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Stenotic effects in a tube of elliptic cross-section at low reynolds numbers
137
Int J Biomed Comput. 30 (1992) 137-146
Elsevier Scientific Publishers Ireland Ltd.
STENOTIC EFFECTS IN A TUBE OF ELLIPTIC REYNOLDS NUMBERS
CROSS-SECTION
AT LOW
V.B. SARIN and R. MEHROTRA Centre for Atmospheric and Fluids Sciences, Indian Institute of Technology. Hauz Khas. New Delhi-110016 (India)
(Received September 26th, 1991) (Accepted January 9th, 1992) With an objective to understanding arteriosclerosis, the blood flow in a cylindrical tube with local constriction is analysed. The cross-section of the tube is an ellipse, the axes of which are in an arbitrary position with respect to the axis of the tube. Blood is taken to be a Newtonian and homogeneous fluid. The cross-sectional area varies slowly with the longitudinal distance and the area change is so adjusted to take account of stenosis. The transverse velocity field and the effects of inertia on the primary velocity and pressure distribution are calculated to a first order in the relevant small parameter and effects of asymmetry on the wall shear stress and impedance are presented. Keywords: Stenosis; Impedance; Shear-stress; Volume flow rate
Introduction It is becoming
increasingly accepted that mean wall shear-stress is an important factor which plays a role in the initiation of atherosclerosis, a disease which refers to the occlusion of the arterial lumen. There is abundant evidence that the existence of abnormal flow conditions due to irregularities at the wall can be an important factor in the development and progression of the arterial disease. Once the local constriction or stenosis becomes large enough to produce a separated flow region, further and possibly more rapid growth of the stenosis could be induced by this flow pattern. The possibility of thrombus formation in such a slowly recirculating flow region also exists. The boundary irregularities in the blood vessels are caused by intravascular plaques or the impingement of ligaments on the vessel wall. The important flow characteristics in the arterial system are the pressure, shear-stress, possible separation and reattachment. These are related to the physiologically important problems of (i) increase in the resistance of the blood flow and existence of low pressure regions, causing a suction effect (ii) possible damage to the red and endothelial cells due to the existence of high shear regions. Therefore, the knowledge of a detailed flow field in a vessel with irregular surface is important in the understanding and prevention of arterial disease. haemodynamic
Correspondence
to: V.B. Satin,
Centre for Atmospheric
and Fluids Sciences, Indian Institute of
Technology,Hauz Khas, New Delhi-l 10016, India. 0020-7101/92/$05.00 0 1992 Elsevier Scientific Publishers Ireland Ltd. Printed and Published in Ireland
138
V.B. Sarin and R. Mehotra
So far, several investigators have studied various aspects of problem of stenosis, in particular, the resistance, flow separation and the distribution of the shear-stress at the wall. Chow and Soda [l] have shown that the flow field in conduits with irregular surfaces is altered considerably due to the irregularity of the wall resulting in the flow separation, increase in the shear stresses, increase in the energy dissipation and non-uniform pressure gradient as compared with the flow in a tube with straight wall. Young [2] has considered the consequences of this problem taking the flow as laminar and steady and considering the artery as a tube of constant diameter. He considered blood as a homogeneous viscous fluid flowing through an axisymmetric constriction. He considered blood as a homogeneous viscous fluid flowing through an axisymmetric constriction. He showed that for mild stenosis, the velocity distribution is approximately parabolic as in the Poiseuille flow and pressure drop varies inversely with Reynolds number. His study was restricted to very small Reynolds numbers. Lee and Fung [3] obtained the numerical solution for a flow in a tube with a bell-shaped constriction for Reynolds numbers between 0 and 25. Forrester and Young [4,5] used approximate techniques to obtain the solution for flow in a tube with a cosine-shaped constriction valid for the amplitude of the wall variation which is compared with the radius and wavelength. Young and Tasi (61, in their experimental study, showed that the geometry of stenosis plays a dominant role in the flow of blood. They studied this problem with varying Reynolds numbers (100-5000) and obtained that for low Reynolds numbers there was no separation and flow was laminar and unidirectional throughout. Morgan and Young [7] considered the axisymmetric incompressible flow through a tube the help of approximate solution. They also described some experiments and compared the findings with the theoretical results which indicated good agreement. MacDonald [8] presented a technique for solution of the approximate equations governing steady flow for models of mild axisymmetric arterial stenosis. In the present study, with an objective to understand arteriosclerosis, the blood flow in a finite segment of a cylindrical tube with constriction is analysed. The flow is assumed to be steady and at fairly low Reynolds numbers. Blood is taken to be a homogeneous fluid, which is good approximation in the vessels of diameter greater than 100 pm, because the scale of microstructure is much smaller than that of flow. The average shear rate S, at the walls of arteries is significantly greater than 0.01 s and on that basis we shall assume blood to be Newtonian, although S is small near the centre of the straight vessel, or in the separated region of recirculation flow. The cross-section of the pipe is an ellipse, the axes of which are in an arbitrary position with respect to the axis of the tube. The cross-sectional area varies slowly with the longitudinal distance and the area change is so adjusted to take account of stenosis. The transverse velocity field and the effects of inertia on the primary velocity and pressure distributions are calculated to a first order in the relevant small parameter and effects of asymmetry on wall shear-stress and impedance are discussed. Mathematical Formulation Figure 1 shows the system of coordinates (x, y’, z’) with reference to Ox’y’z’.
