Skip to main content
SpringerLink
Log in

Lagrangian potential functions of surface forces and their role in fluid mechanics

  • Technical Paper
  • Published:
Journal of the Brazilian Society of Mechanical Sciences and Engineering Aims and scope Submit manuscript

Abstract

In classical fluid mechanics, variational principles have been applied to derive the Navier–Stokes equations with relative success. The common procedure uses indirect and semi-direct methods with nonstandard Lagrangians. In this paper, the standard Lagrangian is used to derive the Navier–Stokes equations. In this regard, a Lagrangian potential function related to an isotropic tensor field is introduced. For compressible flow, another Lagrangian potential function related to the viscosity coefficients is defined. The Navier–Stokes equations are then derived from Lagrange’s equations. It is shown that in derivation of governing equations of viscous flow the standard Lagrangian is more efficient than nonstandard Lagrangians. Energy and Hamiltonian rate equations that may be used in fluid mechanics are also proposed.

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

Access this article

Log in via an institution

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1

Similar content being viewed by others

Data availability

Data sharing is not applicable to this article as no new data were created or analyzed in this study.

Abbreviations

\(\bar{Q}\) :

Monogenic force

\(\varvec{a}\) :

Position vector of a fluid particle in reference configuration

\(\dot{\hat{x}}_i\) :

Component of the velocity vector of fluid particle

\(\dot{q}\) :

Generalized velocity

\(\hat{{\textbf{r}}}\) :

Position vector of fluid particle

\(\hat{{\textbf{v}}}\) :

Velocity vector of fluid particle

\(\hat{{\mathcal{L}}}\) :

Lagrangian density function

\(\hat{{\mathcal{T}}}\) :

Kinetic energy density

\(\hat{{\mathcal{V}}}\) :

Density of Lagrangian potential function

\(\hat{\rho }\) :

Mass density of fluid particle

\(\hat{p}\) :

Pressure of fluid particle

\(\hat{v}_i\) :

Component of the velocity vector of fluid particle

\(\hat{x}_i\) :

Component of the position vector of fluid particle

L :

Lagrangian function

Q :

Generalized force

q :

Generalized coordinate

T :

Kinetic energy

V :

Lagrangian potential function

\(\bar{\tau }_{ij}\) :

Components of deviatoric part of viscous stress tensor

\(\varvec{\nabla }\) :

Gradient vector

\(\varvec{\sigma }\) :

Stress tensor

\(\varvec{\tau }\) :

Viscous stress tensor

\(\varvec{f}\) :

Force density

\(\varvec{f}_{\textrm{mg}}\) :

Monogenic force density

\(\varvec{f}_{\textrm{pg}}\) :

Polygenic force density

\(\varvec{g}\) :

Gravitational acceleration vector

\(\breve{\tau }_{ij}\) :

Components of spherical part of viscous stress tensor

\(\delta _{ij}\) :

Kronecker delta

\(\dot{\bar{{\mathcal{W}}}}_{\textrm{pg}}\) :

Power density of polygenic force

\(\kappa\) :

Coefficient of bulk viscosity

\(\lambda\) :

Second coefficient of viscosity

\({\textbf{p}}\) :

Momentum density

\({\textbf{r}}\) :

Position vector

\({\textbf{v}}\) :

Velocity field

\({\mathcal{E}}\) :

Energy density function

\({\mathcal{H}}\) :

Hamiltonian density function

\({\mathcal{L}}\) :

Lagrangian density function

\({\mathcal{L}}^s\) :

Lagrangian function per unit mass

\({\mathcal{T}}\) :

Kinetic energy density

\({\mathcal{T}}^s\) :

Kinetic energy per unit mass

\({\mathcal{V}}\) :

Lagrangian potential function per unit volume

\({\mathcal{V}}^s\) :

Lagrangian potential function per unit mass

\(\mu\) :

Shear viscosity

\(\nabla ^2\) :

Laplacian operator

\(\nu\) :

