Hostname: page-component-848d4c4894-m9kch Total loading time: 0 Render date: 2024-05-04T06:54:20.528Z Has data issue: false hasContentIssue false
  • English
  • Français

An anisotropic particle in a simple shear flow: an instance of chaotic scattering

Published online by Cambridge University Press:  19 February 2021

Mahan Raj Banerjee  and
Ganesh Subramanian Open the ORCID record for Ganesh Subramanian [Opens in a new window]
Mahan Raj Banerjee
Affiliation:
Jawaharlal Nehru Centre for Advanced Scientific Research, Jakkur, Bangalore
Ganesh Subramanian*
Affiliation:
Jawaharlal Nehru Centre for Advanced Scientific Research, Jakkur, Bangalore
*
†Email address for correspondence: sganesh@jncasr.ac.in
Get access Rights & Permissions [Opens in a new window]

Abstract

In the Stokesian limit, the streamline topology around a single neutrally buoyant sphere is identical to the topology of pair-sphere pathlines, both in an ambient simple shear flow. In both cases there are fore–aft symmetric open and closed trajectories spatially demarcated by an axisymmetric separatrix surface. We show that the topology of the fluid pathlines around a neutrally buoyant freely rotating spheroid, in simple shear flow, is profoundly different, and will have a crucial bearing on transport from such particles in shearing flows. An inertialess non-Brownian spheroid in a simple shear flow rotates indefinitely in any one of a one-parameter family of Jeffery orbits. The parameter is the orbit constant $C$, with $C = 0$ and $C = \infty$ denoting the limiting cases of a spinning (log-rolling) spheroid, and a spheroid tumbling in the flow–gradient plane, respectively. The streamline pattern around a spinning spheroid is qualitatively identical to that around a sphere regardless of its aspect ratio. For a spheroid in any orbit other than the spinning one ($C >0$), the velocity field being time dependent in all such cases, the fluid pathlines may be divided into two categories. Pathlines in the first category extend from upstream to downstream infinity without ever crossing the flow axis; unlike the spinning case, these pathlines are fore–aft asymmetric, suffering a net displacement in both the gradient and vorticity directions. The second category includes primarily those pathlines that loop around the spheroid, and to a lesser extent those that cross the flow axis, without looping around the spheroid, reversing direction in the process. The residence time, in the neighbourhood of the spheroid, is a smooth function of upstream conditions for pathlines belonging to the first category. In contrast, the number of loops, and thence, the residence time associated with pathlines in the second category, is extremely sensitive to upstream conditions. Plots reveal a fractal structure with singularities distributed on a Cantor-like set, suggesting the existence of a chaotic saddle in the vicinity of the spheroid.

JFM classification

Low-Reynolds-number flows: Low-Reynolds-number flows Complex Fluids: Suspensions Mixing: Chaotic advection
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 913 , 25 April 2021 , A2
Check for updates
Copyright
© The Author(s), 2021. Published by 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

REFERENCES

Acrivos, A. 1971 Heat transfer at high Peclet number from a small sphere freely rotating in a simple shear field. J. Fluid Mech. 46 (2), 233–240.CrossRefGoogle Scholar
Acrivos, A. & Goddard, J.D. 1965 Asymptotic expansions for laminar forced-convection heat and mass transfer. J. Fluid Mech. 23 (2), 273–291.CrossRefGoogle Scholar
Amini, H., Lee, W. & Di Carlo, D. 2014 Inertial microfluidic physics. Lab on a Chip 14, 2739–2761.CrossRefGoogle ScholarPubMed
Angilella, J.-R., Vilela, R.D. & Motter, A.E. 2014 Inertial particle trapping in an open vortical flow. J. Fluid Mech. 744, 183–216.CrossRefGoogle Scholar
Aref, H. 1983 Integrable, chaotic, and turbulent vortex motion in two-dimensional flows. Annu. Rev. Fluid Mech. 15, 345–389.CrossRefGoogle Scholar
Aref, H. & Pomphrey, N. 1980 Integrable and chaotic motions of four vortices. Phys. Lett. A 78, 297–300.CrossRefGoogle Scholar
Aref, H. & Pomphrey, N. 1982 Integrable and chaotic motions of four vortices. I. The case of identical vortices. Proc. R. Soc. Lond. A 380, 359–387.Google Scholar
Aref, H. & Stremler, M.A. 1999 Four-vortex motion with zero total circulation and impulse. Phys. Fluids 11, 3704–3715.CrossRefGoogle Scholar
Batchelor, G.K. 1970 Slender-body theory for particles of arbitrary cross-section in Stokes flow. J. Fluid Mech. 44 (3), 419–440.CrossRefGoogle Scholar
Batchelor, G.K. & Green, J.T. 1972 a The hydrodynamic interaction of two small freely-moving spheres in a linear flow field. J. Fluid Mech. 56 (2), 375–400.CrossRefGoogle Scholar
Batchelor, G.K. & Green, J.T. 1972 b The determination of the bulk stress in a suspension of spherical particles to order c 2. J. Fluid Mech. 56, 401–427.CrossRefGoogle Scholar
Bleher, S., Grebogi, C. & Ott, E. 1990 Bifurcation to chaotic scattering. Physica D 46, 87–121.CrossRefGoogle Scholar
Bleher, S., Ott, E. & Grebogi, C. 1989 Routes to chaotic scattering. Phys. Rev. Lett. 63, 919–922.CrossRefGoogle ScholarPubMed
Brenner, H. 1964 a The Stokes resistance of an arbitrary particle—III: Shear fields. Chem. Engng Sci. 19 (9), 631–651.CrossRefGoogle Scholar
Brenner, H. 1964 b The Stokes resistance of an arbitrary particle—IV Arbitrary fields of flow. Chem. Engng Sci. 19 (10), 703–727.CrossRefGoogle Scholar
Brenner, H. 1966 The Stokes resistance of an arbitrary particle—Part V.: Symbolic operator representation of intrinsic resistance. Chem. Engng Sci. 21 (1), 97–109.CrossRefGoogle Scholar
Cox, R.G., Zia, I.Y.Z. & Mason, S.G. 1968 Particle motions in sheared suspensions XXV. Streamlines around cylinders and spheres. J. Colloid Interface Sci. 27 (1), 7–18.CrossRefGoogle Scholar
Da Cunha, F.R. & Hinch, E.J. 1996 Shear-induced dispersion in a dilute suspension of rough spheres. J. Fluid Mech. 309, 211–223.CrossRefGoogle Scholar
Dabade, V., Marath, N.K. & Subramanian, G. 2015 Effects of inertia and viscoelasticity on sedimenting anisotropic particles. J. Fluid Mech. 778, 133–188.CrossRefGoogle Scholar
Dabade, V., Marath, N.K. & Subramanian, G. 2016 The effect of inertia on the orientation dynamics of anisotropic particles in simple shear flow. J. Fluid Mech. 791, 631–703.CrossRefGoogle Scholar
Eckhardt, B. 1988 Irregular scattering of vortex pairs. Europhys. Lett. 5, 107.CrossRefGoogle Scholar
Eckhardt, B. & Aref, H. 1988 Integrable and chaotic motions of four vortices II. Collision dynamics of vortex pairs. Phil. Trans. R. Soc. Lond. A 326, 655–696.Google Scholar
