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Non-linear wave modulation in a prestressed viscoelastic thin tube filled with an inviscid fluid
International Journal of Non-Linear Mechanics 34 (1999) 571—588
Non-linear wave modulation in a prestressed viscoelastic thin tube filled with an inviscid fluid Gu¨ler Akgu¨n*, Hilmi Demiray Istanbul Technical University Faculty of Sciences and Letters, Department of Engineering Sciences, 80626 Maslak, Istanbul, Turkey Received 7 June 1998
Abstract In the present work, the propagation of weakly non-linear waves in a prestressed thin viscoelastic tube filled with an incompressible inviscid fluid is studied. Considering that the arteries are initially subjected to a large static transmural pressure P and an axial stretch j and, in the course of blood flow, a finite time-dependent displacement is added to this X initial field, the governing non-linear equation of motion in the radial direction is obtained. Using the reductive perturbation technique, the propagation of weakly non-linear, dispersive and dissipative waves is examined and the evolution equations are obtained. Utilizing the same set of governing equations the amplitude modulation of weakly non-linear and dissipative but strongly dispersive waves is examined. The localized travelling wave solution to these field equations are also given. 1998 Elsevier Science Ltd. All rights reserved. Keywords: Wave modulation; Viscoelastic tube; Solitary waves
1. Introduction Propagation of harmonic waves in an initially stressed (or unstressed) circularly cylindrical tube filled with a viscous (or inviscid) fluid is a problem of interest since the time of Thomas Young who first obtained the pulse wave speed in human arteries. The historical evolution of the subject may be found in the books by McDonald [1] and Fung [2]. As far as the biological applications are concerned, most of the works on wave propagation in distensible tubes have considered small amplitude
*Corresponding author. Contributed by W.F. Ames.
waves, ignoring the non-linear effects, and concentrated on the dispersive character of waves [3—5]. However, when the non-linear effects arising from the convective terms of fluid and/or the constitutive relations of the tube material are introduced, one has to consider either finite amplitude or, smallbut-finite amplitude waves, depending on the order of non-linearity. The propagation of finite amplitude waves in fluid-filled elastic or viscoelastic tubes has been examined, for instance, by Rudinger [6], Anliker et al. [7], Tait and Moodie [8] by employing 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 [9], Hashizume [10], Yomosa [11], Erbay et al.
0020-7462/98/$19.00 1998 Elsevier Science Ltd. All rights reserved. PII: S 0 0 2 0 - 7 4 6 2 ( 9 8 ) 0 0 0 4 5 - 6
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[12] and Demiray [13] by employing various asymptotic methods. As is known, if the amplitude of a wave is sufficiently small, many non-linear systems admit harmonic wave solutions, in which the non-linear terms are small enough to be neglected, and the amplitude will remain constant in time. If the amplitude of the wave is small-but-finite, the non-linear terms cannot be neglected and give rise to a variation in amplitude in both space and time. When the amplitude varies slowly over a period of oscillation, a stretching transformation allows us to separate the system into a rapidly varying part associated with the oscillation and a 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 non-linear Schro¨dinger 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 non-linearity and dispersion. The problem of non-linear self-modulation of small-but-finite waves in fluid-filled distensible tubes was considered by Ravindran and Prasad [14], in which they showed that for a linear elastic tube wall model the non-linear self-modulation of pressure waves is governed by a NLS equation. Non-linear self-modulation in fluid-filled distensible tubes had been investigated by Erbay and Erbay [15], by employing the non-linear equations of a viscoelastic thin tube and the approximate fluid equations and they showed that the non-linear modulation of pressure waves is governed by a dissipative NLS equation. In the present work, utilizing the constitutive relation proposed by Demiray [16], the relation between the inner pressure and radial displacement and its time and space derivatives is obtained. Then, employing this pressure—displacement relation and the approximate (one-dimensional) fluid equations, the propagation of weakly non-linear waves in such a dispersive and dissipative medium is studied, in the longwave limit through the use of reductive perturbation method. Depending on the order of various coefficients charecterizing the material properties, the governing equation for the lowest-order terms in the expansion is obtained as
the Burgers’ equation, Korteweg—de Vries equation (KdV) and Korteweg—de Vries—Burgers’ equation (KdVB). The localized travelling wave solutions for these evolution equations are also given. Finally, the amplitude modulation of these non-linear governing equations for strongly dispersive case is examined by use of the reductive perturbation method and the dissipative non-linear Schro¨dinger equation is obtained as the governing equation. The solution of this equation under some initial condition is also investigated numerically by employing the split step Fourier method. It is observed that the amplitude of the solitary wave decreases with increasing time and the initial deformation.
2. Theoretical preliminaries 2.1. Equations of membrane The arterial wall material is known to be incompressible, anisotropic and viscoelastic (see [17, 18]). For its simplicity in non-linear analysis the arterial wall material shall be assumed to be incompressible, homogeneous, isotropic and viscoelastic. We consider a circular cylindrical long tube of radius R , which is subjected to a uniform pressure P and an axial stretch j . Let r be the radius of such X a tube after such a finite symmetrical static deformation. Upon this deformation we shall superimpose a finite time-dependent displacement u*(z*, t*), but in view of the external tethering effects, the axial motion is neglected. Hence, the position vector of a sample point after such a motion may be represented by r"(r #u*)e #z*e , z*"j Z, (1) P X X where e , e and e are the unit base vectors in the P F X cylindrical polar coordinates, Z is the axial coordinate of a point in the undeformed configuration and z* is the spatial coordinate after finite static deformation. The vector tangent to the meridional curve is defined by
*u* *r *u* T" " e #j e "j e #e . P X X X X *Z *Z *z* P
(2)
G. Akgu( n, H. Demiray / International Journal of Non-Linear Mechanics 34 (1999) 571–588
Thus, the length of this tangent vector is given by
*u* "T""j 1# "j " , X X X *z*
(3)
where
*u* " " 1# . X *z*
(4)
Hence, the unit tangent vector to the meridional curve and the exterior unit normal vector to the deformed membrane are, respectively, given by
1 *u* t" e #e , X " *z* P X 1 *u* n"e ;t" e! e . (5) F P *z* X " X Thus, the stretches in the meridional and circumferential directions are expressed as
u* j "j " , j "j " , " "1# , (6) X X F F F r where j "r /R is the stretch ratio in the circumF ferential direction after finite initial deformation. Let ¹ and ¹ be the membrane forces acting along the meridional and circumferential curves of the tube, respectively. Then the equation of the tube in the radial direction may be given by
* " *u* ¹" *u* F¹ ! X#" P*"o h , (7) F *t* r *z* " *z* X where h is the deformed thickness of the tube, o is the mass density of the tube after finite static deformation, t* is the time parameter and P* is the value of the fluid pressure on the tube surface. Let t be the Cauchy stress tensor referred to the IJ final configuration. The stress resultants ¹ and ¹ are, then, given by ¹ "h t , ¹ "h t , (8) where h is the final thickness of the tube, t and t are the stress components along the meridional and circumferential directions in the final configuration, respectively. Assuming that the tube material is incompressible, the incompressibility
573
condition requires that h H h " or h " , (9) j j "" F X where H is the initial thickness of the tube. In order to complete the field equations, one must know the value of the fluid pressure P* on the tube surface. Therefore, Eq. (7) is to be supplemented by the equations governing the fluid body. 2.2. Fluid equations Although blood is known to be an incompressible viscous fluid, as pointed out by Rudinger [6], in a number of applications, such as flow in large blood vessels, the effect of viscosity may be neglected, i.e. the blood may be treated as an incompressible inviscid fluid. As a result of this simplifying assumption the variation of field quantities with radial coordinate may be disregarded. However, the radial changes are included by taking the variation of cross-sectional area into consideration. The equation governing the mass conservation of the fluid body may be given by *A * # (Av*)"0, *t* *z*
(10)
where A denotes the internal cross-sectional area of the tube and v* is the axial velocity component of the fluid. Recalling the definition of A in terms of the final inner radius of tube, i.e. A"n(r #u*), the above equation may be written as 2
*u* *u* *v* #2v* #(r #u*) "0. *t* *z* *z*
(11)
The equation of balance of linear momentum in the axial direction may be given by *v* *v* 1 *P* #v* # "0, (12) *t* *z* o *z* where o is the mass density of the fluid. Eqs. (7), (11) and (12) provide sufficient relations to determine the unknowns u*, v* and P*. In order to complete the fluid equations one must know the constitutive relation for the tube material. In this work we shall adopt the following type of constitutive relation proposed by Demiray
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574
This constitutive relation is non-linear in terms of c , but it is linear in terms of d . Since we IJ KL assume that the tube is thin the stress component t may approximately be taken to be zero. By using this result the hydrostatic pressure is expressed as follows:
[16] for soft biological tissues: t "Pd #exp[a(I !3)] IJ IJ R* j (t*!q)cR (q) dq ; c b# IJ T KK \ R* # k (t*!q) [b (t*, q) b (t*, q) T IK JK \
#b (t*, q) b (t*, q)] dq , IK JK
(13)
where an overdot denotes the differentiation with respect to variable q, P is the hydrostatic pressure, t is the Cauchy stress tensor referred to the final IJ configuration, c is the Finger deformation tensor, IJ I is the first invariant of c , a and b are two IJ material constants, j (t*) and k (t*) are two funcT T tions characterizing the viscoelastic effects, c (t*) IJ and b (t*, q) are defined by IJ c (t*)"F (t*)F (t*), IJ I) J) b (t*, q)"F (t*) F (q). (14) IJ I) J) Here x "x (X , t*) defines the motion, and I I ) F ,*x /*X is the deformation gradient. I) I ) In general, it is quite difficult to study a nonlinear problem by use of the constitutive relation given in Eq. (13). Assuming that the memory of the material under consideration is short, the viscoelastic coefficients j (t*) and k (t*) may be approximT T ated as follows: j (t*)" j d(t*), k (t*)"k d(t*). (15) T T T T Here, j and k are constants characterizing visT T coelastic effects, d(t*) is the Dirac-Delta function. Inserting Eq. (15) into Eq. (13) and then integrating the resulting equations with respect to q we obtain the following constitutive relation: t "Pd #[bc #j c d c IJ IJ IJ T KL KL IJ #2k d c c ] exp[a(I !3)], (16) T KL KI LJ where d is the deformation rate tensor defined by IJ d , (» #» ). (17) J I IJ I J with » being the components of the tube velocity. I
P"![bc #j (c d #c d #c d )c T #2k d (c )] exp[a(I !3)]. (18) T Since the Finger deformation tensor is diagonal, its components are given as follows: 1 . (19) c " j j Furthermore, by using the relation CQ " )* 2d F F , the components of deformation rate IJ I) J) tensor are given by
c "j ,
c "j ,
j j j j d " , d " , d "! # . (20) j j j j At this stage it might be convenient to introduce the following non-dimensionalized quantities: r t*" t, z*"r z, u*"r u, v*"r v, c o h P*"co (p #p), m" o r D bh c" , ¹ "bh p , ¹ "bh p , o r D j c k c l " T , l " T . (21) b r b r Inserting Eq. (21) into Eqs. (7), (11) and (12) we obtain the following non-dimensionalized equations:
m *u p 1 * p *u p #p" # ! , " *t " " *z " *z F F F X *u *u *v 2 #2v #(1#u) "0, *t *z *z *v *v *p #v # "0. *t *z *z
(22)
(23)
(24)
G. Akgu( n, H. Demiray / International Journal of Non-Linear Mechanics 34 (1999) 571–588
In these equations the principal stresses p and p are given by
p "
1 1 j! # (l #2l ) j# j j j j
2l j 1 1 ! # l jj ! ! j j j j (l #2l ) j F(j , j ), # j j j p "
1 1 1 j! # l j j! ! j j j j
(l #2l ) j # (l #2l ) # j j j
1 2l j ; j# ! F(j , j ), j j j j
(25)
where the function F(j , j ) is defined by
1 F(j , j ),exp a j#j# !3 j j
.
