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Chaos / Volume 22 / Issue 1 / REGULAR ARTICLES

Chaos 22, 013103 (2012); http://dx.doi.org/10.1063/1.3672510 (8 pages)

Resonance phenomena and long-term chaotic advection in volume-preserving systems

Dmitri L. Vainchtein1,2 and Alimu Abudu1

1Department of Mechanical Engineering, Temple University, Philadelphia, Pennsylvania 19122, USA
2Space Research Institute, Moscow, Russia

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(Received 11 October 2011; accepted 5 December 2011; published online 3 January 2012)

Creating chaotic advection is the most efficient strategy to achieve mixing on microscale or in very viscous fluids. In this paper, we present a quantitative theory of the long-time resonant mixing in 3D near-integrable flows. We use the flow between two coaxial elliptic counter-rotating cylinders as a demonstrative model, where multiple scatterings on resonance result in mixing by causing the jumps of adiabatic invariants. We improve the existing estimates of the width of the mixing domain. We show that the resulting mixing both on short and long time scales can be described in terms of a single diffusion-type equation with a diffusion coefficient depending on the averaged effect of multiple passages through resonances. We discuss the exact location of the boundaries of the chaotic domain and show how it affects the properties of mixing.

© 2012 American Institute of Physics

Article Outline

  1. INTRODUCTION
  2. VOLUME-PRESERVING ACTION-ACTION-ANGLE SYSTEMS WITH RESONANCES
  3. MODEL SYSTEM AND MAIN EQUATIONS
  4. AVERAGING METHOD AND STRUCTURE OF THE RESONANCE
  5. SCATTERING ON RESONANCE
    1. Jumps of AI inside the first boundaries
    2. Jumps of AI between the first and second layer boundaries
  6. LONG-TERM CHAOTIC ADVECTION AND DIFFUSION EQUATION
    1. Numerical results of adiabatic spreading
  7. IMPROVED AI
  8. CONCLUSIONS

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KEYWORDS and PACS

Keywords

chaos, diffusion, mixing, rotational flow, viscosity

PACS

ARTICLE DATA

Digital Object Identifier
http://dx.doi.org/10.1063/1.3672510

PUBLICATION DATA

ISSN

1054-1500 (print)  
1089-7682 (online)

Publisher

$abstract.society.value is a Member of CrossRef American Institute of Physics

For access to fully linked references, you need to log in.
    Cartwright, J. H. E., Feingold, M., and Piro, O., “Global diffusion in a realistic three-dimensional time-dependent nonturbulent fluid flow,” Phys. Rev. Lett. 75, 3669–3672 (1995).

    Neishtadt, A. I., “Change of an adiabatic invariant at a separatrix,” Sov. J. Plasma Phys. 12, 568–573 (1986).

    Vainchtein, D. L., Neishtadt, A. I., and Mezic, I., “On passage through resonances in volume-preserving systems,” Chaos 16, 043123 (2006)CHAOEH000016000004043123000001.

    Vainchtein, D. L., Widloski, J., and Grigoriev, R. O., “Resonant chaotic mixing in a cellular flow,” Phys. Rev. Lett. 99, 094501 (2007).

    Vainchtein, D. L., Widloski, J., and Grigoriev, R. O., “Resonant mixing in perturbed action-action-angle flow,” Phys. Rev. E 78, 026302 (2008).

    Vainshtein, D. L., Vasiliev, A. A., and Neishtadt, A. I., “Adiabatic chaos in a two-dimensional mapping,” Chaos 6, 514–518 (1996)CHAOEH000006000004000514000001.

    Vainshtein, D. L., Vasiliev, A. A., and Neishtadt, A. I., “Changes in the adiabatic invariant and streamline chaos in confined incompressible Stokes flow,” Chaos 6, 67–77 (1996)CHAOEH000006000001000067000001.

    Ward, T., and Homsy, G. M., “Electrohydrodynamically driven chaotic mixing in a translating drop,” Phys. Fluids 13, 3521–3525 (2001)PHFLE6000013000012003521000001.


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