Eckhardt, B., Schneider, T.M., Hof, B. & Westerweel, J. 2007 Turbulence transition in pipe flow. Annu. Rev. Fluid Mech. 39, 447–468.CrossRefGoogle Scholar
Einstein, A. 1906 Eine neue Bestimmung der Molekuldimensionen. Annalen der Physik 324 (2), 289–306.CrossRefGoogle Scholar
Faisst, H. & Eckhardt, B. 2004 Sensitive dependence on initial conditions in transition to turbulence in pipe flow. J. Fluid Mech. 504, 343–352.CrossRefGoogle Scholar
Gottdiener, L. 1975 The multiple-collision region in non-reactive atom-diatom collisions. Mol. Phys. 29 (5), 1585–1595.CrossRefGoogle Scholar
Gouillart, E., Dauchot, O., Thiffeault, J.-L. & Roux, S. 2009 Open-flow mixing: experimental evidence for strange eigenmodes. Phys. Fluids 21 (2), 023603.CrossRefGoogle Scholar
Harlen, O.G. & Koch, D.L. 1993 Simple shear flow of a suspension of fibres in a dilute polymer solution at high Deborah number. J. Fluid Mech. 252, 187–207.CrossRefGoogle Scholar
Ho, B.P. & Leal, L.G. 1974 Inertial migration of rigid spheres in two-dimensional unidirectional flows. J. Fluid Mech. 65 (2), 365–400.CrossRefGoogle Scholar
Ho, B.P. & Leal, L.G. 1976 Migration of rigid spheres in a two-dimensional unidirectional shear flow of a second-order fluid. J. Fluid Mech. 76 (4), 783–799.CrossRefGoogle Scholar
Jánosi, I.M., Tél, T., Wolf, D.E. & Gallas, J.A.C. 1997 Chaotic particle dynamics in viscous flows: the three-particle Stokeslet problem. Phys. Rev. E 56, 2858–2868.CrossRefGoogle Scholar
Jeffery, G.B. 1922 The motion of ellipsoidal particles immersed in a viscous fluid. Proc. R. Soc. Lond. A 102, 161–179.Google Scholar
Jung, C., Tél, T. & Ziemniak, E. 1993 Application of scattering chaos to particle transport in a hydrodynamical flow. Chaos 3, 555–568.CrossRefGoogle Scholar
Jung, C. & Ziemniak, E. 1992 Hamiltonian scattering chaos in a hydrodynamical system. J. Phys. A: Math. Gen. 25, 3929.CrossRefGoogle Scholar
Kao, S.V., Cox, R.G. & Mason, S.G. 1977 Streamlines around single spheres and trajectories of pairs of spheres in two-dimensional creeping flows. Chem. Engng Sci. 32 (12), 1505–1515.CrossRefGoogle Scholar
Kawakami, A. & Funakoshi, M. 1999 Chaotic motion of fluid particles around a rotating elliptic vortex in a linear shear flow. Fluid Dyn. Res. 25 (4), 167–193.CrossRefGoogle Scholar
Krishnamurthy, D. & Subramanian, G. 2018 Heat or mass transport from drops in shearing flows. Part 2. Inertial effects on transport. J. Fluid Mech. 850, 484–524.CrossRefGoogle Scholar
Kushch, V.I. 1997 Microstresses and effective elastic moduli of a solid reinforced by periodically distributed spheroidal particles. Intl J. Solids Struct. 34 (11), 1353–1366.CrossRefGoogle Scholar
Kushch, V.I. 1998 Elastic equilibrium of a medium containing a finite number of arbitrarily oriented spheroidal inclusions. Intl J. Solids Struct. 35 (12), 1187–1198.CrossRefGoogle Scholar
Kushch, V. 2013 Micromechanics of Composites: Multipole Expansion Approach. Elsevier Science.Google Scholar
Lau, Y.-T., Finn, J.M. & Ott, E. 1991 Fractal dimension in nonhyperbolic chaotic scattering. Phys. Rev. Lett. 66, 978–981.CrossRefGoogle ScholarPubMed
Leal, L.G. & Hinch, E.J. 1971 The effect of weak brownian rotations on particles in shear flow. J. Fluid Mech. 46 (4), 685–703.CrossRefGoogle Scholar
