Applied Mathematics and Computation 154 (2004) 747–767
www.elsevier.com/locate/amc
Amplitude modulation of nonlinear waves
in a fluid-filled tapered elastic tube
_
Ilkay
Bakirtasß a, Hilmi Demiray
a
b,*
Department of Engineering Sciences, Faculty of Science and Letters, Istanbul Technical University,
Maslak, Istanbul, Turkey
b
Department of Mathematics, Faculty of Arts and Sciences, Isik University,
Buyukdere Caddesi, 80670 Maslak, Istanbul, Turkey
Abstract
In the present work, treating the arteries as a tapered, thin walled, long and circularly
conical prestressed elastic tube and using the reductive perturbation method, we have
studied the amplitude modulation of nonlinear waves in such a fluid-filled elastic tube.
By considering the blood as an incompressible inviscid fluid the evolution equation is
obtained as the nonlinear Schr€
odinger equation with variable coefficients. It is shown
that this type of equations admit a solitary wave type of solution with variable wave
speed. It is observed that, the wave speed decreases with distance for tubes with descending radius while it increases for tubes with ascending radius.
Ó 2003 Elsevier Inc. All rights reserved.
1. Introduction
The striking feature of the arterial blood flow is its pulsatile character. The
intermittent ejection of blood from the left ventricle produces pressure and flow
pulses in the arterial tree. Experimental studies reveal that the flow velocity in
blood vessels largely depends on the elastic properties of the vessel wall and
they propagate towards the periphery with a characteristic pattern [1].
Due to its applications in arterial mechanics, the propagation of pressure
pulses in fluid-filled distensible tubes has been studied by several researchers
*
Corresponding author.
E-mail address: [email protected] (H. Demiray).
0096-3003/$ - see front matter Ó 2003 Elsevier Inc. All rights reserved.
doi:10.1016/S0096-3003(03)00748-3
748
_ Bakirtasß, H. Demiray / Appl. Math. Comput. 154 (2004) 747–767
I.
[2,3]. Most of the works on wave propagation in compliant tubes have considered small amplitude waves ignoring the nonlinear effects and focused on the
dispersive character of waves (see [4–6]). However, when the nonlinear effects
arising from the constitutive equations and kinematical relations are introduced, one has to consider either finite amplitude, or small-but-finite amplitude
waves, depending on the order of nonlinearity.
The propagation of finite amplitude waves in fluid-filled elastic or viscoelastic tubes has been examined, for instance, by Rudinger [7], Anliker et al. [8]
and Tait and Moodie [9] by using the method of characteristics, in studying the
shock formation. On the other hand, the propagation of small-but-finite amplitude waves in distensible tubes has been investigated by Johnson [10],
Hashizume [11], Yomosa [12], Erbay et al. [13] and Demiray [14] by employing
various asymptotic methods. In all these works [10–14], depending on the
balance between the nonlinearity, dispersion and dissipation, the Korteweg-de
Vries (KdV), BurgerÕs or KdV–BurgerÕs equations are obtained as the evolution equations.
As is well known, when the nonlinear effects are small, the system of
equations that describe the physical phenomenon admit harmonic wave solution with constant amplitude. If the amplitude of the wave is small-but-finite,
the nonlinear terms cannot be neglected and the nonlinearity gives rise to the
variation of amplitude both in space and time variables. Then the amplitude
varies slowly over a period of oscillation, a stretching transformation allows us
to decompose the system into a rapidly varying part associated with the oscillation and slowly varying part such as the amplitude. A formal solution can
be given in the form of an asymptotic expansion, and an equation determining
the modulation of the first-order amplitude can be derived. For instance, the
nonlinear Schr€
odinger equation is the simplest representative equation describing the self-modulation of one-dimensional monochromatic plane waves
in dispersive media. It exhibits a balance between the nonlinearity and dispersion. The problem of self-modulation of small-but-finite amplitude waves in
fluid-filled distensible tubes was considered by Ravindran and Prasad [15], in
which they showed that for a linear elastic tube wall model, the nonlinear selfmodulation of pressure waves is governed by the NLS equation. Nonlinear
self-modulation in fluid-filled distensible tubes had been investigated by Erbay
and Erbay [16], by employing the nonlinear equations of a viscoelastic thin
tube and the approximate fluid equations and they showed that the nonlinear
pressure waves is governed by the dissipative NLS equation. Demiray [17],
employing the exact equations of a viscous fluid and of a prestressed elastic
tube, studied the amplitude modulation of nonlinear waves and obtained the
dissipative nonlinear Schr€
odinger equation as the governing equation. In all
these works, the arteries are considered as a cylindrical tube with constant
radius. In essence, the radius of the arteries is variable along the axis of the tube
(see [18]).
_ Bakirtasß, H. Demiray / Appl. Math. Comput. 154 (2004) 747–767
I.
749
In the present work, treating the arteries as a tapered elastic thin walled long
circularly conical tube and using the reductive perturbation method we have
studied the amplitude modulation of weakly nonlinear waves in such a fluidfilled elastic tubes. By considering the blood as an incompressible inviscid fluid,
the evolution equation is obtained as the nonlinear Schr€
odinger equation with
a variable coefficient. It is shown that this type of equations admit a solitary
wave type of solution with variable wave speed. The numerical calculations
indicate that the wave speed decreases with distance for tubes with descending
radius while it increases for tubes with ascending radius.
2. Basic equations and theoretical preliminaries
2.1. Equations of tube
In this section, we shall derive the basic equations governing the motion of a
pre-stressed thin and tapered elastic tube filled with an inviscid fluid. For that
purpose, we consider a circularly conical tube of radius R0 at the origin of the
coordinate system with tapering angle U. Then, the position vector R of a
generic point on the tube may be described by
R ¼ ðR0 þ UZÞer þ Zez ;
ð1Þ
where er , eh and ez are the unit base vectors in the cylindrical polar coordinates,
Z is the axial coordinate of a material point in the undeformed configuration.
