Hostname: page-component-8448b6f56d-42gr6 Total loading time: 0 Render date: 2024-04-23T14:22:09.026Z Has data issue: false hasContentIssue false
  • English
  • Français

Trapping and sedimentation of inertial particles in three-dimensional flows in a cylindrical container with exactly counter-rotating lids

Published online by Cambridge University Press:  19 November 2009

CRISTIAN ESCAURIAZA  and
FOTIS SOTIROPOULOS
CRISTIAN ESCAURIAZA
Affiliation:
St. Anthony Falls Laboratory, Department of Civil Engineering, University of Minnesota, Minneapolis, MN 55414, USA Departamento de Ing. Hidráulica y Ambiental, Pontificia Universidad Católica de Chile, Av. Vicuña Mackenna 4860, 7820436, Santiago, Chile
FOTIS SOTIROPOULOS*
Affiliation:
St. Anthony Falls Laboratory, Department of Civil Engineering, University of Minnesota, Minneapolis, MN 55414, USA
*
†Email address for correspondence: fotis@umn.edu
Get access Rights & Permissions [Opens in a new window]

Abstract

Stirring and sedimentation of solid inertial particles in low-Reynolds-number flows has acquired great relevance in multiple environmental, industrial and microfluidic systems, but few detailed numerical studies have focused on chaotically advected experimentally realizable flows. We carry out one-way coupling simulations to study the dynamics of inertial particles in the steady three-dimensional flow in a cylindrical container with exactly counter-rotating lids, which was recently studied by Lackey & Sotiropoulos (Phys. Fluids, vol. 18, 2006, paper no. 053601). We elucidate the rich Lagrangian dynamics of the flow in the vicinity of toroidal invariant regions and show that depending on the Stokes number inertial particles could get trapped for long times in different equilibrium positions inside integrable islands. In the chaotically advected region of the flow the balance between inertia and gravity forces (represented by the settling velocity) can produce a striking fractal sedimentation regime, characterized by a sequence of discrete deposition events of seemingly random number of particles separated by hiatuses of random duration. The resulting staircase-like distribution of the time series of the number of particles in suspension is shown to be a devil's staircase whose fractal dimension is equal to the 0.87 value found in multiple dissipative dynamical systems in nature. Our work sheds new light on the complex mechanisms governing the stirring and deposition of inertial particles and provides new information about the parameters that are relevant in the characterization of particle dynamics in different regions of chaotically advected flows.

JFM classification

Mixing: Chaotic advection Vortex Flows: Vortex Flows
Type
Papers
Information
Journal of Fluid Mechanics , Volume 641 , 25 December 2009 , pp. 169 - 193
Copyright
Copyright © Cambridge University Press 2009

