Summary
An oscillatory flow of a viscous incompressible fluid in an elastic tube of variable cross section has been investigated at low Reynolds number. The equations governing, the flow are derived under the assumption that the variation of the cross-section is slow in the axial direction for a tethered tube. The problem is then reduced to that of solving for the excess pressure from a second order ordinary differential equation with complex valued Bessel functions as the coefficients. This equation has been solved numerically for geometries of physiological interest and a comparison is made with some of the known theoretical and experimental results.
Similar content being viewed by others
References
Anliker, M., Raman, R. K.: Korotkoff sounds at diastole. A phenomenon of dynamic instability of fluid filled shells. Int. J. Solids Structures2, 467–491 (1966).
Caro, C. G., Pedley, T. J., Schroter, R. C., Seed, W. A.: The mechanics of circulation. Oxford University Press 1978.
Hall, P.: Unsteady viscous flow in a pipe of slowly varying, cross section. J. Fluid Mech.64, 209–226 (1974).
Kurz, W.: Nichtstationäre Strömung in elastischen Modellen arterieller Gefäße. Dissertation, RWTH Aachen, 1980.
Lee, J. S., Fung, T. C.: Flow in a locally constricted tube at low Reynolds numbers. J. Appl. Mech.29, 9–16 (1970).
Lighthill, M. J.: Mathematical biofluid dynamics. Philadelphia: SIAM 1975.
Manton, M. J.: Low Reynolds number flow in slowly varying axisymmetric tubes. J. Fluid Mech.49, 451–459 (1971).
Morgan, G. W., Kiely, J. P.: Wave propagation in a viscous liquid contained in a flexible tube. J. Acoust. Soc. Amer.26, 323–328 (1954).
Ramachandra Rao, A., Devanathan, R.: Pulsatile flow in tubes of varying crosssection. ZAMP24, 203–213 (1973).
Rubinow, S. I., Keller, J. B.: Flow of a viscous fluid through an elastic tube with application to blood flow. J. Theor. Biol.35, 299–313 (1972).
Schneck, D. J., Ostrach, S.: Pulsatile blood flow in a channel of small exponential divergence. I: The linear approximation for low Reynolds number. J. Fluids Engng.97, 353–360 (1975).
Smith, F. T.: Flow through constricted or dilated pipes and channels. Part I and II. Quart. J. Mech. Appl Math.29, 343–363, 364–376 (1976).
Smith, F. T.: The separating flow through severely constricted symmetric tube. J. Fluid Mech.90, 725–754 (1979).
Tomm, D.: Model investigation of sound generation in vessel stenosis. INSERM-Euromech 92, Cardiovascular and Pulmonary Dynamics.71, 179–191 (1978).
Womersley, J. R.: Oscillatory motion of a viscous liquid in a thin walled elastic tube. I: The linear approximation for long waves. Phil. Mag.46, 199–221 (1955).
Womersley, J. R.: An elastic tube theory of pulse transmission and oscillatory flow in mammalian arteries, WADC-TR-56-614, Wright Air Development Center, Wright-Patterson Air Force Base, Ohio. 1957.
Young, D. F., Tsai, F. Y.: Flow characteristics in models of arterial stenosis. I-Steady flow, II-Unsteady flow, J. Biomechanics6, 395–409, 547–559 (1973).
Zeller, H., Reinecke, J.: Geräuschentwicklung in arteriellen Gefäßverengungen. Abhandlungen Aerodyn. Institut der RWTH Aachen Heft24, 7–13 (1980).
Zeller, H., Reinecke, J., Tomm, D., Rieger, H.: Analysis of the sound caused by pulsatile flow through arterial stenoses, Satellite Symposium, Cardiac System Dynamics: Models and Measurements. Proc. of the 28th International Congress of Physiological Sciences. New York: Plenum 1980 (in press).
Author information
Authors and Affiliations
Additional information
With 7 Figures
Rights and permissions
About this article
Cite this article
Rao Ramachandra, A. Oscillatory flow in an elastic tube of variable cross-section. Acta Mechanica 46, 155–165 (1983). https://doi.org/10.1007/BF01176771
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF01176771