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Pulse propagation in a viscoelastic tube containing a viscous liquid
Pulse propagation in a viscoelastic tube containing a viscous liquid D. W. Barclay
Department of Mathematics and StatL~tics, University of New Brunswick, Fredericton, Canada E3 B 5A3 T . B. M o o d i e
Department of Mathematics and Statistics, University of Alberta. Edmonton, Canada T6G 2GI (Received February 1986; revised September 1986)
Previous attempts at analysing tube propagation in a viscoelastic tube containing a viscous liquid have concentrated on examining sinusoidal wave trains of infinite length. This paper takes viscosity into account in the formulation of initial and boundary value problems which mimic experimental configurations.
Keyworks: pulse propagation, viscosity, boundary value problems, flexible tubes Attempts to analyse blood flow in arteries have generated a substantial theoretical and experimental literature on fluid-filled flexible tubes.~4 For recent developments on fluid filled tubes see reference 5. If large wall displacements, large pressure perturbations, or stress dependent wall properties are to be considered, a nonlinear theory is, in general, required. To reduce complications and focus attention on the important wallfluid interaction, simpler models have been considered. Recent experimental results on pulse generation, propagation, and reflection in fluid filled latex rubber tubes have been reported in reference 5. In order to examine these results a model has been proposed and tested experimentally in references 6-8. Unlike most previous models which have concentrated on examining sinusoidal wave trains of infinite length, the authors have attempted to formulate initial and boundary value problems which mimic experimental configurations. Their results indicate that both dispersion and dissipation are present and that the latter effect is due, principally, to tube wall properties. In this paper they complete their consideration of the effects of the linear terms in the equations governing the model by incorporating fluid viscosity. This complicates the equations and adds extra boundary conditions. Since the parameter 1/m = v/Rc. which determines the viscous effects, is very much smaller than unity, one is
hesitant to introduce these additional complications. Here, R is the mean radius of the tube, u the coefficient of kinematic viscosity, and c, a reference speed specified in the next section. By presenting the first exact results for a transient pulse in a viscoelastic and distensible tube containing a viscous liquid it can be shows that the fluid viscosity introduces additional damping early in the motion (near field) and has a discernable effect upon the waveform as the pulse propagates down the tube. Formulation
Based on the analysis in reference 6 and the subsequent experimental investigation in reference 7, the following set of equations are employed:
0
)
P
"~gt + l W = ~
yct 02W 20 or-
Ow at
Op + l__fO2w low 02w + --- + a,. m \ Or2 r Or Ox2
Ou_ Ot
Op
(1) w~
)
(2)
1 {02u l Ou OZu~
Ox+m~-~r'-+,'Or+-~xQ
(3)
Appl. Math. Modelling, 1987, Vol. 11, June 215
Pulse propagation in a viscoelastic tube: D. W. Barclay and T. B. Moodie
Ow w au --++- = 0 Or r Ox
(4)
20w P = P t - ~ - ~ r (1 ,x, t)
(5)
(1,x,t) + w(1,x,t) = 0
(13)
Equations (1), (2) and (12) are now investigated together with equations (6), (7), (9), (10) and (13). Denote the Fourier sine transform off(r,x,t) by:
f(r,A.t) = I~f(r, x,t) sin (~')dx
subject to the boundary conditions:
pi(x,t) = p(1,x,t)
Ow
(14)
(6) and in turn the Laplace transform off(r,A,t) by:
OW
w(1 ,x,t) = ~
(x,t)
(7)
e-"~(r,At)dt
f(r,h,s) =
(15)
(I
u(1,x,t) = 0
(8)
w(r,O,t) = 0
(9)
p(r,O,t) = poO(t)H(t)
(10)