Stenoticeffects in a tube
139
Fig. 1. The coordinate system.
The equation of ellipse with OXyz as reference axes can be written as
( u@;*J2 +(G&y =l
(1)
where a0 is a characteristic tube radius, a&z’) and aob(z’) are semi-major and semi-minor axes of the elliptical cross-section. The equation of continuity and of motion with Ox’y’z’ as reference axes can be written as v .7=0
(2)
(V.V)V=
- L Vp’ + VV% P
(3)
where
v=--
a ad
i+
a, ay’
-J-I---
a-, aZr ’
and p, Y,p’ are the density, kinematic viscosity and pressure, respectively. In order to obtain the dimensionless form of Eqns. (2) and (3), it is natural to refer the velocity components to the characteristic velocity uo, the coordinate (x’,y’) to a0 and z’ to A, the stenotic length (Fig. 2) and we write
(4)
140
V.B. Sarin and R. Mehotra
Y
I
/_-L/z
A-12
----I
I
Fig. 2. Geometry of stenosis.
where
E=2!?-
(5)
x
For pressure p’, we write P
4 = ---p’
(6)
PU20
where
is the Reynolds number. It is convenient to work in a system which does not vary with z, So we introduce variables [, 7 as follows:
(7)
With the help of Eqns. (4-7), the Eqns. (2) and (3) take the following dimensionless form:
--i a
au
at
+Ldv b all
+Dw=O
1 _-AL+ at2
at
a2 --&
2 +
$$
u +
>
e2D2u
(9)
Stenotic effects in a tube
---1
ap+
__ +bVa+wD a7
w=
bc;*
141
’ +
all
-Dp+
(
a2
I___
62
a#2
E2D2V
(10)
i a2 +
62
__
ar12
w
+
E2D2w
(11)
>
where operator D is defined as
D= and primes denote differentiation with respect to z. The equation of ellipse with Ox’y ‘z ’ as reference axes reduces to
AIt2 + B,q* + 2D,[7 = 1
(12)
where Al = cos20 + c2 sin’cz B, = COS’CY + 1 sin’o C2
D, =
c = a/b
and CY is the angle which Ox makes with Ox’. The associated boundary conditions are u=v=w=O
(13)
on the wall given by Eqn. (12). Solution of the Problem
The solution
of Eqns.
(8-l 1) is based on the assumption
that ao, the
142
V.B. Sarin and R. Mehotra
characteristic tube radius is much less than h, the stenotic length. Consequently, we can write T’= ii, +
ez,+ . , .
p = po
EpI
+
+
. .
(14)
??