The arclengths of the curves along the meridional and circumferential directions are given by
dSZ ¼ ð1 þ U2 Þ
1=2
dZ;
dSh ¼ ðR0 þ UZÞ dh:
ð2Þ
We assume that such a tapered tube is subjected to an inner static pressure
P0 ðZÞ and the deformation is described by
r0 ¼ ðr0 þ /z Þer þ z ez ;
z ¼ kz Z:
ð3Þ
Here, z is the axial coordinate after static deformation, kz is the stretch ratio in
the axial direction, r0 is the deformed end radius at the origin, / is the deformed tapering angle. Then, the arclengths of the corresponding curves in the
meridional and circumferential directions are given by
ds0z ¼ ð1 þ /2 Þ1=2 dz ;
ds0h ¼ ðr0 þ /z Þ dh:
ð4Þ
Then, the corresponding stretch ratios after static deformation are given as
1=2
ds0
1 þ /2
ds0
ðr0 þ /z Þ
k01 ¼ z ¼ kz
:
ð5Þ
; k02 ¼ h ¼ kz
2
ðkz R0 þ Uz Þ
dSZ
dSh
1þU
_ Bakirtasß, H. Demiray / Appl. Math. Comput. 154 (2004) 747–767
I.
750
Now, on this initial static deformation, we shall superimpose a dynamical
radial displacement u ðz ; t Þ but, in view of the external tethering, the axial
displacement is assumed to be zero. Then, the position vector of a generic point
on the membrane after final deformation is given by
r ¼ ðr0 þ /z þ u Þer þ z ez ;
ð6Þ
where t is the time parameter. The arclengths along the deformed meridional
and circumferential curves are given by
"
2 #1=2
ou
dz ; dsh ¼ ðr0 þ /z þ u Þ dh:
ð7Þ
dsz ¼ 1 þ / þ oz
Then, the stretch ratios in the final configurations read
h
i1=2
2
1 þ / þ ou
oz
ðr0 þ /z þ u Þ
k1 ¼ k z
;
k
¼
k
:
2
z
1=2
kz R0 þ Uz
ð1 þ U2 Þ
ð8Þ
The unit tangent vector t along the deformed meridional curve may be given by
er þ ez
/ þ ou
oz
t¼h
ð9Þ
i1=2 :
2
1 þ / þ ou
oz
Similarly, the unit exterior normal vector n to the deformed membrane is
given as
er / þ ou
ez
oz
n¼h
ð10Þ
i1=2 :
2
1 þ / þ ou
oz
Throughout this work, we shall assume that the material under consideration is
incompressible. Let H be the initial thickness of the tube element, and let h and
h0 be the thickness after static and final dynamical deformations, respectively.
The incompressibility of the material requires that
h¼
h0 ¼
H ð1 þ U2 Þ1=2 kz R0 þ Uz
;
1=2 r þ /z
0
k2z ð1 þ /2 Þ
H ð1 þ U2 Þ
k2z ½1
þ ð/ þ
1=2
ou
oz
2 1=2
Þ
ðkz R0 þ Uz Þ
:
½r0 þ /z þ u ð11Þ
ð12Þ
As is seen from (11), even for the static deformation, the thickness changes
along the tube axis. The value of the thickness at the coordinate origin and
infinity are, respectively, given by
_ Bakirtasß, H. Demiray / Appl. Math. Comput. 154 (2004) 747–767
I.
h0 ¼
H ð1 þ U2 Þ1=2
;
kh kz ð1 þ /2 Þ1=2
h1 ¼
H ð1 þ U2 Þ1=2 U
k2z ð1 þ /2 Þ
1=2
/
751
ð13Þ
;
where kh ¼ r0 =R0 is the stretch ratio at the origin in the radial direction. If the
tapering angles before and after static deformation are related to each other by
/ ¼ Ukh =kz , the thickness remains constant throughout the axis and given by
h¼
H ð1 þ U2 Þ
1=2
kh kz ð1 þ /2 Þ1=2
ð14Þ
:
As a matter of fact, in this case the generators before and after static deformation remain parallel to each other.
Let T1 and T2 be the membrane forces along the meridional and circumferential curves, respectively. Then, the equation of the radial motion of a small
tube element placed between the planes z ¼ const, z þ dz ¼ const, h ¼ const
and h þ dh ¼ const may be given by
"
(
)
2 #1=2
ðr0 þ /z þ u Þð/ þ ou
Þ
ou
o
oz
T2 1 þ / þ þ T1
2 1=2
oz
oz
½1 þ ð/ þ ou Þ oz
qH
o2 u
1=2
þ P ðr0 þ /z þ u Þ ¼ 0 2 ð1 þ U2 Þ ðR0 kz þ Uz Þ 2 ;
ot
kz
ð15Þ
where q0 is the mass density of the membrane and P is the fluid pressure.
Let mR be the strain energy density function of the membrane, where m is the
shear modulus of the membrane material. The membrane forces may be
expressed in terms of the stretch ratios as
T1 ¼
mH oR
;
k2 ok1
T2 ¼
mH oR
:
k1 ok2
ð16Þ
Introducing (16) into Eq. (15), the equation of motion in the radial direction
takes the following form
(
)
ou
ðk
R
þ
Uz
Þð/
þ
Þ
m
oR
m
o
oR
z
0
1=2
oz
H ð1 þ U2 Þ
þ H
2 1=2
kz
ok2 kz oz
ok1
½1 þ ð/ þ ou Þ oz
q
o2 u
1=2
þ P ðr0 þ /z þ u Þ ¼ 20 H ð1 þ U2 Þ ðkz R0 þ Uz Þ 2 :
ot
kz
ð17Þ
2.2. Equations of fluid
Blood is known to be an incompressible non-Newtonian fluid. However, in
the course of flow in large blood vessels, the red blood cells in the vicinity of
arteries move to the central region of the artery so that the hematocrit ratio
_ Bakirtasß, H. Demiray / Appl. Math. Comput. 154 (2004) 747–767
I.
