Hostname: page-component-cd9895bd7-p9bg8 Total loading time: 0 Render date: 2024-12-22T16:08:40.942Z Has data issue: false hasContentIssue false
  • English
  • Français

A sphere in a uniformly rotating or shearing flow

Published online by Cambridge University Press:  26 March 2008

J. J. BLUEMINK ,
D. LOHSE ,
A. PROSPERETTI  and
L. VAN WIJNGAARDEN
J. J. BLUEMINK
Affiliation:
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
D. LOHSE
Affiliation:
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
A. PROSPERETTI
Affiliation:
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 Department of Mechanical Engineering, The John Hopkins University Baltimore, MD 21218, USA
L. VAN WIJNGAARDEN
Affiliation:
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
Get access Rights & Permissions [Opens in a new window]

Abstract

It is known that, in a linear shear flow, fluid inertia causes a particle to spin more slowly than the surrounding fluid. The present experiments performed with a sphere with fixed centre, but free to rotate in a fluid undergoing solid-body rotation around a horizontal axis indicate that the spin rate of the sphere can be larger than that of the flow when the sphere is sufficiently far from the axis. Numerical simulations at Reynolds number 5≤Re≤200 confirm this observation. To gain a better understanding of the phenomenon, the rotating flow is decomposed into two shear flows along orthogonal directions. It is found numerically that the cross-stream shear has a much stronger effect on the particle spin rate than the streamwise shear. The region of low stress at the back of the sphere is affected by the shear component of the incident flow. While for the streamwise case the shift is minor, it is significant for cross-stream shear. The results are interpreted on the basis of the effect of the shear flow components on the quasi-toroidal vortex attached in the sphere's near wake. The contributions of streamwise and cross-stream shear to the particle spin can be linearly superposed for Re=20 and 50.

Type
Papers
Information
Journal of Fluid Mechanics , Volume 600 , 10 April 2008 , pp. 201 - 233
Copyright
Copyright © Cambridge University Press 2008

