Skip to main content

Chaotic advection by laminar flow in a twisted pipe

  • Scott W. Jones (a1), Oran M. Thomas (a1) and Hassan Aref (a1)

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.

Hide All
Aref, H. 1984 Stirring by chaotic advection. J. Fluid Mech. 143, 121.
Aref, H. & Balachandar, S. 1986 Chaotic advection in a Stokes flow. Phys. Fluids 29, 35153521.
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. 209216.
Aref, H., Jones, S. W., Mofina, S. & Zawadzki, I 1989 Vortices, kinematics and chaos. Physica D 37, 423440.
Aref, H., Jones, S. W. & Thomas, O. M. 1988 Computing particle motions in fluid flows. Computers in Phys. (USA) 2 (6), 2227.
Arter, W. 1983 Ergodic stream-lines in steady convection. Phys. Lett. A 97, 171174.
Berger, S. A., Talbot, L. & Yao, L.-S. 1983 Flow in curved pipes. Ann. Rev. Fluid Mech. 15, 461512.
Berry, M. V., Balazs, N. L., Tabor, M. & Voros, A. 1979 Quantum maps. Ann. Phys. 122, 2663.
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, 165174.
Chaiken, J., Chu, C. K., Tabor, M. & Tan, Q. M. 1987 Lagrangian turbulence and spatial complexity in a Stokes flow. Phys. Fluids 30, 687694.
Chang, L.-J. & Tarbell, J. M. 1985 Numerical simulation of fully developed sinusoidal and pulsatile (physiological) flow in curved tubes. J. Fluid Mech. 161, 175198.
Chien, W.-L., Rising, H. & Ottino, J. M. 1986 Laminar mixing and chaotic mixing in several cavity flows. J. Fluid Mech. 170, 355377.
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, 208223.
Dean, W. R. 1928 The streamline motion of fluid in a curved pipe. Phil. Mag. 5, 673693.
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, 353391.
Eckhardt, B. 1988 Irregular scattering. Physica D 33, 8998.
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, 465484.
Feingold, M., Kadanoff, L. & Piro, O. 1988 Passive scalars, three dimensional volume-preserving maps and chaos. J. Statist. Phys. 50, 529565.
Hénon, M. 1966 Sur la topologie des lignes de courant dans un cas particulier. C. R. Acad. Sci. Paris 262, 312314.
Janssen, L. A. M. 1976 Axial dispersion in laminar flow through coiled tubes. Chem. Engng Sci. 31, 215218.
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, 329345.
Jones, S. W. & Aref, H. 1988 Chaotic advection in pulsed source—sink systems. Phys. Fluids 31, 469485.
Jones, S. W., Thomas, O. M. & Aref, H. 1987 Chaotic advection by laminar flow in a twisted pipe. Bull Am. Phys. Soc. 32, 2026.
Jones, S. W. & Young, W. R. 1989 Shear dispersion and anomalous diffusion by chaotic advection. J. Fluid Mech. (submitted).
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, 801817.
Kao, H. C. 1987 Torsion effect on fully developed flow in a helical pipe. J. Fluid Mech. 184, 335356.
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, 29092926.
Khakhar, D. V. & Ottino, J. M. 1986 Fluid mixing (stretching) by time periodic sequences for weak flows. Phys. Fluids 29, 35033505.
Khakhar, D. V., Rising, H. & Ottino, J. M. 1986 Analysis of chaotic mixing in two model systems. J. Fluid Mech. 172, 419451.
Lamb, H. 1878 On the conditions for steady motion of a fluid. Proc. Lond. Math. Soc. 9, 9192.
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. 83146. New York: Barnes & Noble.
Mackay, R. S., Meiss, J. D. & Percival, I. C. 1984 Transport in Hamiltonian systems. Physica D 13, 5581.
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, 363383.
Nunge, R. J., Lin, T.-S. & Gill, W. N. 1972 Laminar dispersion in curved tubes and channels. J. Fluid Mech. 51, 363383.
Ottino, J. M. 1989 The mixing of fluids. Sci. Am. 260, 5667.
Prusa, J. & Yao, L.-S. 1982 Numerical solution for fully developed flow in heated curved tubes. J. Fluid Mech. 123, 503522.
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, 243249.
Taylor, G. I. 1953 Dispersion of soluble matter in solvent flowing slowly through a tube. Proc. R. Soc. Lond. A 219, 186203.
Weissman, M. H. & Mockros, L. F. 1968 Gas transfer to blood flowing in coiled circular tubes. Proc. ASCE Engng Mech. Div. J. 94, 857872.
White, C. M. 1929 Streamline flow through curved tubes. Proc. R. Soc. Lond. A 123, 645663.
Zimmerman, J. T. F. 1986 The tidal whirlpool: A review of horizontal dispersion by tidal and residual currents. Netherlands J. Sea Res. 20, 133154.
Recommend this journal

Email your librarian or administrator to recommend adding this journal to your organisation's collection.

Journal of Fluid Mechanics
  • ISSN: 0022-1120
  • EISSN: 1469-7645
  • URL: /core/journals/journal-of-fluid-mechanics
Please enter your name
Please enter a valid email address
Who would you like to send this to? *


Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed