Oscillatory flow of a Newtonian fluid in an elastic tube, which is a model of blood flow in arteries, is analyzed in this paper. For a rigid tube, the steady flow field can be described by Poiseuille’s law and the unsteady flow field by Womersley’s solution. These are the linearized solutions for flow in elastic tubes. To evaluate the importance of nonlinear effects, a perturbation solution is developed realizing that the amplitude of arterial wall movement is small (typically 5–10 percent of the diameter). The nonlinear effects on the amplitude of the wall shear rate, on the amplitude of the pressure gradient, and on the mean velocity profile have been considered. Nonlinear effects on the oscillatory components depend on Womersley’s unsteadiness parameter (α), the ratio between the mean and amplitude of the flow rate, the diameter variation, and the phase difference between the diameter variation and the flow rate (φ) which is indicative of the degree of wave reflection. On the other hand, the mean velocity profile is found to be dependent on the steady-streaming Reynolds number, Rs. When Rs is small, the mean velocity profile is parabolic (1 − ξ2); however, when Rs is large, the velocity profile is distorted by the nonlinear effect and can be described by sin(πξ2). The increase of the amplitude and reduction of the mean of wall shear rate as π changes from 0 to −90 deg suggests an indirect mechanism for the role of hypertension in arterial disease: hypertension → increased wave reflection → wall shear stress is reduced and more oscillatory.

1.
Blennerhassett, P., 1976, “Secondary Motion and Diffusion in Unsteady Flow in a Curved Pipe,” PhD Thesis, Imperial College, London.
2.
Dutta
A.
,
Wang
D. M.
, and
Tarbell
J. M.
,
1992
, “
Numerical Analysis of Flow in an Elastic Artery Model
,”
ASME JOURNAL OF BIOMECHANICAL ENGINEERING
, Vol.
114
, p.
26
26
.
3.
Fung
Y. C.
, and
Yih
C. S.
,
1968
, “
Peristaltic Transport
,”
ASME Journal of Applied Mechanics
, Vol.
35
, p.
669
669
.
4.
Johnson
G. A.
,
Borovetz
H. S.
, and
Anderson
J. L.
,
1992
, “
A Model of Pulsatile Flow in a Uniform Deformable Vessel
,”
J. Biomechanics
, Vol.
25
, p.
91
91
.
5.
Klanchar
M.
,
Tarbell
J. M.
, and
Wang
D. M.
,
1990
, “
In Vitro Study of the Influence of Radial Wall Motion on Wall Shear Stress in an Elastic Tube Model of the Aorta
,”
Circ. Res.
, Vol.
66
, p.
1624
1624
.
6.
Ku
D. N.
,
Giddens
D. P.
,
Zarins
C. K.
, and
Glagov
S.
,
1985
, “
Pulsatile Flow and Atherosclerosis in the Human Carotid Bifurcation
,”
Arteriosclerosis
, Vol.
5
, p.
293
293
.
7.
Lighthill, M. J., 1978, Waves in Fluids, Cambridge University Press.
8.
Ling
S. C.
, and
Atabek
H. B.
,
1972
, “
A Nonlinear Analysis of Pulsatile Flow in Arteries
,”
J. Fluid Meek.
, Vol.
55
, p.
493
493
.
9.
Merillon
J. P.
,
Fontenier
G. J.
,
Lerallut
J. F.
,
Jaffrin
M. Y.
,
Motte
G. A.
,
Genian
C. P.
, and
Gourgon
R. R.
,
1982
, “
Aortic Input Impedance in Normal Man and Arterial Hypertension: Its Modifications During Changes in Aortic Pressure
,”
Cardiovasc. Res.
, Vol.
16
, p.
646
646
.
10.
Nayfeh, A. H., 1973, Perturbation Methods, John Wiley, New York.
11.
Nerem, R. M., and Levesque, M. J., 1987, Handbook of Bioengineering, McGraw-Hill, New York.
12.
Padmanabhan
N.
, and
Pedley
T. J.
,
1987
, “
Three-Dimensional Steady Streaming in an Uniform Tube with an Oscillating Elliptical Cross-Section
,”
J. Fluid Mech.
, Vol.
178
, p.
325
325
.
13.
Pedley, T. J., 1980, The Fluid Mechanics of Large Blood Vessels, Cambridge University Press.
14.
Rayleigh
Lord
,
1884
, “
On the Circulation of Air Observed in Kundt’s Tubes, and on Some Allied Acoustical Problems
,”
Phil. Trans. Roy. Soc.
, Series A. Vol.
175
, p.
1
1
.
15.
Riley
N.
,
1965
, “
Oscillating Viscous Flows
,”
Mathematika
, Vol.
12
, p.
161
161
.
16.
Schlichting
H.
,
1932
, “
Berechnung Ebener Periodischer Grenzschichtstromungen
,”
Phys. Z.
, Vol.
33
, p.
327
327
.
17.
Secomb
T. W.
,
1978
, “
Flow in a Channel with Pulsating Walls
,”
J. Fluid Mech.
, Vol.
88
, p.
273
273
.
18.
Stuart
J. T.
,
1966
, “
Double Boundary Layers in Oscillatory Flow
,”
J. Fluid Mech.
, Vol.
24
, p.
673
673
.
19.
Van Dyke, M., 1964, Perturbation Methods in Fluid Mechanics, Academic, New York.
20.
Wang
D. M.
, and
Tarbell
J. M.
,
1992
, “
Nonlinear Analysis of Flow in an Elastic Tube (Artery): Steady Streaming Effects
,”
J. Fluid Mech.
, Vol.
239
, p.
341
341
.
21.
White, K. C., 1991, “Hemodynamics and Wall Shear Rate Measurements in the Abdominal Aorta of Dogs,” PhD Thesis, The Penn State University.
22.
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.
, Vol.
46
, p.
199
199
.
23.
Womersley
J. R.
,
1957
a, “
Oscillatory Flow in Arteries: The Constrained Elastic Tube as a Model of Arterial Flow and Pulse Transmission
,”
Phys. Med. Biol.
, Vol.
2
, p.
178
178
.
24.
Womersley, J. R., 1957b, “An Elastic Tube Theory of Pulse Transmission and Oscillatory Flow in Mammalian Arteries,” WADC Tech. Rep. 56.
This content is only available via PDF.
You do not currently have access to this content.