A study in three-dimensional chaotic dynamics: Granular flow and transport in a bi-axial spherical tumbler
Abstract
We study three-dimensional (3D) chaotic dynamics through an analysis of transport in a granular flow in a half-full spherical tumbler rotated sequentially about two orthogonal axes (a bi-axial “blinking” tumbler). The flow is essentially quasi-two-dimensional in any vertical slice of the sphere during rotation about a single axis, and we provide an explicit exact solution to the model in this case. Hence, the cross-sectional flow can be represented by a twist map, allowing us to express the 3D flow as a linked twist map (LTM). We prove that if the rates of rotation about each axis are equal, then (in the absence of stochasticity) particle trajectories are restricted to two-dimensional (2D) surfaces consisting of a portion of a hemispherical shell closed by a “cap''; if the rotation rates are unequal, then particles can leave the surface they start on and traverse a volume of the tumbler. The period-one structures of the governing LTM are examined in detail: analytical expressions are provided for the location of period-one curves, their extent into the bulk of the granular material, and their dependence on the protocol parameters (rates and durations of rotations). Exploiting the restriction of trajectories to 2D surfaces in the case ofmore »
- Authors:
-
- Northwestern Univ., Evanston, IL (United States); Princeton Univ., Princeton, NJ (United States); Los Alamos National Lab. (LANL), Los Alamos, NM (United States)
- Northwestern Univ., Evanston, IL (United States)
- Univ. of Leeds, Leeds (United Kingdom)
- Publication Date:
- Research Org.:
- Los Alamos National Laboratory (LANL), Los Alamos, NM (United States)
- Sponsoring Org.:
- USDOE
- OSTI Identifier:
- 1200683
- Report Number(s):
- LA-UR-14-20941
Journal ID: ISSN 1536-0040
- Grant/Contract Number:
- AC52-06NA25396
- Resource Type:
- Accepted Manuscript
- Journal Name:
- SIAM Journal on Applied Dynamical Systems
- Additional Journal Information:
- Journal Volume: 13; Journal Issue: 2; Journal ID: ISSN 1536-0040
- Publisher:
- Society for Industrial and Applied Mathematics
- Country of Publication:
- United States
- Language:
- English
- Subject:
- 42 ENGINEERING; 97 MATHEMATICS AND COMPUTING; Mathematics; chaotic advection; transport; linked twist maps; granular flow
Citation Formats
Christov, Ivan C., Lueptow, Richard M., Ottino, Julio M., and Sturman, Rob. A study in three-dimensional chaotic dynamics: Granular flow and transport in a bi-axial spherical tumbler. United States: N. p., 2014.
Web. doi:10.1137/130934076.
Christov, Ivan C., Lueptow, Richard M., Ottino, Julio M., & Sturman, Rob. A study in three-dimensional chaotic dynamics: Granular flow and transport in a bi-axial spherical tumbler. United States. https://doi.org/10.1137/130934076
Christov, Ivan C., Lueptow, Richard M., Ottino, Julio M., and Sturman, Rob. Thu .
"A study in three-dimensional chaotic dynamics: Granular flow and transport in a bi-axial spherical tumbler". United States. https://doi.org/10.1137/130934076. https://www.osti.gov/servlets/purl/1200683.
