On a reformulation of Navier-Stokes equations based on Helmholtz-Hodge decomposition

TL;DR

This paper introduces a novel formulation of the Navier-Stokes equations using Helmholtz-Hodge decomposition, emphasizing energy conservation and discrete mechanics principles, which preserves classical solutions while offering new analytical properties.

Contribution

It presents a reformulation of Navier-Stokes equations based on Helmholtz-Hodge decomposition aligned with discrete mechanics, maintaining classical solutions and introducing new properties.

Findings

Classical solutions are preserved in the new formulation.

The reformulation emphasizes energy conservation without additional mass conservation laws.

New properties emerge when differential operators are applied to the decomposed inertial terms.

Abstract

The proposal for a new formulation of the Navier-Stokes equations is based on a Helmholtz-Hodge decomposition where all the terms corresponding to the physical phenomena are written as the sum of a divergence-free term and another curl-free term. These transformations are founded on the bases of discrete mechanics, an alternative approach to the mechanics of continuous media, where conservation of the acceleration on a segment replaces that of the momentum on a volume. The equation of motion thus becomes a law of conservation of total mechanical energy per volume unit where the conservation of mass is no longer necessarily an additional law. The new formulation of the Navier-Stokes equations recovers the properties of the discrete approach without altering those of its initial form; the solutions of the classical form are also those of the proposed formulation. Writing inertial terms in two components resulting from the Helmholtz-Hodge decomposition gives the equation of motion new properties when differential operators are applied to it directly.