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

REFERENCES

Abatan, A. A., McCarthy, J. J. & Vargas, W. L. 2006 Particle migration in the rotating flow between co-axial disks. AIChE J. 52, 2039–2045.CrossRefGoogle Scholar
Aref, H. 1984 Stirring by chaotic advection. J. Fluid Mech. 143, 1–21.CrossRefGoogle Scholar
Auton, T. R., Hunt, J. C. R. & Prud'homme, M. 1988 The force exerted on a body in inviscid unsteady non-uniform rotational flow. J. Fluid Mech. 197, 241–257.CrossRefGoogle Scholar
Bak, P. 1986 The devil's staircase. Phys. Today 39, 38–45.CrossRefGoogle Scholar
Bec, J. 2005 Multifractal concentrations of inertial particles in smooth random flows. J. Fluid Mech. 528, 255–277.CrossRefGoogle Scholar
Crowe, C. T., Troutt, T. R. & Chung, J. N. 1998 Multiphase Flows with Droplets and Particles. CRC Press.Google Scholar
Dávila, J. & Hunt, J. C. R. 2001 Settling of small particles near vortices and in turbulence. J. Fluid Mech. 440, 117–145.CrossRefGoogle Scholar
Escauriaza, C. 2008 Three-dimensional unsteady modelling of clear-water scour in the vicinity of hydraulic structures: Lagrangian and Eulerian perspectives. PhD thesis, University of Minnesota, Minneapolis, MN.Google Scholar
King, G. P., Rowlands, G., Rudman, M. & Yannacopoulos, A. N. 2001 Predicting chaotic dispersion with Eulerian symmetry measures: wavy Taylor-vortex flow. Phys. Fluids 13, 2522–2528.CrossRefGoogle Scholar
Lacis, S., Barci, J. C., Cebers, A. & Perzynski, R. 1997 Frequency locking and devil's staircase for a two-dimensional ferrofluid droplet in an elliptically polarized rotating magnetic field. Phys. Rev. E 55, 2640–2648.Google Scholar
Lackey, T. C. & Sotiropoulos, F. 2006 Relationship between stirring rate and Reynolds number in the chaotically advected steady flow in a container with exactly counter-rotating lids. Phys. Fluids 18, 053601.CrossRefGoogle Scholar
Mackay, R. S., Meiss, J. D. & Percival, I. C. 1984 Transport in Hamiltonian systems. Physica D 13, 55–81.CrossRefGoogle Scholar
Malhotra, N., Mezić, I. & Wiggins, S. 1998 Patchiness: a new diagnostic for Lagrangian trajectory analysis in time-dependent fluid flows. Intl J. Bifur. Chaos 8, 1053–1093.CrossRefGoogle Scholar
Maxey, M. R. 1987 The motion of small spherical particles in a cellular flow field. Phys. Fluids 30, 1915–1928.CrossRefGoogle Scholar
McLaughlin, J. B. 1988 Particle size effects on Lagrangian turbulence. Phys. Fluids 31, 2544–2553.CrossRefGoogle Scholar
Mezić, I. 2001 Chaotic advection in bounded Navier–Stokes flows. J. Fluid Mech. 431, 347–370.CrossRefGoogle Scholar
Mezić, I. & Sotiropoulos, F. 2002 Ergodic theory and experimental visualization of invariant sets in chaotically advected flows. Phys. Fluids 14, 2235–2243.CrossRefGoogle Scholar
Mezić, I. & Wiggins, S. 1999 A method for visualization of invariant sets of dynamical systems based on the ergodic partition. Chaos 9, 213–218.CrossRefGoogle ScholarPubMed
Nore, C., Tartar, M., Daube, O. & Tuckerman, L. S. 2004 Survey of instability thresholds of flow between exactly counter-rotating disks. J. Fluid Mech. 511, 45–65.CrossRefGoogle Scholar
Nore, C., Tuckerman, L. S., Daube, O. & Xin, S. 2003 The 1:2 mode interaction in exactly counter-rotating von Kármán swirling flow. J. Fluid Mech. 477, 51–88.CrossRefGoogle Scholar
Reichhardt, C. & Nori, F. 1999 Phase locking, devil's staircase, Farey trees, and Arnold tongues in driven vortex lattices with periodic pinning. Phys. Rev. Lett. 82, 414–417.CrossRefGoogle Scholar
Ridderinkhof, H. & Zimmerman, J. T. F. 1992 Chaotic stirring in a tidal system. Science 258, 1107–1111.CrossRefGoogle Scholar
Rudman, M. 1998 Mixing and particle dispersion in the wavy vortex regime of Taylor–Couette flow. AIChE J. 44, 1015–1026.CrossRefGoogle Scholar
Sadler, P. M. 1981 Sediment accumulation rates and the completeness of stratigraphic sections. J. Geol. 89, 569–584.CrossRefGoogle Scholar
Sadler, P. M. 1999 The influence of hiatuses on sediment accumulation rates. In On the Determination of Sediment Accumulation Rates. GeoResearch Forum (ed. Bruns, P. & Hass, H. C.), pp. 25–60. Trans Tech.Google Scholar
Saffman, P. G. 1965 The lift on a small sphere in a slow shear flow. J. Fluid Mech. 22, 385–400.CrossRefGoogle Scholar
Solomon, T. H. & Mezić, I. 2003 Uniform resonant chaotic mixing in fluid flows. Nature 425, 376–380.CrossRefGoogle ScholarPubMed
Sotiropoulos, F., Ventikos, Y. & Lackey, T. C. 2001 Chaotic advection in three-dimensional stationary vortex-breakdown bubbles: Šil'nikov's chaos and the devil's staircase. J. Fluid Mech. 444, 257–297.CrossRefGoogle Scholar
Sotiropoulos, F., Webster, D. R. & Lackey, T. C. 2002 Experiments on Lagrangian transport in steady vortex-breakdown bubbles in a confined swirling flow. J. Fluid Mech. 466, 215–248.CrossRefGoogle Scholar
Squires, K. & Eaton, J. 1991 Preferential concentration of particles by turbulence. Phys. Fluids A 3, 1169–1178.CrossRefGoogle Scholar
Squires, T. M. & Quake, S. R. 2005 Microfluidics: fluid physics at the nanoliter scale. Rev. Mod. Phys. 77, 977–1026.CrossRefGoogle Scholar
Stommel, H. 1949 Trajectories of small bodies sinking slowly through convection cells. J. Mar. Res. 8, 24–29.Google Scholar
Tsega, Y., Michaelides, E. E. & Eschenazi, E. V. 2001 Particle dynamics and mixing in the frequency driven Kelvin cat eyes flow. Chaos 11, 351–358.CrossRefGoogle ScholarPubMed
Tuval, I., Mezić, I., Bottausci, F., Zhang, Y. T., MacDonald, N. C. & Piro, O. 2005 Control of particles in microelectrode devices. Phys. Rev. Lett. 95, 236002.CrossRefGoogle ScholarPubMed
Voth, G. A., Saint, T. C., Dobler, G. & Gollub, J. P. 2003 Mixing rates and symmetry breaking in two-dimensional chaotic flow. Phys. Fluids 15, 2560–2566.CrossRefGoogle Scholar
Wang, L. P., Burton, T. D. & Stock, D. E. 1990 Chaotic dynamics of heavy particle dispersion: fractal dimension versus dispersion coefficients. Phys. Fluids A 2, 1305–1308.CrossRefGoogle Scholar
Wang, L. P., Burton, T. D. & Stock, D. E. 1991 Quantification of chaotic dynamics for heavy particle dispersion in ABC flow. Phys. Fluids A 3, 1073–1080.CrossRefGoogle Scholar
Wang, L. P., Maxey, M. R., Burton, T. D. & Stock, D. E. 1992 Chaotic dynamics of particle dispersion in fluids. Phys. Fluids A 4, 1789–1804.CrossRefGoogle Scholar
Wereley, S. T., Akonur, A. & Lueptow, R. M. 2002 Particle–fluid velocities and fouling in rotating filtration of a suspension. J. Membr. Sci. 209, 469–484.CrossRefGoogle Scholar
Wereley, S. T. & Lueptow, R. M. 1999 Inertial particle motion in a Taylor Couette rotating filter. Phys. Fluids 11, 325–333.CrossRefGoogle Scholar
Wiggins, S. & Ottino, J. M. 2004 Foundations of chaotic mixing. Phil. Trans. R. Soc. Lond. A 362, 937–970.CrossRefGoogle ScholarPubMed