@article{osti_1200683,
title = {A study in three-dimensional chaotic dynamics: Granular flow and transport in a bi-axial spherical tumbler},
author = {Christov, Ivan C. and Lueptow, Richard M. and Ottino, Julio M. and Sturman, Rob},
abstractNote = {We study three-dimensional (3D) chaotic dynamics through an analysis of transport in a granular flow in a half-full spherical tumbler rotated sequentially about two orthogonal axes (a bi-axial “blinking” tumbler). The flow is essentially quasi-two-dimensional in any vertical slice of the sphere during rotation about a single axis, and we provide an explicit exact solution to the model in this case. Hence, the cross-sectional flow can be represented by a twist map, allowing us to express the 3D flow as a linked twist map (LTM). We prove that if the rates of rotation about each axis are equal, then (in the absence of stochasticity) particle trajectories are restricted to two-dimensional (2D) surfaces consisting of a portion of a hemispherical shell closed by a “cap''; if the rotation rates are unequal, then particles can leave the surface they start on and traverse a volume of the tumbler. The period-one structures of the governing LTM are examined in detail: analytical expressions are provided for the location of period-one curves, their extent into the bulk of the granular material, and their dependence on the protocol parameters (rates and durations of rotations). Exploiting the restriction of trajectories to 2D surfaces in the case of equal rotation rates about the axes, a method is proposed for identifying and constructing 3D Kolmogorov--Arnold--Moser (KAM) tubes around the normally elliptic period-one curves. The invariant manifold structure arising from the normally hyperbolic period-one curves is also examined. When the motion is restricted to 2D surfaces, the structure of manifolds of the hyperbolic points in the bulk differs from that corresponding to hyperbolic points in the flowing layer. Each is reminiscent of a template provided by a non-integrable perturbation to a Hamiltonian system, though the governing LTM is not. This highlights the novel 3D chaotic behaviors observed in this model dynamical system.},
doi = {10.1137/130934076},
journal = {SIAM Journal on Applied Dynamical Systems},
number = 2,
volume = 13,
place = {United States},
year = {Thu May 22 00:00:00 EDT 2014},
month = {Thu May 22 00:00:00 EDT 2014}
}
Free Publicly Available Full Text
Publisher's Version of Record
Other availability
Search WorldCat to find libraries that may hold this journal
Cited by: 18 works
Citation information provided by
Web of Science
Web of Science
Save to My Library
You must Sign In or Create an Account in order to save documents to your library.
Works referenced in this record:
Stirring: The Eckart paradigm revisited
journal, November 2010
- Branicki, M.; Kirwan, A. D.
- International Journal of Engineering Science, Vol. 48, Issue 11
Symmetry concepts for the geometric analysis of mixing flows
journal, February 1992
- Franjione, J. G.; Ottino, J. M.; Smith, F. T.
- Philosophical Transactions of the Royal Society of London. Series A: Physical and Engineering Sciences, Vol. 338, Issue 1650
The reversible umbilic bifurcation
journal, January 1998
- Hanßmann, Heinz
- Physica D: Nonlinear Phenomena, Vol. 112, Issue 1-2
Analysis of mixing in three-dimensional time-periodic cavity flows
journal, May 1999
- Anderson, P. D.; Galaktionov, O. S.; Peters, G. W. M.
- Journal of Fluid Mechanics, Vol. 386
The computation of finite-time Lyapunov exponents on unstructured meshes and for non-Euclidean manifolds
journal, March 2010
- Lekien, Francois; Ross, Shane D.
- Chaos: An Interdisciplinary Journal of Nonlinear Science, Vol. 20, Issue 1
Formation of coherent structures by fluid inertia in three-dimensional laminar flows
journal, June 2010
- Pouransari, Z.; Speetjens, M. F. M.; Clercx, H. J. H.
- Journal of Fluid Mechanics, Vol. 654
Segregation-driven organization in chaotic granular flows
journal, October 1999
- Hill, K. M.; Khakhar, D. V.; Gilchrist, J. F.
- Proceedings of the National Academy of Sciences, Vol. 96, Issue 21
Competition between chaos and order: Mixing and segregation in a spherical tumbler
journal, December 2003
- Gilchrist, J. F.; Ottino, J. M.
- Physical Review E, Vol. 68, Issue 6
Blinking Rolls: Chaotic Advection in a Three-Dimensional Flow with an Invariant
journal, January 2005
- Mullowney, P.; Julien, K.; Meiss, J. D.
- SIAM Journal on Applied Dynamical Systems, Vol. 4, Issue 1
Passive scalars, three-dimensional volume-preserving maps, and chaos
journal, February 1988
- Feingold, Mario; Kadanoff, Leo P.; Piro, Oreste
- Journal of Statistical Physics, Vol. 50, Issue 3-4
An analytical study of transport, mixing and chaos in an unsteady vortical flow
journal, May 1990
- Rom-Kedar, V.; Leonard, A.; Wiggins, S.
