Chaotic motion in classical fluids with scale relativistic methods

TL;DR

This paper applies scale relativity theory to model chaotic and potentially turbulent behavior in classical fluids, deriving a Schrödinger-like equation and exploring implications for fluid dynamics.

Contribution

It extends scale relativity formalism to describe space and time chaos in fluids using a fractal space-time framework, leading to new equations for fluid motion.

Findings

Derivation of a three-fluid velocity field model

Identification of relations between curl components

Proposal of experimental tests for the model

Abstract

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 Schr\"odinger-like equation acting in the macroscopic domain. The fluid interpretation of such a Schr\"odinger 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.