Hostname: page-component-76fb5796d-vfjqv Total loading time: 0 Render date: 2024-04-25T12:56:30.723Z Has data issue: false hasContentIssue false
  • English
  • Français

Chaotic advection by laminar flow in a twisted pipe

Published online by Cambridge University Press:  26 April 2006

Scott W. Jones ,
Oran M. Thomas  and
Hassan Aref
Scott W. Jones
Affiliation:
Department of Applied Mechanics and Engineering Science, University of California at San Diego, La Jolla, CA 92093, USA
Oran M. Thomas
Affiliation:
Department of Applied Mechanics and Engineering Science, University of California at San Diego, La Jolla, CA 92093, USA
Hassan Aref
Affiliation:
Department of Applied Mechanics and Engineering Science, University of California at San Diego, La Jolla, CA 92093, USA
Get access Rights & Permissions [Opens in a new window]

Abstract

The appearance of chaotic particle trajectories in steady, laminar, incompressible flow through a twisted pipe of circular cross-section is demonstrated using standard dynamical systems diagnostics and a model flow based on Dean's perturbation solutions. A study is performed to determine the parameters that control fluid stirring in this mixing device that has no moving parts. Insight into the chaotic dynamics are provided by a simple one-dimensional map of the pipe boundary onto itself. The results of numerical experiments illustrating the stretching of material lines, stirring of blobs of material, and the three-dimensional trajectories of fluid particles are presented. Finally, enhanced longitudinal particle dispersal due to the coupling between chaos in the transverse direction and the non-uniform longitudinal transport of particles is shown.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 209 , December 1989 , pp. 335 - 357
Copyright
© 1989 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. 1986 Chaotic advection in a Stokes flow. Phys. Fluids 29, 3515–3521.Google Scholar
Aref, H. & Jones, S. W. 1987 Chaotic advection: Efficient stirring of viscous liquids. In Proc. Fifth Symp. Energy Engin. Sci., Argonne Nat. Lab. CONF-8706187, pp. 209–216.Google Scholar
Aref, H., Jones, S. W., Mofina, S. & Zawadzki, I 1989 Vortices, kinematics and chaos. Physica D 37, 423–440.Google Scholar
Aref, H., Jones, S. W. & Thomas, O. M. 1988 Computing particle motions in fluid flows. Computers in Phys. (USA) 2 (6), 22–27.Google Scholar
Arter, W. 1983 Ergodic stream-lines in steady convection. Phys. Lett. A 97, 171–174.Google Scholar
Berger, S. A., Talbot, L. & Yao, L.-S. 1983 Flow in curved pipes. Ann. Rev. Fluid Mech. 15, 461–512.Google Scholar
Berry, M. V., Balazs, N. L., Tabor, M. & Voros, A. 1979 Quantum maps. Ann. Phys. 122, 26–63.Google Scholar
Chaiken, J., Chevray, R., Tabor, M. & Tan, Q. M. 1986 Experimental study of Lagrangian turbulence in a 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, 687–694.Google Scholar
Chang, L.-J. & Tarbell, J. M. 1985 Numerical simulation of fully developed sinusoidal and pulsatile (physiological) flow in curved tubes. J. Fluid Mech. 161, 175–198.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
Courant, R. & Robbins, H. 1941 What is Mathematics? An Elementary Approach to Ideas and Methods. Oxford University Press, 521 pp.
Dean, W. R. 1927 Note on the motion of fluid in a curved pipe. Phil. Mag. 4, 208–223.Google Scholar
Dean, W. R. 1928 The streamline motion of fluid in a curved pipe. Phil. Mag. 5, 673–693.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
Eckhardt, B. 1988 Irregular scattering. Physica D 33, 89–98.Google Scholar
Erdogan, M. E. & Chatwin, P. C. 1967 The effects of curvature and buoyancy on the laminar dispersion of solute in a horizontal tube. J. Fluid Mech. 29, 465–484.Google Scholar
