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Chaotic advection in three-dimensional unsteady incompressible laminar flow

Published online by Cambridge University Press:  26 April 2006

Julyan H. E. Cartwright ,
Mario Feingold  and
Oreste Piro
Julyan H. E. Cartwright
Affiliation:
Departament de Física, Universitat de les Illes Balears, 07071 Palma de Mallorca, Spain; email julyan@hp1.uib.es; piro@hp1.uib.es Centre de Càlcul i Informatització, Universitat de les Illes Balears, 07071 Palma de Mallorca, Spain
Mario Feingold
Affiliation:
Department of Physics, Ben-Gurion University, Beer-Sheva 84105, Israel; email mario@bgumail.bgu.ac.il
Oreste Piro
Affiliation:
Departament de Física, Universitat de les Illes Balears, 07071 Palma de Mallorca, Spain; email julyan@hp1.uib.es; piro@hp1.uib.es Institut Mediterrani d’Estudis Avançats (CSIC–UIB), 07071 Palma de Mallorca, Spain
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Abstract

We discuss chaotic advection in three-dimensional unsteady incompressible laminar flow, and analyse in detail the most important novel advection phenomenon in these flows: the global dispersion of passive scalars in flows with two slow and one fast velocity components. We make a comprehensive study of the first model of an experimentally realizable flow to exhibit this resonance-induced dispersion: biaxial unsteady spherical Couette flow is a three-dimensional incompressible laminar flow with periodic time dependence derived analytically from the Navier–Stokes equations in the low-Reynolds-number limit.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 316 , 10 June 1996 , pp. 259 - 284
Copyright
© 1996 Cambridge University Press