752
becomes quite low near the arterial wall and the blood viscosity does not
change much with shear rate. Moreover, due to high shear rate near the arterial
wall, the blood viscosity is further reduced. Therefore, for flow problems in
large blood vessels, the blood may be treated as an incompressible inviscid
fluid. Due to the assumption of smallness of the viscosity, the variation of field
quantities with the radial coordinate may be neglected. Thus, the averaged
equations of motion are given by
oA
o
þ
ðAw Þ ¼ 0;
ot oz
ð18Þ
ow
1 oP ow
þ
w
þ
¼ 0;
ot
oz qf oz
ð19Þ
where w is the averaged axial velocity, P is the pressure and Aðz ; t Þ is the
cross-sectional area of the tube. Noticing the relation between the cross-sec2
tional area and the final radius, i.e., A ¼ pðr0 þ /z þ u Þ , Eq. (18) becomes
ou
ou
1
ow
þ
/
þ
ð20Þ
w þ ðr0 þ /z þ u Þ ¼ 0:
2
ot
oz
oz
At this stage it is convenient to introduce the following nondimensionalized
quantities
R0
qH
t ¼
t; z ¼ R0 z; u ¼ R0 u; m ¼ 0 ;
qf R 0
c0
ð21Þ
mH
:
w ¼ c0 w; r0 ¼ kh R0 ; P ¼ qf c20 p; c20 ¼
qf R0
Introducing (21) into Eqs. (17), (19) and (20) we obtain the following nondimensionalized equations:
ou
ou
1
ow
þ /þ
¼ 0;
ð22Þ
w þ ðkh þ /z þ uÞ
ot
oz
2
oz
ow
ow op
þw
þ
¼ 0;
ot
oz oz
ð23Þ
1=2
p¼
m
ðkz þ UzÞ o2 u
ð1 þ U2 Þ
oR
2 1=2
ð1
þ
U
Þ
þ
2
2
ðkh þ /z þ uÞ ot
kz ðkh þ /z þ uÞ ok2
kz
(
)
Þðkz þ Uz Þ oR
1
o ð/ þ ou
oz
:
kz ðkh þ /z þ uÞ oz ½1 þ ð/ þ ouÞ2 1=2 ok1
ð24Þ
oz
These equations give sufficient relations to determine the field quantities u, w
and p completely.
_ Bakirtasß, H. Demiray / Appl. Math. Comput. 154 (2004) 747–767
I.
753
3. Nonlinear wave modulation
In this section we shall examine the amplitude modulation of weakly nonlinear waves in a fluid filled nonlinear thin elastic tube whose dimensionless
governing equations are given in (22)–(24). The analysis of dispersion relations
of the linearized field equations and considering that the problem of concern
is the boundary value problem, the following stretched coordinates may be
introduced
n ¼ ðz ktÞ;
s ¼ 2 z;
ð25Þ
where is a small parameter measuring the weakness of nonlinearity and k is a
scale constant to be determined from the solution.
In order to take the effect of tapering into account, the order of the tapering
angles should be of order of 3 . Hence, the tapering angles can be expressed as
U ¼ A3 ;
/ ¼ a3 :
ð26Þ
Here A and a are some finite constants describing the tapering angles before
and after static deformation. We shall assume that the generators before and
after finite static deformation remain parallel to each other. As pointed out
before, for this case the relation between a and A becomes
a¼
kh
A:
kz
ð27Þ
Noting the differential relations
o
o
o
o
! þ
þ 2 ;
oz
oz
on
os
o
o
o
! k ;
ot
ot
on
ð28Þ
and expanding the field variables into asymptotic series as
u ¼ u1 þ 2 u2 þ 3 u3 þ ;
w ¼ w1 þ 2 w2 þ 3 w3 þ ;
2
ð29Þ
3
p ¼ p0 þ p1 þ p2 þ p3 þ ;
where u1 ; . . . p3 are functions of the fast variables ðz; tÞ and the slow variables
ðn; sÞ, and introducing (28) and (29) into the field equations (22) and (23) the
following sets of differential equations are obtained:
OðÞ order equations:
ou1 kh ow1
þ
¼ 0;
ot
2 oz
ow1 op1
þ
¼ 0:
ot
oz
ð30Þ
754
_ Bakirtasß, H. Demiray / Appl. Math. Comput. 154 (2004) 747–767
I.
Oð2 Þ order equations:
ou2 kh ow2
ou1 kh ow1
ou1 1 ow1 1 kh
ow1
þ
k
þ
þ w1
þ u1
þ
¼ 0;
As
2 kz
ot
2 oz
on
2 on
oz 2
oz
oz
ow2 op2
ow1 op1
ow1
þ
k
þ
þ w1
¼ 0:
ot
oz
on
on
oz
ð31Þ
Oð3 Þ order equations:
ou3 kh ow3
ou2 kh ow2 kh ow1 ou1
ou2 ou1
þ
k
þ
þ
þ
w2 þ w1
þ
ot
2 oz
on
2 on
2 os
oz
oz
on
1
ow2 ow1
1 ow1 kh
ow2 kh
ow1
þ
þ
¼ 0;
þ u1
As
As
þ
þ u2
2
2
oz
on
oz
2kz
oz
2kz
on
ow3 op3
ow2 op2 op2 o
ow1
þ
k
þ
þ
þ ðw1 w2 Þ þ w1
¼ 0:
ot
oz
on
on
os oz
on
ð32Þ
In order to make the equations self-sufficient, we need the explicit expressions
of p1 , p2 and p3 , which should be obtained from Eq. (24). Noting that under the
coordinate transformations (28), the tapering angle terms in these expressions
become (As) and (as), in this approximation. Hence the following expansions
are permissible
ou1 ou2 ow1 3
þ
þ þ kz
oz
oz
on
!
!
As
As
A2 s2
2
k2 ¼ kh þ u1 þ u2 u1 þ u3 u2 þ 2 u1 3 þ kz
kz
kz
! )
(
!
1
u
A
u
u21 2As
A2 2 2
1
2
1
þ s þ þ 2þ
1
u1 þ 2 s ½kh þ as þ u ¼
kh kz
kh kh kh kz
kh
kz
kz
k1 ¼ kz þ
2
ou1
oz
2
þ
2
!
u3 2u1 u2 2As
u31 3As 2 3A2 s2
A3 3 3
u2 3 u1 2 u1 2 s þ 2 þ
kh
kh kz
kh
kh kz k2h
kz kh
kz
þ "
2 #
1 oR
As
ou1
2
2
¼ b0 þ b1 u1 þ b2 u1 þ b1 u2 u1 þ a1 kz
oz
kh kz ok2
kz
2
!
As 2
As
A2 s2
3
4
þ b3 u1 þ 2b2 u1 u2 2b2 u1 þ b1 u3 u2 þ 2 u1
kz
kz
kz
!2 3
ou1 ou2 ou1
ou1 5 3
þ 2a1 kz
þ
þ a2 kz u1
þ oz
oz
on
oz
"
2 #
1 oR
As
ou1
2
2 þ ;
¼ a0 þ 2a1 u1 þ a2 u1 þ 2a1 u2 u1 þ c1 kz
kh kz ok1
kz
oz
ð33Þ
_ Bakirtasß, H. Demiray / Appl. Math. Comput. 154 (2004) 747–767
I.
