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G. Falkovich Leiden, August 2006

Smooth, rough, broken: From Lyapunov exponents and zero modes to caustics in the description of inertial particles. G. Falkovich Leiden, August 2006. Smooth flow. 1d. H is convex. Multi-dimensional. → singular (fractal) SRB Measure. entropy. Coarse-grained density.

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G. Falkovich Leiden, August 2006

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  1. Smooth, rough, broken:From Lyapunov exponents and zero modes to caustics in the description of inertial particles. G. Falkovich Leiden, August 2006

  2. Smooth flow 1d H is convex

  3. Multi-dimensional

  4. → singular (fractal) SRB Measure entropy

  5. Coarse-grained density An anomalous scaling corresponds to slower divergence of particles to get more weight. Statistical integrals of motion (zero modes) of the backward-in-time evolution compensate the increase in the distances by the concentration decrease inside the volume. Bec, Gawedzki, Horvai, Fouxon

  6. Inertial particles u v Maxey

  7. Spatially smooth flow One-dimensional model Equivalent in 1d to Anderson localization: localization length=Lyapunov exponent

  8. Velocity gradient

  9. Fouxon, Stepanov, GF

  10. Lyapunov exponent

  11. Gawedzki, Turitsyn and GF.

  12. Statistics of inter-particle distance in 1d high-order moments correspond effectively to large Stokes

  13. Continuous flow Piterbarg, Turitsyn, Derevyanko, Pumir, GF

  14. Derevyanko

  15. 2d short-correlated Baxendale and Harris, Chertkov, Kolokolov, Vergassola, Piterbarg, Mehlig and Wilkinson

  16. Coarse-grained density:

  17. -2 n Falkovich, Lukaschuk, Denissenko

  18. 3d Short-correlated flow Duncan, Mehlig, Ostlund, Wilkinson Finite-correlated flow Bec, Biferale, Boffetta, Cencini, Musacchio, Toschi

  19. Clustering versus mixing in the inertial interval: Balkovsky, Fouxon, Stepanov, GF, Horvai, Bec Cencini, Hillerbrand

  20. Fouxon, Horvai

  21. Fluid velocity roughness decreases clustering of particles Pdf of velocity difference has a power tail Bec, Cencini, Hillerbrand

  22. Collision rate Sundaram, Collins; Balkovsky, Fouxon, GF Fouxon, Stepanov, GF Bezugly, Mehlig and Wilkinson Pumir, GF

  23. Main open problems 1. To understand relations between the Lagrangian and Eulerian descriptions. 2. To sort out two contributions into different quantities: i) from a smooth dynamics and multi-fractal spatial distribution, and ii) from explosive dynamics and caustics. 3. Find how collision rate and density statistics depend on the dimensionless parameters (Reynolds, Stokes and Froude numbers).

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