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

Annamalai, P. & Cole, R. 1986 Particle migration in rotating liquids. Phys. Fluids 29, 647–649.CrossRefGoogle Scholar
Auton, T. R. 1987 The lift force on a spherical body in a rotational flow. J. Fluid Mech. 183, 199–218.CrossRefGoogle Scholar
Auton, T. R., Hunt, J. C. R. & Prud'homme, M. 1988 The force excerted on a body in inviscid unsteady non-uniform rotating flow. J. Fluid Mech. 197, 241–257.CrossRefGoogle Scholar
Bagchi, P. 2002 Particle dynamics in inhomogeneous flows at moderate-to-high Reynolds number. PhD thesis, University of Illinois.Google Scholar
Bagchi, P. & Balachandar, S. 2002 a Effect of free rotation on the motion of a solid sphere in linear shear flow at moderate Re. Phys. Fluids 14, 2719–2737.CrossRefGoogle Scholar
Bagchi, P. & Balachandar, S. 2002 b Shear versus vortex-induced lift force on a rigid sphere at moderate Re. J. Fluid Mech. 473, 379–388.CrossRefGoogle Scholar
Bagchi, P. & Balachandar, S. 2003 Inertial and viscous force on a rigid sphere in straining flows at moderate Reynolds numbers. J. Fluid Mech. 481, 105–148.CrossRefGoogle Scholar
Barkla, H. M. & Auchterlonie, L. J. 1971 The Magnus or Robins effect on rotating spheres. J. Fluid Mech. 47, 437–447.CrossRefGoogle Scholar
Bluemink, J. J., van Nierop, E., Luther, S., Deen, N., Magnaudet, J., Prosperetti, A. & Lohse, D. 2005 Asymmetry-induced particle drift in a rotating flow. Phys. Fluids 17, 072106.CrossRefGoogle Scholar
Brown, D., Cortez, R. & Minion, M. 2001 Accurate projection methods for the incompressible Navier-Stokes equations. J. Comput. Phys. 168, 196–216.CrossRefGoogle Scholar
Bugayevskiy, L. M. & Snyder, J. P. 1995 Map Projections – A Reference Manual. Taylor & Francis.Google Scholar
Candelier, F., Angilella, J. R., & Souhar, M. 2004 On the effect of the Boussinesq-Basset force on the radial migration of a stokes particle in a vortex. Phys. Fluids 16, 1765–1776.CrossRefGoogle Scholar
Candelier, F., Angilella, J.-R. & Souhar, M. 2005 On the effect of inertia and history forces on the slow motion of a spherical solid or gaseous inclusion in a solid-body rotation flow. J. Fluid Mech. 545, 113–139.CrossRefGoogle Scholar
Childress, S. 1964 The slow motion of a sphere in a rotating, viscous fluid. J. Fluid Mech. 20, 305–314.CrossRefGoogle Scholar
Clift, R., Grace, J. R. & Weber, M. E. 1978 Bubbles, Drops and Particles. Academic.Google Scholar
Coimbra, C. F. M. & Kobayashi, M. H. 2002 On the viscous motion of a small particle in a rotating cylinder. J. Fluid Mech. 469, 257–286.CrossRefGoogle Scholar
Dandy, D. S. & Dwyer, H. A. 1990 A sphere in shear flow at finite Reynolds number: effect of shear on particle lift, drag, and heat transfer. J. Fluid Mech. 216, 381–410.CrossRefGoogle Scholar
Degani, A. T., Walker, J. D. A. & Smith, F. T. 1998 Unsteady separation past moving surfaces. J. Fluid Mech. 375, 1–38.CrossRefGoogle Scholar
Délery, J. M. 2001 Robert Legendre and Henri Werlé: Toward the elucidation of three-dimensional separation. Annu. Rev. Fluid Mech. 33, 129–154.CrossRefGoogle 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, 257–279.CrossRefGoogle Scholar
Dennis, S. C. R. & Walker, J. D. A. 1971 Calculation of the steady flow past a sphere at low and moderate {R}eynolds numbers. J. Fluid Mech. 48, 771–789.CrossRefGoogle Scholar
Drew, D. A. & Lahey, R. T. 1987 The virtual mass and lift force on a sphere in rotating and straining inviscid flow. Intl J. Multiphase Flow 13, 113–121.CrossRefGoogle Scholar
Gao, H., Ayyaswamy, P. S. & Ducheyne, P. 1997 Dynamics of a microcarrier particle in the simulated microgravity environment of a rotating-wall vessel. Microgravity Sci. Technol. 10, 154–165.Google ScholarPubMed
