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Peristaltic pumping in water waves

Published online by Cambridge University Press:  20 April 2006

M. S. Longuet-Higgins
M. S. Longuet-Higgins
Affiliation:
Department of Applied Mathematics and Theoretical Physics, Silver Street, Cambridge, and Institute of Oceanographic Sciences, Wormley, Surrey
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Abstract

In this paper we calculate the streaming induced by gravity waves passing over a thin fluid layer, one side of which is rigid while the other is a flexible, inextensible membrane. The problem is relevant to some recent laboratory experiments by Allison (1983) on the pumping action of water waves.

On the assumption that the flow is laminar and that the lateral displacement b of the membrane is small compared with the thickness Δ of the fluid layer, we calculate the velocity profile of the streaming U within the layer. This depends on the ratio Δ/δ, where δ is the thickness of the Stokes layers at the upper and lower boundaries. When 0 < Δ/δ > 6 the boundary layers interact strongly and the velocity profile is unimodal. At large values of Δ/δ the profile of U exhibits thin ‘jets’ near the boundaries.

The calculated drift velocities agree as regards order of magnitude with those observed. However, the pressure gradients observed were larger than those calculated, due possibly to turbulence, but probably also to finite-amplitude and end-effects.

The theory given here can be considered as an extension of the theory of peristaltic pumping to flows at higher Reynolds number.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 137 , December 1983 , pp. 393 - 407
Copyright
© 1983 Cambridge University Press

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References

Allison, H. 1983 Streaming of fluid under a near-bottom membrane for utilization of sea-wave energy J. Fluid Mech. 137, 385–392.Google Scholar
Bagnold, R. A. 1947 Sand movement by waves: some small-scale experiments with sand of very low density. J. Inst. Civ. Engrs Lond. 27, 447–810.Google Scholar
Barton, C. & Raynor, S. 1968 Peristaltic flow in tubes Bull. Math. Biophys. 30, 663–683.Google Scholar
Fung, Y. C. & Yih, C. S. 1968 Peristaltic transport. Trans. ASME E: J. Appl. Mech. 35, 669–810.Google Scholar
Jaffrin, M. Y. & Shapiro, A. H. 1971 Peristaltic pumping Ann. Rev. Fluid Mech. 3, 13–36.Google Scholar
Lamb, H. 1932 Hydrodynamics, 6th edn. Cambridge University Press.
Longuet-Higgins, M. S. 1953 Mass transport in water waves. Phil. Trans. R. Soc. Lond A 245, 535–581.Google Scholar
Longuet-Higgins, M. S. 1956 The mechanics of the boundary-layer near the bottom in a progressive wave. In Proc. 6th Intl Conf. on Coastal Engng, C. 10, pp. 184–193.
Rayleigh, Lord 1884 On the circulation of air observed in Kundt's tubes, and on some allied acoustical problems Phil. Trans. R. Soc. Lond. 175, 1–21.Google Scholar
Russell, R. C. H. & Osorio, J. D. C. 1956 An experimental investigation of drift profiles in a closed channel. In Proc. 6th Intl Conf. on Coastal Engng, C. 10, pp. 171–183.