- Journal of Fluid Mechanics, Vol. 214, Issue -1
On a class of steady confined Stokes flows with chaotic streamlines
journal, March 1990
- Bajer, K.; Moffatt, H. K.
- Journal of Fluid Mechanics, Vol. 212, Issue -1
Mixing and particle dispersion in the wavy vortex regime of Taylor–Couette flow
journal, May 1998
- Rudman, Murray
- AIChE Journal, Vol. 44, Issue 5
Capturing patterns and symmetries in chaotic granular flow
journal, September 2006
- Meier, Steven W.; Cisar, Stephen E.; Lueptow, Richard M.
- Physical Review E, Vol. 74, Issue 3
Exit times and transport for symplectic twist maps
journal, April 1993
- Easton, R. W.; Meiss, J. D.; Carver, S.
- Chaos: An Interdisciplinary Journal of Nonlinear Science, Vol. 3, Issue 2
Chaotic advection in a Rayleigh-Bénard flow
journal, January 1991
- Camassa, R.; Wiggins, S.
- Physical Review A, Vol. 43, Issue 2
Hamiltonian formulation of the equations of streamlines in three-dimensional steady flows
journal, June 1994
- Bajer, K.
- Chaos, Solitons & Fractals, Vol. 4, Issue 6
Chaotic mixing and transport in wavy Taylor–Couette flow
journal, July 2002
- Akonur, Alp; Lueptow, Richard M.
- Physica D: Nonlinear Phenomena, Vol. 167, Issue 3-4
Segregated regions in continuous laminar stirred tank reactors
journal, April 2004
- Arratia, P. E.; Lacombe, J. P.; Shinbrot, T.
- Chemical Engineering Science, Vol. 59, Issue 7
Streamline topology in eccentric Taylor vortex flow
journal, February 1995
- Ashwin, P.; King, G. P.
- Journal of Fluid Mechanics, Vol. 285
Hamiltonian formulation of the equations of streamlines in three-dimensional steady flows
journal, June 1994
- Bajer, K.
- Chaos, Solitons & Fractals, Vol. 4, Issue 6
Invariant manifold templates for chaotic advection
journal, June 1994
- Beigie, Darin; Leonard, Anthony; Wiggins, Stephen
- Chaos, Solitons & Fractals, Vol. 4, Issue 6
Chaotic streamlines in pre-turbulent states
journal, January 1989
- Beloshapkin, V. V.; Chernikov, A. A.; Natenzon, M. Ya.
- Nature, Vol. 337, Issue 6203
Stirring: The Eckart paradigm revisited
journal, November 2010
- Branicki, M.; Kirwan, A. D.
- International Journal of Engineering Science, Vol. 48, Issue 11
�ber Abbildung von Mannigfaltigkeiten
journal, March 1911
- Brouwer, L. E. J.
- Mathematische Annalen, Vol. 71, Issue 1
Mass-transfer enhancement via chaotic laminar
flow within a droplet
journal, January 1999
- Bryden, Michelle D.; Brenner, Howard
- Journal of Fluid Mechanics, Vol. 379
Chaotic advection in three-dimensional unsteady incompressible laminar flow
journal, June 1996
- Cartwright, Julyan H. E.; Feingold, Mario; Piro, Oreste
- Journal of Fluid Mechanics, Vol. 316
Tailored mixing inside a translating droplet
journal, March 2008
- Chabreyrie, R.; Vainchtein, D.; Chandre, C.
- Physical Review E, Vol. 77, Issue 3
An Illustrative Visualization Framework for 3D Vector Fields
journal, September 2011
- Chen, Cheng-Kai; Yan, Shi; Yu, Hongfeng
- Computer Graphics Forum, Vol. 30, Issue 7
Streamline jumping: A mixing mechanism
journal, April 2010
- Christov, Ivan C.; Ottino, Julio M.; Lueptow, Richard M.