(26)
As can be seen from Eq. (22) the pressure function p depends upon u, *u/*z, *u/*z, *u/*t*z, *u/*t*z, *u/*t and *u/*t. If the pressure function p is expanded into a power series of u and its derivatives we obtain *u *u *u p"b u#b #b #b #b u *z *t *t
*u *u *u #b #b u #b u *z *z *t
575
*u *u *u #2b *z *z *z*t
*u *u #b #2 . *z *z*t
(27)
Here, b (i"1, 2, . . . , 19) coefficients are some G functions of the initial deformation and b , b , b , b , b , b , b , b and b depend on the elastic effects, b , b , b , b , b —b characterize the viscous effects and b , b and b are related to the inertial effects. The explicit expression of these coefficients are given in the appendix. Thus, Eqs. (23), (24) and (27) give sufficient relations to determine u, v and p.
3. Long-wave approximation In this section we shall examine the propagation of small-but-finite amplitude waves in a fluid filled and prestressed non-linear viscoelastic thin tube whose dimensionless governing equations are given by the Eqs. (23), (24) and (27). For this we adopt the long-wave approximation and employ the reductive perturbation method [19]. For that purpose we would like to see the dispersive character of our model equations. Linearizing the field Eqs. (23), (24) and (27) and seeking a harmonic wave type of solution to these equations we obtain the following dispersion relation: (2#b k)u!ib ku!b k#b k"0,
(28)
*u *u *u *u #b u #b #b u#b u *t *t *z *t
where u is the angular frequency and k is the wave number. Assuming that the wavelength is large as compared to the radius of the tube, one can expand the dispersion relation, u(k), into a power series of k around k"0 and obtain
*u *u #b u #b u *z *t
u"gk(1#p k#p k#2),
*u *u *u #b #b u *z *z *z *u *u *u *u #b #b u *t *z *t *z
(29)
where g, p and p are defined by
b ib b b b # ! . g" , p " , p "! 4g 2 2b 4 16b (30)
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576
The form of the dispersion relation (29) suggests us to introduce the following coordinate stretching: m"e?(z!gt), q"e?>gt,
*u *u *u p"b u#b e? #b e?g !2e *m *m *m*q #e
*u *u *u !b ge? !e #2 . *q *q *m
u" eLu (m, q), v" eLv (m, q), L L L L p" eLp (m, q), (33) L L where u , v and p (n"1, 2, . . .) are unknown L L L functions to be determined from the solution of the field equations. Introducing expansion (33) into Eq. (32) and setting the like powers of e equal to zero we obtain the following sets of differential equations. O(e) order equations:
!g
*u *v # "0, *m *m
*v *p # "0, *m *m
From the solution of the set (34) we obtain
!2g
*u *º *º *v #2g #6gº # "0, *m *q *m *m
!g
*v *º *º *p #2g #4gº # "0, *q *m *m *m
*º p "b u #(b #gb )e?\ *m *º !b ge?\ #b º. *m
p "b u .
(34)
*u *u *u *v *v # #u "0, !2g #2g #2v *m *m *q *m *m *v *v *v *p #g #v # "0, *m *q *m *m
(37)
Eliminating u , v and p between these equations we obtain the following master equation: *º *º *º #c º #c e?\ *m *m *q *º !c e?\ "0, *m
(38)
where the coefficients c , c and c are defined by 5 b 1 2b b c " # , c " b # , c " . (39) 4 2 b b 4g From this general evolution equation various wellknown equations may be obtained as some special cases.
O(e) order equations:
!g
3.1. Solution of the field equations
(32)
Assuming that the field quantities can be expanded into asymptotic series of e we have
!2g
(35)
u "º(m, q), v "2gº, p "2gº, g"b /2, (36) where º(m, q) is an unknown function whose governing equation will be obtained later. Introducing Eq. (36) into the set of equations (35) we get
*u *u *v *u #2eg #2v #(1#u) "0, *m *q *m *m
*v *v *p *v #eg #v # "0, *q *m *m *m
!g
*u !b ge?\ #b u. *m
(31)
where e is a small parameter measuring the weakness of dispersion and/or non-linearity and a is a positive constant whose values will be specified later. Introducing Eq. (31) into the field Eqs. (23), (24) and (27) we obtain !2g
*u *u #b ge?\ p "b u #b e?\ *m *m
(i) b "O(1) and a"1: In this case the coefficient of *º/*m is of the order of e and the evolution equation reduces to the Burgers’ equation *º *º *º #c º !c "0, *q *m *m
(40)
G. Akgu( n, H. Demiray / International Journal of Non-Linear Mechanics 34 (1999) 571–588
which results from the balance of non-linearity with dissipation. As is well known, the Burgers’ equation has a steady solution of the form
3c [sech f!2 tanh f!2], º(m, q)" 25c c
º(m, q)" (º\#º>)# (º\!º>) tanh f,
c 3c q. f" m# 10c 125c
c c f" (º\!º>) m! (º\#º>)q , 4c 2
(41)
where º\ and º> are, respectively, the uniform states of º at GR. This solution may be regarded as ‘‘shock’’ wave since it relates two different states denoted by º\ and º> . (ii) b "O(e) and a": In this special case the coefficient of *º/*m is of the order of e and the resulting evolution equation reduces to the wellknown Korteweg—de Vries (KdV) equation *º *º *º #c º #c "0, *m *m *q
(42)
which results from the balance of non-linearity with dispersion. The KdV equation has a steady solution of the form º(m, q)"º #b sech f,
f"
c b b m!c º # q , 3 12c
(43)
where º '0 is the value of º as fP$Rand b is the amplitude of the wave relative to the constant solution º at infinity. (iii) b "O(e ) and a" : For this special case the evolution equation reduces to the following Korteweg—de Vries—Burgers’ (KdVB) equation *º *º *º *º #c º !c #c "0, *m *m *m *q
(44)
which is obtained by balancing the non-linearity with dispersion and dissipation. The KdVB equation has a travelling wave solution of the following form [20]: 3c [sech f#2 tanh f#2], º(m, q)" 25c c c 3c f"! m# q, 10c 125c
577
(45)
The variation of these solution profiles with initial deformation and other characteristics will be discussed later.
4. Non-linear wave modulation In this section we shall examine the amplitude modulation of weakly non-linear waves in a fluidfilled non-linear viscoelastic tube whose dimensionless governing equations are given in Eqs. (23), (24) and (27). For this, we again employ the reductive perturbation method [19] and introduce the following coordinate stretching: m"e(z!jt), q"et,
(46)
where j is a constant which will be shown to be the group velocity. We further assume that our field variables are functions of fast variables (z, t) and also slow variables (m, q). Thus, the following substitution is permissible: * * * P #e , *z *z *m
* * * !ej #e . *t *m *q
(47)
We shall further assume that the field variables are expressible as asymptotic series in e as follows: u" eLu , v" eLv , p" eLp . L L L L L L
(48)
where u , v and p are some unknown functions of L L L fast (z, t) and slow (m, q) variables. For our future purposes we shall assume that the viscoelastic coefficients are of the following order in terms of e: b "bM e. G G
(49)
Introducing the expansions (47) and (48) into the field Eqs. (23), (24) and (27) and setting the coefficients of like powers of e equal to zero we obtain the following set of differential equations.
