Why Navier-Stokes equations are wrong
Résumé
The completeness of the Navier-Stokes equations is addressed using two examples, a simple rotational
motion and the turbulent Taylor-Green vortex; in both cases the solutions are not physical. The main
artefacts and paradoxes of the Navier-Stokes equations are intrinsic to the notion of a continuous medium
and to the representation of the velocity field in three-dimensional space in terms of components. The
main error is inherent in the form of the inertia of these equations, more precisely in the Lamb vector,
which induces the existence of fictitious forces, the Coriolis and centrifugal forces. In the presence of
strong rotational effects, as in turbulence, the solutions of the Navier-Stokes equations are incorrect. On
the other hand, the discrete law of motion, considered as an alternative to the Navier-Stokes equations
[7], makes it possible to find the vast majority of solutions of the classical formalism, but excludes from
the outset non-physical solutions. The inertia is represented as the sum of two contributions, a first
curl-free component and a second component divergence-free, a Helmholtz-Hodge decomposition. The
main difference between the two approaches is the abandonment of the concepts of classical mechanics,
where the global frame of reference is replaced by Maxwell’s local frame of reference. Rigid rotation
thus becomes inertial motion in the same way as uniform translational motion in the Galilean frame
of reference. Taylor-Green turbulent flow, the only interesting case where the two solutions diverge, is
reinterpreted in the light of the conclusions drawn from the in-depth analysis of inertial effects.
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