- Physical Review E, Vol. 81, Issue 4
Chaotic streamlines in the ABC flows
journal, June 1986
- Dombre, T.; Frisch, U.; Greene, J. M.
- Journal of Fluid Mechanics, Vol. 167, Issue -1
Experimental characterization of the chaotic dynamics of cohesionless particles: application to a V-blender
journal, December 2007
- Doucet, J.; Bertrand, F.; Chaouki, J.
- Granular Matter, Vol. 10, Issue 2
Exit times and transport for symplectic twist maps
journal, April 1993
- Easton, R. W.; Meiss, J. D.; Carver, S.
- Chaos: An Interdisciplinary Journal of Nonlinear Science, Vol. 3, Issue 2
Granular flows in a rotating drum: the scaling law
between velocity and thickness of the flow
journal, January 2007
- Félix, G.; Falk, V.; D'Ortona, U.
- The European Physical Journal E, Vol. 22, Issue 1
Chaotic mixing in a bounded three-dimensional flow
journal, August 2000
- Fountain, G. O.; Khakhar, D. V.; MeziĆ, I.
- Journal of Fluid Mechanics, Vol. 417
Symmetry concepts for the geometric analysis of mixing flows
journal, February 1992
- Franjione, J. G.; Ottino, J. M.; Smith, F. T.
- Philosophical Transactions of the Royal Society of London. Series A: Physical and Engineering Sciences, Vol. 338, Issue 1650
Analysis and Optimization of Kenics Static Mixers
journal, May 2003
- Galaktionov, O. S.; Anderson, P. D.; Peters, G. W. M.
- International Polymer Processing, Vol. 18, Issue 2
Volume-preserving maps with an invariant
journal, April 2002
- Gómez, A.; Meiss, J. D.
- Chaos: An Interdisciplinary Journal of Nonlinear Science, Vol. 12, Issue 2
Reduction of three-dimensional, volume-preserving flows with symmetry
journal, March 1998
- Haller, G.; Mezic, I.
- Nonlinearity, Vol. 11, Issue 2
Analysis on Laminar Chaotic Mixing Based on Configuration of Streak Lobes in an Impeller-Agitated Vessel
journal, May 2012
- Hashimoto, Shunsuke; Osaka, Ryota; Kawamata, Makiko
- Industrial & Engineering Chemistry Research, Vol. 51, Issue 19
Segregation-driven organization in chaotic granular flows
journal, October 1999
- Hill, K. M.; Khakhar, D. V.; Gilchrist, J. F.
- Proceedings of the National Academy of Sciences, Vol. 96, Issue 21
Zero‐helicity Lagrangian kinematics of three‐dimensional advection
journal, May 1991
- Holm, Darryl D.; Kimura, Yoshifumi
- Physics of Fluids A: Fluid Dynamics, Vol. 3, Issue 5
Dynamical systems in pin mixers of single-screw extruders
journal, January 2004
- Hwang, W. R.; Kang, K. W.; Kwon, T. H.
- AIChE Journal, Vol. 50, Issue 7
Effect of interstitial fluid on a granular flowing layer
journal, June 2004
- Jain, Nitin; Ottino, J. M.; Lueptow, R. M.
- Journal of Fluid Mechanics, Vol. 508
Chaotic advection by laminar flow in a twisted pipe
journal, December 1989
- Jones, Scott W.; Thomas, Oran M.; Aref, Hassan
- Journal of Fluid Mechanics, Vol. 209
Mixing by cutting and shuffling 3D granular flow in spherical tumblers
journal, May 2012
- Juarez, Gabriel; Christov, Ivan C.; Ottino, Julio M.
- Chemical Engineering Science, Vol. 73
Mixing by cutting and shuffling
journal, July 2010
- Juarez, G.; Lueptow, R. M.; Ottino, J. M.
- EPL (Europhysics Letters), Vol. 91, Issue 2
Rheology and Mixing of Granular Materials: Rheology and Mixing of Granular Materials
journal, February 2011
- Khakhar, Devang V.
