
  17. Mottaghi S, Benaroya H (2016) Reduced-order modeling of fluid-structure interaction and vortex-induced vibration systems using an extension of Jourdain’s principle. J Sound Vib 382(1):193–212 ArticleGoogle Scholar
  18. Papastavridis JG (1992) On Jourdain’s principle. Int J Eng Sci 30(2):135–140 ArticleMathSciNetGoogle Scholar
  19. Price JF (2006) Lagrangian and Eulerian representations of fluid flow: kinematics and the equations of motion. Woods Hole Technical Report Google Scholar
  20. Riewe F (1997) Mechanics with fractional derivatives. Phys Rev E 55:3581–3592 ArticleMathSciNetGoogle Scholar
  21. Stokes GG (1847) On the theory of oscillatory waves. Trans Camb Philos Soc 8:441–455 Google Scholar
  22. Vujanovic B, Atanackovic T (1978) On the use of Jourdain’s variational principle in nonlinear mechanics and transport phenomena. Acta Mech 29:229–238 ArticleGoogle Scholar
  23. Wang L-S, Pao Y-H (2003) Jourdain’s variational equation and Appell’s equation of motion for nonholonomic dynamical systems. Am J Phys 71(1):72–82 ArticleGoogle Scholar
  24. Xing JT, Price WG (2000) The theory of non-linear elastic ship-water interaction dynamics. J Sound Vib 230(4):877–914 ArticleGoogle Scholar

Author information

Authors and Affiliations

  1. FAA William J. Hughes Technical Center, Atlantic City International Airport, Atlantic City, NJ, USA Sohrob Mottaghi
  2. Pratt & Whitney, East Hartford, CT, USA Rene Gabbai
  3. Mechanical and Aerospace Engineering, Rutgers University, Piscataway, NJ, USA Haym Benaroya
  1. Sohrob Mottaghi