The role of homology in fluid vortices I: non-relativistic flow

TL;DR

This paper explores how homology and cohomology theories, especially singular and de Rham homology, underpin the fundamental laws of vortex dynamics in non-relativistic fluid flows, emphasizing their role in integral invariants.

Contribution

It applies homology and cohomology methods to fluid vortex theory, highlighting their foundational role in formulating vortex laws through integral invariants.

Findings

Homology methods clarify vortex structure and motion.

Integral invariants are rooted in homology theory.

Homology provides a unifying framework for vortex laws.

Abstract

The methods of singular and de Rham homology and cohomology are reviewed to the extent that they are applicable to the structure and motion of vortices. In particular, they are first applied to the concept of integral invariants. After a brief review of the elements of fluid mechanics, when expressed in the language of exterior differential forms and homology theory, the basic laws of vortex theory are shown to be statements that are rooted in the homology theory of integral invariants.