and quiescent initial conditions. These equations are in nondimensional form. All lengths are divided by R, the mean radius of the tube, a = h/R, where /7 is the thickness of the wall, y and p are wall and fluid densities, respectively, and W the radial wall displacement. If Ge is the equilibrium shear modulus for the incompressible, isotropic, linear viscoelastic wall, a speed is introduced as follows:
and is used to divide all velocities. Time is divided by R/co and pressure by pc~. The transmural pressure P takes the radial viscous stress into account and the fluid velocity 0 has physical components w,u in.the radial and longitudinal directions. The problem is cylindrically symmetric. The parameter:
02p
and since re = 0.15 in the numerical calculations with m ~- 9000, the term (2/m)sl~' is dropped. In transformed form equations (1), (2) and (12) now read:
s~=
d/~
1]-d2ff
ld~,
(17)
[,
1/
]
- d---2+ m [~r_~ + r -d-7 - [A- + ~5} Jff
dr 2
+Sd
r d r - A'~/~= -Ap00(s)
(18)
(19)
where equations (9) and (10) have been used. These equations are subject to the boundary conditions
/~(1,A,s) =/Ji(A,s)
(20)
6,(1,A,s) = sl~'(A,s)
(21)
0¢,
3--7(l'A's) + ~(1 ,A,s) = 0
(12)
Consider the initial-boundary value problem formulated in the previous section. At r = 1, rewrite equation (4) as:
Appl. Math. Modelling, 1987, Vol. 11, June
(22)
obtained from equations (6), (7) and (13). The general solutions to equations (18) and (19), finite at r = 0, are:
~(r, 2, s) = A(2, s)lo(2r ) + rpo ~(s)/2]
(23)
~(r, 2, s) = BO, s)ll(kr) - [A(2, s)2I, (¢r)/s]
(24)
where: k
1 Op . O~-p
Transform solution
216
16,
(11 )
where v is the coefficient of kinematic viscosity of the fluid. For the system considered in reference 3 with R = 0.4cm, co = 833 cm s -*, and for a liquid similar to whole blood, assuming the shear rates are such that non-Newtonian behaviour is negligible, m would have a value of approximately 9000. p denotes the internal pressure of the fluid, H(t) is the Heaviside step function, and qb(t) models the experimental input of reference 7. Clearly, in dealing with wave propagation problems, it is appropriate to introduce time scales related to the wave speeds involved and the parameter m is preferable to a Reynolds number which one would normally introduce in flow problems. In the present context, u/co 1, and nonlinear terms in equations (2) and (3) are omitted. Note also that equations (2)-(4) may be combined to give: -0--~+ r a-7 + aT.2= 0
ls+:l
/~, = [ 7 s 2 + 2(1 + r j ) l I~
co = (2Gca/p),/2
m = Rco/v
Transforming equation (1) and noting equations (12) and (7):
= (A 2 +
ms) I/2
(25)
and 1,,(z) denotes the modified Bessel function of the first kind. A and B are to be determined from the boundary conditions. Once A and B have been fixed using equations (19)-(21), expressions for wall displacement, radial fluid velocity and fluid pressure follow. The axial fluid velocity is then determined from equation (4). The fluid
Pulse propagation in a viscoelastic tube: D. W. Barclay and T. B. Moodie
+(t)
pressure is the quantity normally measured experimentally and attention is confined to determining p. It follows that"
_ po@(s)/,,(Ar) + poCk(s______)) AD(A,s)I.(A) 2
s2 +
[ ll(k) X Lkl,,(k )
l.(k) ] _
2(1 + r~)
s
(27)
A/~--~-)J 1
The mean fluid pressure
{-f~(2)@(t)
+ 2f~2(2)[f2(2)T¢@¢(,;t, t) + (1 -- fl2(2)~)~,(2, t)-I}
(38) where b ° is the leading term in an expansion of,b,, which is valid for large values of the parameter m. However, even though ,~o can be inverted analytically, the following terms in the series cannot, and the expansion is of limited use. In the next section numerical results from equations (29) and (38) are obtained. A graphical comparison of these results will then provide a useful indication of the importance of fluid viscosity on pulse propagation in fluid-filled tubes.