Ghidersa, B. & Dusek, J. 2000 Breaking of axisymmetry and onset of unsteadiness in the wake of a sphere. J. Fluid Mech. 423, 33–69.CrossRefGoogle Scholar
Gotoh, T. 1990 {Brownian} motion in a rotating flow. J. Statist. Phys. 59, 371–402.CrossRefGoogle Scholar
Happel, J. & Brenner, H. 1965 Low Reynolds Number Hydrodynamics. Prentice Hall.Google Scholar
Harper, E. Y. & Chang, I-Dee 1968 Maximum dissipation resulting from lift in a slow viscous shear flow. J. Fluid Mech. 33, 209–225.CrossRefGoogle Scholar
Herron, I. H., Davis, S. D. & Bretherton, F. P. 1975 On the sedimentation of a sphere in a centrifuge. J. Fluid Mech. 68, 209–234.CrossRefGoogle Scholar
Jenny, M., Bouchet, G. & Dusek, J. 2003 Nonvertical ascension or fall of a free sphere in a newtonian fluid. Phys. Fluids 15, L9–L12.CrossRefGoogle Scholar
Jenny, M., Dusek, J. & Bouchet, G. 2004 Instabilities and transition of a sphere falling or ascending freely in a newtonian fluid. J. Fluid Mech. 508, 201–239.CrossRefGoogle Scholar
Johnson, T. A. & Patel, V. C. 1999 Flow past a sphere up to a Reynolds number of 300. J. Fluid Mech. 378, 19–70.CrossRefGoogle Scholar
Kim, D. & Choi, H. 2002 Laminar flow past a sphere rotating in the streamwise direction. J. Fluid Mech. 461, 365–386.CrossRefGoogle Scholar
Kim, S. & Karrila, S. 1991 Microhydrodynamics. Butterworth-Heinemann.Google Scholar
Kobayashi, M. H. & Coimbra, C. F. M. 2005 On the stability of the Maxey-Riley equation in nonuniform linear flows. Phys. Fluids 17, 113301.CrossRefGoogle Scholar
Kurose, R. & Komori, S. 1999 Drag and lift forces on a rotating sphere in a linear shear flow. J. Fluid Mech. 384, 183–206.CrossRefGoogle Scholar
Lamb, H. 1932 Hydrodynamics, 6th Edn. Dover.Google Scholar
Lee, S. S. 2000 A numerical study of the unsteady wake behind a sphere in a uniform flow at moderate Reynolds numbers. Comput. Fluids 29, 639–667.CrossRefGoogle Scholar
Lee, S. S. & Wilczak, J. M. 2000 The effects of shear flow on the unsteady wakes behind a sphere at moderate Reynolds numbers. Fluid Dyn. Res. 27, 1–22.CrossRefGoogle Scholar
Lighthill, M. J. 1956 Drift. J. Fluid Mech. 1, 31–53.CrossRefGoogle Scholar
Lin, C. J., Peery, J. H. & Schowalter, W. R. 1970 Simple shear flow round a rigid sphere: inertial effects and suspension rheology. J. Fluid Mech. 44, 1–17.CrossRefGoogle Scholar
Lohse, D. & Prosperetti, A. 2003 Controlling bubbles. J. Phys.: Condens. Matter 15, S415–S420.Google Scholar
Magnaudet, J., Rivero, M. & Fabre, J. 1995 Accelerated flows past a rigid sphere or a spherical bubble. J. Fluid Mech. 284, 97–135.CrossRefGoogle Scholar
McLaughlin, J. B. 1991 Inertial migration of a small sphere in linear shear flows. J. Fluid Mech. 224, 261–274.CrossRefGoogle Scholar
Mei, R. 1992 An approximate expression for the shear lift force on a spherical bubble at finite Reynolds number. Intl J. Multiphase Flow 18, 145–147.CrossRefGoogle Scholar
Mikulencak, D. R. & Morris, J. F. 2004 Stationary shear flow around fixed and free bodies at finite {R}eynolds number. J. Fluid Mech. 520, 215–242.CrossRefGoogle Scholar
Milne-Thompson, L. M. 1968 Theoretical hydrodynamics. Macmillan.CrossRefGoogle Scholar
Mortensen, P. H., Andersson, H. I., Gillissen, J. J. J. & Boersma, B. J. 2007 Particle spin in a turbulent shear flow. Phys. Fluids 19, 078109.CrossRefGoogle Scholar
Mullin, T., Li, Y., del Pino, C. & Ashmore, J. 2005 An experimental study of fixed points and chaos in the motion of spheres in a stokes flow. IMA J. Appl. Maths 70, 666–676.CrossRefGoogle Scholar