Li, C., Thomases, B. & Guy, R.D. 2019 Orientation dependent elastic stress concentration at tips of slender objects translating in viscoelastic fluids. Phys. Rev. Fluids 4, 031301.CrossRefGoogle Scholar
Marath, N.K., Dwivedi, R. & Subramanian, G. 2017 An orientational order transition in a sheared suspension of anisotropic particles. J. Fluid Mech. 811, R3.CrossRefGoogle Scholar
Marath, N.K. & Subramanian, G. 2017 The effect of inertia on the time period of rotation of an anisotropic particle in simple shear flow. J. Fluid Mech. 830, 165–210.CrossRefGoogle Scholar
Marath, N.K. & Subramanian, G. 2018 The inertial orientation dynamics of anisotropic particles in planar linear flows. J. Fluid Mech. 844, 357–402.CrossRefGoogle Scholar
Noid, D.W., Gray, S.K. & Rice, S.A. 1986 Fractal behavior in classical collisional energy transfer. J. Chem. Phys. 84 (5), 2649–2652.CrossRefGoogle Scholar
Poe, G.G. & Acrivos, A. 1976 Closed streamline flows past small rotating particles: heat transfer at high Peclet numbers. Intl J. Multiphase Flow 2 (4), 365–377.CrossRefGoogle Scholar
Powell, R.L. 1983 External and internal streamlines and deformation of drops in linear two-dimensional flows. J. Colloid Interface Sci. 95 (1), 148–162.CrossRefGoogle Scholar
Rom-Kedar, V., Leonard, A. & Wiggins, S. 1990 An analytical study of transport, mixing and chaos in an unsteady vortical flow. J. Fluid Mech. 214, 347–394.CrossRefGoogle Scholar
Schneider, Tobias M., Eckhardt, B. & Yorke, J.A. 2007 Turbulence transition and the edge of chaos in pipe flow. Phys. Rev. Lett. 99, 034502.CrossRefGoogle ScholarPubMed
Skufca, J.D., Yorke, J.A. & Eckhardt, B. 2006 Edge of chaos in a parallel shear flow. Phys. Rev. Lett. 96, 174101.CrossRefGoogle Scholar
Subramanian, G. & Brady, J.F. 2006 Trajectory analysis for non-brownian inertial suspensions in simple shear flow. J. Fluid Mech. 559, 151–203.CrossRefGoogle Scholar
Subramanian, G. & Koch, D.L. 2005 Inertial effects on fibre motion in simple shear flow. J. Fluid Mech. 535, 383–414.CrossRefGoogle Scholar
Subramanian, G. & Koch, D.L. 2006 a Centrifugal forces alter streamline topology and greatly enhance the rate of heat and mass transfer from neutrally buoyant particles to a shear flow. Phys. Rev. Lett. 96, 134503.CrossRefGoogle ScholarPubMed
Subramanian, G. & Koch, D.L. 2006 b Inertial effects on the orientation of nearly spherical particles in simple shear flow. J. Fluid Mech. 557, 257–296.CrossRefGoogle Scholar
Subramanian, G. & Koch, D.L. 2006 c Inertial effects on the transfer of heat or mass from neutrally buoyant spheres in a steady linear velocity field. Phys. Fluids 18 (7), 073302.CrossRefGoogle Scholar
Torza, S., Henry, C.P., Cox, R.G. & Mason, S.G. 1971 Particle motions in sheared suspensions. XXVI. Streamlines in and around liquid drops. J. Colloid Interface Sci. 35 (4), 529–543.CrossRefGoogle Scholar
Wiggins, S. 2003 Introduction to Applied Nonlinear Dynamical Systems and Chaos, vol. 2. Springer Science & Business Media.Google Scholar
Yang, S., Kim, J.Y., Lee, S.J., Lee, S.S. & Kim, J.M. 2011 Sheathless elasto-inertial particle focusing and continuous separation in a straight rectangular microchannel. Lab on a Chip 11, 266–273.CrossRefGoogle Scholar
Ziemniak, E.M., Jung, C. & Tél, T. 1994 Tracer dynamics in open hydrodynamical flows as chaotic scattering. Physica D 76, 123–146.CrossRefGoogle Scholar