755
where the coefficients a0 , a1 , a2 , b0 , b1 , b2 , b3 and c1 are defined by
1 oR
1 o2 R
1
o3 R
; a1 ¼
; a2 ¼
kh kz okz
2kh kz okh kz
2kh kz ok2h okz
1 oR
1 o2 R
1 o2 R
b0 ¼
; b1 ¼
; b2 ¼
2
kh kz okz
kh kz ok0
2kh kz ok3h
1 o4 R
1 o2 R
b3 ¼
; c1 ¼
:
4
6kh kz okh
2kh kz ok2z
a0 ¼
ð34Þ
Introducing the expansion (33) into the expression of the pressure term given
in (24), we obtain various order pressure terms as
m o2 u1
o2 u1
b0
bA
p1 ¼
a
k
þ
b
ð35Þ
u1 0 s:
0 z
1
kh kz ot2
kz
oz2
kh
!
!
m
o2 u2
o2 u1 u1 o2 u1
b1 b0
p2 ¼
2k
þ b2 : þ 2 u21
kh kz ot2
oton kh ot2
kh kh
2
2
b
o u2
ou1
o2 u1
þ b1 0 u 2 a 0 k z 2 a 1 k z
2a0 kz
kh
oz
oz
ozon
a0
o2 u1 2A
b0
b 0 A2 2
þ kz
b1 ð36Þ
2a1 u1 2 su1 þ 2 s :
kz
kh
oz
kh
kz
"
2
m o2 u3
o2 u2
1
o2 u2
o2 u1
2 o u1
2k
p3 ¼
u1 2 þ u2 2
þk
kh kz ot2
oton
ot
ot
on2 kh
# !
2
2 2
2k o u1 u1 o u1
b0
b1 b0
þ u1
þ
u3 þ 2 b2 þ 2 u1 u2
þ b1 kh
kh kh
kh oton k2h ot2
!
b2 b1 b0
ou1 ou2 ou1 ou1
3
þ
þ b3 : þ 2 3 u1 2a1 kz
kh kh kh
oz oz
oz on
2
a1
ou1
o2 u3
o2 u2
o2 u1
2a0 kz
u1
kz a2 a0 kz 2 2a0 kz
kh
oz
oz
ozon
ozos
2 2 2
2 2
3
o u1 o u1
a0
ou
o2 u1
þ kz
2a1
u1 2 þ u2 2
a0 3c1 kz
þ kz
2
oz
oz2
kh
oz
oz
!
!
2
2
2
a0
o u1
a0 2a1
o u1
o u1
þ kz 2 þ
2a1 u1
a2 u21 2 a0 kz
þ 2kz
kh
ozon
k
oz
k
on2
h
"
!h
2
m
o2 u1
b1 b0 u21
b0 u2
ou1
þ 2 2 u1 2 3 b2 þ 2
2 b1 þ a1
ot
kh kh kz
kh kz
oz
kh kz
" !
#
#
a0
o2 u1
b u1 2 2 A3 s3
u1 2 As þ 3 b1 0
As 3 :
þ 2a1 ð37Þ
kh
oz
kh k2z
kz
_ Bakirtasß, H. Demiray / Appl. Math. Comput. 154 (2004) 747–767
I.
756
3.1. Solution of the field equations
In this sub-section we shall present a solution to the field equations given in
(30)–(32) and (35)–(37).
OðÞ order equations:
The form of Eqs. (30) and (35) suggests us to seek the following type of
solution
u1 ¼ U0 ðn; sÞ þ fU ðn; sÞ exp½iðxt kzÞ þ c:c:g;
ð1Þ
w1 ¼ W0 ðn; sÞ þ fW1 ðn; sÞ exp½iðxt kzÞ þ c:c:g;
p1 ¼ P0 ðn; sÞ þ
ð1Þ
fP1 ðn; sÞ exp½iðxt
ð38Þ
kzÞ þ c:c:g;
with
b
bA
b1 0 U0 0 s
kz
kh
mx2
b
¼ þ a 0 k z k 2 þ b1 0 U ;
kh kz
kh
P0 ¼
ð1Þ
P1
ð39Þ
ð1Þ
where x is the angular frequency, k is the wave number, U0 ðn; sÞ; . . . ; P1 are
some unknown functions of their arguments and c.c. stands for the complex
conjugate of the corresponding expressions.
Introducing (38) and (39) into Eq. (30) we obtain
ð1Þ
W1
¼
2 x
U;
kh k
ð1Þ
P1 ¼
2 x2
U;
kh k 2
ð40Þ
provided that the following dispersion relation holds true
mk 2
2þ
x2 ðkh b1 b0 Þk 2 a0 kz kh k 4 ¼ 0:
kz
ð41Þ
Oð2 Þ order equations:
For this order of equations we shall seek a solution of the form
(
)
2
X
ð0Þ
ð‘Þ
U2 exp½i‘ðxt kzÞ þ c:c: ;
u2 ¼ U 2 þ
(
w2 ¼
ð0Þ
W2
þ
(
p2 ¼
ð0Þ
P2
þ
‘¼1
2
X
)
ð‘Þ
W2
‘¼1
2
X
‘¼1
ð‘Þ
P2
exp½i‘ðxt kzÞ þ c:c: ;
)
exp½i‘ðxt kzÞ þ c:c:
ð42Þ
_ Bakirtasß, H. Demiray / Appl. Math. Comput. 154 (2004) 747–767
I.
757
with
ð0Þ
P2
ð1Þ
P2
ð2Þ
P2
!
x2
2
2
¼
þ 4 2 þ 2b2 þ 2a1 kz k jU j
kh k 2
!
b1 b0
2A
b0
b A2 s 2
2
þ b2 þ 2 U0 b1 sU0 þ 0 2 ;
kz
kh kh
kh
kz
!
2x2 ð1Þ
mxk oU
b
b
þ
2b2 1 þ 20
¼
U
þ
2i
a
k
k
0 z
kh kz on
kh k 2 2
kh kh
2x2
2A
b
2 þ 2a1 kz k 2 U0 U b1 0 sU ;
kz
kh
kh k 2
"
!#
x2
b0
2x2
ð2Þ
2
¼ 8
þ
þ
b
þ
3a
k
k
U 2:
3
b
U
1 z
1
2
2
kh k 2
kh
k2h k 2
b
b1 0
kh
!