G. Akgu( n, H. Demiray / International Journal of Non-Linear Mechanics 34 (1999) 571–588
578
2
*v *v *p *u # "0, # "0, *z *t *z *t
(50)
*u *u #b u #b u#b u *z *t
*u *v *u *v *u # !2j # #2v *z *t *z *m *m *v "0, #u *z
#b u
*v *p *v *p *v # !j # #v "0, *z *z *m *m *t
*u *u #b u . #b u *z *t
(51)
¹hird-order, O(e), equations: *u *v *u *v *u *u # !2j # #2 #2v *z *z *m *m *q *t *u *v *v *u #u #u #2v #2v *z *z *z *m *v "0, #u *m *v *p *v *p *v * # !j # # # (v v ) *t *z *m *m *q *z
*u *u #b p "b u #b *z *t
u "º exp[i(ut!kz)]#c.c., v "» exp[i(ut!kz)]#c.c., p "P exp[i(ut!kz)]#c.c., (53) where u is the angular frequency, k is the wave number º, » and P are amplitude functions which depend on the slow variables m and q and c.c. stands for the complex conjugate of the corresponding quantity. Introducing Eq. (53) into Eq. (50) and requiring the non-vanishing solution for º, » and P we obtain 2u 2u º"º, »" º, P" º (54) k k
4.1.2. The solution of O(e) order equations Introducing Eqs. (53) and (54) into Eq. (51) one obtains
*u *u # #b 2 *z*m *m
(52)
provided that the dispersion relation (28) holds true. Here º(m, q) is an unknown function whose governing equation will be obtained later.
*v "0, #v *m
*u *u *u # j #2 #b !2j *t*m *m *t*q
*u *u *u #b . *z *z *z
4.1.1. The solution of O(e) order equations The form of the differential equations given in Eq. (50) suggests us to seek the following type of solution to these differential equations:
*u *u #b u#b !2b j *t*m *z
4.1. The solution of field equations
*u *u *u #b #2b p "b u #b *z *t *z*m
*u *u *u #u !2ju #b u *t *t *t*m
Second-order, O(e), equations:
2
*u *u *u #u #2u #b u *z *z *z*m
*u *u . #b p "b u #b *z *t
2
*u *u *u *u #2b u u #2b # #b *t *z *z *m
First-order, O(e), equations:
2
*u *v u *º # #2 !j e (#6iuºe ( *t *z k *m #c.c."0,
G. Akgu( n, H. Demiray / International Journal of Non-Linear Mechanics 34 (1999) 571–588
*p 2u u *v *º # # !j e (#4iuºe ( *t *z k k *m #c.c."0, *u *u #b p "b u #b *t *z #2[b #(b !b )k!b u]"º" *º !2i(b k#b uj) e ( *m #[b !(b #b )k!b u]ºe (, (55) where, for convenience, we have defined "º"" ºº*, where º* is the complex conjugate of º and the phasor is defined by "ut!kz. The form of Eq. (55) suggests us to seek the following type of solution for u , v and p : u "º# º? e ?(#c.c., ? v "»# »? e ?(#c.c., ? p "P# P? e ?(#c.c., (56) ? where º , . . . , P are functions of slow variables m and q. Comparing third of Eq. (55) and third of (56) we obtain the following expressions: P"b º#2[b #(b !b )k !b u]"º", P"(b !b k!b u)º *º !2i(b k#b uj) , *m P"(b !4b k!4b u)º #[b !(b #b )k!b u]º. (57) Introducing Eq. (56) into Eq. (55) yields the following sets of equations:
u *º 2uº!k»"2i !j , k *m 2iu u *º u»!kP" !j , k k *m
(b k#b u !b )º#P *º "!2i(b k#b uj) , *m
579
(58)
and 2uº!k»"3uº, 2u º, u»!kP" k (4b k#4b u!b )º#P "[b !(b #b )k!b u]º. (59) The solution of the set in Eq. (58) gives the following result:
2u *º 2i u »" º! !j , k k k *m 2u 4iu u *º !j . P" º! k k k *m
(60)
In order to have a non-zero solution for º the following condition must be satisfied: 2u!b k . j"v " (61) E uk(2#b k) Hence, the solution of the set of Eq. (59) gives [6u#b k!(b #b )k!b uk] º" º, 3(b k!2u) 2u 3u »" º! º. (62) k k 4.1.3. The solution of O(e) order equations Introducing the solutions (53) and (56) into Eq. (52) we obtain the following equations: 2
*v *» *º *u # # !2j #2iu (º* º *z *m *m *t !ºº*)#ik(»º*!»*º) 6u * *» *º !2j # "º" e # k *m *m *m
#2
*º !iº(2k»#2uº *q
G. Akgu( n, H. Demiray / International Journal of Non-Linear Mechanics 34 (1999) 571–588
580
Fig. 1. The variation of Burgers’ schok with circumferential stretch.
!6iuº º*) e (#
*» *º !2j *m *m
6u *º !3i (2uº#k» )º# º e ( k *m !iº(10uº#4k» )e ("0, (63) *v *p *P *» 4u * # # !j # "º" e *t *z *m *m k *m
#
*» 2u *º *P # !j !2iu(»º *m *m k *q
#» º*) e (#
*P *» !j *m *m
4u *º !4iu»º# º e ( k *m !6i»ºe ("0.
(64)
The forms of the differential Eqs. (63) and (64) suggest us to seek the following type of solution for u , v and p : u "º# º?e ?(#c.c., ? v "»# »? e ?(#c.c., ? p "P# P? e ?(#c.c., ?
(65)
where º , . . . , P are functions of slow variables m and q. Here we need to solve the equations corresponding to the zeroth (a"0) and first (a"1) modes only. Introducing expression (65) into Eqs. (63) and (64) the following equations are obtained.
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581
Fig. 2. The variation of solitary wave profile for KdV equation with circumferential stretch.
a"0 equations:
»"2jº!2
*» *º !2j #2iu(º* º!ºº*) *m *m 6u * #ik(»º*!»*º)# "º""0, k *m
(67)
*» *º iuº!ik»# !2j *m *m #2
(66)
º 4u 4u #2j !j #2[b #(b !b )k!b u] k k " 2j!b ;º,
4u !j "º". k
a"1 equations:
*» 4u * *P # !j "º""0, *m *m k *m P"º#2[b #(b !b )k !b u]"º""0. The solution of the set of Eq. (66) gives
*º !2iº(k»#uº ) *q
!6iuºº*"0, *P *» 2u *º # iu»!ikP# !j *m *m k *q !2iu(»º#ºº*)"0, P"(b !b k!b u)º *º *º #(b #jb ) !2i(kb #jub ) *m *m
582
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Fig. 3. The variation of solution profile for KdVB equation.
*º #2ib u #iub º#(2b !b k *q !b u)ºº#[2b !(4b #5b )k !5b u]ºº*#[3(b !b k !b u)#b k!b k]"º"º.
(68)
Eliminating º, » and P between Eq. (68) we obtain the following dissipative non-linear Schro¨dinger equation: i
*º *º #k #k "º"º#ik º"0, *m *q
(69)
where the coefficients k , k and k are defined by 1 k " [(2#b k)j#4b ukj 2u(2#b k) #6b k#ub !b ],
1 2u 4u k " #j !j #b 2u(2#b k) k k #(b !b )k!b u [2u#8juk#2b k!(b k#b u)k] # b j ! 2 [6u#b k!(b #b )k!b uk] # 3(b k!2u) [10u#2b k#(4b !5b )k!5b uk] ; 3(b k!2u) !38u#8ukj#3b k#(b !3b )k
!3b uk!b k , b k k " . 2(2#b k)
(70)
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583
Fig. 4. The variation of solution profile for dissipative NLS equation with time.
If the coefficient b vanishes, Eq. (69) reduces to the well-known non-linear Schro¨dinger equation i
*º *º #k #k "º"º"0. *m *q
(71)
The NLS equation which arises in a variety of fields as an equation describing the self-modulation of the one-dimensional monocromatic plane waves in dispersive media. The steady-state solution of the NLS equation, which generally represents the wavetrains expressible in terms of Jacobian elliptic functions, include a bright and a dark envelope solitons, a phase jump, and a plane wave with a constant amplitude as special cases. The criterion whether k k '0 or k k (0 is important in determining how the given initial data will evolve for long times for the asymptotic field governed by the NLS equation. Now let as seek the travelling wave solutions of the NLS equation in the
form º(m, q)"»(f) exp[i(Km!)q)], f"m!v q, v "const., (72) where »(f) is a real function of f. In this case, if "º" approaches a constant » at infinity, the solution is given by a non-linear plane wave º(f)"» exp[i(Km!)q)], (73) where )"k K!k ». In general, the solution can be obtained in terms of the Jacobian elliptic functions under the assumption of v "2k K. The explicit functional form of these solutions will not be given here, expect for the following limiting cases. Assuming that »P0 and d»/dfP0 as "f"PR, for k k '0, the solution is given by k »(f)"A sech Af , (74) 2k
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584
Fig. 5. The variation of solution profile for dissipative NLS equation with circumferential stretch.
where )"k K!k A/2. For k k (0, if ºP» and dº/dfP0 as fPR, the solution be comes k »f , (75) »"» tanh ! 2k where )"k K!k » . These solutions corres pond to an envelop solitary wave and a phase jump, respectively. On the other hand, it is wellknown that the plane wave solution of NLS equation is modulationally unstable if k k '0 or stable if k k (0.
