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A new technique for treating multiparticle slow viscous flow: axisymmetric flow past spheres and spheroids

Published online by Cambridge University Press:  29 March 2006

Michael J. Gluckman ,
Robert Pfeffer  and
Sheldon Weinbaum
Michael J. Gluckman
Affiliation:
The City College of The City University of New York
Robert Pfeffer
Affiliation:
The City College of The City University of New York
Sheldon Weinbaum
Affiliation:
The City College of The City University of New York
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Abstract

This paper is the first in a series of investigations having the overall objective of developing a new technique for treating the slow viscous motion past finite assemblages of particles of arbitrary shape. The new method, termed the multi-pole representation technique, is based on the theory that any object conforming to a natural co-ordinate system in a particle assemblage can be approximated by a truncated series of multi-lobular disturbances in which the accuracy of the representation is systematically improved by the addition of higher order multipoles. The essential elements of this theory are illustrated by examining the flows past finite line arrays of axisymmetric bodies such as spheres and spheroids which conform to special natural co-ordinate systems. It is demonstrated that this new procedure converges more rapidly and is simpler to use than the method of reflexions and represents the desired boundaries more precisely than the point-force approximation even when the objects are touching one another. Comparison of these solutions with the exact solutions of Stimson & Jeffery (1926) for the two sphere problem demonstrates the rapidity of convergence of this multipole procedure even when the spheres are touching. Drag results are also presented for flows past chains containing up to 101 spheres as well as for chains containing up to 15 prolate or oblate spheroids. The potential value of the technique is suggested by the rapidity with which the drag calculations were made, the 101 sphere problem requiring about 10 seconds on an IBM 360–65 computer to determine both the fluid flow and the drag coefficient.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 50 , Issue 4 , 29 December 1971 , pp. 705 - 740
Copyright
© 1971 Cambridge University Press

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Footnotes

This paper was presented at the International Symposium on Two-Phase Systems, 29 August-2 September 1971, Technion City, Haifa, Israel.

References

Broersma, S. 1960 J. Chem. Phys. 32, 1632.
Burgers, J. M. 1938 Second Report on Viscosity and Plasticity. Amsterdam: North Holland Publishing Co.
Burgers, J. M. 1940 Proc. Konigl. Akad. Wetenschap. (Amsterdam), 43, 425, 646.
Burgers, J. M. 1941 Proc. Konigl. Wetenschap. (Amsterdam), 44, 1045.
Burgers, J. M. 1942 Proc. Konigl. Akad. Wetenschap. (Amsterdam), 45, 9.
Chen, T. C. & Skalak, R. 1970 Appl. Sci. Res. 22, 403.
Davis, M. H. 1969 Chem. Engng. Sci. 24, 1769.
Dean, W. R. & O'Neill, M. E. 1963 Mathematika, 10, 13.
Faxen, H. (with appendix by Dahl) 1925 Ark. Mat. Astr. Fys. 19A, no. 13.
Faxen, H. 1927 Z. angew. Math. Mech. 7, 79.
Gluckman, M. J., Weinbaum, S. & Pfeffer, R. 1971a A new technique for treating slow viscous flows: axisymmetric flow past arbitrary bodies of revolution. To be published.
Gluckman, M. J., Pfeffer, R. & Weinbaum, S. 1971b A theoretical analysis of three spheres falling along their line of centres in an infinite viscous fluid. To be published.
Goldman, A. J., Cox, R. G. & Brenner, H. 1966 Chem. Engng. Sci. 21, 1151.
Haberman, W. L. & Sayre, R. M. 1958 David W. Taylor model basin Rep. no. 1143, Washington, D.C.
Happel, J. & Brenner, H. 1965 Low Reynolds Number Hydrodynamics. Prentice-Hall.
Kynch, G. J. 1959 J. Fluid Mech. 5, 193.
Mcnown, J. S. & Lin, P. N. 1952 Proc. Second Midwestern Conf. Fluid Mech. Iowa State University. Reprint in Eng. 109.
O'Brien, V. 1968 A.I.Ch.E. J. 14 (6), 870.
Payne, L. E. & Pell, W. H. 1960 J. Fluid Mech. 7, 529.
Sampson, R. A. 1891 Phil. Trans. Roy. Soc. A 182, 449.
Savic, P. 1953 Nat. Res. Counc. Canada Rep. no. MT-22.
Slack, G. W. & Matthews, H. W. 1961 Porton Tech. Paper, Chemical Defence Exptl. Establishment, Porton, no. 797.
Smoluchowski, M. 1911 Bull Int. Acad. Polonaise Sci. Lett. 1A, 28.
Smoluchowski, M. 1912 Proc. Fifth Int. Cong. Math. 2, 192.
Stimson, M. & Jeffery, O. B. 1926 Proc. Roy. Soc. A 111, 110.
Tam, C. K. W. 1969 J. Fluid Mech. 38, 537.
Tchen, C. 1954 J. Appl. Phys. 25, 463.
Wakiya, S. 1965 J. Phys. Soc. Japan, 20 (8), 1502.
Wang, H. & Skalak, R. 1969 J. Fluid Mech. 38, 75.