- Macromolecular Materials and Engineering, Vol. 296, Issue 3-4
A case study of chaotic mixing in deterministic flows: The partitioned-pipe mixer
journal, January 1987
- Khakhar, D. V.; Franjione, J. G.; Ottino, J. M.
- Chemical Engineering Science, Vol. 42, Issue 12
Chaotic streamlines in steady bounded three-dimensional Stokes flows
journal, June 1999
- Kroujiline, Dimitri; Stone, H. A.
- Physica D: Nonlinear Phenomena, Vol. 130, Issue 1-2
Experiments on mixing in continuous chaotic flows
journal, March 1992
- Kusch, H. A.; Ottino, J. M.
- Journal of Fluid Mechanics, Vol. 236
Experimental and computational investigation of the laminar flow structure in a stirred tank
journal, April 1999
- Lamberto, D. J.; Alvarez, M. M.; Muzzio, F. J.
- Chemical Engineering Science, Vol. 54, Issue 7
The computation of finite-time Lyapunov exponents on unstructured meshes and for non-Euclidean manifolds
journal, March 2010
- Lekien, Francois; Ross, Shane D.
- Chaos: An Interdisciplinary Journal of Nonlinear Science, Vol. 20, Issue 1
Comparative study of the mixing of free-flowing particles in a V-blender and a bin-blender
journal, March 2007
- Lemieux, M.; Bertrand, F.; Chaouki, J.
- Chemical Engineering Science, Vol. 62, Issue 6
Resonance zones and lobe volumes for exact volume-preserving maps
journal, June 2009
- Lomelí, H. E.; Meiss, J. D.
- Nonlinearity, Vol. 22, Issue 8
Mixing in the Stokes flow in a cylindrical container
journal, August 2002
- Malyuga, V. S.; Meleshko, V. V.; Speetjens, M. F. M.
- Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences, Vol. 458, Issue 2024
A dynamical systems approach to mixing and segregation of granular materials in tumblers
journal, September 2007
- Meier, Steven W.; Lueptow, Richard M.; Ottino, Julio M.
- Advances in Physics, Vol. 56, Issue 5
Composing chaos: An experimental and numerical study of an open duct mixing flow
journal, January 2005
- Metcalfe, Guy; Rudman, M.; Brydon, A.
- AIChE Journal, Vol. 52, Issue 1
On the integrability and perturbation of three-dimensional fluid flows with symmetry
journal, December 1994
- Mezić, I.; Wiggins, S.
- Journal of Nonlinear Science, Vol. 4, Issue 1
Transport in Transitory Dynamical Systems
journal, January 2011
- Mosovsky, B. A.; Meiss, J. D.
- SIAM Journal on Applied Dynamical Systems, Vol. 10, Issue 1
Blinking Rolls: Chaotic Advection in a Three-Dimensional Flow with an Invariant
journal, January 2005
- Mullowney, P.; Julien, K.; Meiss, J. D.
- SIAM Journal on Applied Dynamical Systems, Vol. 4, Issue 1
Parameters and scalings for dry and immersed granular flowing layers in rotating tumblers
journal, July 2012
- Pignatel, Florent; Asselin, Caroline; Krieger, Lucas
- Physical Review E, Vol. 86, Issue 1
Mixing and particle dispersion in the wavy vortex regime of Taylor–Couette flow
journal, May 1998
- Rudman, Murray
- AIChE Journal, Vol. 44, Issue 5
Chaotic particle transport in time-dependent Rayleigh-Bénard convection
journal, December 1988
- Solomon, T. H.; Gollub, J. P.
- Physical Review A, Vol. 38, Issue 12
A numerical and experimental study on advection in three-dimensional Stokes flows
journal, September 2004
- Speetjens, M. F. M.; Clercx, H. J. H.; Heijst, G. J. F. Van
- Journal of Fluid Mechanics, Vol. 514
Inertia-induced coherent structures in a time-periodic viscous mixing flow
journal, August 2006
- Speetjens, M. F. M.; Clercx, H. J. H.; van Heijst, G. J. F.