5. Numerical results and discussion As stated before, the coefficients c (i"1, 2, 3) G defined in Eq. (39) and k (i"1, 2, 3) defined in Eq. G (70) depend on the initial deformation, mechanical and geometrical characteristics of the wall material.
For the static case, the present mechanical model is compared by Demiray [21] with the experimental measurements by Simon et al. [22] on canine abdominal artery with geometrical characteristics R "0.31 cm, R "0.38 cm and j "1.53, and the G X value of the material constant a was found to be a"1.948. Also choosing l "2l , the profile of the travelling wave solutions given in Eqs. (41), (43) and (45) are evaluated numerically for various values of circumferential stretch and the variable f and the results are depicted in Figs. 1—3. Fig. 1 gives the variation of the solution profile for Burgers’ equation with circumferential stretch ratio and the variable f. Since, with increasing initial deformation, the degree of non-linearity of the material is weakened, the shock front with increasing initial deformation is flattened. The variation of solitary wave profile of the KdV equation with f and the initial deformation is given in Fig. 2. As is seen from this figure the profile of solitary wave gets
G. Akgu( n, H. Demiray / International Journal of Non-Linear Mechanics 34 (1999) 571–588
steepened with increasing circumferential stretch ratio. The solution profile of the KdVB equation for a set of initial deformation is shown in Fig. 3. The branch KdVB-1 in the figure corresponds to the solution given in the first line of Eq. (45) and KdVB-2 to the solution in the second line in Eq. (45). As to the variation the solution profile of the non-linear Schro¨dinger equation it might be interesting to study the dissipative non-linear Schro¨dinger equation by numerical means. By utilizing split step Fourier method [23] and the initial condition º(m, 0)"exp[!i(2m#n/2)] sech m, the evolution equation solved numerically and the results are depicted in Figs. 4 and 5, respectively. Fig. 4 shows the variation of solution profile with time. As is seen from the figure due to dissipative term the amplitude of the solitary wave decreases with increasing time, which is to be expected. Moreover, the amount of decrement depends on the value of the dissipative coefficient k which varies with in itial deformation. The variation of solution profile with initial deformation is shown in Fig. 5. As is seen from the figure the solitary wave profile decreases with increasing initial deformation. Indeed, our numerical calculations show that the dissipation coefficient k increases with initial circum ferential stretch.
Appendix. Polynomial approximation of the pressure function In this subsection, we shall try to give the polynomial approximation to the pressure function in terms of the radial displacement and its space and time derivatives. In Eq. (22), pressure was found to be
1 * p *u m *u p # ! , (A.1) p #p" " *t " " *z " *z F F X F in which the stress components p and p are defined by
j p " G (j , j ) #G (j , j j p " G (j , j ) #G (j , j
j j ) F(j , j ), j j j ) F(j , j ), j (A.2)
585
where G (j , j ) (a"1, 2, 3) and F(j , j ) are de? fined by
1 2l ! , G "(l #2l ) j# j j j (l #2l ) 1 1 #l jj! ! G " , j j jj 1 2l ! , G "(l #2l ) j# jj j 1 F"exp a j#j# !3 . (A.3) jj For our future purposes we need the non-linearity up to and including cubic terms. Considering the following expansions:
j *u *u j *u *u *u : , : !u #u , j *z *z*t j *t *t *t *u #2 , j"j 1#2 X *z
j"j [1#4u#6u#4u#2] , F 1 *u 1 #2 , " 1! j j *z X 1 1 " [1!2u#3u!4u#2], j j F *u jj"jj 1#2u#u# F X *z
#2u
*u #2 , *z
1 1 *u " 1!2u#3u! !4u jj jj *z X F *u #2u , *z one can write G "G#Gu,
G "G#Gu, *u G "G#Gu#Gu#G , *z
(A.4)
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586
*u F" 1#F u#F u#F #F u *z
*u , #F u *z
(A.5)
where G , . . . , F are defined by 1 2l G"(l #2l ) j# ! , X jj j X F F (l #2l ) l , G"4 ! j j j F F X 1 1 (l #2l ) , G"l jj! ! # F X j j jj F X F X 1 (l #2l ) , !4 G"2l jj# F X j jj F F X 1 2l G"(l #2l ) j# ! , F jj j X F X 1 , G"4(l #2l ) j! F jj X F 10 G"(l #2l ) 6j# , F jj X F (l #2l ) l , G"2 ! jj j F X X 1 F "2a j! , F jj F X 1 3 #a j# F "2a j! , F jj F jj F X F X 1 F "a j! , X jj F X 4 1 1 F " a j! #2a j! 3 F jj F jj F X F X 3 4a ; j# ! , F jj jj F X X F a 1 1 F "2 #a j! j! F X jj jj jj F X F X F X
pressure may be expressed as follows:
*u *u *u *u #b #b u #b u #b u *z *t *t *z
*u *u *u *u #b #b u#b u #b u *t *z *t *z
*u *u *u *u #b u #b u #b *t *z *z *z
*u *u *u *u #b u #b *t *z *t *z
*u *u *u *u *u #2b #b #2 , *z *z *z*t *z *z*t
*u *u *u #b #b #b u p"b u#b *t *t *z
(A.7)
.
(A.6) Introducing the expansions (A.3) and (A.4) into Eq. (A.1) and keeping up to the cubic terms, the
where the coefficients b are defined by G 4 1 #2a j! F, b " F jj j j X F F X 1 b " !j F , X jj X F b "m, 1 2l ! F, b " (l #2l ) j# F jj j X F X 10 1 11 b "! #a j! j# F jj F jj jj X F F X F X 1 #2a j! F, F jj F X 1 1 1 b "! #a j! j! X F jj jj jj X F F X F X 3 1 b " j! !2a j! X jj F jj F X F X 1 ; j! F, X jj F X b "!m, 6l 7 b " #(l #2l ) j! F j jj X F X
F,
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1 1 #2a j! (l #2l ) j# F jj F jj F X F X 2l ! F, j X 1 1 (l #2l ) F, # !jj ! b " l j j X F jj X F X F 4a 20 9 b " # !5j F jj jj jj X F X F F X 7 1 #2a j# j! F jj F jj F X F X 1 4 F, # a j! F jj 3 F X 12l 28 b " ! # (l #2l ) j jj X F X 1 1 3j# #a (l #2l ) j! F jj F jj F X X F 9 2 2l # !5j (l #2l )! F j jj jj X F F X X 1 #2a j! F jj F X 1 2l ; (l #2l ) j# ! F, F jj j F X X 6 5 5 b " !j #a jj! ! X X F j j jj X F F X 9 1 # !2a j! X jj jj X F F X 1 ; j! F, F jj F X b "m, 4 2 1 1 b " #4a ! ! jj jj j j X F X F X F 1 1 !2a j! j! F, X jj F jj F X F X 6 1 b "! #3a j! F, X jj jj X F F X
587
3 3(l #2l ) 1 b " l ! !jj # j j F X jj X F X F 1 1 #a (l #2l ) j# j! F jj X jj F X F X 2 1 # !j F jj jj X F F X 1 1 1 #2l j # ! !jj F, F j j jj F X F X F X 1 6(l #2l ) 2 !2l b " # j j jj X X F F 1 1 1 !2a j! l jj! ! F jj X F j j F X X F (l #2l ) # F, jj X F 2l 1 F . (A.8) b " !(l #2l ) j# X jj j F X F This expression of the pressure is used in studying the field equations in the long-wave limit and in the amplitude modulation.
Acknowledgements One of the authors (HD) was supported by the Turkish Academy of Sciences.
References [1] D.A. Mc Donald, Blood Flow in Arteries, The Williams and Wilkins Co., Baltimore, MD, 1960. [2] Y.C. Fung, Biodynamics: Circulation, Springer, New York, 1984. [3] J.R. Womersley, Oscillatory motion of a viscous liquid in a thin walled elastic tube I: The linear approximation for long waves, Phil. Mag. 46 (1955) 199—219. [4] A.I. Rachev, Effects of transmural pressure and muscular activity on pulse waves in arteries, J. Biomech. Engng ASME 102 (1980) 119—123. [5] H. Demiray, Wave propagation through a viscous fluid contained in a prestressed thin elastic tube, Int. J. Engng Sci. 30 (1992) 1607—1620. [6] G. Rudinger, Shock waves in mathematical models of the aorta, J. Appl. Mech. 37 (1970) 34—37.
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[7] M. Anliker, R.L. Rockwell, E. Ogden, Nonlinear analysis of flow pulses and shock waves in arteries, ZAMP 22 (1971) 217—246. [8] R.J. Tait, T.B. Moodie, Waves in nonlinear fluid filled tubes, Wave Motion 6 (1984) 197—203. [9] R.S. Johnson, A nonlinear equation incorparating damping and dispersion, J. Fluid Mech. 42 (1970) 49—60. [10] Y. Hashizume, Nonlinear pressure waves in a fluid filled elastic tube, J. Phys. Soc. Japan 54 (1985) 3305—3312. [11] S. Yomosa, Solitary waves in large blood vessels, J. Phys. Soc. Japan 56 (1987) 506—520. [12] H.A. Erbay, S. Erbay, S. Dost, Wave propagation in fluid filled nonlinear viscoelastic tubes, Acta Mechanica 95 (1992) 87—102. [13] H. Demiray, Solitary waves in prestressed elastic tubes, Bull. Math. Biol. 58 (1996) 939—955. [14] A.E. Ravindran, P. Prassad, A mathematical analysis of nonlinear waves in a fluid-filled viscoelastic tube, Acta Mechanica 31 (1979) 253—280. [15] H.A. Erbay, S. Erbay, Nonlinear wave modulation in fluid filled distensible tubes, Acta Mechanica 104 (1994) 201—214.
