**March 15, 2021 at 10:00 - March 26, 2021**
- BCAM

**Itzhak Fouxon**

**DATES:** March 15-26, 2021, from Monday to Friday (10 sessions)

March 19 there will be no course because is Holiday in Basque Country

**TIME:** 10:00 - 13:00, from 15th to 23rd, and 10:00 - 14:00, from 24th to 26th (a total of 30 hours)

**LOCATION:** Online

**ABSTRACT:**
In this course we will start with the theoretical description of Lagrangian chaos. This is the chaos of fluid particle trajectories that holds below the Kolmogorov scale in a developed turbulent flow. We will consider the general characterization of chaos and its peculiarities in turbulence such as dependence on the Reynolds number. We will describe the applications of the theory to particle collisions in turbulence, underlying the formation of rain, and the growth of the seed magnetic field in turbulence. Then we will concentrate on the problem of turbulent behavior of dilute polymer solutions, known to exhibit remarkable properties. Turbulence in these solutions was first discovered at large Reynolds numbers via the famous, still not completely explained, observation of drag reduction with the help of minute amounts of dissolved polymers. We will also consider the regime of elastic turbulence that holds in polymer solutions at zero Reynolds number. The solutions provide us with prototype example of turbulence in non-Newtonian fluids. The lectures aim at coherent introduction that starts with fundamentals and brings the participants to state-of-the-art. Open research problems that need theoretical, numerical and experimental inputs will be presented.

**PROGRAMME:**
1. Lagrangian chaos

2. Large deviations theory

3. Collisions

4. Magnetic field

5. Single polymer molecule dynamics in the random flow. Universal probability distribution of size and its derivation from statistics of finite-time Lyapunov exponents and large deviations. Experimental confirmation of the theory. Degradation and non- linear elasticity. Implications for dilute solutions.

6. Rheology of polymer solutions. Magnetohydrodynamic type formulation and its implications.

7. Changes in the nature of turbulence in dilute polymer solutions as the Reynolds number increases from zero to infinity. Maximum drag reduction asymptote and its current understanding.

8. Theory of elastic turbulence and its experimental confirmations.

9. Recent discoveries: Alfven waves and drag reduction in elastic turbulence. Open problems.

** *Registration is free, but mandatory before March 10th.** To sign-up go to

https://forms.gle/Fs2RacH9KtidQDMx7 and fill the registration form.