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Wall effects on a rotating sphere

Published online by Cambridge University Press:  26 May 2010

QIANLONG LIU
Affiliation:
Department of Mechanical Engineering, Johns Hopkins University, Baltimore, MD 21218, USA
ANDREA PROSPERETTI*
Affiliation:
Department of Mechanical Engineering, Johns Hopkins University, Baltimore, MD 21218, USA Faculty of Science and Technology and J. M. Burgers Centre for Fluid Dynamics, University of Twente, P.O. Box 217, 7500 AE Enschede, The Netherlands
*
Email address for correspondence: prosperetti@jhu.edu

Abstract

The flow induced by a spherical particle spinning in the presence of no-slip planar boundaries is studied by numerical means. In addition to the reference case of an infinite fluid, the situations considered include a sphere rotating near one or two infinite plane walls parallel or perpendicular to the axis of rotation and a sphere centred within a cube. The hydrodynamic force and couple acting on the sphere exhibit a complex behaviour under the sometimes competing, sometimes cooperating action of viscous, inertial and centrifugal effects.

Type
Papers
Copyright
Copyright © Cambridge University Press 2010

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References

REFERENCES

Banks, W. H. H. 1976 The laminar boundary layer on a rotating sphere. Acta Mech. 24, 273287.CrossRefGoogle Scholar
Barrett, K. E. 1967 On the impulsively started rotating sphere. J. Fluid Mech. 27, 779788.CrossRefGoogle Scholar
Besseris, G. J., Miller, I. F. & Yeates, D. B. 1999 Rotational magnetic particle microrheometry: the Newtonian case. J. Rheol. 43, 591608.Google Scholar
Bickley, W. G. 1938 The secondary flow due to a sphere rotating in a viscous fluid. Phil. Mag. 25, 746752.CrossRefGoogle Scholar
Brenner, H. 1964 Slow viscous rotation of an axisymmetric body within a circular cylinder of finite length. Appl. Sci. Res. A 13, 81120.CrossRefGoogle Scholar
Bush, J. W. M., Stone, H. A. & Tanzosh, J. P. 1994 Particle motion in rotating viscous fluids: historical survey and recent developments. Curr. Top. Phys. Fluids 1, 337355.Google Scholar
Chaoui, M. & Feuillebois, F. 2003 Creeping flow around a small sphere in a shear flow close to a wall. Q. J. Mech. Appl. Nath. 56, 381410.CrossRefGoogle Scholar
Collins, W. D. 1955 On the steady rotation of a sphere in a viscous fluid. Mathematika 2, 4247.CrossRefGoogle Scholar
Cox, R. G. & Brenner, H. 1967 a Effect of boundaries on the Stokes resistance of an arbitrary particle. Part 3. Translation and rotation. J. Fluid Mech. 28, 391411.Google Scholar
Cox, R. G. & Brenner, H. 1967 b The slow motion of a sphere through a viscous fluid towards a plane surface. II. Small gap widths, including inertial effects. Chem. Engng Sci. 22, 17531777.CrossRefGoogle Scholar
Damiano, E. R., Long, D. S., El-Khatib, F. H. & Stace, T. M. 2004 On the motion of a sphere in a Stokes flow parallel to a Brinkman medium. J. Fluid Mech. 500, 75101.Google Scholar
Dean, W. R. & O'Neill, M. E. 1963 A slow rotation of viscous liquid caused by the rotation of a solid sphere. Mathematika 10, 1324.CrossRefGoogle Scholar
Dennis, S. C. R. & Duck, P. W. 1988 Unsteady flow due to an impulsively started rotating sphere. Comput. Fluids 16, 291310.CrossRefGoogle Scholar
Dennis, S. C. R., Ingham, D. B. & Singh, S. N. 1981 The steady flow of a viscous fluid due to a rotating sphere. Q. J. Mech. Appl. Math. 34, 361381.Google Scholar
Dennis, S. C. R., Singh, S. N. & Ingham, D. B. 1980 The steady flow due to a rotating sphere at low and moderate Reynolds numbers. J. Fluid Mech. 101, 257279.CrossRefGoogle Scholar
Feuillebois, F. & Lasek, A. 1978 On the rotational historic term in non-stationary Stokes flow. Q. J. Mech. Appl. Math. 31, 435443.CrossRefGoogle Scholar
Fletcher, C. 1988 Computational Techniques for Fluid Dynamics. Springer.Google Scholar
Ganatos, P., Pfeffer, R. & Weinbaum, S. 1980 a Strong interaction theory for the creeping motion of a sphere between plane parallel boundaries. Part 2. Parallel motion. J. Fluid Mech. 99, 755783.CrossRefGoogle Scholar
Ganatos, P., Weinbaum, S. & Pfeffer, R. 1980 b Strong interaction theory for the creeping motion of a sphere between plane parallel boundaries. Part 1. Perpendicular motion. J. Fluid Mech. 99, 739754.CrossRefGoogle Scholar
Garrett, S. J. & Peake, N. 2002 The stability of the boundary layer on a rotating sphere. J. Fluid Mech. 456, 199218.CrossRefGoogle Scholar