[16] H. Demiray, Int. J. Engng Sci. 29 (1991) 575—585. [17] W.O. Fenn, Changes in length of blood vessels on inflation, in: Remington (Ed.), Tissue Elasticity, American Physiological Society, Washington, DC, 1957, pp. 154—167. [18] B.S. Gow, M.G. Taylor, Measurements of viscoelastic properties of arteries in living dog, Circulation Res. 23 (1968) 111—122. [19] A. Jeffrey, T. Kawahara, Asymptotic Methods in Nonlinear Wave Theory, London, 1982. [20] A. Jeffrey, S. Xu, Exact solutions to the Korteweg— de Vries—Burgers equation, Wave Motion 11 (1989) 559—564. [21] H. Demiray, Large deformation analysis of some basic problems in biophysics, Bull. Math. Biol. 38 (1976) 701—711. [22] B.R. Simon, A.S. Kobayashi, D.E. Stradness, C.A. Wiederhielm, Re-evaluation of arterial constitutive laws, Circulation Res. 30 (1972) 491—500. [23] T.R. Taha, M.J. Ablowitz, Analytical and numerical aspects of certain nonlinear evolution equations. II. Numerical nonlinear Schro¨dinger equation, J. Comput. Phys. 55 (1984) 203—230.
Non-linear wave modulation in a prestressed viscoelastic thin tube filled with an inviscid fluid Gu¨ler Akgu¨n*, Hilmi Demiray Istanbul Technical University Faculty of Sciences and Letters, Department of Engineering Sciences, 80626 Maslak, Istanbul, Turkey Received 7 June 1998
Abstract In the present work, the propagation of weakly non-linear waves in a prestressed thin viscoelastic tube filled with an incompressible inviscid fluid is studied. Considering that the arteries are initially subjected to a large static transmural pressure P and an axial stretch j and, in the course of blood flow, a finite time-dependent displacement is added to this X initial field, the governing non-linear equation of motion in the radial direction is obtained. Using the reductive perturbation technique, the propagation of weakly non-linear, dispersive and dissipative waves is examined and the evolution equations are obtained. Utilizing the same set of governing equations the amplitude modulation of weakly non-linear and dissipative but strongly dispersive waves is examined. The localized travelling wave solution to these field equations are also given. 1998 Elsevier Science Ltd. All rights reserved. Keywords: Wave modulation; Viscoelastic tube; Solitary waves
1. Introduction Propagation of harmonic waves in an initially stressed (or unstressed) circularly cylindrical tube filled with a viscous (or inviscid) fluid is a problem of interest since the time of Thomas Young who first obtained the pulse wave speed in human arteries. The historical evolution of the subject may be found in the books by McDonald [1] and Fung [2]. As far as the biological applications are concerned, most of the works on wave propagation in distensible tubes have considered small amplitude
*Corresponding author. Contributed by W.F. Ames.
waves, ignoring the non-linear effects, and concentrated on the dispersive character of waves [3—5]. However, when the non-linear effects arising from the convective terms of fluid and/or the constitutive relations of the tube material are introduced, one has to consider either finite amplitude or, smallbut-finite amplitude waves, depending on the order of non-linearity. The propagation of finite amplitude waves in fluid-filled elastic or viscoelastic tubes has been examined, for instance, by Rudinger [6], Anliker et al. [7], Tait and Moodie [8] by employing 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 [9], Hashizume [10], Yomosa [11], Erbay et al.
0020-7462/98/$19.00 1998 Elsevier Science Ltd. All rights reserved. PII: S 0 0 2 0 - 7 4 6 2 ( 9 8 ) 0 0 0 4 5 - 6
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[12] and Demiray [13] by employing various asymptotic methods. As is known, if the amplitude of a wave is sufficiently small, many non-linear systems admit harmonic wave solutions, in which the non-linear terms are small enough to be neglected, and the amplitude will remain constant in time. If the amplitude of the wave is small-but-finite, the non-linear terms cannot be neglected and give rise to a variation in amplitude in both space and time. When the amplitude varies slowly over a period of oscillation, a stretching transformation allows us to separate the system into a rapidly varying part associated with the oscillation and a 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 non-linear Schro¨dinger 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 non-linearity and dispersion. The problem of non-linear self-modulation of small-but-finite waves in fluid-filled distensible tubes was considered by Ravindran and Prasad [14], in which they showed that for a linear elastic tube wall model the non-linear self-modulation of pressure waves is governed by a NLS equation. Non-linear self-modulation in fluid-filled distensible tubes had been investigated by Erbay and Erbay [15], by employing the non-linear equations of a viscoelastic thin tube and the approximate fluid equations and they showed that the non-linear modulation of pressure waves is governed by a dissipative NLS equation. In the present work, utilizing the constitutive relation proposed by Demiray [16], the relation between the inner pressure and radial displacement and its time and space derivatives is obtained. Then, employing this pressure—displacement relation and the approximate (one-dimensional) fluid equations, the propagation of weakly non-linear waves in such a dispersive and dissipative medium is studied, in the longwave limit through the use of reductive perturbation method. Depending on the order of various coefficients charecterizing the material properties, the governing equation for the lowest-order terms in the expansion is obtained as
the Burgers’ equation, Korteweg—de Vries equation (KdV) and Korteweg—de Vries—Burgers’ equation (KdVB). The localized travelling wave solutions for these evolution equations are also given. Finally, the amplitude modulation of these non-linear governing equations for strongly dispersive case is examined by use of the reductive perturbation method and the dissipative non-linear Schro¨dinger equation is obtained as the governing equation. The solution of this equation under some initial condition is also investigated numerically by employing the split step Fourier method. It is observed that the amplitude of the solitary wave decreases with increasing time and the initial deformation.
2. Theoretical preliminaries 2.1. Equations of membrane The arterial wall material is known to be incompressible, anisotropic and viscoelastic (see [17, 18]). For its simplicity in non-linear analysis the arterial wall material shall be assumed to be incompressible, homogeneous, isotropic and viscoelastic. We consider a circular cylindrical long tube of radius R , which is subjected to a uniform pressure P and an axial stretch j . Let r be the radius of such X a tube after such a finite symmetrical static deformation. Upon this deformation we shall superimpose a finite time-dependent displacement u*(z*, t*), but in view of the external tethering effects, the axial motion is neglected. Hence, the position vector of a sample point after such a motion may be represented by r"(r #u*)e #z*e , z*"j Z, (1) P X X where e , e and e are the unit base vectors in the P F X cylindrical polar coordinates, Z is the axial coordinate of a point in the undeformed configuration and z* is the spatial coordinate after finite static deformation. The vector tangent to the meridional curve is defined by
*u* *r *u* T" " e #j e "j e #e . P X X X X *Z *Z *z* P
(2)
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Thus, the length of this tangent vector is given by
*u* "T""j 1# "j " , X X X *z*
(3)
where
*u* " " 1# . X *z*
(4)
Hence, the unit tangent vector to the meridional curve and the exterior unit normal vector to the deformed membrane are, respectively, given by
1 *u* t" e #e , X " *z* P X 1 *u* n"e ;t" e! e . (5) F P *z* X " X Thus, the stretches in the meridional and circumferential directions are expressed as
u* j "j " , j "j " , " "1# , (6) X X F F F r where j "r /R is the stretch ratio in the circumF ferential direction after finite initial deformation. Let ¹ and ¹ be the membrane forces acting along the meridional and circumferential curves of the tube, respectively. Then the equation of the tube in the radial direction may be given by
* " *u* ¹" *u* F¹ ! X#" P*"o h , (7) F *t* r *z* " *z* X where h is the deformed thickness of the tube, o is the mass density of the tube after finite static deformation, t* is the time parameter and P* is the value of the fluid pressure on the tube surface. Let t be the Cauchy stress tensor referred to the IJ final configuration. The stress resultants ¹ and ¹ are, then, given by ¹ "h t , ¹ "h t , (8) where h is the final thickness of the tube, t and t are the stress components along the meridional and circumferential directions in the final configuration, respectively. Assuming that the tube material is incompressible, the incompressibility
573
condition requires that h H h " or h " , (9) j j "" F X where H is the initial thickness of the tube. In order to complete the field equations, one must know the value of the fluid pressure P* on the tube surface. Therefore, Eq. (7) is to be supplemented by the equations governing the fluid body. 2.2. Fluid equations Although blood is known to be an incompressible viscous fluid, as pointed out by Rudinger [6], in a number of applications, such as flow in large blood vessels, the effect of viscosity may be neglected, i.e. the blood may be treated as an incompressible inviscid fluid. As a result of this simplifying assumption the variation of field quantities with radial coordinate may be disregarded. However, the radial changes are included by taking the variation of cross-sectional area into consideration. The equation governing the mass conservation of the fluid body may be given by *A * # (Av*)"0, *t* *z*
