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

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.