Anomalous Transport in Steady Plane Viscous Flows: Simple Models

Chapter
Part of the Nonlinear Systems and Complexity book series (NSCH, volume 20)

Abstract

Steady laminar plane flows of incompressible viscous fluids past lattices of solid bodies or arrays of soft obstacles (steady vortices) are less trivial as they may seem to be. Dynamics of tracers in regular patterns of streamlines is often aperiodic: repeated slow passages near stagnation points and/or solid surfaces provide a mechanism for eventual decorrelation. Velocity spectra for such tracers are known to possess fractal component, whereas correlation functions, in spite of integrability of the underlying dynamical system, display algebraic decay. Singularities of passage times near the obstacles depend on the shape and boundary conditions (stress-free/no-slip); this has implications for existence and quantitative characteristics of subdiffusive and superdiffusive regimes of transport. Here, we study the transport properties with the help of the simple and computationally efficient model: the special flow. Characteristics of anomalous transport, computed for such flows, match well the theoretical estimates.

Keywords

Viscous flow Stagnation points Anomalous transport Dynamic diffusion 

Notes

Acknowledgements

The authors feel deeply obliged to M.I. Rabinovich for his tireless proliferation of dynamical system approach to various fields of science. This research was supported by a Grant from the GIF, the German-Israeli Foundation for Scientific Research and Development, Grant I-1271-303.7/2014.

References

  1. 1.
    Aranson, I.S., Rabinovich, M.I., Tsimring, L.Sh.: Anomalous diffusion of particles in regular flows. Phys. Lett. A 151, 523–528 (1990)CrossRefGoogle Scholar
  2. 2.
    Bouchaud, P., George, A.: Anomalous diffusion in disordered media: statistical mechanisms, models and physical applications. Phys. Rep. 195, 127–293 (1990)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Cornfeld, I.P., Fomin, S.V., Sinai, Ya.G.: Ergodic Theory. Springer, New York (1982)Google Scholar
  4. 4.
    Hasimoto, H.: On the periodic fundamental solution of the Stokes equations and their application to viscous flow past a cubic array of spheres. J. Fluid Mech. 5, 317–328 (1959)Google Scholar
  5. 5.
    Havlin, S., Ben-Abraham, D.: Diffusion in disordered media. Adv. Phys. 36, 695–798 (1987)CrossRefGoogle Scholar
  6. 6.
    Kochergin, A.V.: On mixing in special flows over translation of intervals and in smooth flows on surfaces. Math. Sbornik 96, 471–502 (1975)Google Scholar
  7. 7.
    Kochergin, A.V.: Nondegenerate saddle-points and absence of mixing. Math. Notes 19, 453–468 (1976)CrossRefGoogle Scholar
  8. 8.
    Kolmogorov, A.N.: On dynamical systems with an integral invariant on a torus. Dokl. Akad. Nauk. SSSR Ser. Mat. 93, 763–766 (1953)MathSciNetMATHGoogle Scholar
  9. 9.
    Majda, A.J., Kramer, P.R., Simplified models for turbulent diffusion: theory, numerical modelling, and physical phenomena. Phys. Rep. 314, 237–254 (1999)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Meshalkin, L.D., Sinai, Ya,G.: Investigation of the stability of a stationary solution of a system of equations for the plane movement of an incompressible viscous liquid. J. Appl. Math. Mech. (Prikl. Mat. Mekh.) 25, 1700–1705 (1961)Google Scholar
  11. 11.
    Nepomnyashchy, A., Zaks, M.A.: in preparationGoogle Scholar
  12. 12.
    Pikovsky, A.S., Zaks, M.A., Feudel, U., Kurths, J.: Singular continuous spectra in dissipative dynamical systems. Phys. Rev. E 52, 285–296 (1995)Google Scholar
  13. 13.
    Pöschke, P., Sokolov, I.M., Nepomnyashchy, A.A., Zaks, M.A.: Anomalous transport in cellular flows: the role of initial conditions and aging. Phys. Rev. E 94, 032128 (2016)CrossRefGoogle Scholar
  14. 14.
    Sangani A.S., Yao, C.: Transport processes in random arrays of cylinders. II. Viscous flow. Phys. Fluids 31, 2435–2444 (1988)Google Scholar
  15. 15.
    Sinai, Ya.G., Khanin, K.M.: Mixing for certain classes of special flows over the circle shift. Funct. Anal. Appl. 26, 155–169 (1992)Google Scholar
  16. 16.
    Sommeria, J.: Experimental study of the two-dimensional inverse energy cascade in a square box. J. Fluid Mech. 170, 139–168 (1986)Google Scholar
  17. 17.
    Taylor, G.I.: Diffusion by continuous movement. Proc. Lond. Math. Soc. 2, 196–212 (1921)MathSciNetMATHGoogle Scholar
  18. 18.
    Taylor, G.I.: Dispersion of soluble matter in solvent flowing slowly through a tube. Proc. R. Soc. A 219, 186–203 (1953)CrossRefGoogle Scholar
  19. 19.
    von Neumann, J.: Zur Operatorenmethode in der Klassichen Mechanik. Ann. Math. 33, 587–642 (1932)MathSciNetCrossRefMATHGoogle Scholar
  20. 20.
    Young, W., Pumir, A., Pomeau, Y.: Anomalous diffusion of tracer in convection rolls. Phys. Fluids A 1, 462–469 (1989)MathSciNetCrossRefMATHGoogle Scholar
  21. 21.
    Zaks, M.A.: Anomalous transport in steady plane flows of viscous fluids. In: Collet P., Courbage M., et al. (eds.) Chaotic Dynamics and Transport in Classical and Quantum Systems, pp. 401–412. Kluwer Academic Publishers, Dordrecht (2005)CrossRefGoogle Scholar
  22. 22.
    Zaks, M.A., Straube, A.V.: Steady Stokes flow with long-range correlations, fractal Fourier spectrum and anomalous transport. Phys. Rev. Lett. 89, 244101 (2002)CrossRefGoogle Scholar
  23. 23.
    Zaks, M.A., Pikovsky, A.S., Kurths, J.: Steady viscous flow with fractal power spectrum. Phys. Rev. Lett. 77, 4338–4341 (1996)CrossRefGoogle Scholar

Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  1. 1.Humboldt University BerlinBerlinGermany
  2. 2.TechnionHaifaIsrael

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