ð0Þ
U2
ð43Þ
ð44Þ
ð45Þ
Introducing the solutions (38) and (42) into Eq. (31) the following sets of
equations are obtained
oU0 kh oW0
oW0 oP0
þ
¼ 0; k
þ
¼ 0:
on
2 on
on
on
kh k ð1Þ
x
oU
x
xAs
ð1Þ
W i
kW0 þ U0 þ
xU2 k
U ¼ 0;
2 2
k
on
kh
kz
2 x x
oU 2x
ð1Þ
ð1Þ
W0 U ¼ 0;
k
xW2 kP2 i
kh k k
on
kh
k
ð2Þ
ð2Þ
2xU2 kh kW2 3
ð2Þ
ð2Þ
xW2 kP2 x 2
U ¼ 0;
kh
ð46Þ
ð47Þ
ð48Þ
2 x2 2
U ¼ 0:
k2h k
From the solution of (46) together with Eq. (39)1 we obtain
W0 ¼ 2
k
U0 þ CðsÞ;
kh
½2k2 þ ðkh b1 b0 Þ
oU0
¼ 0;
on
ð49Þ
where CðsÞ is an arbitrary function of the variable s. In order to have a nonzero
solution for oU0 =on the following condition must be satisfied
k2 ðkh b1 b0 Þ=2 ¼ 0:
ð50Þ
As might be noticed, the constant ðkh b1 b0 Þ=2 is just the square of the phase
velocity in the long-wave limit. Eq. (50) shows that for the long-wave limit the
scaling factor k is equal to the phase velocity.
_ Bakirtasß, H. Demiray / Appl. Math. Comput. 154 (2004) 747–767
I.
758
First, we would like to discuss the case where oU0 =on is different from zero,
i.e., k2 ðkh b1 b0 Þ=2 ¼ 0. For this case it is readily seen that U ¼ 0, and the
other unknowns become
2 x ð1Þ
2 x2 ð1Þ
ð1Þ
ð1Þ
ð2Þ
ð2Þ
ð2Þ
U2 ; P2 ¼
W2 ¼
U ; U2 ¼ W2 ¼ P2 ¼ 0:
ð51Þ
kh k
kh k 2 2
For the long-wave limit Oð3 Þ order equations become
ð0Þ
ð0Þ
oU2
kh oW2
oU0 kh dCðsÞ
þ
þk
þ
¼ 0;
on
2 on
os
2 ds
ð0Þ
ð0Þ
oP2
oW2
b oU0 b0 A
¼ 0;
k
þ b1 0
on
on
kh os
kz
!
b0
b1 b0
A
b0
b A2
ð0Þ
ð0Þ
2
U2 þ b2 þ 2 U0 2
sU0 þ 0 2 s2 :
P 2 ¼ b1 b1 kh
kh kh
kh
kz
kz
k
ð52Þ
Eliminating
is obtained
oU0
þ
os
ð0Þ
W2
between these equations the following evolution equation
kh b2 2
oU0 A oU0
s
¼ 0:
U0
2
kh
kz on
on
2k
ð53Þ
Here we have chosen CðsÞ as CðsÞ ¼ ðb0 As=kkh Þ, in order to remove the constant term from the evolution equation (53). Eq. (53) is the degenerate form
of the KdV equation with variable coefficient.
The other possibility in Eq. (49) is that k2 ðkh b1 b0 Þ=2 6¼ 0, that is, U0 is
an arbitrary function of the variable s. Thus, the arbitrary function CðsÞ an be
taken to be zero and the solution takes the following form
k
b
bA
W0 ¼ 2 U0 ðsÞ; P0 b1 0 U0 ðsÞ 0 s:
ð54Þ
kh
kz
kh
ð1Þ
ð1Þ
Introducing the expression of P2 in (44) into Eq. (47) and eliminating W2
between these equations one obtains.
x
mxk
oU
i 2x
k þ kh k 2 a0 kz k k
kh kz
on
4
b1 b0
2
4
þ
kxk þ kh b2 þ
k þ a1 kh kz k U0
kh
2 2kh
As
þ ½x2 ðkh b1 b0 Þk 2 U ¼ 0:
ð55Þ
kz
As is seen from this equation, the coefficient of oU =on is purely imaginary and
of U is real. Therefore in order to have a nonzero solution for U , these coefficients must vanish. Thus, the following equations are obtained.
_ Bakirtasß, H. Demiray / Appl. Math. Comput. 154 (2004) 747–767
I.
x
mxk
2
2x
k þ kh k a0 kz k ¼ 0;
k
kh kz
4
b1 b0
2
4
kxk þ kh b2 þ
k þ a1 kh kz k U 0
kh
2 2kh
As
þ ½x2 ðkh b1 b0 Þk 2 ¼ 0:
kz
759
ð56Þ
ð57Þ
The solution of Eq. (56) gives the parameter k as the group velocity, i.e.,
k¼
2x2 þ a0 kh kz k 4
:
xkð2 þ mk 2 =kz Þ
ð58Þ
From the solution of (57) we have
U0 ¼ U0
A
s;
kz
U0 ¼ h
½x2 þ ðkh b1 b0 Þk 2 i:
b0
b1
4
2 þ a k k k4
kxk
þ
k
b
þ
k
h
1
h
z
2
kh
2
2kh
ð59Þ
Then, the solution of Eq. (47) yields
2 x ð1Þ
2i x
oU 2 1 x
x A
ð1Þ
U 2k þ
sU ;
k
U0 þ
W2 ¼
kh k 2
kh k k
on kh kh
k
k kz
2 x2 ð1Þ 4ix x
oU 2 x 1
x
x A
ð1Þ
P2 ¼
U 4k þ
sU :
k
U0 þ
kh k 2 2
kh k 2 k
on kh k kh
k
k kz
ð60Þ
Finally, from the solution of (45) and (48) we have
ð2Þ
U2 ¼
½3x2 =kh þ kh b2 k 2 þ 3a1 kh kz k 4 U 2
;
3½ðkh b1 b0 Þk 2 2x2 ð2Þ
W2
¼
2 x ð2Þ 3 x 2
U 2 U :
kh k 2
kh k
ð61Þ
Oð3 Þ order equations:
For this order of equations we shall seek a solution of the form
(
)
3
X
ð0Þ
ð‘Þ
U3 exp½i‘ðxt kzÞ þ c:c: ;
u3 ¼ U 3 þ
(
w3 ¼
ð0Þ
W3
þ
(
p3 ¼
ð0Þ
P3
þ
‘¼1
3
X
)
ð‘Þ
W3
‘¼1
3
X
‘¼1
ð‘Þ
P3
exp½i‘ðxt kzÞ þ c:c: ;
)
exp½i‘ðxt kzÞ þ c:c: ;
ð62Þ
_ Bakirtasß, H. Demiray / Appl. Math. Comput. 154 (2004) 747–767
I.