where
O(P) terms
Substituting Eqn. (14) into Eqns. (8) and (11) and comparing coefficients of e” terms, we obtain i au0 --
1
aver + Dwo = 0
at +b arl
a
--i a2wo + a2 at2
L
a2wo
b2 p
--
ho
=
dz
(15) (16)
The solution of differential Eqn. (16) satisfying the boundary condition Eqn. (13) is given as wo = (1 - A#
- B,v2 - 2D,WGo(z)G
(17)
where
dpo Go(z)= - x and
c, = -
a2b2 2(Ala2
+ Blb2)
If u. is defined as the average velocity of the flow when the tube cross-section is a circle of radius ao, and if volume flow rate is non-dimensionalised with respect to ao2uo, we have w ab d[ dq = ?r
o= ss
cnxssection
(18)
Stenotic effects in a tube
143
Hence,
Go(z)=
2a(Ara* + B,b2)
(19)
a3b31
where I = Z(a,b,cY) O(E)
terms
Substituting Eqn. (14) into Eqns. (9) and (10) and comparing coeffkients of c and eliminating pressure terms, we obtain i a3uo -___ a a&
i + 7
a3vo av2at
(20)
The solution of differential Eqn. (20) satisfying the boundary condition (13) and the continuity Eqn. (15) is written as uo =
s t[
1 - Art2 - Blg2 - 2D1hl
*b’ vg = abl rl[l - -hi2 - B,rl’ -
(21)
2D1Esl
(22)
Substituting Eqn. (14) into Eqn. (11) and comparing coefficients of E terms, we obtain the differential equation for wI as follows:
a2w, i azwl __ = - Cl (z) a2 at2 + bz aq2 1
+R,
w@wo---
u.
awe
a
at
v.
+ T
awe __
a7
1
(23)
where
4-a
Cl(z) = - __ dz
The solution of differential Eqn. (23) satisfying the boundary condition (13) is written as WI = R,(l - Arc2 - B,q2 - 2D&j) x rz, + I, + 1242 + Z3[4 + z4g
+ h4
+ &2r12
+ M3tl
+ h&l3
+ 195rll
(24)
V. B. Sarin and R. Mehotra
144
where 2, =
a2b2 2(A12b2 + B12a2)R,
(25)
G(z)
and I,, 12, . . ., I9 involve solution of linear algebraic relations in a, b, A,, B, and D,. G,(z) is obtained from the volume flux condition in the form
(26) For cr = 0 (27) A = nab If n is the normal at any point on the wall, the shear-stress T,,x can be written as
G33)
Tnx = cos tan-’ Ttan8)T,,+sin(tan-‘ftan8)T,
3Q-
2.5-
I A
2,0-
l.S-
’
-1.2
I -05
I
,
I
0
0.5
1.0
-x
Fig. 3. Variation of area (A) along the stenotic length.
Stenotic effects in a tube
145
-2
-lO.O/Fig. 4. Variation of shear-stress with distance (fJ = O”, 180’).
where TX,
=
!b!!?+
E
Tyz
=
aw, ax
ax
’
!!!k + !!% E
ay
ay
and [ = r
sin 8, 7j = r COs 8.
Fig. 5. Variation of shear-stress with distance (0 = 90”).
(29)
146
V.B. Sarin and R. Mehotra
The resistive impedance, which is a measure of resistance to flow, is defined as
where Ap is the pressure drop across the stenosis and Q is the discharge through the tube.
Figure 3 depicts the variation of area of cross-section of the tube along the stenotic length for a particular choice of a and b. For different values of 0, variation of wall shear stress with axial distance are presented in Figs. 4 and 5. For all values of 8, the wall shear-stress is maximum near the point of maximum constriction for the symmetrical case (u = 0), while in asymmetrical case, the maximum value is obtained near z = -0.5. Due to asymmetry, the minimum value of wall shear-stress gets increased as compared to that obtained for the symmetrical case. Impedance is an important characteristic of stenotic blood flow. In the present model, we find that the resistance to the flow in an elliptic tube is approximately ten times less than the flow in the tube of circular cross-section. References Chow JCF and Soda K: Laminar flow in tubes with constriction, Phys. Fluids, 13 (1972) 1701- 1706. Young DF: Effect of a time dependent stenosis of flow through a tube, J Eng Ind, Tram ASME, 90 (1968) 248-254.
Lee JS and Fung YC: Flow in locally constricted tube at low Reynolds numbers. J Appl Mech., Trans ASME,
37 (1970) 9-16.
Forrester JH and Young DF: Flow through a converging-diverging tube and its applications in occlusive vascular disease - I, J Biomech, 3 (1970) 297-306. Forrester JH and Young DF: Flow through a converging-diverging tube and its applications in occlusive vascular disease - II, J Biomech, 3 (1970) 307-316. Young DF and Tasi FY: Flow characteristics in models of arterial stenosis - I, J Biomech, 6 (1973) 395-410.
Morgan BE and Young DF: An integral method for the analysis of flow in arterial stenoses, Bull Math Biol, 36 (1974) 39-53.
MacDonald DA: On steady flow through modelled vascular stenoses, J Biomech, 12 (1979) 13-20.
