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Three-dimensional steady streaming in a uniform tube with an oscillating elliptical cross-section

Published online by Cambridge University Press:  21 April 2006

N. Padmanabhan  and
T. J. Pedley
N. Padmanabhan
Affiliation:
Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Silver Street, Cambridge CB3 9EW, UK Permanent address: Centre for Atmospheric and Fluids Science, Indian Institute of Technology-Delhi, Hauz Khas, New Delhi 110016, India.
T. J. Pedley
Affiliation:
Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Silver Street, Cambridge CB3 9EW, UK
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Abstract

We analyse the steady streaming generated in an infinite elliptical tube containing a viscous, incompressible fluid when the boundary oscillates in such a way that the area and ellipticity of the cross-section vary with time but remain independent of the longitudinal coordinate. The parameters α−1 = (ν/Ωa02)½ and ε = U0/a0Ω, where ν is the kinematic viscosity, Ω is the oscillation frequency, a0 is the undisturbed semi-major axis and U0 is a typical wall velocity, are taken to be small, so that the Stokes layer is thin and the interaction which leads to the steady streaming can be analysed as a small perturbation. Coupled axial and transverse velocities, both oscillatory and steady, are generated. A complication is the need to specify the tangential as well as the normal velocity component on the tube wall, which requires an assumption concerning its elastic properties. We have considered two cases: (i) constant major axis, in which all boundary points move parallel to the minor axis, and (ii) an inextensible wall. The three-dimensional steady streaming in the core of the tube is computed only in the limit of small steady-streaming Reynolds number, Rs = ε2α2.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 178 , May 1987 , pp. 325 - 343
Copyright
© 1987 Cambridge University Press

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References

Cancelli, C. & Pedley, T. J. 1985 A separated-flow model for collapsible-tube oscillations. J. Fluid Mech. 157, 375–404.Google Scholar
Conrad, W. A. 1969 Pressure—flow relationships in collapsible tubes. IEEE Trans. Bio-med. Engng. BME-16, 284–295.Google Scholar
Gupta, M. M. & Manohar, R. P. 1979 Boundary approximations and accuracy in viscous flow computations. J. Comp. Phys. 31, 265–288.Google Scholar
Longuet-Higgins, M. S. 1953 Mass transport in water waves. Phil. Trans. R. Soc. Lond. A 245, 535–581.Google Scholar
Lyne, W. H. 1971 Unsteady viscous flow in a curved pipe. J. Fluid Mech. 45, 13–31.Google Scholar
Moreno, A. H., Katz, A. I., Gold, L. D. & Reddy, R. V. 1970 Mechanics of distension of dog veins and other very thin-walled tubular structures. Circ. Res. 27, 1069–1080.Google Scholar
Pedley, T. J. 1980 The Fluid Mechanics of Large Blood Vessels. Cambridge University Press.
Rayleigh, Lord 1883 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
Riley, N. 1967 Oscillatory viscous flows. Review and extension. J. Inst. Maths Applics 3, 419–434.Google Scholar
Roache, P. J. 1972 Computational Fluid Dynamics. Albuquerque, Hermosa.
Secomb, T. W. 1978 Flow in a channel with pulsating walls. J. Fluid Mech. 88, 273–288.Google Scholar