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In the framework of the scale relativity theory, the chaotic behavior in timeonly of a number of macroscopic systems corresponds to motion in a space withgeodesics of fractal dimension 2 and leads to its representation by aSchr\"odinger-like equation acting in the macroscopic domain. The fluidinterpretation of such a Schr\"odinger equation yields Euler and Navier-Stokesequations. We therefore choose to extend this formalism to study the propertiesof a system exhibiting a chaotic behavior both in space and time which amountsto consider them as issued from the geodesic features of a mathematical objectexhibiting all the properties of a fractal `space-time'. Starting with thesimplest Klein-Gordon-like form that can be given to the geodesic equation inthis case, we obtain a motion equation for a `three fluid' velocity field andthree continuity equations, together with parametric expressions for the threevelocity components which allow us to derive relations between theirnon-vanishing curls. At the non relativistic limit and owing to the physicalproperties exhibited by this solution, we suggest that it could represent somekind of three-dimensional chaotic behavior in a classical fluid, tentativelyturbulent if particular conditions are fulfilled. The appearance of atransition parameter ${\cal D}$ in the equations allows us to considerdifferent ways of testing experimentally our proposal.

Chaotic motion in classical fluids with scale relativistic methods