Hostname: page-component-78c5997874-g7gxr Total loading time: 0 Render date: 2024-11-13T01:26:46.315Z Has data issue: false hasContentIssue false
  • English
  • Français

Wall effects on a sphere translating at constant velocity

Published online by Cambridge University Press:  20 April 2006

A. Ambari ,
B. Gauthier-Manuel  and
E. Guyon
A. Ambari
Affiliation:
Laboratoire d'Hydrodynamique et de Mécanique Physique, ERA No 1000, ESPCI, 10 rue Vauquelin, 75231 Paris Cedex 05
B. Gauthier-Manuel
Affiliation:
Laboratoire d'Hydrodynamique et de Mécanique Physique, ERA No 1000, ESPCI, 10 rue Vauquelin, 75231 Paris Cedex 05
E. Guyon
Affiliation:
Laboratoire d'Hydrodynamique et de Mécanique Physique, ERA No 1000, ESPCI, 10 rue Vauquelin, 75231 Paris Cedex 05
Get access Rights & Permissions [Opens in a new window]

Abstract

We present experimental results on the modified Stokes force F exerted on a sphere in magnetic levitation whose position is kept fixed by an optical feedback system. A Newtonian liquid moves at a constant velocity U relative to the sphere. We consider the motion in two different situations.

(i) When the sphere approaches a wall perpendicular to U, the increase in |F| due to lubrication agrees quantitatively with theoretical results such as those of Brenner (1961) and Maude (1961), obtained neglecting the unsteadiness of the flow field.

(ii) In the complementary situation of a sphere moving along the axis of a cylindrical tube, our results expressed as a function of the eccentricity of the trajectory and of the ratio of the two radii confirm and extend previous theoretical analyses. They show in particular the existence of a minimum of |F| away from the axis of the cylinder and a sharp increase in |F| when the sphere approaches the sidewall. By comparing with the results for a sphere moving parallel to a flat wall, we analyse the effect of the curvature of the cylindrical tube.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 149 , December 1984 , pp. 235 - 253
Copyright
© 1984 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

Ambari, A., Gauthier-Manuel, B. & Guyon, E. 1983 Effect of a plane wall on a sphere moving parallel to it. J. Phys. Lettres 44, L143–L146.Google Scholar
Batchelor, G. K. 1972 Sedimentation in a dilute dispersion of spheres. J. Fluid Mech. 52, 245–268.Google Scholar
Brenner, H. 1961 The slow motion of a sphere through a viscous fluid towards a plane surface. Chem. Engng Sci. 16, 242–251.Google Scholar
Brenner, H. & Happel, J. 1958 Slow viscous flow past a sphere in a cylindrical tube. J. Fluid Mech. 4, 195–213.Google Scholar
Bungay, P. M. & Brenner, H. 1973 The motion of a closely-fitting sphere in a fluid-filled tube. Intl J. Multiphase Flow 1, 25–56.Google Scholar
Christopherson, D. G. & Dowson, D. 1959 An example of minimum energy dissipation in viscous flow. Proc. R. Soc. Lond. A 251, 550–564.Google Scholar
Cooley, M. D. A. & O'Neill, M. E. 1969 On the slow motion generated in a viscous fluid by the approach of a sphere to a plane wall or stationary sphere. Mathematika 16, 37–49. (Note that in their calculations a factor of has been omitted.)Google Scholar
Coutanceau, M. 1968 Mouvement uniforme d'une sphère dans l'axe d'un cylindre contenant un liquide visqueux. J. Méc. 7, 49–67.Google Scholar
Cox, R. G. & Brenner, H. 1967 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, 1753–1777.CrossRefGoogle Scholar
de Gennes, P. G. 1981 Dynamics of concentrated dispersions: a list of problems. Physico-Chem. Hydrodyn. 2, 31–44.Google Scholar
Famuloro, J. 1962 D.Eng.Sci. thesis. New York University.
Faxén, H. 1921 Einwirkung der Gefäßwände auf den Widerstand gegen die Bewegung einer kleinen Kugel in einer zähen Flüssigkeit. Diss. Uppsala, pp. 55–128.
Gauthier-Manuel, B. & Guyon, E. 1980 C.R. Acad. Sci. Paris B 290, 465–467.
Gauthier-Manuel, B., Meyer, R. & Pieranski, P. 1984 J. Phys. E: Sci. Instrum. (to appear).
Goldman, A. J., Cox, R. G. & Brenner, H. 1967 Slow viscous motion of a sphere parallel to a plane wall—I. Motion through a quiescent fluid. Chem. Engng Sci. 22, 637–651.Google Scholar
Haberman, W. & Sayre, R. M. 1958 David Taylor Model Basin Rep. 1143 (US Navy Dept, Washington DC).
Happel, J. & Brenner, H. 1973 Low Reynolds Number Hydrodynamics. Noordhoff.
Hasimoto, H. 1976 Slow motion of a sphere in a cylindrical domain. J. Phys. Soc. Japan, Lett. 41, 2143–2144.Google Scholar
Lee, S. H., Chadwick, R. S. & Leal, L. G. 1979 Motion of a sphere in the presence of a plane interface. Part 1. An approximate solution by generalization of the method of Lorentz. J. Fluid Mech. 93. 705–726 (and Corrigendum, J. Fluid Mech. 104 (1981) 534).Google Scholar
Lee, S. H. & Leal, L. G. 1980 Motion of a sphere in the presence of a plane interface. Part 2. An exact solution in bipolar co-ordinates. J. Fluid Mech. 98, 193–224.Google Scholar
Lorentz, H. A. 1907 Abhandl. theor. Phys. (Leipzig) 1, 23.
MacKay, G. D. M. & Mason, S. G. 1961 Approach of a solid sphere to a rigid plane interface. J. Colloid Sci. 16, 632–635.Google Scholar
MacKay, G. D. M., Suzuki, M. & Mason, S. G. 1963 Approach of a solid sphere to a rigid plane interface—Part 2. J. Colloid Sci. 18, 103–104.Google Scholar
Maude, A. D. 1961 End effects in a falling sphere viscometer. Brit. J. Appl. Phys. 12, 293–295.Google Scholar
O'Neill, M. E. 1964 A slow motion of viscous liquid caused by a slowly moving solid sphere. Mathematika 121, 67–74.Google Scholar
O'Neill, M. E. & Ranger, K. B. 1983 The approach of a sphere to an interface. Phys. Fluids 26, 2035–2042.Google Scholar
O'Neill, M. F. & Stewartson, K. 1967 On the slow motion of a sphere parallel to a nearby plane wall. J. Fluid Mech. 27, 705–724.Google 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.Google Scholar
Shinohara, M. & Hasimoto, H. 1979 The lateral force on a small sphere sedimenting in a viscous fluid bounded by a cylindrical wall. J. Phys. Soc. Japan 46, 320–327.Google Scholar
Stimson, M. & Jeffery, G. B. 1926 The motion of two spheres in viscous fluid. Proc. R. Soc. Lond. A 111, 110–116.Google Scholar
Tözeren, H. 1983 Drag on eccentrically positioned spheres translating and rotating in tubes. J. Fluid Mech. 129, 77–90.Google Scholar
Yuu, S. & Fukui, Y. 1981 Measurement of fluid resistance correction factor for a sphere moving through a viscous fluid toward a plane surface. AIChE J. 27, 168–170.Google Scholar