p,,,(x,t) is given by:
pm (X,t) = l I IAPdA = 2 Itlfp(r,x,t)dr
Results and c o n c l u s i o n s
Equations (29) and (38) have been inverted numerically. In equation (38) the Fourier transform has been handled by applying the Gaussian quadrature method of Hurwitz and Zweifel 9 for the evaluation of Fourier integrals. For the double transform in equation (29) this method is
(28)
so that it follows immediately from equation (26) that: ,.~<,@(s)l. (A)
2fl(,~)
= Po ~
(26)
where: D(A,s) =
.b°(2, t) = Po - ~
po@(s)
(29)
Since inversion of the double transform (29) appears intractable analytically, the solution proceeds numerically. As a check, note that as m --* ov in equation (29) the double transform of the mean pressure b ° is recovered for the inviscid case, namely:
pl,~(s)
2po~(s)s 2
;~
,~3C(;~,s)
.b°(2, s ) - - where:
C(A,s) = 2(1 + rcs) + s 2
(_~ /,,(A)/ + A~I(A)]
A
(30)
(31) 0
The Laplace transform can be inverted in terms of the convolutions:
I
0
to
2t o t
Figure 1 Pressure input function
@,O.,t) = fo@(Z)~b,(;t,t - z)dz
(32)
and: • c(A,t) =
fl
qb(r)qJc(A,t- r)dr
(33)
1.4
o
1.2
where: ~bs(2,t) =
125
1.0
e-m~o'sin(f~aU2t)/a u2
q4(h,t) = e -n-k' cos
(f~a'/2t)
(34)
o 0.8
(35)
,~
"
0.6
and introducing: a(A) = 2 -
/
0.4 .O.2(h)"rc 2
(36)
0.2 0
Ft(A) =
+ AIj(A)]
(37)
-°"20
I
i
I
I
I
i
I
I
I
I
20
40
60
80
100
120
140
160
180
200
t
Equation (30) then gives the Fourier transform of the inviscid mean fluid pressure as:
Figure 2 V a r i a t i o n o f Pm/Pow i t h t i m e : ( (. . . .
) inviscid f l u i d ;
) viscous fluid
Appl. Math. Modelling, 1987, Vol. 11, June
217
Pulse propagation in a viscoelastic tube: D. W. Barclay and T. B. Moodie applied in tandem with Crump's ~° Fourier series approximation of the inverse Laplace transform. Numerical results are obtained for a semi-infinite tube with physical parameters corresponding to tube B of reference 8. The values are h = 0.035 cm, R = 0.4 cm, ~/p = 1.1, and Co = 833cm s - t . Based upon a previous study v z~ is taken as 0.15, which corresponds to a characteristic time of 0.72 x 10 -4 s. As indicated previously, if the liquid in the tube is whole blood, then m = 9000. This value is used for m in the numerical work. The pressure input is determined by the choice: "/7
• (t) = ~{1 -4- cos to (t - to)}H(2t o - t)
t >1 0
The graph of this function is shown in Figure 1, and the function has been chosen to mimic the pressure input of the experiments in reference 8. Again, based on reference 8, t, = 6. The mean pressure for both the inviscid and viscous cases is displayed graphically in Figure 2 for various stations along the tube. As expected, both pulses broaden and attenuate as they propagate down the tube. There is a definite increase in the damping of the pulse due to the fluid viscosity and this is plainly visible. It is clear then that pulse attenuation in a fluid-filled tube is due not only to the viscoelastic effect of the wall but is significantly influenced by the fluid viscosity. The results reported here are distinctive in the field of wave propagation in fluid-filled tubes in that the exact
218
Appl. Math. Modelling, 1987, Vol. 11, June
solution for a pulse wave in a viscous fluid contained in a viscoelastic tube has been obtained. The results reported concerning the effects of viscosity are surprising in the light of the values of the p a r a m e t e r m employed. This effect upon pulses could not have been anticipated on the basis of frequency plots alone in the absence of Fourier synthesis. The results also have applications in the transient response of fluid lines. ~