- Physics of Fluids, Vol. 18, Issue 8
Merger of coherent structures in time-periodic viscous flows
journal, December 2006
- Speetjens, M. F. M.; Clercx, H. J. H.; van Heijst, G. J. F.
- Chaos: An Interdisciplinary Journal of Nonlinear Science, Vol. 16, Issue 4
Chaotic advection, transport and patchiness in clouds of pollution in an estuarine flow
journal, January 2003
- Stirling, J. R.
- Discrete & Continuous Dynamical Systems - B, Vol. 3, Issue 2
Imaging and quantifying mixing in a model droplet micromixer
journal, June 2005
- Stone, Z. B.; Stone, H. A.
- Physics of Fluids, Vol. 17, Issue 6
Linked twist map formalism in two and three dimensions applied to mixing in tumbled granular flows
journal, April 2008
- Sturman, R.; Meier, S. W.; Ottino, J. M.
- Journal of Fluid Mechanics, Vol. 602
Existence of invariant tori in three dimensional maps with degeneracy
journal, July 2012
- Vaidya, Umesh; Mezić, Igor
- Physica D: Nonlinear Phenomena, Vol. 241, Issue 13
On passage through resonances in volume-preserving systems
journal, December 2006
- Vainchtein, D. L.; Neishtadt, A. I.; Mezic, I.
- Chaos: An Interdisciplinary Journal of Nonlinear Science, Vol. 16, Issue 4
Resonant mixing in perturbed action-action-angle flow
journal, August 2008
- Vainchtein, Dmitri L.; Widloski, John; Grigoriev, Roman O.
- Physical Review E, Vol. 78, Issue 2
Coherent structures and chaotic advection in three dimensions
journal, June 2010
- Wiggins, Stephen
- Journal of Fluid Mechanics, Vol. 654
Simulation of flow and mixing of particles in a rotating and rocking cylinder
journal, June 1998
- Wightman, Carolyn; Moakher, Maher; Muzzio, Fernando J.
- AIChE Journal, Vol. 44, Issue 6
Slow axial drift in three-dimensional granular tumbler flow
journal, July 2013
- Zaman, Zafir; D'Ortona, Umberto; Umbanhowar, Paul B.
- Physical Review E, Vol. 88, Issue 1
Chaotic and Resonant Streamlines in the ABC Flow
journal, February 1993
- Zhao, Xiao-Hua; Kwek, Keng-Huat; Li, JI-Bin
- SIAM Journal on Applied Mathematics, Vol. 53, Issue 1
Observability of periodic lines in three-dimensional lid-driven cylindrical cavity flows
journal, June 2012
- Znaien, J.; Speetjens, M. F. M.; Trieling, R. R.
- Physical Review E, Vol. 85, Issue 6
Works referencing / citing this record:
Topological Dynamics of Volume-Preserving Maps Without an Equatorial Heteroclinic Curve
text, January 2020
- Arenson, Joshua G.; Mitchell, Kevin A.
- arXiv
Identifying invariant ergodic subsets and barriers to mixing by cutting and shuffling: Study in a birotated hemisphere
journal, January 2020
- Lynn, Thomas F.; Ottino, Julio M.; Umbanhowar, Paul B.
- Physical Review E, Vol. 101, Issue 1
Persistent structures in a three-dimensional dynamical system with flowing and non-flowing regions
journal, August 2018
- Zaman, Zafir; Yu, Mengqi; Park, Paul P.
- Nature Communications, Vol. 9, Issue 1
Persistent structures in a 3D dynamical system with solid and fluid regions
text, January 2017
- Zaman, Zafir; Yu, Mengqi; Park, Paul P.
- arXiv
Lagrangian Transport and Chaotic Advection in Three-Dimensional Laminar Flows
journal, April 2021
- Speetjens, Michel; Metcalfe, Guy; Rudman, Murray
- Applied Mechanics Reviews, Vol. 73, Issue 3
Optimized mixing by cutting-and-shuffling
preprint, January 2018
- Smith, Lachlan D.; Umbanhowar, Paul B.; Ottino, Julio M.
- arXiv