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Theoretical study of the slow motion of a sphere and a fluid in a cylindrical tube

Published online by Cambridge University Press:  28 March 2006

T. Greenstein  and
J. Happel
T. Greenstein
Affiliation:
Department of Chemical Engineering, New York University, Bronx, New York 10453 Present address: Newwark College of Engineering, Newark, N.J.
J. Happel
Affiliation:
Department of Chemical Engineering, New York University, Bronx, New York 10453
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Abstract

The results of a theoretical treatment are presented for the slow flow of a viscous fluid through a circular cylinder within which a small spherical particle is confined. The sphere is situated in an arbitrary position within the cylinder, rotates with an arbitrary constant angular velocity and moves at constant velocity parallel to the wall. Approximate expressions are presented which give the frictional force, torque, and permanent pressure drop caused by the presence of this obstacle in the original Poiseuillian field of flow.

An eccentricity function for the torque on a sphere in a circular cylinder was evaluated numerically. It can be used to predict the wall-effect for the torque as well as the angular velocity with which a ‘dense’ spherical particle will rotate. Expressions are presented which predict the angular velocity of ‘dense’ as well as neutrally buoyant hydrodynamically supported spherical particles.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 34 , Issue 4 , 23 December 1968 , pp. 705 - 710
Copyright
© 1968 Cambridge University Press

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References

Brenner, H. & Happel, J. 1958 J. Fluid Mech. 4, 195.
Famularo, J. 1962 D.Eng.Sci. thesis, New York University.
Greenstein, T. 1967 Ph.D. Thesis, New York University.
Happel, J. & Brenner, H. 1957 A.I.Ch.E. J. 3, 506; also supplement deposited as document 5441 with the American Documentation Institute, Library of Congress, Washington, D.C.
Happel, J. & Brenner, H. 1965 Low Reynolds Number Hydrodynamics. Englewood Cliffs, N.J.: Prentice-Hall.
Jeffrey, R. C. & Pearson, J. R. A. 1965 J. Fluid Mech. 22, 721.