An alternative to the Navier–Stokes equation based on the conservation of acceleration

Abstract

The derivation of the Navier–Stokes equation in continuum mechanics leads to a number of consequences which are discussed in depth. In spite of its very high representativity of real flows, this equation presents some artefacts due to the whole notion of the continuous medium. An alternative to the Navier–Stokes equation is proposed, based on the conservation of energy per unit mass instead of momentum. The classical inertial frame of reference is replaced by a set of local frames of reference where interactions are treated as cause and effect. Invoking the principle of equivalence between energy and mass, the latter is eliminated from the quantities used in this new formalism. All quantities, variables and physical properties are thus expressed in units of mass. The law of motion is established in the form of the conservation of acceleration, an energy per unit of mass and length. The acceleration is thus written in the form of a Helmholtz–Hodge decomposition, in two terms, the first curl-free and the second divergence-free as a function of two potentials, scalar and vector. Maxwell’s idea of federating the laws of electrodynamics and magnetism to establish electromagnetism is taken up here to establish the new law of motion as a nonlinear wave equation. This approach makes it possible to demonstrate that this law is relativistic from the start. The form of the equation of motion in two Lagrangians gives access to symmetries related to the conservation of certain quantities according to Noether’s theorem.