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Extremum principles for slow viscous flows with applications to suspensions

Published online by Cambridge University Press:  28 March 2006

Joseph B. Keller ,
Lester A. Rubenfeld  and
John E. Molyneux
Joseph B. Keller
Affiliation:
Courant Institute of Mathematical Sciences, New York University, New York, N.Y.
Lester A. Rubenfeld
Affiliation:
Courant Institute of Mathematical Sciences, New York University, New York, N.Y.
John E. Molyneux
Affiliation:
Department of Mechanical and Aerospace Sciences, University of Rochester, Rochester, N.Y.
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Abstract

Helmholtz stated and Korteweg proved that of all divergenceless velocity fields in a domain, with prescribed values on the boundary, the solution of the Stokes equation minimizes the rate of viscous energy dissipation. Hill & Power and also Kearsley proved the corresponding reciprocal maximum principle involving the stress tensor. We prove generalizations of both these principles to the flow of a liquid containing one or more solid bodies and drops of another liquid. The essential point in doing this is to take account of the motion of the solids or drops, which must be determined along with the flow. We illustrate the use of these principles by deducing several consequences from them. In particular we obtain upper and lower bounds on the effective coefficient of viscosity and a lower bound on the sedimentation velocity of suspensions of any concentration. The results involve the statistical properties of the distribution of suspended particles or drops. Graphs of the bounds are shown for special cases. For very low concentrations of spheres, both bounds on the effective viscosity coefficient are the same, and agree with the results of Einstein and Taylor.

Information

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 30 , Issue 1 , 17 October 1967 , pp. 97 - 125
Copyright
© 1967 Cambridge University Press

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References

Brenner, H. & Cox, R. G. 1963 J. Fluid Mech. 17, 561.
Courant, R. & Hilbert, D. 1953 Methods of Mathematical Physics, 1, 234. New York: Interscience.
Einstein, A. 1906 Annln Phys. 19, 289.
Einstein, A. 1911 Annln Phys. 34, 591.
Friedrichs, K. O. 1929 Nachr. Ges. Wiss., Göttingen, pp. 13–20. See also Courant & Hilbert (1953).
Hashin, Z. 1962 Bounds for viscosity coefficients of fluid mixtures by variational methods. Proceedings of the International Symposium on Second-Order Effects in Elasticity, Plasticity and Fluid Dynamics, Haifa.
Hashin, Z. 1967 Viscosity of Rigid Particle Suspensions. Contributions to Mechanics, Reiner Anniversary Volume. New York: Pergamon Press.
Happel, J. & Brenner, H. 1965 Low Reynolds Number Hydrodynamics with Special Applications to Particulate Media. New Jersey: Prentice-Hall.
Helmholtz, H. 1868 Wiss. Abh. 1, 223–30.
Hill, R. & Power, G. 1956 Q. Jl Mech. Appl. Math. 9, 314–19.
Kearsley, E. A. 1960 Archs ration. Mech. Analysis 5, 347–54.
Korteweg, D. J. 1883 Phil. Mag. (5), 16, 112–18.
Lamb, H. 1945 Hydrodynamics. New York: Dover.
Moore, D. 1959 J. Fluid Mech. 6, 113–30.
Prager, S. 1963 Physica, 29, 129–39.
Rayleigh, Lord 1913 Phil. Mag. (6) 26, 776.
Taylor, G. I. 1932 Proc. Roy. Soc. A, 138, 41.
Thomas, T. Y. 1942 Am. J. Math. 64, 754–67.