760
For our future purposes we need the zeroth and first-order equations in
terms of the phasor u ¼ xt kz. From (32), these equations take the following
form
ð0Þ
ð0Þ
oU2
kh oW2
1
x
o
A
2
jU j þ kU0 ¼ 0;
þ
þ
k
4 k
kh
k
on
kz
on
2 on
ð0Þ
ð0Þ
oP2
oW2
4 x2 o
b
A
2
jU j þ b1 0 U0 b0
þk
þ 2 2
¼ 0;
kz
on
on
kz
kh k on
!
!
b0
4x2
ð0Þ
ð0Þ
2
P2 ¼ b1 U2 þ 2b2 2 þ 2a1 kz k jU 2 j
kh
kh k 2
"
þ
b
b
b2 1 þ 20
kh kh
!
U20
ð63Þ
#
b0
A2
2 b1 U0 þ b0 2 s2 :
kh
kz
2
oU ð1Þ i x
ikkh ð1Þ x oU x
oU
2
W3 þ
k
k
2
on
k os
k
k k
on2
x ð2Þ x ð0Þ
2kk x
A
ð0Þ
ð1Þ
þ
U0 þ x
3i U2 U ikW2 U i U2 U i
sU
kh
kh
kh
kh
kz 2
2
U0
x
A oU
U0
2kk x
A
þ
U0 þ x 2 s2 U
1þ
s
k
þi 1þ
kh
kh
k
kz on
kh
kh
kz
ð1Þ
ixU3
¼ 0;
ð1Þ
ð1Þ
ixW3 ikP3 þ
ð1Þ
2 x x
oU2
2 x2 oU
2i x
2x
þ
k
k
k
kh k k
kh k 2 os kh k k
k
on
o2 U
2x ð0Þ
4 x2 ð2Þ 6 x2
2
U
jU j U
i
W
U
i
U
þ
i
2
2
kh
on2
k2h k
k3h k
4xk A ð1Þ 2 x
4k x
x A oU
i 2 U0 sU2 s
k
þ
U0 þ
kz
kh k
kh kh k
k kz on
kh
2
4k
2kk x
A
þ i 2 U0
þ
U0 þ x 2 s2 U ¼ 0;
kh kh
kh
kz
_ Bakirtasß, H. Demiray / Appl. Math. Comput. 154 (2004) 747–767
I.
ð1Þ
P3 ¼
761
ð1Þ
2 x2 ð1Þ
oU
mxk oU2
U
þ
2ia
k
k
k
k
þ
2i
a
0
z
0
z
3
on
kh k 2
os
kh kz
"
!
!
#
mk2
o2 U
10x2
b1 b0
ð2Þ
2
þ
þ 2 þ 2b2 þ 3
a0 kz
þ 6a1 kz k U2 U kh kz
kh k2h
on2
kh k 2
"
#
"
2x2
b1 b0
6x2
b
ð0Þ
2
þ 2 þ 2b2 2 þ 2a1 kz k U2 U 3 þ 3 b3 2
kh kh
kh
kh k 2
kh k 2
!#
3
5a1
a0 3c1 kz k 4 þ kz k 2 2a2 jU j2 U
kz
kh
2
"
!#
x2
b1 b0
b0
As ð1Þ
4
U
þ
2 2 þ 2b2 þ 2 þ 2a1 kz k U0 2 b1 kh kh
kh
kz 2
kh k 2
"
!#
("
!
kmx
a0
A oU
2x2
b2
þ
þ 2i 2 kz k
2a1 U0 s
þ 3 b3 kh
kh
kz on
kh kz
k3h k 2
!
!#
b
b
2a1
þ 2 21 30 kz k 2
a2 U20
kh
kh kh
!
) 2
2x2
b1
b0
b0
A 2
2
þ
s U:
6b2 þ 5 5 2 2a1 kz k U0 þ 3 b1 kh
kh
k2h k 2
kh
k2z
ð64Þ
The solution of the set given in Eq. (59) yields
h
i
2
2
2
2
1
x
2
k
þ
4k
þ
k
b
þ
a
k
k
k
jU j
1
z
h 2
h
kh
k
ð0Þ
U2 ¼
½k2 ðkh b1 b0 Þ=2
kh b2 b1 þ bkh0 U20 2ðkh b1 b0 ÞU0 þ kh b0 A2 2
s
þ
k2z
2½k2 ðkh b1 b0 Þ=2
2
k þ kh b21 b20 U0 kh b20 A
þ
n;
kz
½k2 ðkh b1 b0 Þ=2
2k ð0Þ 2 4x
k
A
ð0Þ
2
W2 ¼ U2 2
k jU j 2 U0 n:
kh
kh kz
kh k
ð1Þ
ð1Þ
ð65Þ
ð1Þ
Eliminating U3 , W3 and P3 between Eq. (64) we obtain the following
nonlinear Schr€
odinger equation with variable coefficients
i
oU
o2 U
oU
2
þ l1
þ l4 s2 U þ l5 nU ¼ 0;
þ l2 jU j U þ il3 s
os
on
on2
ð66Þ
762
_ Bakirtasß, H. Demiray / Appl. Math. Comput. 154 (2004) 747–767
I.
where the coefficients l1 ; l2 ; . . . ; l5 are defined by
x x 1 m 2 2
k 3 k k þ 2 kz k k a0 kh kz k 2
k
;
l1 ¼
½a0 kh kz k 3 þ2x2 =k
(
x2 4kxk 3
l2 ¼ ½a0 kh kz k 3 þ2x2 =k1 16 2 þ 2 þ ðkh b3 b2 ÞK 2
2
kh
kh
5 a1 4
3
3
k kh kz a0 c1 kz k 6
þkh kz a2 2 kh
4
2
3
b
þ b2 kh k 2 þ b1 0 k 2 þ3a1 kh kz k 4
2
kh
½3x2 =kh þkh b2 k 2 þ3a1 kh kz k 4 3½ðkhta1 b0 Þk 2 2x2 "
!
