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Consistent section-averaged equations of quasi-one-dimensional laminar flow

Published online by Cambridge University Press:  01 July 2010

PAOLO LUCHINI  and
FRANÇOIS CHARRU
PAOLO LUCHINI*
Affiliation:
Dipartimento di Ingegneria Meccanica, Università di Salerno, 84084 Fisciano (SA), Italy
FRANÇOIS CHARRU
Affiliation:
Institut de Mécanique des Fluides de Toulouse, CNRS – Université de Toulouse, 31400 Toulouse, France
*
†Email address for correspondence: [email protected]
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Abstract

Section-averaged equations of motion, widely adopted for slowly varying flows in pipes, channels and thin films, are usually derived from the momentum integral on a heuristic basis, although this formulation is affected by known inconsistencies. We show that starting from the energy rather than the momentum equation makes it become consistent to first order in the slowness parameter, giving the same results that have been provided until today only by a much more laborious two-dimensional solution. The kinetic-energy equation correctly provides the pressure gradient because with a suitable normalization the first-order correction to the dissipation function is identically zero. The momentum equation then correctly provides the wall shear stress. As an example, the classical stability result for a free falling liquid film is recovered straightforwardly.

JFM classification

Mathematical Foundations: General fluid mechanics Instability: Instability Geophysical and Geological Flows: Shallow water flows
Type
Papers
Information
Journal of Fluid Mechanics , Volume 656 , 10 August 2010 , pp. 337 - 341
Copyright
Copyright © Cambridge University Press 2010

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References

REFERENCES

Batchelor, G. K. 1967 An Introduction to Fluid Dynamics. Cambridge University Press.Google Scholar
Benjamin, T. B. 1957 Wave formation in laminar flow down an inclined plane. J. Fluid Mech. 2, 554–574.CrossRefGoogle Scholar
Benney, D. J. 1966 Long waves on liquid films. J. Math. Phys. 45, 150–155.CrossRefGoogle Scholar
Chang, H.-C. & Demehkin, E. A. 2002 Complex Wave Dynamics on Thin Films. Elsevier Scientific.Google Scholar
Grotberg, J. B. & Jensen, O. E. 2004 Biofluid mechanics in flexible tubes. Annu. Rev. Fluid Mech. 36, 121–147.CrossRefGoogle Scholar
Ruyer-Quil, C. & Manneville, P. 1998 Modeling film flows down inclined planes. Eur. Phys. J. B 6, 277–292.CrossRefGoogle Scholar
Shkadov, V. Ya. 1967 Wave modes in the gravity flow of a thin layer of a viscous fluid. Izv. Akad. Nauk. SSSR, Mekh. Zhidk. Gaza 3, 43–51.Google Scholar
Yih, C. S. 1963 Stability of liquid flow down an inclined plane. Phys. Fluids 6, 321–334.CrossRefGoogle Scholar