(10)
where A denotes the internal cross-sectional area of the tube and v* is the axial velocity component of the fluid. Recalling the definition of A in terms of the final inner radius of tube, i.e. A"n(r #u*), the above equation may be written as 2
*u* *u* *v* #2v* #(r #u*) "0. *t* *z* *z*
(11)
The equation of balance of linear momentum in the axial direction may be given by *v* *v* 1 *P* #v* # "0, (12) *t* *z* o *z* where o is the mass density of the fluid. Eqs. (7), (11) and (12) provide sufficient relations to determine the unknowns u*, v* and P*. In order to complete the fluid equations one must know the constitutive relation for the tube material. In this work we shall adopt the following type of constitutive relation proposed by Demiray
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574
This constitutive relation is non-linear in terms of c , but it is linear in terms of d . Since we IJ KL assume that the tube is thin the stress component t may approximately be taken to be zero. By using this result the hydrostatic pressure is expressed as follows:
[16] for soft biological tissues: t "Pd #exp[a(I !3)] IJ IJ R* j (t*!q)cR (q) dq ; c b# IJ T KK \ R* # k (t*!q) [b (t*, q) b (t*, q) T IK JK \
#b (t*, q) b (t*, q)] dq , IK JK
(13)
where an overdot denotes the differentiation with respect to variable q, P is the hydrostatic pressure, t is the Cauchy stress tensor referred to the final IJ configuration, c is the Finger deformation tensor, IJ I is the first invariant of c , a and b are two IJ material constants, j (t*) and k (t*) are two funcT T tions characterizing the viscoelastic effects, c (t*) IJ and b (t*, q) are defined by IJ c (t*)"F (t*)F (t*), IJ I) J) b (t*, q)"F (t*) F (q). (14) IJ I) J) Here x "x (X , t*) defines the motion, and I I ) F ,*x /*X is the deformation gradient. I) I ) In general, it is quite difficult to study a nonlinear problem by use of the constitutive relation given in Eq. (13). Assuming that the memory of the material under consideration is short, the viscoelastic coefficients j (t*) and k (t*) may be approximT T ated as follows: j (t*)" j d(t*), k (t*)"k d(t*). (15) T T T T Here, j and k are constants characterizing visT T coelastic effects, d(t*) is the Dirac-Delta function. Inserting Eq. (15) into Eq. (13) and then integrating the resulting equations with respect to q we obtain the following constitutive relation: t "Pd #[bc #j c d c IJ IJ IJ T KL KL IJ #2k d c c ] exp[a(I !3)], (16) T KL KI LJ where d is the deformation rate tensor defined by IJ d , (» #» ). (17) J I IJ I J with » being the components of the tube velocity. I
P"![bc #j (c d #c d #c d )c T #2k d (c )] exp[a(I !3)]. (18) T Since the Finger deformation tensor is diagonal, its components are given as follows: 1 . (19) c " j j Furthermore, by using the relation CQ " )* 2d F F , the components of deformation rate IJ I) J) tensor are given by
c "j ,
c "j ,
j j j j d " , d " , d "! # . (20) j j j j At this stage it might be convenient to introduce the following non-dimensionalized quantities: r t*" t, z*"r z, u*"r u, v*"r v, c o h P*"co (p #p), m" o r D bh c" , ¹ "bh p , ¹ "bh p , o r D j c k c l " T , l " T . (21) b r b r Inserting Eq. (21) into Eqs. (7), (11) and (12) we obtain the following non-dimensionalized equations:
m *u p 1 * p *u p #p" # ! , " *t " " *z " *z F F F X *u *u *v 2 #2v #(1#u) "0, *t *z *z *v *v *p #v # "0. *t *z *z
(22)
(23)
(24)
G. Akgu( n, H. Demiray / International Journal of Non-Linear Mechanics 34 (1999) 571–588
In these equations the principal stresses p and p are given by
p "
1 1 j! # (l #2l ) j# j j j j
2l j 1 1 ! # l jj ! ! j j j j (l #2l ) j F(j , j ), # j j j p "
1 1 1 j! # l j j! ! j j j j
(l #2l ) j # (l #2l ) # j j j
1 2l j ; j# ! F(j , j ), j j j j
(25)
where the function F(j , j ) is defined by
1 F(j , j ),exp a j#j# !3 j j
.
(26)
As can be seen from Eq. (22) the pressure function p depends upon u, *u/*z, *u/*z, *u/*t*z, *u/*t*z, *u/*t and *u/*t. If the pressure function p is expanded into a power series of u and its derivatives we obtain *u *u *u p"b u#b #b #b #b u *z *t *t
*u *u *u #b #b u #b u *z *z *t
575
*u *u *u #2b *z *z *z*t
*u *u #b #2 . *z *z*t
(27)
Here, b (i"1, 2, . . . , 19) coefficients are some G functions of the initial deformation and b , b , b , b , b , b , b , b and b depend on the elastic effects, b , b , b , b , b —b characterize the viscous effects and b , b and b are related to the inertial effects. The explicit expression of these coefficients are given in the appendix. Thus, Eqs. (23), (24) and (27) give sufficient relations to determine u, v and p.
3. Long-wave approximation In this section we shall examine the propagation of small-but-finite amplitude waves in a fluid filled and prestressed non-linear viscoelastic thin tube whose dimensionless governing equations are given by the Eqs. (23), (24) and (27). For this we adopt the long-wave approximation and employ the reductive perturbation method [19]. For that purpose we would like to see the dispersive character of our model equations. Linearizing the field Eqs. (23), (24) and (27) and seeking a harmonic wave type of solution to these equations we obtain the following dispersion relation: (2#b k)u!ib ku!b k#b k"0,
(28)
*u *u *u *u #b u #b #b u#b u *t *t *z *t
where u is the angular frequency and k is the wave number. Assuming that the wavelength is large as compared to the radius of the tube, one can expand the dispersion relation, u(k), into a power series of k around k"0 and obtain
*u *u #b u #b u *z *t
u"gk(1#p k#p k#2),
*u *u *u #b #b u *z *z *z *u *u *u *u #b #b u *t *z *t *z
(29)
where g, p and p are defined by
b ib b b b # ! . g" , p " , p "! 4g 2 2b 4 16b (30)
G. Akgu( n, H. Demiray / International Journal of Non-Linear Mechanics 34 (1999) 571–588
576
The form of the dispersion relation (29) suggests us to introduce the following coordinate stretching: m"e?(z!gt), q"e?>gt,
*u *u *u p"b u#b e? #b e?g !2e *m *m *m*q #e
*u *u *u !b ge? !e #2 . *q *q *m
u" eLu (m, q), v" eLv (m, q), L L L L p" eLp (m, q), (33) L L where u , v and p (n"1, 2, . . .) are unknown L L L functions to be determined from the solution of the field equations. Introducing expansion (33) into Eq. (32) and setting the like powers of e equal to zero we obtain the following sets of differential equations. O(e) order equations:
!g
*u *v # "0, *m *m
*v *p # "0, *m *m
From the solution of the set (34) we obtain
!2g
*u *º *º *v #2g #6gº # "0, *m *q *m *m
!g
*v *º *º *p #2g #4gº # "0, *q *m *m *m
*º p "b u #(b #gb )e?\ *m *º !b ge?\ #b º. *m
p "b u .
(34)
*u *u *u *v *v # #u "0, !2g #2g #2v *m *m *q *m *m *v *v *v *p #g #v # "0, *m *q *m *m
(37)
Eliminating u , v and p between these equations we obtain the following master equation: *º *º *º #c º #c e?\ *m *m *q *º !c e?\ "0, *m
(38)
where the coefficients c , c and c are defined by 5 b 1 2b b c " # , c " b # , c " . (39) 4 2 b b 4g From this general evolution equation various wellknown equations may be obtained as some special cases.
O(e) order equations:
!g
3.1. Solution of the field equations
(32)
Assuming that the field quantities can be expanded into asymptotic series of e we have
!2g
(35)
u "º(m, q), v "2gº, p "2gº, g"b /2, (36) where º(m, q) is an unknown function whose governing equation will be obtained later. Introducing Eq. (36) into the set of equations (35) we get
*u *u *v *u #2eg #2v #(1#u) "0, *m *q *m *m
*v *v *p *v #eg #v # "0, *q *m *m *m
!g
*u !b ge?\ #b u. *m
(31)
where e is a small parameter measuring the weakness of dispersion and/or non-linearity and a is a positive constant whose values will be specified later. Introducing Eq. (31) into the field Eqs. (23), (24) and (27) we obtain !2g
*u *u #b ge?\ p "b u #b e?\ *m *m
(i) b "O(1) and a"1: In this case the coefficient of *º/*m is of the order of e and the evolution equation reduces to the Burgers’ equation *º *º *º #c º !c "0, *q *m *m
(40)
G. Akgu( n, H. Demiray / International Journal of Non-Linear Mechanics 34 (1999) 571–588
which results from the balance of non-linearity with dissipation. As is well known, the Burgers’ equation has a steady solution of the form
3c [sech f!2 tanh f!2], º(m, q)" 25c c
º(m, q)" (º\#º>)# (º\!º>) tanh f,
c 3c q. f" m# 10c 125c
c c f" (º\!º>) m! (º\#º>)q , 4c 2
(41)
where º\ and º> are, respectively, the uniform states of º at GR. This solution may be regarded as ‘‘shock’’ wave since it relates two different states denoted by º\ and º> . (ii) b "O(e) and a": In this special case the coefficient of *º/*m is of the order of e and the resulting evolution equation reduces to the wellknown Korteweg—de Vries (KdV) equation *º *º *º #c º #c "0, *m *m *q
(42)
which results from the balance of non-linearity with dispersion. The KdV equation has a steady solution of the form º(m, q)"º #b sech f,
f"
c b b m!c º # q , 3 12c
(43)
where º '0 is the value of º as fP$Rand b is the amplitude of the wave relative to the constant solution º at infinity. (iii) b "O(e ) and a" : For this special case the evolution equation reduces to the following Korteweg—de Vries—Burgers’ (KdVB) equation *º *º *º *º #c º !c #c "0, *m *m *m *q
(44)
which is obtained by balancing the non-linearity with dispersion and dissipation. The KdVB equation has a travelling wave solution of the following form [20]: 3c [sech f#2 tanh f#2], º(m, q)" 25c c c 3c f"! m# q, 10c 125c
577
(45)
The variation of these solution profiles with initial deformation and other characteristics will be discussed later.
