Study of chaotic motion in fluid mechanics with the scale relativity methods

Abstract

Chaotic motion in time of a number of macroscopic systems has been analyzed, in the framework of scale relativity, as motion in a fractal space with topological dimension 3 and geodesics with fractal dimension 2. The motion equation is then Schr\"odinger-like and its interpretation in fluid mechanics gives the well-known Euler and Navier-Stokes equations. We generalize here this formalism to the study of a system exhibiting a chaotic behavior both in space and time. We are thus lead to consider macroscopic fluid properties as issuing from the geodesic features of a fractal `space-time' with topological dimension 4 and geodesics with fractal dimension 2. This allows us to obtain both a motion equation for the fluid velocity field, which exhibits then three components while only one is necessary for the description of an ordinary fluid, and a relation between their three curls. The physical properties of this solution suggest it could represent some three-dimensional chaotic behavior for a classical fluid, tentatively turbulent if particular conditions are fulfilled. Different ways of testing experimentally these assumptions are proposed.