Int J Biomed Comput. 30 (1992) 137-146
Elsevier Scientific Publishers Ireland Ltd.
STENOTIC EFFECTS IN A TUBE OF ELLIPTIC REYNOLDS NUMBERS
CROSS-SECTION
AT LOW
V.B. SARIN and R. MEHROTRA Centre for Atmospheric and Fluids Sciences, Indian Institute of Technology. Hauz Khas. New Delhi-110016 (India)
(Received September 26th, 1991) (Accepted January 9th, 1992) With an objective to understanding arteriosclerosis, the blood flow in a cylindrical tube with local constriction is analysed. The cross-section of the tube is an ellipse, the axes of which are in an arbitrary position with respect to the axis of the tube. Blood is taken to be a Newtonian and homogeneous fluid. The cross-sectional area varies slowly with the longitudinal distance and the area change is so adjusted to take account of stenosis. The transverse velocity field and the effects of inertia on the primary velocity and pressure distribution are calculated to a first order in the relevant small parameter and effects of asymmetry on the wall shear stress and impedance are presented. Keywords: Stenosis; Impedance; Shear-stress; Volume flow rate
Introduction It is becoming
increasingly accepted that mean wall shear-stress is an important factor which plays a role in the initiation of atherosclerosis, a disease which refers to the occlusion of the arterial lumen. There is abundant evidence that the existence of abnormal flow conditions due to irregularities at the wall can be an important factor in the development and progression of the arterial disease. Once the local constriction or stenosis becomes large enough to produce a separated flow region, further and possibly more rapid growth of the stenosis could be induced by this flow pattern. The possibility of thrombus formation in such a slowly recirculating flow region also exists. The boundary irregularities in the blood vessels are caused by intravascular plaques or the impingement of ligaments on the vessel wall. The important flow characteristics in the arterial system are the pressure, shear-stress, possible separation and reattachment. These are related to the physiologically important problems of (i) increase in the resistance of the blood flow and existence of low pressure regions, causing a suction effect (ii) possible damage to the red and endothelial cells due to the existence of high shear regions. Therefore, the knowledge of a detailed flow field in a vessel with irregular surface is important in the understanding and prevention of arterial disease. haemodynamic
Correspondence
to: V.B. Satin,
Centre for Atmospheric
and Fluids Sciences, Indian Institute of
Technology,Hauz Khas, New Delhi-l 10016, India. 0020-7101/92/$05.00 0 1992 Elsevier Scientific Publishers Ireland Ltd. Printed and Published in Ireland
138
V.B. Sarin and R. Mehotra
So far, several investigators have studied various aspects of problem of stenosis, in particular, the resistance, flow separation and the distribution of the shear-stress at the wall. Chow and Soda [l] have shown that the flow field in conduits with irregular surfaces is altered considerably due to the irregularity of the wall resulting in the flow separation, increase in the shear stresses, increase in the energy dissipation and non-uniform pressure gradient as compared with the flow in a tube with straight wall. Young [2] has considered the consequences of this problem taking the flow as laminar and steady and considering the artery as a tube of constant diameter. He considered blood as a homogeneous viscous fluid flowing through an axisymmetric constriction. He considered blood as a homogeneous viscous fluid flowing through an axisymmetric constriction. He showed that for mild stenosis, the velocity distribution is approximately parabolic as in the Poiseuille flow and pressure drop varies inversely with Reynolds number. His study was restricted to very small Reynolds numbers. Lee and Fung [3] obtained the numerical solution for a flow in a tube with a bell-shaped constriction for Reynolds numbers between 0 and 25. Forrester and Young [4,5] used approximate techniques to obtain the solution for flow in a tube with a cosine-shaped constriction valid for the amplitude of the wall variation which is compared with the radius and wavelength. Young and Tasi (61, in their experimental study, showed that the geometry of stenosis plays a dominant role in the flow of blood. They studied this problem with varying Reynolds numbers (100-5000) and obtained that for low Reynolds numbers there was no separation and flow was laminar and unidirectional throughout. Morgan and Young [7] considered the axisymmetric incompressible flow through a tube the help of approximate solution. They also described some experiments and compared the findings with the theoretical results which indicated good agreement. MacDonald [8] presented a technique for solution of the approximate equations governing steady flow for models of mild axisymmetric arterial stenosis. In the present study, with an objective to understand arteriosclerosis, the blood flow in a finite segment of a cylindrical tube with constriction is analysed. The flow is assumed to be steady and at fairly low Reynolds numbers. Blood is taken to be a homogeneous fluid, which is good approximation in the vessels of diameter greater than 100 pm, because the scale of microstructure is much smaller than that of flow. The average shear rate S, at the walls of arteries is significantly greater than 0.01 s and on that basis we shall assume blood to be Newtonian, although S is small near the centre of the straight vessel, or in the separated region of recirculation flow. The cross-section of the pipe is an ellipse, the axes of which are in an arbitrary position with respect to the axis of the tube. The cross-sectional area varies slowly with the longitudinal distance and the area change is so adjusted to take account of stenosis. The transverse velocity field and the effects of inertia on the primary velocity and pressure distributions are calculated to a first order in the relevant small parameter and effects of asymmetry on wall shear-stress and impedance are discussed. Mathematical Formulation Figure 1 shows the system of coordinates (x, y’, z’) with reference to Ox’y’z’.
