A fully observer-covariant formulation of the fluid dynamics of simple fluids: derivation, simple examples and a generalized Orr-Sommerfeld equation

TL;DR

This paper introduces a fully covariant formalism for fluid dynamics applicable to arbitrary observers, including derivations of stability equations and a generalized Orr-Sommerfeld equation, enhancing the analysis of fluid perturbations.

Contribution

It develops a novel covariant framework for fluid dynamics that is observer-independent, allowing analysis from any observer's perspective, including purely Lagrangian viewpoints.

Findings

Derived a generalized Orr-Sommerfeld equation for arbitrary observers.

Formulated covariant prognostic equations using exterior algebra.

Applied the formalism to stability analysis in fluid flows.

Abstract

We present a formalism to describe the motion of a fluid that is fully covariant with respect to arbitrary observers. To achieve full covariance, we write prognostic equations for quantities that belong to the graded exterior algebra of the cotangent bundle of the manifold occupied by the fluid. In particular, equations are that are fully covariant can be written for a purely Lagrangian observer, for which the fluid velocity (qua section of the tangent bundle) is not a meaningful concept. With the new formalism, we consider problems of stability, and we derive a generalization of the Orr-Sommerfeld equation that describes the evolution of perturbations relative to an arbitrary observer. The latter is applied to cases where the observer is the Lagrangian observer comoving with the background flow.