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References

Aref, H. 1984 Stirring by chaotic advection. J. Fluid Mech. 143, 1–21.Google Scholar
Aref, H. & Balachandar, S. 1986 Chaotic advection in a Stokes flow. Phys. Fluids 29, 3515–3521.Google Scholar
Aref, H., Jones, S. W., Mofina, S. & Zawadzki, I. 1989 Vortices, kinematics and chaos. Physica D 37, 423–440.Google Scholar
Arnold, V. I. 1965 Sur la topologie des écoulements stationnaires des fluides parfaits. C. R. Acad. Sci. Paris A 261, 17–20.Google Scholar
Arnold, V. I. 1978 Mathematical Methods of Classical Mechanics. Springer.
Arrowsmith, D. K. & Place, C. M. 1990 An Introduction to Dynamical Systems. Cambridge University Press.
Bajer, K. 1994 Hamiltonian formulation of the equations of streamlines in three-dimensional steady flows. Chaos, Solitons & Fractals 4, 895–911.Google Scholar
Bajer, K. & Moffatt, H. K. 1990 On a class of steady confined Stokes flows with chaotic streamlines. J. Fluid Mech. 212, 337–363.Google Scholar
Bajer, K. & Moffatt, H. K. 1992 Chaos associated with fluid inertia. In Topological Aspects of the Dynamics of Fluids and Plasmas (ed. H. K. Moffatt, G. M. Zaslavsky, P. Comte & M. Tabor), pp. 517–534. Kluwer.
Bühler, K. 1992 Pattern formation of instabilities in spherical Couette flow. In Pattern Formation in Complex Dissipative Systems: Fluid Patterns, Liquid Crystals, Chemical Reactions (ed. S. Kai), pp. 298–302. World Scientific.
Cartwright, J. H. E., Feingold, M. & Piro, O. 1994a Dynamically diffusive Lagrangian trajectories in time-periodic three-dimensional flows. In Heat Transfer Enhancement by Lagrangian Chaos and Turbulence (ed. H. Peerhossaini & A. Provenzale), pp. 101–107. ISITEM. Univ. Nantes.
Cartwright, J. H. E., Feingold, M. & Piro, O. 1994b Passive scalars and three-dimensional Liouvillian maps. Physica D 76, 22–33.Google Scholar
Cartwright, J. H. E., Feingold, M. & Piro, O. 1995 Global diffusion in a realistic three-dimensional time-dependent nonturbulent fluid flow. Phys. Rev. Lett. 75, 3669–3672.Google Scholar
Cartwright, J. H. E. & Piro, O. 1992 The dynamics of Runge–Kutta methods. Intl J. Bifurcation Chaos 2, 427–449.Google Scholar
Chaiken, J., Chevray, R., Tabor, M. & Tan, Q. M. 1986 Experimental study of Lagrangian turbulence in Stokes flow. Proc. R. Soc. Lond. A 408, 165–174.Google Scholar
Chaiken, J., Chu, C. K., Tabor, M. & Tan, Q. M. 1987 Lagrangian turbulence and spatial complexity in a Stokes flow. Phys. Fluids 30, 687–699.Google Scholar
Cheng, C.-Q. & Sun, Y.-S. 1990 Existence of invariant tori in three-dimensional measure preserving mappings. Celest. Mech. 47, 275–292.Google Scholar
Chien, W.-L., Rising, H. & Ottino, J. M. 1986 Laminar mixing and chaotic mixing in several cavity flows. J. Fluid Mech. 170, 355–377.Google Scholar
Dombre, T., Frisch, U., Greene, J. M., Hénon, M., Mehr, A. & Soward, A. M. 1986 Chaotic streamlines in the ABC flows. J. Fluid Mech. 167, 353–391.Google Scholar
Feingold, M., Kadanoff, L. P. & Piro, O. 1987 A way to connect fluid dynamics to dynamical systems: Passive scalars. In Fractal Aspects of Materials: Disordered Systems (ed. A. J. Hurd, D. A. Weitz & B. B. Mandelbrot), pp. 203–205. Materials Research Society.
Feingold, M., Kadanoff, L. P. & Piro, O. 1988a Diffusion of passive scalars in fluid flows: Maps in three dimensions. In Universalities in Condensed Matter (ed. R. Jullien, L. Peliti, R. Rammal & N. Boccara), pp. 236–241. Les Houches. Springer.
Feingold, M., Kadanoff, L. P. & Piro, O. 1988b Passive scalars, three-dimensional volume-preserving maps, and chaos. J. Statist. Phys. 50, 529–565.Google Scholar
Feingold, M., Kadanoff, L. P. & Piro, O. 1989 Transport of passive scalars: KAM surfaces and diffusion in three-dimensional Liouvillian maps. In Instabilities and Nonequilibrium Structures II (ed. P. Collet, E. Tirapegui & D. Villarroel). Reidel.
Haberman, W. L. 1962 Secondary flow about a sphere rotating in a viscous liquid inside a coaxially rotating spherical container. Phys. Fluids 5, 625–626.Google Scholar
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
Jones, S. W., Thomas, O. M. & Aref, H. 1989 Chaotic advection by laminar flow in a twisted pipe. J. Fluid Mech. 209, 335–357.Google Scholar
Khakhar, D. V., Franjione, J. G. & Ottino, J. M. 1987 A case study of chaotic mixing in deterministic flows. The partitioned pipe mixer. Chem. Engng Sci. 42, 2909–2926.Google Scholar
Khakhar, D. V., Rising, H. & Ottino, J. M. 1986 An analysis of chaotic mixing in two chaotic flows. J. Fluid Mech. 172, 419–451.Google Scholar
Koschmieder, E. L. 1993 Bénard Cells and Taylor Vortices. Cambridge University Press.
Kusch, H. A. & Ottino, J. M. 1992 Experiments on mixing in continuous chaotic flows. J. Fluid Mech. 236, 319–348.Google Scholar
Le Guer, Y., Castelain, C. & Peerhossaini, H. 1996 Experimental study of chaotic advection regime in a twisted duct flow. J. Fluid Mech. submitted.Google Scholar
MacKay, R. S. 1994 Transport in 3D volume-preserving flows. J. Nonlinear Sci. 4, 329–354.Google Scholar
Mezić, I. & Wiggins, S. 1994 On the integrability and perturbation of three-dimensional fluid flows with symmetry. J. Nonlinear Sci. 4, 157–194.Google Scholar
Munson, B. R. 1974 Viscous incompressible flow between eccentric coaxially rotating spheres. Phys. Fluids 17, 528–531.Google Scholar
Munson, B. R. & Joseph, D. D. 1971 Viscous incompressible flow between concentric rotating spheres. Part 1. Basic flow. J. Fluid Mech. 49, 289–303.Google Scholar
Newhouse, S. E., Ruelle, D. & Takens, F. 1978 Occurrence of strange axiom A attractors near quasiperiodic flows on Tm, m ≥ 3. Commun. Math. Phys. 64, 35–40.Google Scholar
Ottino, J. M. 1989 The Kinematics of Mixing: Stretching, Chaos, and Transport. Cambridge University Press.
Parker, T. S. & Chua, L. O. 1989 Practical Numerical Algorithms for Chaotic Systems. Springer.
Pearson, C. E. 1967 A numerical study of the time-dependent viscous flow between two rotating spheres. J. Fluid Mech. 28, 323–336.Google Scholar
Piro, O. & Feingold, M. 1988 Diffusion in three-dimensional Liouvillian maps. Phys. Rev. Lett. 61, 1799–1802.Google Scholar
Ruelle, D. & Takens, F. 1971 On the nature of turbulence. Commun. Math. Phys. 20, 167–192; 23, 343–344.Google Scholar
Solomon, T. H. & Gollub, J. P. 1988 Chaotic particle transport in time dependent Rayleigh–Bénard convection. Phys. Rev. A 38, 6280–6286.Google Scholar
Stone, H. A., Nadim, A. & Strogatz, S. H. 1991 Chaotic streaklines inside drops immersed in steady linear flows. J. Fluid Mech. 232, 629–646.Google Scholar
Swanson, P. D. & Ottino, J. M. 1990 A comparative computational and experimental study of chaotic mixing of viscous fluids. J. Fluid Mech. 213, 227–249.Google Scholar
Thyagaraja, A. & Haas, F. A. 1985 Representation of volume-preserving maps induced by solenoidal vector fields. Phys. Fluids 28, 1005–1007.Google Scholar
Wannier, G. H. 1950 A contribution to the hydrodynamics of lubrication. Q. Appl. Maths 8, 1–32.Google Scholar
Yavorskaya, I. M., Belyaev, Y. N., Monakhov, A. A., Astaf'eva, N. M., Scherbakov, S. A. & Vvedenskaya, N. D. 1980 Stability, non-uniqueness and transition to turbulence in the flow between two rotating spheres. In Proc. XV Intl Congress of Theoretical and Applied Mechanics, pp. 431–443. IUTAM.
Zheligovsky, V. A. 1993 A kinematic magnetic dynamo sustained by a Beltrami flow in a sphere. Geophys. Astrophys. Fluid Dyn. 73, 217–254.Google Scholar