Hostname: page-component-745bb68f8f-l4dxg Total loading time: 0 Render date: 2025-01-09T22:05:04.877Z Has data issue: false hasContentIssue false
  • English
  • Français

Nonlinear analysis of flow in an elastic tube (artery): steady streaming effects

Published online by Cambridge University Press:  26 April 2006

D. M. Wang  and
J. M. Tarbell
D. M. Wang
Affiliation:
Department of Chemical Engineering, The Pennsylvania State University, 155 Fenske Laboratory, University Park, PA 16802, USA
J. M. Tarbell
Affiliation:
Department of Chemical Engineering, The Pennsylvania State University, 155 Fenske Laboratory, University Park, PA 16802, USA
Get access Rights & Permissions [Opens in a new window]

Abstract

We analyse the nonlinear flow of a Newtonian fluid in an elastic tube when subjected to an oscillatory pressure gradient with motivation from the problem of blood flow in arteries. Two parameters: the unsteadiness, α = R0(ω/ν)½ and the diameter variation, ε = (Rmax − R0)/R0, are important in characterizing the flow problem. The diameter variation (ε) is taken to be small so that the perturbation method is valid, and asymptotic solutions for two limiting cases of the steady-streaming Reynolds number, Rs = (αε)2 (either small or large), are derived.

The results indicate that nonlinear convective acceleration induces finite mean pressure gradient and mean wall shear rate even when no mean flow occurs. The magnitude of this effect depends on the amplitude of the diameter variation and the flow rate waveforms and the phase angle difference between them, which can be related to the impedance (pressure/flow) phase angle. Changes in the impedance phase angle, which is indicative of the degree of wave reflection, can change the direction of the induced mean flow. It is also shown that the induced mean wall shear rate is proportional to α when α is large. In addition, it is observed that the steady flow structure in the core can be influenced by wave reflection. The streamlines in the core are always parallel to the tube wall when there is no reflection. However, with total reflection, the induced mean flow recirculates between the nodes and points of maximum amplitude in a closed streamline pattern. Implications of the steady-streaming phenomena for physiological flow applications are discussed in a concluding section.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 239 , June 1992 , pp. 341 - 358
Copyright
© 1992 Cambridge University Press

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Batchelor, G. K.: 1967 An Introduction to Fluid Dynamics. Cambridge University Press.
Caro, C. G., Pedley, T. J., Schroter, R. C. & Seed, W. A., 1978 The Mechanics of the Circulation. Oxford University Press.
Dragon, G. A. & Grotberg, J. B., 1991 Oscillatory flow and mass transport in a flexible tube. J. Fluid Mech. 231, 135–155.Google Scholar
Dutta, A. & Tarbell, J. M., 1989 Numerical simulation of sinusoidal flow in a straight elastic tube: effects of phase angles. Biorheol. 26, 1–22.Google Scholar
Fung, Y. C. & Yih, C. S., 1968 Peristaltic transport. Trans. ASME E: J. Appl. Mech. 35, 669–675.Google Scholar
Grotberg, J. B.: 1984 Volume-cycled oscillatory flow in a tapered channel. J. Fluid Mech. 141, 249–264.Google Scholar
Hall, P.: 1974 Unsteady viscous flow in a pipe of slowly varying cross-section. J. Fluid Mech. 64, 209–226.Google Scholar
Ku, D. N., Giddens, D. P., Zarins, C. K. & Glagov, S., 1985 Pulsatile flow and atherosclerosis in the human carotid bifurcation. Arteriosclerosis 5, 293–302.Google Scholar
Ling, S. C. & Atabek, H. B., 1972 A nonlinear analysis of pulsatile flow in arteries. J. Fluid Mech. 55, 493–511.Google Scholar
Longuet-Higgins, M. S.: 1953 Mass transport in water waves. Phil. Trans. R. Soc. Lond. A 245, 535–581.Google Scholar
Mcdonald, W. A.: 1974 Blood Flow in Arteries. Baltimore: Williams and Wilkins Co.
Mercant, G. J. & Davis, S. H., 1989 Modulated stagnation-point flow and steady streaming. J. Fluid Mech. 198, 543–555.Google Scholar
Merillon, J. P., Fontenier, G. J., Lerallut, J. F., Jaffrin, M. Y., Motte, G. A., Genain, C. P. & Gourgon, R. R., 1982 Aortic input impedance in normal man and arterial hypertension: Its modifications during changes in aortic pressure. Cardiovasc. Res. 16, 646–656.Google Scholar
Nerem, R. M. & Levesque, M. J., 1987 In Handbook of Bioengineering, pp. 21.1–21.22. McGraw-Hill.
Nishimura, T., Arakawa, S., Murakami, S. & Kawamura, Y., 1989 Oscillatory viscous flow in symmetric wavy-walled channels. Chem. Eng. Sci. 44, 2137–2148.Google Scholar
Padmanabhan, N. & Pedley, T. J., 1987 Three-dimensional steady streaming in a uniform tube with an oscillating elliptical cross-section. J. Fluid Mech. 178, 325–343.Google Scholar
Riley, N.: 1965 Oscillating viscous flows. Mathematika 12, 161–175.Google Scholar
Riley, N.: 1975 Unsteady laminar boundary layers. SIAM Rev. 17, 274–297.Google Scholar
Schlichting, H.: 1932 Berechnung Ebener Periodischer Grenzschichtstromungen. Phys. Z. 33, 327–335.Google Scholar
Schlichting, H.: 1979 Boundary Layer Theory, Ch. 15, pp. 431–432. McGraw-Hill.
Secomb, T. W.: 1978 Flow in a channel with pulsating walls. J. Fluid Mech. 8, 273–288.Google Scholar
Stuart, J. T.: 1966 Double boundary layers in oscillatory flow. J. Fluid Mech. 24, 673–687.Google Scholar
Taylor, M. G.: 1957 An approach to an analysis of the arterial pulse wave – I. Oscillations in an attenuating line. Phys. Med. Biol. 1, 258–269.Google Scholar
Van Dyke, M. 1964 Perturbation Methods in Fluid Mechanics. Academic.
Womersley, J. R.: 1955 Oscillatory motion of a viscous liquid in a thin-walled elastic tube – I. The linear approximation for long waves. Phil. Mag. (7) 46, 199–221.Google Scholar
Womersley, J. R.: 1957a Oscillatory flow in arteries: the constrained elastic tube as a model of arterial flow and pulse transmission. Phys. Med. Biol. 2, 178–187.Google Scholar
Womersley, J. R.: 1957b An elastic tube theory of pulse transmission and oscillatory flow in mammalian arteries. WADC Tech. Rep. 56–614.Google Scholar