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Mixing and segregation in two-phase flows

Mixing and segregation in two-phase flows. Gregory Falkovich Weizmann Institute of Science, Israel. Turbulent Mixing and Beyond, ICTP 2007. Mixing versus segregation in terms of an infinitesimal element. Lyapunov exponents. → singular (fractal) SRB Measure. entropy.

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Mixing and segregation in two-phase flows

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  1. Mixing and segregation in two-phase flows Gregory Falkovich Weizmann Institute of Science, Israel Turbulent Mixing and Beyond, ICTP 2007

  2. Mixing versus segregation in terms of an infinitesimal element.Lyapunov exponents.

  3. → singular (fractal) SRB Measure entropy

  4. Density in random compressible flows Analogy: statistical distribution in phase spaces (Sinai-Ruelle-Bowen measures) Balkovsky, Fouxon, GF, Gawedzki, Bec, Horvai

  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 mass decrease inside the volume.

  6. Inertial particles u v

  7. Spatially smooth flow Stokes number

  8. One-dimensional model Equivalent in 1d to Anderson localization: localization length = Lyapunov exponent

  9. Super-symmetry broken Lyapunov exponent

  10. DNS, Bec et al

  11. Fouxon, Stepanov, GF

  12. -2 n Falkovich, Lukaschuk, Denissenko, Nature 2005

  13. 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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