References 1 Skalak, R. Wave Propagation in blood flow, in Biomechanics, Proc. Syrup. Appl. Mech. Div. A S M E (ed. Fung. Y. C.), 1966. pp. 20-46 2 Skalak, R., Keller, S.R. and Secomb, T.W. Mechanics of blood flow, J. Biomed. Eng., 1981, 103, pp. 102-115 3 Anliker, M. Towards a nontraumatie study of the circulatory system, in Biomechanics (eds. Fung, Y.C., Perrone, N. and Anliker, M.), Prentice-Hall, New Jersey, 1972, pp. 337-379 4 Fung, Y.C. Biomechanics: a survey of the blood flow problem, Advances in Applied Mechanics Vol. II (ed. Yih, C-S.), Academic Press, New York, 1971, pp. 65-130 5 Waves in fluid filled tubes, Europ. Mech. Coll. 179 (eds. Buggisch, H. and Mainardi, F.), Bologna, 1985, pp. 24-25 6 Moodie, T.B., Mainardi, F, and Tait, R J . Pressure pulses in fluid filled distensible tubes, Mecc., 1985, 29, pp. 33-37 7 Moodie, T.B., Barclay, D.W. and Greenwald, S.E. Impulse propagation in liquid-filled distensible tubes: theory and experiment for intermediate to long wavelengths, Acta Mech. 1986, 59, 47-58 8 Newman, D.J., Greenwald, S.E. and Moodie, T.B. Reflections from elastic discontinuities, Med. Biol. Eng. Comput., 1983, 21,697-701 9 Hurwitz, H. and Zweifel, R.F. Numerical quadrature of Fourier transform integrals, M.T.A.C., 1956, 10, 140-149 l0 Crump, K.S. Numerical inversion or Laplace transforms using a Fourier series approximation, J.A.C.M., 1976, 23, pp. 89-96 11 Rieutford, E. Transient response o1 fluid viscoelastic lines, J. Fhdds Eng., 1982, 104, pp. 335-341
Department of Mathematics and StatL~tics, University of New Brunswick, Fredericton, Canada E3 B 5A3 T . B. M o o d i e
Department of Mathematics and Statistics, University of Alberta. Edmonton, Canada T6G 2GI (Received February 1986; revised September 1986)
Previous attempts at analysing tube propagation in a viscoelastic tube containing a viscous liquid have concentrated on examining sinusoidal wave trains of infinite length. This paper takes viscosity into account in the formulation of initial and boundary value problems which mimic experimental configurations.
Keyworks: pulse propagation, viscosity, boundary value problems, flexible tubes Attempts to analyse blood flow in arteries have generated a substantial theoretical and experimental literature on fluid-filled flexible tubes.~4 For recent developments on fluid filled tubes see reference 5. If large wall displacements, large pressure perturbations, or stress dependent wall properties are to be considered, a nonlinear theory is, in general, required. To reduce complications and focus attention on the important wallfluid interaction, simpler models have been considered. Recent experimental results on pulse generation, propagation, and reflection in fluid filled latex rubber tubes have been reported in reference 5. In order to examine these results a model has been proposed and tested experimentally in references 6-8. Unlike most previous models which have concentrated on examining sinusoidal wave trains of infinite length, the authors have attempted to formulate initial and boundary value problems which mimic experimental configurations. Their results indicate that both dispersion and dissipation are present and that the latter effect is due, principally, to tube wall properties. In this paper they complete their consideration of the effects of the linear terms in the equations governing the model by incorporating fluid viscosity. This complicates the equations and adds extra boundary conditions. Since the parameter 1/m = v/Rc. which determines the viscous effects, is very much smaller than unity, one is
