An alternative model of Euler equations based on conservation of acceleration
Résumé
The conservation of acceleration, one energy per unit of mass and length, leads to a drastic reduction
in the number of unknowns in the Euler model or the Navier-Stokes equations for compressible flows.
The equations for the conservation of momentum, mass and energy, associated with constitutive laws,
are replaced by a single law for the conservation of total energy, whose variable is acceleration, the only
absolute quantity considered. Abandoning the notion of mass or density and the concepts of discrete
mechanics leads to the derivation of a purely kinematic law. Intrinsic acceleration is expressed by two
orthogonal terms, two accelerations, one divergence-free and the other curl-free, according to a Helmholtz-
Hodge decomposition. The scalar and vector potentials of the acceleration embody the compression and
rotation energies respectively, the two components of total energy per unit mass. The law of motion
thus obtained possesses the remarkable properties of conservation and symmetries enshrined in Noether’s
theorem; it is essentially a naturally relativistic non-linear wave equation.
This physical model, whose formalism has already been validated for incompressible Navier-Stokes
equations, is extended to compressible flows with or without shock. Some analytical solutions of compressible
flows are selected to demonstrate the formal reduction in the number of quantities needed to
describe a problem, whereas they are redundant in a classical description. This parsimonious view of the
laws of fluid mechanics is presented as an alternative for simulating compressible flows.
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