Hostname: page-component-8448b6f56d-xtgtn Total loading time: 0 Render date: 2024-04-23T09:10:04.475Z Has data issue: false hasContentIssue false
  • English
  • Français

Analysis of chaotic mixing in two model systems

Published online by Cambridge University Press:  21 April 2006

D. V. Khakhar ,
H. Rising  and
J. M. Ottino
D. V. Khakhar
Affiliation:
Department of Chemical Engineering, Goessmann Laboratory, University of Massachusetts, Amherst, MA 01003, USA
H. Rising
Affiliation:
Department of Mathematics, University of Massachusetts, Amherst, MA 01003, USA
J. M. Ottino
Affiliation:
Department of Chemical Engineering, Goessmann Laboratory, University of Massachusetts, Amherst, MA 01003, USA
Get access Rights & Permissions [Opens in a new window]

Abstract

We study the chaotic mixing in two periodic model flows, the ‘tendril–whorl’ flow and the ‘Aref-blinking-vortex’ flow, with the objective of supplying evidence for the primary mechanisms responsible for mixing in two-dimensional deterministic flows. The analysis is based on tools of dynamical systems theory but it is clear that the mixing problem generates several questions of its own: low periodic points and horseshoes dominate the picture, since we want to achieve mixing quickly; Poincaré sections, popular in dynamical systems analyses, might give misleading information with regard to dispersion at short times. Our analysis shows that both flows are able to stretch and fold material lines well below the lengthscale of the flows themselves. The inner workings of the two systems are revealed by studying the local and global bifurcations. Computations for the blinking-vortex system indicate the existence of an optimum period at which the average efficiency is maximized, whereas the intensity of segregation – a classical parameter in mixing studies – decays rapidly to an asymptotic value in the globally chaotic region. Even though our flows are not turbulent the results might have some implications for pointing to the limits of similar studies in actual turbulent flows (e.g. line stretching).

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 172 , November 1986 , pp. 419 - 451
Copyright
© 1991 Cambridge University Press

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Aref H.1984 Stirring by chaotic advection. J. Fluid Mech. 143, 1–21.Google Scholar
Aref, H. & Balachandar S.1985 Chaotic advection in a Stokes flow. 38th Annual Meeting of APS/Division of Fluid Dynamics, Tucson, Arizona, USA. (Abstract, Bull. Amer. Phys. Soc. 30, 1703.)
Aref, H. & Tryggvason G.1984 Vortex dynamics of passive and active interfaces. Physica 12D, 59–70.Google Scholar
Arnold V. I.1983 Geometrical Methods in the Theory of Ordinary Differential Equations. Springer.
Bennetin G., Galgani L., Giorgilli, A. & Strelcyn J.-M.1980a Lyapunov characteristic exponents for smooth dynamical systems and for Hamiltonian systems; a method for computing all of them. Part 1: theory. Meccanica March, 11–20.
Bennetin G., Galgani L., Giorgilli, A. & Strelcyn J.-M.1980b Lyapunov characteristic exponents for smooth dynamical systems and for Hamiltonian systems; a method for computing all of them. Part 2: numerical application. Meccanica March, 21–30.Google Scholar
Berry M. V., Balzacs N. L., Tabor, M. & Voros A.1979 Quantum maps. Ann. Phys. 122, 26–63.Google Scholar
Chella, R. & Ottino J. M.1985 Stretching in some classes of fluid motions and asymptotic flow efficiencies as a measure of flow classification. Arch. Rat. Mech. Anal. 90, 15–42.Google Scholar
Chien W.-L., Rising, H. & Ottino J. M.1986 Laminar mixing and chaotic mixing in several cavity flows. J. Fluid Mech. (in press).Google Scholar
Danckwerts P. V.1952 The definition and measurement of some characteristics of mixtures. Appl. Sci. Res. A3, 279–296.Google Scholar
Greene J.1979 A method for determining a stochastic transition. J. Math. Phys. 20, 1183–1201.Google Scholar
Guckenheimer, J. & Holmes P.1983 Non-linear Oscillations, Dynamical Systems, and Bifurcation of Vector Fields. Springer.
Helleman R. H. G.1980 Self-generated chaotic behavior in non-linear mechanics. In Fundamentals Problems in Statistical Mechanics, vol. 5 (ed. E. G. D. Cohen). North-Holland.
Hénon M.1966 Sur la topologie des lignes de courant dans un cas particulier C. R. Acad. Sci. Paris A 262, 312–314.Google Scholar
Khakhar D. V.1986 Fluid mechanics of laminar mixing: dispersion and chaotic flows. Ph.D. thesis, University of Massachusetts, Amherst, MA 01003, USA.
Khakhar D. V., Chella, R. & Ottino J. M.1984 Stretching, chaotic motion and elongated droplets in time dependent flows. In Advances in Rheology, vol. 2, Fluids (ed. B. Mena, A. Garcia-Rejon & C. Rangel-Naifel), pp. 81–88.
Lichtenberg, A. J. & Lieberman M. A.1982 Regular and Stochastic Motion. Springer.
Lieberman, M. A. & Tennyson J. L.1982 Chaotic motion along resonance layers in near-integrable Hamiltonian systems with three or more degrees of freedom. In Non-equilibrium Problems in Statistical Mechanics, vol. 2 (ed. W. Horton, L. Reichl & V. Szebehely). Wiley.
Mackav R. S., Meiss, J. D. & Percival J. C.1984 Transport in Hamiltonian systems. Physica 13D, 55–81.Google Scholar
Ottino J. M.1982 Description of mixing with diffusion and reaction in terms of the concept of material surfaces. J. Fluid Mech. 114, 83–104.Google Scholar
Ottino, J. M. & Chella R.1983 Laminar mixing of polymeric liquids: brief review and recent theoretical developments. Polymer Engng Sci. 23, 357–379.Google Scholar
Osledec V. I.1968 A multiplicative ergodic theorem. Ljapunov characteristic numbers for dynamical systems. Trans. Moscow Math. Soc. 18, 187–231.Google Scholar
Rising H.1986 Mixing and chaos in model flows. Ph.D. in progress. Department of Mathematics, University of Massachusetts, Amherst, MA 01003, USA.
Truesdell, C. & Toupin R. A.1960 The classical field theories, vol. III/1. In Handbuch der Physik (ed. S. Flügge). Springer.