hesitant to introduce these additional complications. Here, R is the mean radius of the tube, u the coefficient of kinematic viscosity, and c, a reference speed specified in the next section. By presenting the first exact results for a transient pulse in a viscoelastic and distensible tube containing a viscous liquid it can be shows that the fluid viscosity introduces additional damping early in the motion (near field) and has a discernable effect upon the waveform as the pulse propagates down the tube. Formulation
Based on the analysis in reference 6 and the subsequent experimental investigation in reference 7, the following set of equations are employed:
0
)
P
"~gt + l W = ~
yct 02W 20 or-
Ow at
Op + l__fO2w low 02w + --- + a,. m \ Or2 r Or Ox2
Ou_ Ot
Op
(1) w~
)
(2)
1 {02u l Ou OZu~
Ox+m~-~r'-+,'Or+-~xQ
(3)
Appl. Math. Modelling, 1987, Vol. 11, June 215
Pulse propagation in a viscoelastic tube: D. W. Barclay and T. B. Moodie
Ow w au --++- = 0 Or r Ox
(4)
20w P = P t - ~ - ~ r (1 ,x, t)
(5)
(1,x,t) + w(1,x,t) = 0
(13)
Equations (1), (2) and (12) are now investigated together with equations (6), (7), (9), (10) and (13). Denote the Fourier sine transform off(r,x,t) by:
f(r,A.t) = I~f(r, x,t) sin (~')dx
subject to the boundary conditions:
pi(x,t) = p(1,x,t)
Ow
(14)
(6) and in turn the Laplace transform off(r,A,t) by:
OW
w(1 ,x,t) = ~
(x,t)
(7)
e-"~(r,At)dt
f(r,h,s) =
(15)
(I
u(1,x,t) = 0
(8)
w(r,O,t) = 0
(9)
p(r,O,t) = poO(t)H(t)
(10)
and quiescent initial conditions. These equations are in nondimensional form. All lengths are divided by R, the mean radius of the tube, a = h/R, where /7 is the thickness of the wall, y and p are wall and fluid densities, respectively, and W the radial wall displacement. If Ge is the equilibrium shear modulus for the incompressible, isotropic, linear viscoelastic wall, a speed is introduced as follows:
and is used to divide all velocities. Time is divided by R/co and pressure by pc~. The transmural pressure P takes the radial viscous stress into account and the fluid velocity 0 has physical components w,u in.the radial and longitudinal directions. The problem is cylindrically symmetric. The parameter:
02p
and since re = 0.15 in the numerical calculations with m ~- 9000, the term (2/m)sl~' is dropped. In transformed form equations (1), (2) and (12) now read:
s~=
d/~
1]-d2ff
ld~,
(17)
[,
1/
]
- d---2+ m [~r_~ + r -d-7 - [A- + ~5} Jff
dr 2
+Sd
r d r - A'~/~= -Ap00(s)
(18)
(19)
where equations (9) and (10) have been used. These equations are subject to the boundary conditions
/~(1,A,s) =/Ji(A,s)
(20)
6,(1,A,s) = sl~'(A,s)
(21)
0¢,
3--7(l'A's) + ~(1 ,A,s) = 0
(12)
Consider the initial-boundary value problem formulated in the previous section. At r = 1, rewrite equation (4) as:
Appl. Math. Modelling, 1987, Vol. 11, June
(22)
obtained from equations (6), (7) and (13). The general solutions to equations (18) and (19), finite at r = 0, are:
~(r, 2, s) = A(2, s)lo(2r ) + rpo ~(s)/2]
(23)
~(r, 2, s) = BO, s)ll(kr) - [A(2, s)2I, (¢r)/s]
(24)
where: k
1 Op . O~-p
Transform solution
216
16,
(11 )
