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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.

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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