Kinematic viscosity

\(\phi\) :

Velocity potential

\(\rho\) :

Mass density

\(\sigma _{ij}\) :

Components of stress tensor

\(\tau _{ij}\) :

Components of viscous stress tensor

\(\textrm{D}/\textrm{D}t\) :

Material derivative

\(d_{ij}\) :

Components of rate of strain tensor

p :

Pressure field

u, v, w :

Cartesian components of velocity field

References

  1. Lanczos C (1970) The variational principles of mechanics. University of Toronto Press, Toronto

    MATH  Google Scholar 

  2. Goldstein H, Poole CP, Safko JL (2001) Classical mechanics. Addison Wesley, Boston

    MATH  Google Scholar 

  3. Lemos NA (2018) Analytical mechanics. Cambridge University Press, Cambridge

    Book  MATH  Google Scholar 

  4. Sadeghi K, Incecik A (2021) Damping potential, generalized potential, and D’Alembert’s principle. Iran J Sci Technol Trans Mech Eng 45:311–319. https://doi.org/10.1007/s40997-020-00350-z

    Article  Google Scholar 

  5. Serrin J, Flugge S (ed.) (1959) Mathematical principles of classical fluid mechanics. In: Flugge S (eds) Handbuch der Physik, vol VIII/1. Springer, Berlin, pp 125–263

  6. Finlayson BA (1972) The method of weighted residuals and variational principles: with application in fluid mechanics, heat and mass transfer. Academic Press, Philadelphia

    MATH  Google Scholar 

  7. Salmon R (1988) Hamiltonian fluid mechanics. Annu Rev Fluid Mech 20:225–256

    Article  Google Scholar 

  8. Salmon R (1998) Lecture notes on geophysical fluid dynamics. Oxford University Press, Oxford

    Google Scholar 

  9. Morrison P (1998) Hamiltonian description of the ideal fluid. Proc R Soc A Math Phys Eng Sci 70(2):467

    MathSciNet  MATH  Google Scholar 

  10. Berdichevsky VL (2009) Variational principles of continuum mechanics I. Fundamentals. Springer, Berlin

    Book  MATH  Google Scholar 

  11. Webb G (2018) Magnetohydrodynamics and fluid dynamics: action principles and conservation laws, vol 946. Springer, Heidelberg

    Book  MATH  Google Scholar 

  12. Arnaudon M, Cruzeiro AB (2015) Stochastic Lagrangian flows and the Navier–Stokes equations. In: Stochastic analysis: a series of lectures. Springer, Berlin, pp 55–75

  13. Chen X, Cruzeiro AB, Ratiu TS (2018) Stochastic variational principles for dissipative equations with advected quantities. Math Phys arXiv:1506.05024

  14. Seliger RL, Whitham GB (1968) Variational principles in continuum mechanics. Proc R Soc A Math Phys Eng Sci 305(1480):1–25

    MATH  Google Scholar 

  15. Kerswell RR (1999) Variational principle for the Navier–Stokes equations. Phys Rev E 59(5):5482

    Article  MathSciNet  Google Scholar 

  16. Fukagawa H, Fujitani Y (2012) A variational principle for dissipative fluid dynamics. Progress Theoret Phys 127(5):921–935

    Article  MATH  Google Scholar 

  17. Galley CR, Tsang D, Stein LC (2014) The principle of stationary nonconservative action for classical mechanics and field theories. 127 arXiv:1412.3082

  18. Gay-Balmaz F, Youshimura H (2017) A Lagrangian variational formulation for nonequilibrium thermodynamics. Part II: continuum systems. J Geom Phys 111:194–212

    Article  MathSciNet  MATH  Google Scholar 

  19. Zuckerwar AJ, Ash RL (2006) Variational approach to the volume viscosity of fluids. Phys. Fluids18: 047101. https://doi.org/10.1063/1.2180780

