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Extensions of extremum principles for slow viscous flows

  • Richard Skalak (a1)

Abstract

Several generalizations of theorems of the types originally stated by Helmholtz concerning the dissipation of energy in slow viscous flow have been given recently by Keller, Rubenfeld & Molyneux (1967). These generalizations included cases in which the fluid contains one or more solid bodies and drops of another liquid assuming the drops do not change shape. Some further extensions are given herein which allow for drops which may be deformed by the flow and include the effect of surface tension. The admissible boundary conditions have also been extended and particular theorems applicable to infinite domains, spatially periodic flows and to flows in infinite cylindrical pipes are derived. Uniqueness theorems are also proved.

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References

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Courant, R. & Hilbert, D. 1962 Methods of Mathematic Physics, vol. 2, New York: Interscience.
Duffin, R. J. 1956 J. Rat. Mech. and Anal. 5, 939950.
Gurtin, M. E. & Sternberg, E. 1961 Arch. Rat. Mech. and Anal. 8, 99119.
Keller, J. B., Rubenfeld, L. A. & Molyneux, J. E. 1967 J. Fluid Mech. 30, 97125.
Landau, L. D. & Lifshitz, E. M. 1959 Fluid Mechanics. London: Pergamon.
Thomas, T. Y. 1942 Am. J. Math. 64, 75467
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