where v is the coefficient of kinematic viscosity of the fluid. For the system considered in reference 3 with R = 0.4cm, co = 833 cm s -*, and for a liquid similar to whole blood, assuming the shear rates are such that non-Newtonian behaviour is negligible, m would have a value of approximately 9000. p denotes the internal pressure of the fluid, H(t) is the Heaviside step function, and qb(t) models the experimental input of reference 7. Clearly, in dealing with wave propagation problems, it is appropriate to introduce time scales related to the wave speeds involved and the parameter m is preferable to a Reynolds number which one would normally introduce in flow problems. In the present context, u/co 1, and nonlinear terms in equations (2) and (3) are omitted. Note also that equations (2)-(4) may be combined to give: -0--~+ r a-7 + aT.2= 0
ls+:l
/~, = [ 7 s 2 + 2(1 + r j ) l I~
co = (2Gca/p),/2
m = Rco/v
Transforming equation (1) and noting equations (12) and (7):
= (A 2 +
ms) I/2
(25)
and 1,,(z) denotes the modified Bessel function of the first kind. A and B are to be determined from the boundary conditions. Once A and B have been fixed using equations (19)-(21), expressions for wall displacement, radial fluid velocity and fluid pressure follow. The axial fluid velocity is then determined from equation (4). The fluid
Pulse propagation in a viscoelastic tube: D. W. Barclay and T. B. Moodie
+(t)
pressure is the quantity normally measured experimentally and attention is confined to determining p. It follows that"
_ po@(s)/,,(Ar) + poCk(s______)) AD(A,s)I.(A) 2
s2 +
[ ll(k) X Lkl,,(k )
l.(k) ] _
2(1 + r~)
s
(27)
A/~--~-)J 1
The mean fluid pressure
{-f~(2)@(t)
+ 2f~2(2)[f2(2)T¢@¢(,;t, t) + (1 -- fl2(2)~)~,(2, t)-I}
(38) where b ° is the leading term in an expansion of,b,, which is valid for large values of the parameter m. However, even though ,~o can be inverted analytically, the following terms in the series cannot, and the expansion is of limited use. In the next section numerical results from equations (29) and (38) are obtained. A graphical comparison of these results will then provide a useful indication of the importance of fluid viscosity on pulse propagation in fluid-filled tubes.
p,,,(x,t) is given by:
pm (X,t) = l I IAPdA = 2 Itlfp(r,x,t)dr
Results and c o n c l u s i o n s
Equations (29) and (38) have been inverted numerically. In equation (38) the Fourier transform has been handled by applying the Gaussian quadrature method of Hurwitz and Zweifel 9 for the evaluation of Fourier integrals. For the double transform in equation (29) this method is
(28)
so that it follows immediately from equation (26) that: ,.~<,@(s)l. (A)
2fl(,~)
= Po ~
(26)
where: D(A,s) =
.b°(2, t) = Po - ~
po@(s)
(29)
Since inversion of the double transform (29) appears intractable analytically, the solution proceeds numerically. As a check, note that as m --* ov in equation (29) the double transform of the mean pressure b ° is recovered for the inviscid case, namely:
pl,~(s)
2po~(s)s 2
;~
,~3C(;~,s)
.b°(2, s ) - - where:
C(A,s) = 2(1 + rcs) + s 2
(_~ /,,(A)/ + A~I(A)]
A
(30)
(31) 0
The Laplace transform can be inverted in terms of the convolutions:
I
0
to
2t o t
Figure 1 Pressure input function
@,O.,t) = fo@(Z)~b,(;t,t - z)dz
(32)
and: • c(A,t) =
fl
qb(r)qJc(A,t- r)dr
(33)
1.4
o
1.2
where: ~bs(2,t) =
125
1.0
e-m~o'sin(f~aU2t)/a u2
q4(h,t) = e -n-k' cos
(f~a'/2t)
(34)
o 0.8
(35)
,~
"
0.6
and introducing: a(A) = 2 -
/
0.4 .O.2(h)"rc 2
(36)
0.2 0
Ft(A) =
+ AIj(A)]
(37)
-°"20
I
i
I
I
I
i
I
I
I
I
20
40
60
80
100
120
140
160
180
200
t
Equation (30) then gives the Fourier transform of the inviscid mean fluid pressure as:
Figure 2 V a r i a t i o n o f Pm/Pow i t h t i m e : ( (. . . .
