A new form of governing equations of fluids arising from Hamilton's principle

TL;DR

This paper derives a new form of fluid governing equations from Hamilton's principle, incorporating conserved quantities and their derivatives, which unify non-dispersive and dispersive cases and include stability analysis.

Contribution

It introduces a novel Hamiltonian-based formulation of fluid equations that encompasses both dispersive and non-dispersive regimes, extending to two-fluid systems.

Findings

Conservation laws are derived for both dispersive and non-dispersive fluids.

The equations can be expressed in symmetric Godunov-Friedrichs-Lax form.

Linear stability and dispersion relations are obtained using Hermitian matrices.

Abstract

A new form of governing equations is derived from Hamilton's principle of least action for a constrained Lagrangian, depending on conserved quantities and their derivatives with respect to the time-space. This form yields conservation laws both for non-dispersive case (Lagrangian depends only on conserved quantities) and dispersive case (Lagrangian depends also on their derivatives). For non-dispersive case the set of conservation laws allows to rewrite the governing equations in the symmetric form of Godunov-Friedrichs-Lax. The linear stability of equilibrium states for potential motions is also studied. In particular, the dispersion relation is obtained in terms of Hermitian matrices both for non-dispersive and dispersive case. Some new results are extended to the two-fluid non-dispersive case.