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

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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.

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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

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

    Keyvan Sadeghi

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  1. Keyvan Sadeghi
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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

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