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Chaotic motion in classical fluids with scale relativistic methods

In the framework of the scale relativity theory, the chaotic behavior in time only of a number of macroscopic systems corresponds to motion in a space with geodesics of fractal dimension 2 and leads to its representation by a Schrödinger-like equation acting in the macroscopic domain. The fluid interpretation of such a Schrödinger equation yields Euler and Navier-Stokes equations. We therefore choose to extend this formalism to study the properties of a system exhibiting a chaotic behavior both in space and time which amounts to consider them as issued from the geodesic features of a mathematical object exhibiting all the properties of a fractal `space-time'. Starting with the simplest Klein-Gordon-like form that can be given to the geodesic equation in this case, we obtain a motion equation for a `three fluid' velocity field and three continuity equations, together with parametric expressions for the three velocity components which allow us to derive relations between their non-vanishing curls. At the non relativistic limit and owing to the physical properties exhibited by this solution, we suggest that it could represent some kind of three-dimensional chaotic behavior in a classical fluid, tentatively turbulent if particular conditions are fulfilled. The appearance of a transition parameter ${\cal D}$ in the equations allows us to consider different ways of testing experimentally our proposal.