#
b1 b0
4kxk
2
4
k þa1 kh kz k þ
þ kh b2 þ
kh
2 2kh
)
2
2
2
½k þ 4kx
þkh b2 þa1 kh kz k 2 k
;
2
kh ½k ðkh b1 b0 =2Þ
x
2 k k ½xþðxþ2kkÞU0 =kh þðmxk=kh kz a0 kz k þ2a1 kh kz kÞU0 k 2 A
l3 ¼
;
kz
½a0 kh kz k 3 þ2x2 =k
l4 ¼ ½a0 kh kz k 3 þ2x2 =k1
("
!
#
x2 3
b1 b0 2
kh a2 4 2
2
k U0
þ ðkh b3 b2 Þk þ
k kz a1 kh k2h
2
k2h 2
2
x
5
b0 2
2
4
þ
3kh b2 k þ b1 k a1 kh kz k U0
2
kh
kh
3
2
þ ðkh b1 b0 Þk 2 ½xþðxþ2kkÞU0 =kh 2
b1 b0
4kxk
2
4
k þa1 kh kz k þ
þ kh b2 þ
kh
2 2kh
h
i)
b0
2
kh b2 b1 þ kh U0 2ðkh b1 b0 ÞU0 þkh b0 A2
;
k2 ðkh b1 b0 Þ=2
k2z
l5 ¼ ½a0 kh kz k 3 þ2x2 =k1
h
i
(
b0
b1
2
4
k
b
þ
þa
k
k
k
þ4xkk=k
k
h
1
h
z
h
2
2
2kh
4kxk
U0 þ
kh
k2 ðkh b1 b0 Þ=2
)
b1 b0
b0 kh
A
2
U0 :
k :þkh kz
2 2
2
ð67Þ
As is seen from Eq. (67), the coefficients l3 , l4 and l5 are functions of the
tapering angle parameter A and they become equal to zero with vanishing
_ Bakirtasß, H. Demiray / Appl. Math. Comput. 154 (2004) 747–767
I.
763
tapering angle parameter. In this case the resulting equation reduces to the well
known nonlinear Schr€
odinger equation
i
oU
o2 U
þ l1
þ l2 jU j2 U ¼ 0:
os
on2
ð68Þ
Thus, the coefficients l3 , l4 and l5 stands for the contributions of tapering
angle to the evolution equation.
Progressive wave solution
Now we shall seek a solution to Eq. (67) of the following form
U ¼ V ðfÞ expfi½ðb0 s þ KÞn uðsÞg;
f ¼ n b1 s2 b2 s;
uðsÞ ¼ c0 s3 þ c1 s2 þ Xs;
ð69Þ
where b0 ,. . ., b2 , c0 , c1 ; K and X are some constants to be determined from
the solution and V ðfÞ is an unknown real function of its argument. Introducing (69) into Eq. (67) we have
2
l1 V 00 þ ðl5 b0 ÞnV þ ½l1 ðb0 s þ KÞ l3 ðb0 s2 þ KsÞ þ 3c0 s2
þ 2c1 s þ XV þ i½2l1 b0 s þ 2l1 K þ l3 s 2b1 s b2 V 0 þ l2 V 3 ¼ 0:
ð70Þ
In order to reduce this equation formally to the progressive wave solution of
the classical Schr€
odinger equation with constant coefficients we must have
b0 ¼ l5 ;
c0 ¼
b1 ¼ l1 l5 þ 12l3 ;
1
ðl l2
3 1 5
þ l3 l5 Þ;
b2 ¼ 2l1 K;
c1 ¼ ðl1 l5 þ 12l3 K:
ð71Þ
The remaining parts of Eq. (70) reduce to
l1 V 00 þ ðX l1 K 2 ÞV þ l2 v3 ¼ 0:
ð72Þ
This is indeed the same with the result of the classical nonlinear Schr€
odinger
equation under progressive wave solution (see, for instance [19]). Eq. (72)
admits the following type of solitary wave solution
"
1=2 #
l2
V ðfÞ ¼ Csech
Cf ;
ð73Þ
2l1
when l1 l2 > 0, where X ¼ l1 K 2 l2 C 2 =2, and
"
1=2 #
l2
V ðfÞ ¼ C tanh
Cf ;
2l1
ð74Þ
when l1 l2 < 0, where X ¼ l1 K 2 l2 C 2 =2. Here, in both cases C stands for the
amplitude of the solitary waves. The speed vp of the progressive and the speed
ve of enveloping waves are, respectively, given by
_ Bakirtasß, H. Demiray / Appl. Math. Comput. 154 (2004) 747–767
I.
764
vp ¼
1
;
2l1 K þ ð2l1 l5 þ l3 Þs
ve ¼
K þ l5 s
:
l5 ðl1 l5 þ l3 Þs2 þ ð2l1 l5 þ l3 ÞKs þ X l5 n
ð75Þ
Here, it is seen from Eq. (75) that these speeds are variable in terms of the
stretched coordinates n and s. In order to see the contributions of the tapering
parameter A, the coefficients a0 ; . . . ; c1 must be known. For that reason the
constitutive equation of the tube material must be known. In the present work
we shall utilize the constitutive relation proposed by Demiray [20] for soft
biological tissues, i.e.,
1
R ¼ fexp½aðI1 3Þ 1g;
ð76Þ
2a
where a is a material constant and I1 is the first invariant of Finger deformation
tensor and defined by I1 ¼ k2z þ k2h þ 1=k2z k2h . Introducing (76) into Eq. (34),
the coefficients a0 , b0 , b1 and b2 are obtained as
a0 ¼
"
a1 ¼
"
1
1
kh k3h k4z
!
F ðkh ; kz Þ;
!#
1
1 2 4
F ðkh ; kz Þ;
kh kz
!
!2 1#
1
1
1
1
2
A F ðkh ; kz Þ;
kh 3 2 þ 2a
kh kh k3h k4z
kh kz
kh3 k2z
1
1
þa 1 4 2
k4h k4z
kh kx
!
3
4a
þ
k5h k4z k4h k4z
!
1
1
b0 ¼
F ðkh ; kz Þ;
kz k4h k3z
"
!
!2 #
1
3
2a
1
b1 ¼
kh 3 2
þ
þ
F ðkh ; kz Þ;
kh kz k5h k3z
kh kz
kh kz
"
!
!
!3 #
6
3a
3
1
2a2
1
b2 ¼ 6 3 þ
1þ 4 2
kh 3 2
kh 3 2 þ
F ðkh ; kz Þ;
kh kz
kh kz
kk
kh kz
kh kz
kh kz
8 h z "
!2
!#
< 10
1
3
8
1
b3 ¼
þa
1þ 4 2
6 3 kh 3 2
: k7h k3z
kh kz
kh kz
kh kz
kh kz
!
