Non-linear behaviours in complex fluid dynamics via non-differentiability. Separation control of the solid components from heterogeneous mixtures

• The geodesic equations are identified with streamlines of the complex fluid.
• The complex fluid acquires energy to jump from its own streamline to another.
• Thermal hysteresis type effects can be obtained by numerical simulations.
• The dissipation anisotropy can induce rotational effects on the solid components.

In the framework of the Scale Relativity Theory, non-linear effects in complex fluids and, implicitly, the separation control of the solid components from heterogeneous mixtures are analysed. Assuming that the movements of the complex fluid entities occur on continuous but non-differentiable curves, the specific momentum and the local energy density conservation law equations are obtained. For potential movements at a fractal scale, as well as for non-potential ones at a differentiable scale, the non-differentiable hydrodynamic model is established. In such context, for different curve motions at various resolution scales, the non-differentiable hydrodynamic model is reduced to either a quantum hydrodynamic model, a standard hydrodynamic model, a random walk model, or a Stokes model. By numerical simulations using the non-differentiable hydrodynamic equations and the internal energy density conservation law with adequate initial and boundary conditions, some non-linear effects are obtained. Moreover, eliminating the time between the viscosity stress tension type and temperature field, for various given positions, thermal hysteresis type effects can be obtained. For each of the models mentioned above, the separation control processes are discussed.

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