Stenoticeffects in a tube
139
Fig. 1. The coordinate system.
The equation of ellipse with OXyz as reference axes can be written as
( u@;*J2 +(G&y =l
(1)
where a0 is a characteristic tube radius, a&z’) and aob(z’) are semi-major and semi-minor axes of the elliptical cross-section. The equation of continuity and of motion with Ox’y’z’ as reference axes can be written as v .7=0
(2)
(V.V)V=
- L Vp’ + VV% P
(3)
where
v=--
a ad
i+
a, ay’
-J-I---
a-, aZr ’
and p, Y,p’ are the density, kinematic viscosity and pressure, respectively. In order to obtain the dimensionless form of Eqns. (2) and (3), it is natural to refer the velocity components to the characteristic velocity uo, the coordinate (x’,y’) to a0 and z’ to A, the stenotic length (Fig. 2) and we write
(4)
140
V.B. Sarin and R. Mehotra
Y
I
/_-L/z
A-12
----I
I
Fig. 2. Geometry of stenosis.
where
E=2!?-
(5)
x
For pressure p’, we write P
4 = ---p’
(6)
PU20
where
is the Reynolds number. It is convenient to work in a system which does not vary with z, So we introduce variables [, 7 as follows:
(7)
With the help of Eqns. (4-7), the Eqns. (2) and (3) take the following dimensionless form:
--i a
au
at
+Ldv b all
+Dw=O
1 _-AL+ at2
at
a2 --&
2 +
$$
u +
>
e2D2u
(9)
Stenotic effects in a tube
---1
ap+
__ +bVa+wD a7
w=
bc;*
141
’ +
all
-Dp+
(
a2
I___
62
a#2
E2D2V
(10)
i a2 +
62
__
ar12
w
+
E2D2w
(11)
>
where operator D is defined as
D= and primes denote differentiation with respect to z. The equation of ellipse with Ox’y ‘z ’ as reference axes reduces to
AIt2 + B,q* + 2D,[7 = 1
(12)
where Al = cos20 + c2 sin’cz B, = COS’CY + 1 sin’o C2
D, =
c = a/b
and CY is the angle which Ox makes with Ox’. The associated boundary conditions are u=v=w=O
(13)
on the wall given by Eqn. (12). Solution of the Problem
The solution
of Eqns.
(8-l 1) is based on the assumption
that ao, the
142
V.B. Sarin and R. Mehotra
characteristic tube radius is much less than h, the stenotic length. Consequently, we can write T’= ii, +
ez,+ . , .
p = po
EpI
+
+
. .
(14)
??