Gavze, E. 1990 The accelerated motion of rigid bodies in non-steady Stokes flow. Intl J. Multiphase Flow 16, 153166.CrossRefGoogle Scholar
Goldman, A. J., Cox, R. G. & Brenner, H. 1967 Slow viscous motion of a sphere parallel to a plane wall. Part I. Motion through a quiescent fluid. Chem. Engng Sci. 22, 637651.CrossRefGoogle Scholar
Happel, J. & Brenner, H. 1973 Low-Reynolds Number Hydrodynamics, with Special Applications to Particulate Media, 2nd edn. Noordhoff.Google Scholar
Hollerbach, R., Wiener, R. J., Sullivan, I. S., Donnelly, R. J. & Barenghi, C. F. 2002 The flow around a torsionally oscillating sphere. Phys. Fluids 14, 41924205.CrossRefGoogle Scholar
Jeffery, G. B. 1915 On the steady rotation of a solid of revolution in a viscous fluid. Proc. Lond. Math. Soc. 14, 327338.CrossRefGoogle Scholar
Keh, H. J. & Chen, P. Y. 2001 Slow motion of a droplet between two parallel plane walls. Chem. Engng Sci. 56, 68636871.CrossRefGoogle Scholar
Kim, S. & Karrila, S. 1991 Microhydrodynamics. Butterworth (reprinted by Dover 2005).Google Scholar
Kohama, Y. & Kobayashi, R. 1983 Boundary-layer transition and the behaviour of spiral vortices on rotating spheres. J. Fluid Mech. 137, 153164.CrossRefGoogle Scholar
Lamb, H. 1932 Hydrodynamics, 6th edn. Cambridge University Press.Google Scholar
Landau, L. & Lifshitz, E. 1987 Fluid Mechanics, 2nd edn. Pergamon.Google Scholar
Leach, J., Mushfique, H., Keen, S., Di Leonardo, R., Ruocco, G., Cooper, J. M. & Padgett, M. 2009 Comparison of Faxn's correction for a microsphere translating or rotating near a surface. Phys. Rev. E 79, 026301.Google Scholar
Loper, D. E. 2001 On the structure of a Taylor column driven by a buoyant parcel in an unbounded rotating fluid. J. Fluid Mech. 427, 131165.Google Scholar
Malysa, K. & van de Ven, T. G. M. 1986 Rotational and translational motion of a sphere parallel to a wall. Intl J. Multiphase Flow 12, 459468.CrossRefGoogle Scholar
Milne-Thomson, L. 1968 Theoretical Hydrodynamics, 5th edn. MacMillan.CrossRefGoogle Scholar
Minkov, E., Ungarish, M. & Israeli, M. 2000 The motion generated by a rising particle in a rotating fluid – numerical solutions. Part 1. A short container. J. Fluid Mech. 413, 111148.Google Scholar
Minkov, E., Ungarish, M. & Israeli, M. 2002 The motion generated by a rising particle in a rotating fluid – numerical solutions. Part 2. The long container case. J. Fluid Mech. 454, 345364.CrossRefGoogle Scholar
O'Neill, M. E. 1964 A slow motion of viscous liquid caused by a slowly moving solid sphere. Mathematika 11, 6774.CrossRefGoogle Scholar
Parkin, S. J. W., Knner, G., Nieminen, T. A., Heckenberg, N. R. & Rubinsztein-Dunlop, H. 2007 Microrheology of microlitre samples: probed with rotating optical tweezers. Proc. SPIE 6644, 66440O.CrossRefGoogle Scholar
Peyret, R. & Taylor, T. 1983 Computational Methods for Fluid Flow. Springer.CrossRefGoogle Scholar
Sawatzki, O. 1970 Das strömungsfeld um eine rotierende kugel. Acta Mech. 9, 159214.Google Scholar
Shail, R. 1997 Some regular perturbation solutions in fluid mechanics. Q. J. Mech. Appl. Math. 50, 128147.Google Scholar
Takagi, H. 1974 Slow rotation of two touching spheres in viscous fluid. J. Phys. Soc. Japan 36, 875877.CrossRefGoogle Scholar
Taniguchi, H., Kobayashi, T. & Fukunishi, Y. 1998 Stability of the boundary layer on a sphere rotating in still fluid. Acta Mech. 129, 243253.CrossRefGoogle Scholar
Walters, K. & Waters, N. D. 1963 On the use of a rotating sphere in the measurement of elasto-viscous parameters. Brit. J. Appl. Phys. 14, 667671.CrossRefGoogle Scholar
Walters, K. & Waters, N. D. 1964 The interpretation of experimental results obtained from a rotating-sphere elasto-viscometer. Brit. J. Appl. Phys. 15, 989991.Google Scholar
Wimmer, M. 1988 Viscous flows and instabilities near rotating bodies. Prog. Aerospace Sci. 25, 43103.Google Scholar
Wu, X., Cen, K., Luo, Z., Wang, Q. & Fang, M. 2008 a Measurement on particle rotation speed in gas–solid flow based on identification of particle rotation axis. Exp. Fluids 45, 11171128.Google Scholar
Wu, X., Wang, Q., Luo, Z., Fang, M. & Cen, K. 2008 b Theoretical and experimental investigations on particle rotation speed in a CFB riser. Chem. Engng Sci. 63, 39793987.Google Scholar
Xin, J. & Megaridis, C. 1996 Droplet spindown in a high-temperature gas environment. Intl J. Heat Fluid Flow 17, 567578.CrossRefGoogle Scholar
Zhang, Z. Z. & Prosperetti, A. 2005 Sedimentation of 1.024 particles. In Proceedings of the ASME Fluids Engineering Division Summer Conference, Paper no. FEDSM2005–77133. ASME.Google Scholar