!2
!4 9
=
4a2
3
1
4a3
1
þ
F ðkh ; kz Þ;
1þ 4 2
þ
kh 3 2
kh 3 2
kh kz
3kh kz
kh kz
kh kz
kh kz ;
"
!
!2 #
1
1
3
a
1
kz 2 3
þ
þ
F ðkh ; kz Þ;
c1 ¼
2 kh kz k3h k5z
kh kz
kh kz
a2 ¼
ð77Þ
_ Bakirtasß, H. Demiray / Appl. Math. Comput. 154 (2004) 747–767
I.
where the function F ðkh ; kz Þ is defined by
"
!#
1
2
2
F ðkh ; kz Þ ¼ exp a kh þ kz þ 2 2 3 :
kh kz
765
ð78Þ
In order to study the variation of wave speeds with the tapering parameter
A, we need the numerical value of the material constant a. For the static case,
the present model was compared by Demiray [21] with the experimental
measurement s by Simon et al. [22] on canine abdominal artery with the
characteristics Ri ¼ 0:31 cm; Ro ¼ 0:38 cm and kz ¼ 1:53, and the value of the
material constant a was found to be a ¼ 1:948.
Using the numerical value of a, the variation of the wave speeds with
variables n and s for the initial deformation kz ¼ 1:6 and kh ¼ 1:6 are calculated in computer and the results are illustrated in Fig. 1. In general, the speeds
are less sensitive to the variation of the variable n. The variation of the wave
speed ve for the enveloping wave in tubes with ascending radius is given in Fig.
1. It is seen that the wave speed increases with increasing distance parameter s.
The variation of ve with s for tubes with descending radius is given in Fig. 2.
The examination of this figure reveals that the wave speed decreases with increasing distance parameter s. The variation of progressive wave speed vp with
s for tubes with ascending radius is depicted in Fig. 3. This figure also reveals
that the wave speed decreases with increasing s. Finally, the variation of
progressive wave speed vp with s for tubes with descending radius is depicted
on Fig. 4. This figure shows that the speed decreases with increasing distance
parameter s. As a matter of fact, these numerical results are consistent with
physical observations conducted on such problems.
Fig. 1. The variation of the wave speed ve with s for tubes of ascending radius.
766
_ Bakirtasß, H. Demiray / Appl. Math. Comput. 154 (2004) 747–767
I.
Fig. 2. The variation of the wave speed ve with s for tubes of descending radius.
Fig. 3. The variation of the wave speed vp with s for tubes of ascending radius.
Fig. 4. The variation of the wave speed vp with s for tubes of descending radius.
_ Bakirtasß, H. Demiray / Appl. Math. Comput. 154 (2004) 747–767
I.
767
Acknowledgements
In conducting this research, H.D. was supported by the Turkish Academy
of Sciences.
References
[1] S.C. Ling, H.B. Atabek, A nonlinear analysis of pulsatile blood flow in arteries, J. Fluid Mech.
55 (1972) 492–511.
[2] T.J. Pedley, Fluid Mechanics of Large Blood Vessels, Cambridge University Press, Cambridge,
1980.
[3] Y.C. Fung, Biodynamics: Circulation, Springer Verlag, New York, 1981.
[4] H.B. Atabek, H.S. Lew, Wave propagation through a viscous incompressible fluid contained
in an initially stressed elastic tube, Biophys. J. 7 (1966) 486–503.
[5] A.J. Rachev, Effects of transmural pressure and muscular activity on pulse waves in arteries,
J. Biomech. Engng., ASME 102 (1980) 119–123.
[6] H. Demiray, Wave propagation through a viscous fluid contained in a prestressed thin elastic
tube, Int. J. Engng. Sci. 30 (1992) 1607–1620.
[7] G. Rudinger, Shock waves in a mathematical model of aorta, J. Appl. Mech. 37 (1970) 34–37.
[8] M. Anliker, R.L. Rockwell, E. Ogden, Nonlinear analysis of flow pulses and shock waves in
arteries, Z. Angew. Math. Phys. 22 (1968) 217–246.
[9] R.J. Tait, T.B. Moodie, Waves in nonlinear fluid filled tubes, Waves Motion 6 (1984) 197–203.
[10] R.S. Johnson, A nonlinear equation incorporating damping and dispersion, J. Fluid Mech. 42
(1970) 49–60.
[11] Y. Hashizume, Nonlinear pressure waves in a fluid-filled elastic tube, J. Phys. Soc. Jpn. 54
(1985) 3305–3312.
[12] Y. Yomosa, Solitary waves in large blood vessels, J. Phys. Soc. Jpn. 56 (1987) 506–520.
[13] H.A. Erbay, S. Erbay, S. Dost, Wave propagation in fluid-filled nonlinear viscoelastic tubes,
Acta Mech. 95 (1992) 87–102.
[14] H. Demiray, Solitary waves in a prestressed elastic tube, Bull. Math. Biology 58 (1996) 939–
955.
[15] A.E. Ravindran, P. Prasad, A mathematical analysis of nonlinear waves in a fluid-filled
viscoelastic tube, Acta Mech. 31 (1979) 253–280.
[16] H.A. Erbay, S. Erbay, Nonlinear wave modulation in a fluid-filled distensible tubes, Acta
Mech. 104 (1994) 201–214.
[17] H. Demiray, Modulation of non-linear waves in a viscous fluid contained in an elastic tube,
Int. J. Nonlinear Mech. 36 (2001) 649–661.
_ Bakirtasß, N. Antar, Evolution equations for nonlinear waves in a tapered elastic tube filled
[18] I.
with a viscous fluid, Int. J. Engng. Sci. 41 (2003) 1163–1176.
[19] H. Demiray, An analytical solution to dissipative nonlinear Schr€
odinger equation, Appl.
Math. Comput. 145 (2003) 179–184.
[20] H. Demiray, On the elasticity of soft biological tissues, J. Biomech. 5 (1972) 309–311.
[21] H. Demiray, Large deformation analysis of some basic problems in biophysics, Bull. Math.
Biology 38 (1976) 701–711.
[22] B.R. Simon, A.S. Kobayashi, D.E. Straddness, C.A. Wiederhielm, Re-evaluation of arterial
constitutive laws, Circ. Res. 30 (1972) 491–500.