  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.