Feingold, M., Kadanoff, L. & Piro, O. 1988 Passive scalars, three dimensional volume-preserving maps and chaos. J. Statist. Phys. 50, 529–565.Google Scholar
Hénon, M. 1966 Sur la topologie des lignes de courant dans un cas particulier. C. R. Acad. Sci. Paris 262, 312–314.Google Scholar
Janssen, L. A. M. 1976 Axial dispersion in laminar flow through coiled tubes. Chem. Engng Sci. 31, 215–218.Google Scholar
Johnson, M. & Kamm, R. D. 1986 Numerical studies of steady flow dispersion at low Dean number in a gently curving tube. J. Fluid Mech. 172, 329–345.Google Scholar
Jones, S. W. & Aref, H. 1988 Chaotic advection in pulsed source—sink systems. Phys. Fluids 31, 469–485.Google Scholar
Jones, S. W., Thomas, O. M. & Aref, H. 1987 Chaotic advection by laminar flow in a twisted pipe. Bull Am. Phys. Soc. 32, 2026.Google Scholar
Jones, S. W. & Young, W. R. 1989 Shear dispersion and anomalous diffusion by chaotic advection. J. Fluid Mech. (submitted).Google Scholar
Kalb, C. E. & Seader, J. D. 1972 Heat and mass transfer phenomena for viscous flow in curved circular tubes. Intl J. Heat Mass Transfer 15, 801–817.Google Scholar
Kao, H. C. 1987 Torsion effect on fully developed flow in a helical pipe. J. Fluid Mech. 184, 335–356.Google Scholar
Khakhar, D. V., Franjione, J. G. & Ottino, J. M. 1987 A case study of chaotic mixing in determinstic flows: the partitioned pipe mixer. Chem. Engng Sci. 42, 2909–2926.Google Scholar
Khakhar, D. V. & Ottino, J. M. 1986 Fluid mixing (stretching) by time periodic sequences for weak flows. Phys. Fluids 29, 3503–3505.Google Scholar
Khakhar, D. V., Rising, H. & Ottino, J. M. 1986 Analysis of chaotic mixing in two model systems. J. Fluid Mech. 172, 419–451.Google Scholar
Lamb, H. 1878 On the conditions for steady motion of a fluid. Proc. Lond. Math. Soc. 9, 91–92.Google Scholar
Lichtenberg, A. J. & Lieberman, M. A. 1983 Regular and Stochastic Motion. Springer.
Lighthill, M. J. 1970 Turbulence. In Osborne Reynolds and Engineering Science Today (ed. D. M. McDowell & J. D. Jackson), pp. 83–146. New York: Barnes & Noble.
Mackay, R. S., Meiss, J. D. & Percival, I. C. 1984 Transport in Hamiltonian systems. Physica D 13, 55–81.Google Scholar
Moser, J. 1973 Stable and Random Motions in Dynamical Systems. Princeton University Press.
Noid, D. W., Gray, S. K. & Rice, S. A. 1986 Fractal behavior in classical collisional energy transfer. J. Chem. Phys. 51, 363–383.Google Scholar
Nunge, R. J., Lin, T.-S. & Gill, W. N. 1972 Laminar dispersion in curved tubes and channels. J. Fluid Mech. 51, 363–383.Google Scholar
Ottino, J. M. 1989 The mixing of fluids. Sci. Am. 260, 56–67.Google Scholar
Prusa, J. & Yao, L.-S. 1982 Numerical solution for fully developed flow in heated curved tubes. J. Fluid Mech. 123, 503–522.Google Scholar
Serrin, J. 1959 The mathematical principles of fluid mechanics. In Handbuch der Physik, vol. VIII/1 (ed. S. Flügge). Springer.
Taylor, G. I. 1929 The criterion for turbulence in curved pipes. Proc. R. Soc. Lond. A 124, 243–249.Google Scholar
Taylor, G. I. 1953 Dispersion of soluble matter in solvent flowing slowly through a tube. Proc. R. Soc. Lond. A 219, 186–203.Google Scholar
Weissman, M. H. & Mockros, L. F. 1968 Gas transfer to blood flowing in coiled circular tubes. Proc. ASCE Engng Mech. Div. J. 94, 857–872.Google Scholar
White, C. M. 1929 Streamline flow through curved tubes. Proc. R. Soc. Lond. A 123, 645–663.Google Scholar
Zimmerman, J. T. F. 1986 The tidal whirlpool: A review of horizontal dispersion by tidal and residual currents. Netherlands J. Sea Res. 20, 133–154.Google Scholar