4. Non-linear wave modulation In this section we shall examine the amplitude modulation of weakly non-linear waves in a fluidfilled non-linear viscoelastic tube whose dimensionless governing equations are given in Eqs. (23), (24) and (27). For this, we again employ the reductive perturbation method [19] and introduce the following coordinate stretching: m"e(z!jt), q"et,
(46)
where j is a constant which will be shown to be the group velocity. We further assume that our field variables are functions of fast variables (z, t) and also slow variables (m, q). Thus, the following substitution is permissible: * * * P #e , *z *z *m
* * * !ej #e . *t *m *q
(47)
We shall further assume that the field variables are expressible as asymptotic series in e as follows: u" eLu , v" eLv , p" eLp . L L L L L L
(48)
where u , v and p are some unknown functions of L L L fast (z, t) and slow (m, q) variables. For our future purposes we shall assume that the viscoelastic coefficients are of the following order in terms of e: b "bM e. G G
(49)
Introducing the expansions (47) and (48) into the field Eqs. (23), (24) and (27) and setting the coefficients of like powers of e equal to zero we obtain the following set of differential equations.
G. Akgu( n, H. Demiray / International Journal of Non-Linear Mechanics 34 (1999) 571–588
578
2
*v *v *p *u # "0, # "0, *z *t *z *t
(50)
*u *u #b u #b u#b u *z *t
*u *v *u *v *u # !2j # #2v *z *t *z *m *m *v "0, #u *z
#b u
*v *p *v *p *v # !j # #v "0, *z *z *m *m *t
*u *u #b u . #b u *z *t
(51)
¹hird-order, O(e), equations: *u *v *u *v *u *u # !2j # #2 #2v *z *z *m *m *q *t *u *v *v *u #u #u #2v #2v *z *z *z *m *v "0, #u *m *v *p *v *p *v * # !j # # # (v v ) *t *z *m *m *q *z
*u *u #b p "b u #b *z *t
u "º exp[i(ut!kz)]#c.c., v "» exp[i(ut!kz)]#c.c., p "P exp[i(ut!kz)]#c.c., (53) where u is the angular frequency, k is the wave number º, » and P are amplitude functions which depend on the slow variables m and q and c.c. stands for the complex conjugate of the corresponding quantity. Introducing Eq. (53) into Eq. (50) and requiring the non-vanishing solution for º, » and P we obtain 2u 2u º"º, »" º, P" º (54) k k
4.1.2. The solution of O(e) order equations Introducing Eqs. (53) and (54) into Eq. (51) one obtains
*u *u # #b 2 *z*m *m
(52)
provided that the dispersion relation (28) holds true. Here º(m, q) is an unknown function whose governing equation will be obtained later.
*v "0, #v *m
*u *u *u # j #2 #b !2j *t*m *m *t*q
*u *u *u #b . *z *z *z
4.1.1. The solution of O(e) order equations The form of the differential equations given in Eq. (50) suggests us to seek the following type of solution to these differential equations:
*u *u #b u#b !2b j *t*m *z
4.1. The solution of field equations
*u *u *u #b #2b p "b u #b *z *t *z*m
*u *u *u #u !2ju #b u *t *t *t*m
Second-order, O(e), equations:
2
*u *u *u #u #2u #b u *z *z *z*m
*u *u . #b p "b u #b *z *t
2
*u *u *u *u #2b u u #2b # #b *t *z *z *m
First-order, O(e), equations:
2
*u *v u *º # #2 !j e (#6iuºe ( *t *z k *m #c.c."0,
G. Akgu( n, H. Demiray / International Journal of Non-Linear Mechanics 34 (1999) 571–588
*p 2u u *v *º # # !j e (#4iuºe ( *t *z k k *m #c.c."0, *u *u #b p "b u #b *t *z #2[b #(b !b )k!b u]"º" *º !2i(b k#b uj) e ( *m #[b !(b #b )k!b u]ºe (, (55) where, for convenience, we have defined "º"" ºº*, where º* is the complex conjugate of º and the phasor is defined by "ut!kz. The form of Eq. (55) suggests us to seek the following type of solution for u , v and p : u "º# º? e ?(#c.c., ? v "»# »? e ?(#c.c., ? p "P# P? e ?(#c.c., (56) ? where º , . . . , P are functions of slow variables m and q. Comparing third of Eq. (55) and third of (56) we obtain the following expressions: P"b º#2[b #(b !b )k !b u]"º", P"(b !b k!b u)º *º !2i(b k#b uj) , *m P"(b !4b k!4b u)º #[b !(b #b )k!b u]º. (57) Introducing Eq. (56) into Eq. (55) yields the following sets of equations:
u *º 2uº!k»"2i !j , k *m 2iu u *º u»!kP" !j , k k *m
(b k#b u !b )º#P *º "!2i(b k#b uj) , *m
579
(58)
and 2uº!k»"3uº, 2u º, u»!kP" k (4b k#4b u!b )º#P "[b !(b #b )k!b u]º. (59) The solution of the set in Eq. (58) gives the following result:
2u *º 2i u »" º! !j , k k k *m 2u 4iu u *º !j . P" º! k k k *m
(60)
In order to have a non-zero solution for º the following condition must be satisfied: 2u!b k . j"v " (61) E uk(2#b k) Hence, the solution of the set of Eq. (59) gives [6u#b k!(b #b )k!b uk] º" º, 3(b k!2u) 2u 3u »" º! º. (62) k k 4.1.3. The solution of O(e) order equations Introducing the solutions (53) and (56) into Eq. (52) we obtain the following equations: 2
*v *» *º *u # # !2j #2iu (º* º *z *m *m *t !ºº*)#ik(»º*!»*º) 6u * *» *º !2j # "º" e # k *m *m *m
#2
*º !iº(2k»#2uº *q
G. Akgu( n, H. Demiray / International Journal of Non-Linear Mechanics 34 (1999) 571–588
580
Fig. 1. The variation of Burgers’ schok with circumferential stretch.
!6iuº º*) e (#
*» *º !2j *m *m
6u *º !3i (2uº#k» )º# º e ( k *m !iº(10uº#4k» )e ("0, (63) *v *p *P *» 4u * # # !j # "º" e *t *z *m *m k *m
#
*» 2u *º *P # !j !2iu(»º *m *m k *q
#» º*) e (#
*P *» !j *m *m
4u *º !4iu»º# º e ( k *m !6i»ºe ("0.
(64)
The forms of the differential Eqs. (63) and (64) suggest us to seek the following type of solution for u , v and p : u "º# º?e ?(#c.c., ? v "»# »? e ?(#c.c., ? p "P# P? e ?(#c.c., ?
(65)
where º , . . . , P are functions of slow variables m and q. Here we need to solve the equations corresponding to the zeroth (a"0) and first (a"1) modes only. Introducing expression (65) into Eqs. (63) and (64) the following equations are obtained.
G. Akgu( n, H. Demiray / International Journal of Non-Linear Mechanics 34 (1999) 571–588
581
Fig. 2. The variation of solitary wave profile for KdV equation with circumferential stretch.
a"0 equations:
»"2jº!2
*» *º !2j #2iu(º* º!ºº*) *m *m 6u * #ik(»º*!»*º)# "º""0, k *m
(67)
*» *º iuº!ik»# !2j *m *m #2
(66)
º 4u 4u #2j !j #2[b #(b !b )k!b u] k k " 2j!b ;º,
4u !j "º". k
a"1 equations:
*» 4u * *P # !j "º""0, *m *m k *m P"º#2[b #(b !b )k !b u]"º""0. The solution of the set of Eq. (66) gives
*º !2iº(k»#uº ) *q
!6iuºº*"0, *P *» 2u *º # iu»!ikP# !j *m *m k *q !2iu(»º#ºº*)"0, P"(b !b k!b u)º *º *º #(b #jb ) !2i(kb #jub ) *m *m
582
G. Akgu( n, H. Demiray / International Journal of Non-Linear Mechanics 34 (1999) 571–588
Fig. 3. The variation of solution profile for KdVB equation.
*º #2ib u #iub º#(2b !b k *q !b u)ºº#[2b !(4b #5b )k !5b u]ºº*#[3(b !b k !b u)#b k!b k]"º"º.
(68)
Eliminating º, » and P between Eq. (68) we obtain the following dissipative non-linear Schro¨dinger equation: i
*º *º #k #k "º"º#ik º"0, *m *q
(69)
where the coefficients k , k and k are defined by 1 k " [(2#b k)j#4b ukj 2u(2#b k) #6b k#ub !b ],
1 2u 4u k " #j !j #b 2u(2#b k) k k #(b !b )k!b u [2u#8juk#2b k!(b k#b u)k] # b j ! 2 [6u#b k!(b #b )k!b uk] # 3(b k!2u) [10u#2b k#(4b !5b )k!5b uk] ; 3(b k!2u) !38u#8ukj#3b k#(b !3b )k
!3b uk!b k , b k k " . 2(2#b k)
(70)
G. Akgu( n, H. Demiray / International Journal of Non-Linear Mechanics 34 (1999) 571–588
583
Fig. 4. The variation of solution profile for dissipative NLS equation with time.
