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Aggregation of inertial particles in random flows

B. Mehlig1, M. Wilkinson2, K. Duncan2, T. Weber1, and M. Ljunggren1

  • 1Department of Physics, Göteborg University, 41296 Göteborg, Sweden
  • 2Department of Applied Mathematics, The Open University, Walton Hall, Milton Keynes, MK7 6AA, England

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Phys. Rev. E 72, 051104 – Published 8 November, 2005

DOI: https://doi.org/10.1103/PhysRevE.72.051104

Abstract

We consider the trajectories of particles suspended in a randomly moving fluid. If the Lyapunov exponent of these trajectories is negative, the paths of these particles coalesce, so that particles aggregate. Here we give a detailed account of a method [B. Mehlig and M. Wilkinson, Phys. Rev. Lett. 92, 250602 (2004)] for calculating this exponent: it is expressed as the expectation value of a random variable evolving under a stochastic differential equation. We analyze this equation in detail in the limit where the correlation time of the velocity field of the fluid is very short, such that the stochastic differential equation is a Langevin equation. We derive an asymptotic perturbation expansion of the Lyapunov exponent for particles suspended in three-dimensional flows in terms of a dimensionless measure of the inertia of the particles, ϵ, and a measure of the relative intensities of potential and solenoidal components of the velocity field, Γ. We determine the phase diagram in the ϵ−Γ plane.

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References (22)

  1. J. Sommerer and E. Ott, Science 359, 334 (1993).
  2. J. R. Cressman, W. I. Goldburg, and J. Schuhmacher, Europhys. Lett. 66, 219 (2004).
  3. J. R. Fessler, J. D. Kulick, and J. K. Eaton, Phys. Fluids 6, 3742 (1994).
  4. B. van Haarlem, B. J. Boersma, and F. T. M Nieuwstadt, Phys. Fluids 10, 2608 (1998).
  5. V. I. Klyatskin and D. Gurarie, Phys. Usp. 42, 165 (1999).
  6. M. R. Maxey, J. Fluid Mech. 174, 441 (1987).
  7. T. Elperin, N. Kleeorin, and I. Rogachevskii, Phys. Rev. Lett. 77, 5373 (1996).
  8. E. Balkovsky, G. Falkovich, and A. Fouxon, Phys. Rev. Lett. 86, 2790 (2001).
  9. J. Bec, Phys. Fluids 15, L81 (2003).
  10. H. Sigurgeirsson and A. M. Stuart, Phys. Fluids 14, 4352 (2002).
  11. J. M. Deutsch, J. Phys. A 18, 1457 (1985).
  12. B. Mehlig and M. Wilkinson, Phys. Rev. Lett. 92, 250602 (2004).
  13. M. Wilkinson and B. Mehlig, Phys. Rev. E 68, 040101(R) (2003).
  14. L. Piterbarg, SIAM J. Appl. Math. 62, 777 (2001).
  15. J.-P. Eckmann and D. Ruelle, Rev. Mod. Phys. 57, 617 (1985).
  16. Y. Le Jan, Z. Wahrscheinlichkeitstheor. Verwandte Geb. 70, 609 (1985).
  17. P. Baxendale and T. Harris, Ann. Prob. 14, 1155 (1986).
  18. L. D. Landau and E. M. Lifshitz, Fluid Mechanics (Pergamon, Oxford, 1958).
  19. U. Frisch, Turbulence, (Cambridge University Press, Cambridge, 1997).
  20. N. G. van Kampen, Stochastic Processes in Physics and Chemistry (North-Holland, Amsterdam, 1992).
  21. P. A. M. Dirac, The Principles of Quantum Mechanics (Oxford University Press, Oxford, 1930).
  22. R. B. Dingle, Asymptotic Expansions: Their Derivation and Interpretation (Academic, New York, 1973).

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Phys. Rev. E 72, 051104– Published 8 November, 2005
  • Received 25 April 2005

© 2005 American Physical Society

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