) inviscid f l u i d ;
) viscous fluid
Appl. Math. Modelling, 1987, Vol. 11, June
217
Pulse propagation in a viscoelastic tube: D. W. Barclay and T. B. Moodie applied in tandem with Crump's ~° Fourier series approximation of the inverse Laplace transform. Numerical results are obtained for a semi-infinite tube with physical parameters corresponding to tube B of reference 8. The values are h = 0.035 cm, R = 0.4 cm, ~/p = 1.1, and Co = 833cm s - t . Based upon a previous study v z~ is taken as 0.15, which corresponds to a characteristic time of 0.72 x 10 -4 s. As indicated previously, if the liquid in the tube is whole blood, then m = 9000. This value is used for m in the numerical work. The pressure input is determined by the choice: "/7
• (t) = ~{1 -4- cos to (t - to)}H(2t o - t)
t >1 0
The graph of this function is shown in Figure 1, and the function has been chosen to mimic the pressure input of the experiments in reference 8. Again, based on reference 8, t, = 6. The mean pressure for both the inviscid and viscous cases is displayed graphically in Figure 2 for various stations along the tube. As expected, both pulses broaden and attenuate as they propagate down the tube. There is a definite increase in the damping of the pulse due to the fluid viscosity and this is plainly visible. It is clear then that pulse attenuation in a fluid-filled tube is due not only to the viscoelastic effect of the wall but is significantly influenced by the fluid viscosity. The results reported here are distinctive in the field of wave propagation in fluid-filled tubes in that the exact
218
Appl. Math. Modelling, 1987, Vol. 11, June
solution for a pulse wave in a viscous fluid contained in a viscoelastic tube has been obtained. The results reported concerning the effects of viscosity are surprising in the light of the values of the p a r a m e t e r m employed. This effect upon pulses could not have been anticipated on the basis of frequency plots alone in the absence of Fourier synthesis. The results also have applications in the transient response of fluid lines. ~
References 1 Skalak, R. Wave Propagation in blood flow, in Biomechanics, Proc. Syrup. Appl. Mech. Div. A S M E (ed. Fung. Y. C.), 1966. pp. 20-46 2 Skalak, R., Keller, S.R. and Secomb, T.W. Mechanics of blood flow, J. Biomed. Eng., 1981, 103, pp. 102-115 3 Anliker, M. Towards a nontraumatie study of the circulatory system, in Biomechanics (eds. Fung, Y.C., Perrone, N. and Anliker, M.), Prentice-Hall, New Jersey, 1972, pp. 337-379 4 Fung, Y.C. Biomechanics: a survey of the blood flow problem, Advances in Applied Mechanics Vol. II (ed. Yih, C-S.), Academic Press, New York, 1971, pp. 65-130 5 Waves in fluid filled tubes, Europ. Mech. Coll. 179 (eds. Buggisch, H. and Mainardi, F.), Bologna, 1985, pp. 24-25 6 Moodie, T.B., Mainardi, F, and Tait, R J . Pressure pulses in fluid filled distensible tubes, Mecc., 1985, 29, pp. 33-37 7 Moodie, T.B., Barclay, D.W. and Greenwald, S.E. Impulse propagation in liquid-filled distensible tubes: theory and experiment for intermediate to long wavelengths, Acta Mech. 1986, 59, 47-58 8 Newman, D.J., Greenwald, S.E. and Moodie, T.B. Reflections from elastic discontinuities, Med. Biol. Eng. Comput., 1983, 21,697-701 9 Hurwitz, H. and Zweifel, R.F. Numerical quadrature of Fourier transform integrals, M.T.A.C., 1956, 10, 140-149 l0 Crump, K.S. Numerical inversion or Laplace transforms using a Fourier series approximation, J.A.C.M., 1976, 23, pp. 89-96 11 Rieutford, E. Transient response o1 fluid viscoelastic lines, J. Fhdds Eng., 1982, 104, pp. 335-341