where
O(P) terms
Substituting Eqn. (14) into Eqns. (8) and (11) and comparing coefficients of e” terms, we obtain i au0 --
1
aver + Dwo = 0
at +b arl
a
--i a2wo + a2 at2
L
a2wo
b2 p
--
ho
=
dz
(15) (16)
The solution of differential Eqn. (16) satisfying the boundary condition Eqn. (13) is given as wo = (1 - A#
- B,v2 - 2D,WGo(z)G
(17)
where
dpo Go(z)= - x and
c, = -
a2b2 2(Ala2
+ Blb2)
If u. is defined as the average velocity of the flow when the tube cross-section is a circle of radius ao, and if volume flow rate is non-dimensionalised with respect to ao2uo, we have w ab d[ dq = ?r
o= ss
cnxssection
(18)
Stenotic effects in a tube
143
Hence,
Go(z)=
2a(Ara* + B,b2)
(19)
a3b31
where I = Z(a,b,cY) O(E)
terms
Substituting Eqn. (14) into Eqns. (9) and (10) and comparing coeffkients of c and eliminating pressure terms, we obtain i a3uo -___ a a&
i + 7
a3vo av2at
(20)
The solution of differential Eqn. (20) satisfying the boundary condition (13) and the continuity Eqn. (15) is written as uo =
s t[
1 - Art2 - Blg2 - 2D1hl
*b’ vg = abl rl[l - -hi2 - B,rl’ -
(21)
2D1Esl
(22)
Substituting Eqn. (14) into Eqn. (11) and comparing coefficients of E terms, we obtain the differential equation for wI as follows:
a2w, i azwl __ = - Cl (z) a2 at2 + bz aq2 1
+R,
w@wo---
u.
awe
a
at
v.
+ T
awe __
a7
1
(23)
where
4-a
Cl(z) = - __ dz
The solution of differential Eqn. (23) satisfying the boundary condition (13) is written as WI = R,(l - Arc2 - B,q2 - 2D&j) x rz, + I, + 1242 + Z3[4 + z4g
+ h4
+ &2r12
+ M3tl
+ h&l3
+ 195rll
(24)
V. B. Sarin and R. Mehotra
144
where 2, =
a2b2 2(A12b2 + B12a2)R,
(25)
G(z)
and I,, 12, . . ., I9 involve solution of linear algebraic relations in a, b, A,, B, and D,. G,(z) is obtained from the volume flux condition in the form
(26) For cr = 0 (27) A = nab If n is the normal at any point on the wall, the shear-stress T,,x can be written as
G33)
Tnx = cos tan-’ Ttan8)T,,+sin(tan-‘ftan8)T,
3Q-
2.5-
I A
2,0-
l.S-
’
-1.2
I -05
I
,
I
0
0.5
1.0
-x
Fig. 3. Variation of area (A) along the stenotic length.
Stenotic effects in a tube
145
-2
-lO.O/Fig. 4. Variation of shear-stress with distance (fJ = O”, 180’).
where TX,
=
!b!!?+
E
Tyz
=
aw, ax
ax
’
!!!k + !!% E
ay
ay
and [ = r
sin 8, 7j = r COs 8.
Fig. 5. Variation of shear-stress with distance (0 = 90”).
(29)
146
V.B. Sarin and R. Mehotra
The resistive impedance, which is a measure of resistance to flow, is defined as
where Ap is the pressure drop across the stenosis and Q is the discharge through the tube.
Figure 3 depicts the variation of area of cross-section of the tube along the stenotic length for a particular choice of a and b. For different values of 0, variation of wall shear stress with axial distance are presented in Figs. 4 and 5. For all values of 8, the wall shear-stress is maximum near the point of maximum constriction for the symmetrical case (u = 0), while in asymmetrical case, the maximum value is obtained near z = -0.5. Due to asymmetry, the minimum value of wall shear-stress gets increased as compared to that obtained for the symmetrical case. Impedance is an important characteristic of stenotic blood flow. In the present model, we find that the resistance to the flow in an elliptic tube is approximately ten times less than the flow in the tube of circular cross-section. References Chow JCF and Soda K: Laminar flow in tubes with constriction, Phys. Fluids, 13 (1972) 1701- 1706. Young DF: Effect of a time dependent stenosis of flow through a tube, J Eng Ind, Tram ASME, 90 (1968) 248-254.
Lee JS and Fung YC: Flow in locally constricted tube at low Reynolds numbers. J Appl Mech., Trans ASME,
37 (1970) 9-16.
Forrester JH and Young DF: Flow through a converging-diverging tube and its applications in occlusive vascular disease - I, J Biomech, 3 (1970) 297-306. Forrester JH and Young DF: Flow through a converging-diverging tube and its applications in occlusive vascular disease - II, J Biomech, 3 (1970) 307-316. Young DF and Tasi FY: Flow characteristics in models of arterial stenosis - I, J Biomech, 6 (1973) 395-410.
Morgan BE and Young DF: An integral method for the analysis of flow in arterial stenoses, Bull Math Biol, 36 (1974) 39-53.
MacDonald DA: On steady flow through modelled vascular stenoses, J Biomech, 12 (1979) 13-20.