Hostname: page-component-8448b6f56d-c4f8m Total loading time: 0 Render date: 2024-04-19T11:54:59.221Z Has data issue: false hasContentIssue false
  • English
  • Français

Flow in a channel with pulsating walls

Published online by Cambridge University Press:  19 April 2006

T. W. Secomb
T. W. Secomb
Affiliation:
Department of Applied Mathematics and Theoretical Physics, University of Cambridge
Get access Rights & Permissions [Opens in a new window]

Abstract

In this paper calculations are made of the two-dimensional flow field of an incompressible viscous fluid in a long parallel-sided channel whose walls pulsate in a prescribed way. The study covers all values of the unsteadiness parameter α and the steady-streaming Reynolds number. The wall motion is, in general, assumed to be of small amplitude and sinusoidal. Particular attention is given to the steady component of the flow at second order in the amplitude parameter ε. The results for the corresponding problem in axisymmetric geometry are given in an appendix.

Next the following problem is considered: the calculation of the wall motion which will result, in response to prescribed unsteady pressures imposed at the ends of the channel and outside its walls, if the walls are assumed to respond elastically to variations in transmural pressure. It is found that the system has a natural frequency of oscillation, and that resonance will occur if this frequency is close to a multiple of the frequency of the external pressure fluctuations. Finally the preceding work is applied in a discussion of blood flow in the coronary arteries of large mammals.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 88 , Issue 2 , 27 September 1978 , pp. 273 - 288
Copyright
© 1978 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

Abramowitz, M. & Stegun, I. A. (eds) 1965 Handbook of Mathematical Functions. Dover.
Batchelor, G. K. 1967 An Introduction to Fluid Mechanics. Cambridge University Press.
Longuet-Higgins, M. S. 1953 Mass transport in water waves. Phil. Trans. Roy. Soc. A 245, 535.Google Scholar
Nerem, R. M., Rumberger, J. A., Gross, D. R., Muir, W. W. & Geiger, G. L. 1976 Hot-film coronary artery velocity measurements in horses. Cardiovasc. Res. 10, 301.Google Scholar
Proudman, I. 1960 An example of steady laminar flow at large Reynolds number. J. Fluid Mech. 9, 593.Google Scholar
Rayleigh, Lord 1884 On the circulation of air observed in Kundt's tubes, and on some allied acoustical problems. Phil. Trans. Roy. Soc. A 175, 1.Google Scholar
Riley, N. 1965 Oscillating viscous flows. Mathematika 12, 161.Google Scholar
Riley, N. 1967 Oscillatory viscous flows. Review and extension. J. Inst. Math. Appl. 3, 419.Google Scholar
Rumberger, J. A. & Nerem, R. M. 1977 A method-of-characteristics calculation of coronary blood flow. J. Fluid Mech. 82, 429.Google Scholar
Stuart, J. T. 1963 Unsteady boundary layers. In Laminar Boundary Layers (ed. L. Rosenhead), p. 349. Oxford: Clarendon Press.
Stuart, J. T. 1966 Double boundary layers in oscillatory viscous flow. J. Fluid Mech. 24, 673.Google Scholar
Uchida, S. & Aoki, I. 1977 Unsteady flows in a semi-infinite contracting or expanding pipe. J. Fluid Mech. 82, 371.Google Scholar
Wells, M. K., Winter, D. C., Nelson, A. W. & McCARTHY, T. C. 1974 Hemodynamic patterns in coronary arteries. In Fluid Dynamic Aspects of Arterial Disease (ed. R. M. Nerem), p. 36. Ohio State University.