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 semidirect 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.
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
Lanczos C (1970) The variational principles of mechanics. University of Toronto Press, Toronto
Goldstein H, Poole CP, Safko JL (2001) Classical mechanics. Addison Wesley, Boston
Lemos NA (2018) Analytical mechanics. Cambridge University Press, Cambridge
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/s4099702000350z
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
Finlayson BA (1972) The method of weighted residuals and variational principles: with application in fluid mechanics, heat and mass transfer. Academic Press, Philadelphia
Salmon R (1988) Hamiltonian fluid mechanics. Annu Rev Fluid Mech 20:225–256
Salmon R (1998) Lecture notes on geophysical fluid dynamics. Oxford University Press, Oxford
Morrison P (1998) Hamiltonian description of the ideal fluid. Proc R Soc A Math Phys Eng Sci 70(2):467
Berdichevsky VL (2009) Variational principles of continuum mechanics I. Fundamentals. Springer, Berlin
Webb G (2018) Magnetohydrodynamics and fluid dynamics: action principles and conservation laws, vol 946. Springer, Heidelberg
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
Chen X, Cruzeiro AB, Ratiu TS (2018) Stochastic variational principles for dissipative equations with advected quantities. Math Phys arXiv:1506.05024
Seliger RL, Whitham GB (1968) Variational principles in continuum mechanics. Proc R Soc A Math Phys Eng Sci 305(1480):1–25
Kerswell RR (1999) Variational principle for the Navier–Stokes equations. Phys Rev E 59(5):5482
Fukagawa H, Fujitani Y (2012) A variational principle for dissipative fluid dynamics. Progress Theoret Phys 127(5):921–935
Galley CR, Tsang D, Stein LC (2014) The principle of stationary nonconservative action for classical mechanics and field theories. 127 arXiv:1412.3082
GayBalmaz F, Youshimura H (2017) A Lagrangian variational formulation for nonequilibrium thermodynamics. Part II: continuum systems. J Geom Phys 111:194–212
Zuckerwar AJ, Ash RL (2006) Variational approach to the volume viscosity of fluids. Phys. Fluids18: 047101. https://doi.org/10.1063/1.2180780
Zuckerwar AJ, Ash RL (2009) Volume viscosity in fluids with multiple dissipative processes. Phys Fluids 21:033105. https://doi.org/10.1063/1.3084814
Scholle M, Marner F (2017) A nonconventional Lagrangian for viscous flow. R Soc Open Sci (4) https://doi.org/10.1098/rsos.160447
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/S03770257(00)(001877)
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
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
Scholle M, Gaskell PH, Marner F (2018) Exact integration of the unsteady incompressible Navier–Stokes equations, gauge criteria, and applications. J Math Phys 59
Sciubba E (2004) Exergy as a Lagrangian for the Navier–Stokes equations for incompressible flow. Int J Thermodyn 7(3):115–122
Taha HE, Gonzalez C (2022) What does nature minimize in every incompressible flow? arXiv:2112.12261v4 [physics.fludyn]
Taha HE, Gonzalez C (2023) A variational principle for Navier–Stokes equations. AIAA SCiTECH Forum
Bistafa SR (2023) Lagrangians for variational formulations of the Navier–Stokes equation. arXiv:2302.14716
Lai WM, Rubin D, Krempl E (2009) Introduction to continuum mechanics. Elsevier
Mase GT, Mase GE, Smelser RE (2010) Continuum mechanics for engineers. CRC Press, Boca Raton
Cassel KW (2013) Variational methods with applications in science and engineering. Cambridge University Press, New York
Kundu PK, Cohen IM, Dowling DR (2015) Fluid mechanics. Academic Press, Cambridge
White F (2010) Fluid mechanics. McGrawHill, New York
Panton RL (2013) Incompressible flow. Wiley, New Jersey
Ginsberg JH (1995) Advanced engineering dynamics. Cambridge University Press, Cambridge
Hu W, T L, & Z H (2022) Dynamical symmetry breaking of infinitedimensional Stocastic system. Symmetry 14:1627. https://doi.org/10.3390/sym14081627
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/s0070701004422
Funding
No funding was received to assist with the preparation of this manuscript.
Author information
Authors and Affiliations
Corresponding author
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 selfarchiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.
About this article
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/s40430023043325
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/s40430023043325