  20. Zuckerwar AJ, Ash RL (2009) Volume viscosity in fluids with multiple dissipative processes. Phys Fluids 21:033105. https://doi.org/10.1063/1.3084814

  21. Scholle M, Marner F (2017) A non-conventional Lagrangian for viscous flow. R Soc Open Sci (4) https://doi.org/10.1098/rsos.160447

  22. Anthony KH (2001) Hamilton’s action principle and thermodynamics of irreversible processes—a unifying procedure for reversible and irreversible processes. J Nonnewton Fluid Mech 96:291–339. https://doi.org/10.1016/S0377-0257(00)(00187-7)

    Article  MATH  Google Scholar 

  23. Marner F, Scholle M, Hermmann D, Gaskell PH (2018) Competing Lagrangians for incompressible and compressible viscous flow. J. Math. Phys. 59 https://doi.org/10.1098/rsos.160447

  24. Scholle M, Marner F, Gaskell PH (2020) Potential fields in fluid mechanics: a review of two classical approaches and related recent advances. Water 12:1241. https://doi.org/10.3390/w12051241

  25. Scholle M, Gaskell PH, Marner F (2018) Exact integration of the unsteady incompressible Navier–Stokes equations, gauge criteria, and applications. J Math Phys 59

  26. Sciubba E (2004) Exergy as a Lagrangian for the Navier–Stokes equations for incompressible flow. Int J Thermodyn 7(3):115–122

    Google Scholar 

  27. Taha HE, Gonzalez C (2022) What does nature minimize in every incompressible flow? arXiv:2112.12261v4 [physics.flu-dyn]

  28. Taha HE, Gonzalez C (2023) A variational principle for Navier–Stokes equations. AIAA SCiTECH Forum

  29. Bistafa SR (2023) Lagrangians for variational formulations of the Navier–Stokes equation. arXiv:2302.14716

  30. Lai WM, Rubin D, Krempl E (2009) Introduction to continuum mechanics. Elsevier

  31. Mase GT, Mase GE, Smelser RE (2010) Continuum mechanics for engineers. CRC Press, Boca Raton

    MATH  Google Scholar 

  32. Cassel KW (2013) Variational methods with applications in science and engineering. Cambridge University Press, New York

    Book  MATH  Google Scholar 

  33. Kundu PK, Cohen IM, Dowling DR (2015) Fluid mechanics. Academic Press, Cambridge

    Google Scholar 

  34. White F (2010) Fluid mechanics. McGraw-Hill, New York

    Google Scholar 

  35. Panton RL (2013) Incompressible flow. Wiley, New Jersey

    Book  MATH  Google Scholar 

  36. Ginsberg JH (1995) Advanced engineering dynamics. Cambridge University Press, Cambridge

    MATH  Google Scholar 

  37. Hu W, T L, & Z H (2022) Dynamical symmetry breaking of infinite-dimensional Stocastic system. Symmetry 14:1627. https://doi.org/10.3390/sym14081627

  38. Casetta L, Pesce CP (2011) On Seliger and Whitham’s variational principle for hydrodynamic systems from the point of view of fictitious particles. Acta Mech 219:181–184. https://doi.org/10.1007/s00707-010-0442-2

    Article  MATH  Google Scholar 

Download references

Funding

No funding was received to assist with the preparation of this manuscript.

Author information

Authors and Affiliations

  1. Buein Zahra Technical University, Buein Zahra, Qazvin, Iran

    Keyvan Sadeghi

Authors
  1. Keyvan Sadeghi
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Keyvan Sadeghi.

Ethics declarations

Conflict of interest

The author has no conflict to disclose.

Additional information

Technical Editor: Daniel Onofre de Almeida Cruz.

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Sadeghi, K. Lagrangian potential functions of surface forces and their role in fluid mechanics. J Braz. Soc. Mech. Sci. Eng. 45, 417 (2023). https://doi.org/10.1007/s40430-023-04332-5

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s40430-023-04332-5

Keywords

Search

Navigation