If the coefficient b vanishes, Eq. (69) reduces to the well-known non-linear Schro¨dinger equation i
*º *º #k #k "º"º"0. *m *q
(71)
The NLS equation which arises in a variety of fields as an equation describing the self-modulation of the one-dimensional monocromatic plane waves in dispersive media. The steady-state solution of the NLS equation, which generally represents the wavetrains expressible in terms of Jacobian elliptic functions, include a bright and a dark envelope solitons, a phase jump, and a plane wave with a constant amplitude as special cases. The criterion whether k k '0 or k k (0 is important in determining how the given initial data will evolve for long times for the asymptotic field governed by the NLS equation. Now let as seek the travelling wave solutions of the NLS equation in the
form º(m, q)"»(f) exp[i(Km!)q)], f"m!v q, v "const., (72) where »(f) is a real function of f. In this case, if "º" approaches a constant » at infinity, the solution is given by a non-linear plane wave º(f)"» exp[i(Km!)q)], (73) where )"k K!k ». In general, the solution can be obtained in terms of the Jacobian elliptic functions under the assumption of v "2k K. The explicit functional form of these solutions will not be given here, expect for the following limiting cases. Assuming that »P0 and d»/dfP0 as "f"PR, for k k '0, the solution is given by k »(f)"A sech Af , (74) 2k
G. Akgu( n, H. Demiray / International Journal of Non-Linear Mechanics 34 (1999) 571–588
584
Fig. 5. The variation of solution profile for dissipative NLS equation with circumferential stretch.
where )"k K!k A/2. For k k (0, if ºP» and dº/dfP0 as fPR, the solution be comes k »f , (75) »"» tanh ! 2k where )"k K!k » . These solutions corres pond to an envelop solitary wave and a phase jump, respectively. On the other hand, it is wellknown that the plane wave solution of NLS equation is modulationally unstable if k k '0 or stable if k k (0.
5. Numerical results and discussion As stated before, the coefficients c (i"1, 2, 3) G defined in Eq. (39) and k (i"1, 2, 3) defined in Eq. G (70) depend on the initial deformation, mechanical and geometrical characteristics of the wall material.
For the static case, the present mechanical model is compared by Demiray [21] with the experimental measurements by Simon et al. [22] on canine abdominal artery with geometrical characteristics R "0.31 cm, R "0.38 cm and j "1.53, and the G X value of the material constant a was found to be a"1.948. Also choosing l "2l , the profile of the travelling wave solutions given in Eqs. (41), (43) and (45) are evaluated numerically for various values of circumferential stretch and the variable f and the results are depicted in Figs. 1—3. Fig. 1 gives the variation of the solution profile for Burgers’ equation with circumferential stretch ratio and the variable f. Since, with increasing initial deformation, the degree of non-linearity of the material is weakened, the shock front with increasing initial deformation is flattened. The variation of solitary wave profile of the KdV equation with f and the initial deformation is given in Fig. 2. As is seen from this figure the profile of solitary wave gets
G. Akgu( n, H. Demiray / International Journal of Non-Linear Mechanics 34 (1999) 571–588
steepened with increasing circumferential stretch ratio. The solution profile of the KdVB equation for a set of initial deformation is shown in Fig. 3. The branch KdVB-1 in the figure corresponds to the solution given in the first line of Eq. (45) and KdVB-2 to the solution in the second line in Eq. (45). As to the variation the solution profile of the non-linear Schro¨dinger equation it might be interesting to study the dissipative non-linear Schro¨dinger equation by numerical means. By utilizing split step Fourier method [23] and the initial condition º(m, 0)"exp[!i(2m#n/2)] sech m, the evolution equation solved numerically and the results are depicted in Figs. 4 and 5, respectively. Fig. 4 shows the variation of solution profile with time. As is seen from the figure due to dissipative term the amplitude of the solitary wave decreases with increasing time, which is to be expected. Moreover, the amount of decrement depends on the value of the dissipative coefficient k which varies with in itial deformation. The variation of solution profile with initial deformation is shown in Fig. 5. As is seen from the figure the solitary wave profile decreases with increasing initial deformation. Indeed, our numerical calculations show that the dissipation coefficient k increases with initial circum ferential stretch.
Appendix. Polynomial approximation of the pressure function In this subsection, we shall try to give the polynomial approximation to the pressure function in terms of the radial displacement and its space and time derivatives. In Eq. (22), pressure was found to be
1 * p *u m *u p # ! , (A.1) p #p" " *t " " *z " *z F F X F in which the stress components p and p are defined by
j p " G (j , j ) #G (j , j j p " G (j , j ) #G (j , j
j j ) F(j , j ), j j j ) F(j , j ), j (A.2)
585
where G (j , j ) (a"1, 2, 3) and F(j , j ) are de? fined by
1 2l ! , G "(l #2l ) j# j j j (l #2l ) 1 1 #l jj! ! G " , j j jj 1 2l ! , G "(l #2l ) j# jj j 1 F"exp a j#j# !3 . (A.3) jj For our future purposes we need the non-linearity up to and including cubic terms. Considering the following expansions:
j *u *u j *u *u *u : , : !u #u , j *z *z*t j *t *t *t *u #2 , j"j 1#2 X *z
j"j [1#4u#6u#4u#2] , F 1 *u 1 #2 , " 1! j j *z X 1 1 " [1!2u#3u!4u#2], j j F *u jj"jj 1#2u#u# F X *z
#2u
*u #2 , *z
1 1 *u " 1!2u#3u! !4u jj jj *z X F *u #2u , *z one can write G "G#Gu,
G "G#Gu, *u G "G#Gu#Gu#G , *z
(A.4)
G. Akgu( n, H. Demiray / International Journal of Non-Linear Mechanics 34 (1999) 571–588
586
*u F" 1#F u#F u#F #F u *z
*u , #F u *z
(A.5)
where G , . . . , F are defined by 1 2l G"(l #2l ) j# ! , X jj j X F F (l #2l ) l , G"4 ! j j j F F X 1 1 (l #2l ) , G"l jj! ! # F X j j jj F X F X 1 (l #2l ) , !4 G"2l jj# F X j jj F F X 1 2l G"(l #2l ) j# ! , F jj j X F X 1 , G"4(l #2l ) j! F jj X F 10 G"(l #2l ) 6j# , F jj X F (l #2l ) l , G"2 ! jj j F X X 1 F "2a j! , F jj F X 1 3 #a j# F "2a j! , F jj F jj F X F X 1 F "a j! , X jj F X 4 1 1 F " a j! #2a j! 3 F jj F jj F X F X 3 4a ; j# ! , F jj jj F X X F a 1 1 F "2 #a j! j! F X jj jj jj F X F X F X
pressure may be expressed as follows:
*u *u *u *u #b #b u #b u #b u *z *t *t *z
*u *u *u *u #b #b u#b u #b u *t *z *t *z
*u *u *u *u #b u #b u #b *t *z *z *z
*u *u *u *u #b u #b *t *z *t *z
*u *u *u *u *u #2b #b #2 , *z *z *z*t *z *z*t
*u *u *u #b #b #b u p"b u#b *t *t *z
(A.7)
.
(A.6) Introducing the expansions (A.3) and (A.4) into Eq. (A.1) and keeping up to the cubic terms, the
where the coefficients b are defined by G 4 1 #2a j! F, b " F jj j j X F F X 1 b " !j F , X jj X F b "m, 1 2l ! F, b " (l #2l ) j# F jj j X F X 10 1 11 b "! #a j! j# F jj F jj jj X F F X F X 1 #2a j! F, F jj F X 1 1 1 b "! #a j! j! X F jj jj jj X F F X F X 3 1 b " j! !2a j! X jj F jj F X F X 1 ; j! F, X jj F X b "!m, 6l 7 b " #(l #2l ) j! F j jj X F X
F,
G. Akgu( n, H. Demiray / International Journal of Non-Linear Mechanics 34 (1999) 571–588
1 1 #2a j! (l #2l ) j# F jj F jj F X F X 2l ! F, j X 1 1 (l #2l ) F, # !jj ! b " l j j X F jj X F X F 4a 20 9 b " # !5j F jj jj jj X F X F F X 7 1 #2a j# j! F jj F jj F X F X 1 4 F, # a j! F jj 3 F X 12l 28 b " ! # (l #2l ) j jj X F X 1 1 3j# #a (l #2l ) j! F jj F jj F X X F 9 2 2l # !5j (l #2l )! F j jj jj X F F X X 1 #2a j! F jj F X 1 2l ; (l #2l ) j# ! F, F jj j F X X 6 5 5 b " !j #a jj! ! X X F j j jj X F F X 9 1 # !2a j! X jj jj X F F X 1 ; j! F, F jj F X b "m, 4 2 1 1 b " #4a ! ! jj jj j j X F X F X F 1 1 !2a j! j! F, X jj F jj F X F X 6 1 b "! #3a j! F, X jj jj X F F X
587
3 3(l #2l ) 1 b " l ! !jj # j j F X jj X F X F 1 1 #a (l #2l ) j# j! F jj X jj F X F X 2 1 # !j F jj jj X F F X 1 1 1 #2l j # ! !jj F, F j j jj F X F X F X 1 6(l #2l ) 2 !2l b " # j j jj X X F F 1 1 1 !2a j! l jj! ! F jj X F j j F X X F (l #2l ) # F, jj X F 2l 1 F . (A.8) b " !(l #2l ) j# X jj j F X F This expression of the pressure is used in studying the field equations in the long-wave limit and in the amplitude modulation.
Acknowledgements One of the authors (HD) was supported by the Turkish Academy of Sciences.
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