Naciri, M. A. 1992 Contribution à l'étude des forces exercées par un liquide sur une bulle de gaz: portance, masse ajoutée et interactions hydrodynamiques. PhD thesis, L'Ecole Central de Lyon.Google Scholar
Natarajan, R. & Acrivos, A. 1993 The instability of the steady flow past spheres and disks. J. Fluid Mech. 254, 323–344.CrossRefGoogle Scholar
van Nierop, E. A., Luther, S., Bluemink, J. J., Magnaudet, J., Prosperetti, A. & Lohse, D. 2007 Drag and lift forces on bubbles in a rotating flow. J. Fluid Mech. 571, 439–454.CrossRefGoogle Scholar
Oesterlé, B. & Dinh, T. Bui 1998 Experiments on the lift of a spinning sphere in a range of intermediate Reynolds numbers. Exps. Fluids 25, 16–22.Google Scholar
Paradisi, P. & Tampieri, F. 2001 Stability analysis of solid particle motion in rotational flows. Nuovo Cimento C 24c, 407–420.Google Scholar
Poe, G. G. & Acrivos, A. 1975 Closed-streamline flows past rotating single cylinders and spheres: inertia effects. J. Fluid Mech. 72, 605–623.CrossRefGoogle Scholar
Raju, N. & Meiburg, E. 1997 Dynamics of small, spherical particles in vortical and stagnation point flow fields. Phys. Fluids 9, 299–314.CrossRefGoogle Scholar
Roberts, G. O., Kornfeld, D. M. & Fowlis, W. W. 1991 Particle orbits in a rotating liquid. J. Fluid Mech. 229, 555–567.CrossRefGoogle Scholar
Rubinow, S. I. & Keller, J. B. 1961 The transverse force on a spinning sphere moving in a viscous fluid. J. Fluid Mech. 11, 447–459.CrossRefGoogle Scholar
Saffman, P. G. 1965 The lift on a small sphere in a slow shear flow. J. Fluid Mech. 22, 385–400; and Corrigendum, 31, p. 624 (1968).CrossRefGoogle Scholar
Sridhar, G. & Katz, J. 1995 Drag and lift forces on microscopic bubbles entrained by a vortex. Phys. Fluids. 7, 389–399.CrossRefGoogle Scholar
Surana, A., Grunberg, O. & Haller, G. 2006 Exact theory of three-dimensional flow separation. Part 1. Steady separation. J. Fluid Mech. 564, 57–103.CrossRefGoogle Scholar
Taneda, S. 1978 Visual observations of the flow past a sphere at Reynolds numbers between 104 and 106. J. Fluid Mech. 85, 178–192.CrossRefGoogle Scholar
Thompson, M. C., Leweke, T. & Provansal, M. 2001 Kinematics and dynamics of sphere wake transition. J. Fluids Struct. 15, 575–585.CrossRefGoogle Scholar
Tomboulides, A. G. & Orszag, S. A. 2000 Numerical investigation of transitional and weak turbulent flow past a sphere. J. Fluid Mech. 416, 45–73.CrossRefGoogle Scholar
Van Dommelen, L. L. & Cowley, S. J. 1990 On the Lagrangian description of unsteady boundary layer separation. Part 1. General theory. J. Flui Fluid Mech. 210, 593–626.CrossRefGoogle Scholar
Wang, Y., Lu, X. & Zhuang, L. 2004 Numerical analysis of the rotating viscous flow approaching a solid sphere. Intl J. Numer. Meth. Fluids 44, 905–925.CrossRefGoogle Scholar
Wedemeyer, E. H. 1964 The unsteady flow within a spinning cylinder. J. Fluid Mech. 20, 383–399.CrossRefGoogle Scholar
Weisenborn, A. J. 1985 Drag on a sphere moving slowly in a rotating viscous fluid. J. Fluid Mech. 153, 215–227.CrossRefGoogle Scholar
Williams, J. C. III 1977 Incompressible boundary-layer separation. Annu. Rev. Fluid Mech. 9, 113–144.CrossRefGoogle Scholar
Ye, J. & Rocco, M. C. 1992 Particle rotation in a Couette flow. Phys. Fluids A 4, 220–224.CrossRefGoogle Scholar
Zhang, Z. & Prosperetti, A. 2005 A second-order method for three-dimensional particle simulation. J. Comput. Phys. 210, 292–324.CrossRefGoogle Scholar
Zhang, Z. Z., Botto, L. & Prosperetti, A. 2006 Microstructural effects in a fully-resolved simulation of 1,024 sedimenting spheres. In IUTAM Symposium on Computational Approaches to Multiphase Flow, Fluid Mechanics and Its Applications, vol. 81.Google Scholar