Action Principle for Hydrodynamics and Thermodynamics including general, rotational flows

TL;DR

This paper develops a comprehensive action principle for hydrodynamics that encompasses general rotational flows and thermodynamics, extending classical theories with a Hamiltonian structure and conservation laws.

Contribution

It introduces a unified action principle combining Eulerian and Lagrangian hydrodynamics with thermodynamics, including viscous dissipation and rotational flows.

Findings

Provides a Hamiltonian framework for hydrodynamics with rotational flows.

Extends thermodynamic principles to include dissipative processes.

Offers a natural energy concept as a conserved first integral.

Abstract

The restriction of hydrodynamics to non-viscous, potential (gradient, irrotational) flows is a theory both simple and elegant; a favorite topic of introductory textbooks. It is known that this theory can be formulated as an action principle and expanded to include thermodynamicics. This paper presents an action principle for hydrodynamics that includes general, rotational flows. The new theory is a combination of Eulerian and Lagrangian hydrodynamics, with an extension to thermodynamics that includes all the elements of the Gibbsean variational principle. In the first place it is an action principle for adiabatic systems, including the usual conservation laws. Viscosity can be introduced in the usual way, by adding a dissipative term to the momentum equation. The equation for energy dissipation then follows. It is an ideal framework for the description of quasi-static processes, including dissipation. It is a major development of the Navier-Stokes-Fourier approach, the principal advantage being a hamiltonian structure with a natural concept of energy in the form of a first integral of the motion, conserved by virtue of the Euler-Lagrange equations.