It is recognized that the addition of polymers is very efficient in reducing the friction drag in turbulent regimes. My talk is about the effects of small-scale turbulence on polymers distribution by using a stochastic scaling and singular limits. Many works have been done in recent years using the scaling limit in both scalar and vector cases. The second one is characterized by the presence of stretching, which adds complications over the scalar case.

In [1], we investigate the stretching mechanism of stochastic models of turbulence acting on a simple model of polymer. Namely, we investigate a scaling limit problem, under suitable intensity assumption. The polymer density equation, initially an SPDE converges (in the first step) weakly to a limit deterministic equation with a new degenerate term with some singular parameter. Recently, in [2] we investigate the singular limit in the spirit of the hydrodynamic limit techniques. One consequence is that the limiting density shows a power-law decay in the polymer length, which is consistent with physical predictions.

The activities mentioned herein were performed in the framework of the project: EU-HORIZON EUROPE ERC-2021-ADG “Noise in Fluids” (NoisyFluid), no. 101053472.

[1] Flandoli, F. and Tahraoui Y.: Stretching of polymers and turbulence: Fokker Planck equation, special stochastic scaling limit and stationary law. Journal of Differential Equations 452 : 113789 (2026)

[2] Tahraoui, Y.: Small-scale turbulence limit of Fokker-Planck equation for polymers in turbulent flow. Mathematical Models and Methods in Applied Sciences (2026)