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Modulation of nonlinear waves in a viscous fluid contained in a tapered elastic tube
International Journal of Engineering Science 40 (2002) 1897–1918 www.elsevier.com/locate/ijengsci
Modulation of nonlinear waves in a viscous fluid contained in a tapered elastic tube Hilmi Demiray
*
Department of Mathematics, Faculty of Arts and Sciences, Isik University, B€ uy€ ukdere Caddesi Maslak, 80670 Istanbul, Turkey Received 15 May 2002; accepted 10 June 2002
Abstract In the present work, treating the arteries as a tapered, thin walled, long and circularly conical prestressed elastic tube and the blood as a Newtonian fluid, we have studied the amplitude modulation of nonlinear waves in such a fluid-filled elastic tube, by use of the reductive perturbation method. The governing evolution equation is obtained as the dissipative nonlinear Schr€ odinger equation with variable coefficients. It is shown that this type of equations admit solitary wave solutions with variable wave amplitude and speed. It is observed that, the wave speed increases with distance for tubes of descending radius while it decreases for tubes of ascending radius. The dissipative effects cause a decay in wave amplitude and wave speed. Ó 2002 Elsevier Science Ltd. All rights reserved. Keywords: Solitary waves; Tapered tubes; Wave modulation
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 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 their applications in arterial mechanics, the propagation of pressure pulses in fluid-filled distensible tubes has been studied by several researchers (Pedley [2] and Fung [3]). Most of the
*
Tel.: +90-212-286-29-60; fax: +90-212-286-57-96. E-mail address: [email protected] (H. Demiray).
0020-7225/02/$ - see front matter Ó 2002 Elsevier Science Ltd. All rights reserved. PII: S 0 0 2 0 - 7 2 2 5 ( 0 2 ) 0 0 1 1 3 - 1
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H. Demiray / International Journal of Engineering Science 40 (2002) 1897–1918
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 terms 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 Moodie and his co-workers [9–12] 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 [13], Hashizume [14], Yomasa [15], Erbay et al. [16] and Demiray [17] by employing various asymptotic methods. In all these works [13–17], 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 phenomena admit harmonic wave solutions 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. When 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 selfmodulation 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-butfinite amplitude waves in fluid-filled distensible tubes was considered by Ravindran and Prasad [18], in which they showed that for a linear elastic tube wall model, the nonlinear self-modulation of pressure waves is governed by the NLS equation. Nonlinear self-modulation in fluid-filled distensible tubes had been investigated by Erbay and Erbay [19], by employing the nonlinear equations of a viscoelastic thin tube and the approximate fluid equations and they showed that the nonlinear pressure waves are governed by the dissipative NLS equation. Demiray [20], 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. In the present work, treating the arteries as a tapered elastic thin walled long circularly conical tube and the blood as a Newtonian fluid, we have studied the amplitude modulation of weakly nonlinear waves in such a fluid-filled elastic tube by use of the reductive perturbation method. The governing evolution equation is found to be the dissipative nonlinear Schr€ odinger equation with variable coefficients. It is shown that this type of equations admit solitary wave solutions with variable wave speed and amplitude. It is observed that, the wave speed increases with distance for tubes of descending radius while it decreases for tubes of ascending radius. It is further observed that the dissipation causes to decay in wave amplitude and the speed of enveloping wave with the distance parameter s.
H. Demiray / International Journal of Engineering Science 40 (2002) 1897–1918
1899
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 prestressed 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 Þ1=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 k01
1=2 ds0z 1 þ /2 ¼ ¼ kz ; dSZ 1 þ U2
k02 ¼
ds0h ðr0 þ /z Þ ¼ kz : ðkz R0 þ Uz Þ dSh
ð5Þ
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 "
ou dsz ¼ 1 þ / þ oz
2 #1=2
dz ;
dsh ¼ ðr0 þ /z þ u Þ dh:
ð7Þ
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H. Demiray / International Journal of Engineering Science 40 (2002) 1897–1918
Then, the stretch ratios in the final configurations read " 2 #1=2 ou 1þ /þ oz ðr0 þ /z þ u Þ k1 ¼ kz ; k ¼ k : 2 z kz R0 þ Uz ð1 þ U2 Þ1=2 Thengent vector t along the deformed meridional curve may be given by ! ou / þ er þ ez oz t¼" 2 #1=2 : ou 1þ /þ oz Similarly, the unit exterior normal vector n to the deformed membrane is given as ! ou er / þ ez oz n¼2 !2 31=2 : 41 þ / þ ou 5 oz
ð8Þ
ð9Þ
ð10Þ
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
H ð1 þ U2 Þ1=2 kz R0 þ Uz ; k2z ð1 þ /2 Þ1=2 r0 þ /z
ð11Þ
H ð1 þ U2 Þ1=2 ðkz R0 þ Uz Þ 2 3 !2 1=2 ½r0 þ /z þ u : ou k2z 41 þ / þ 5 oz
h0 ¼
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 h0 ¼
H ð1 þ U2 Þ1=2 ; kh kz ð1 þ /2 Þ1=2
h1 ¼
H ð1 þ U2 Þ1=2 U 1=2
k2z ð1 þ /2 Þ
/
;
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
Hð1 þ U2 Þ1=2 kh kz ð1 þ /2 Þ1=2
:
ð14Þ
H. Demiray / International Journal of Engineering Science 40 (2002) 1897–1918
1901
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 8 9 #1=2 " > > 2 = ou o < ðr0 þ /z þ u Þ / þ ou oz þ T1 h T2 1 þ / þ þ G ðr0 þ /z þ u Þ i 1=2 > oz > oz ou 2 : ; 1 þ / þ oz ¼
2 q0 H 2 1=2 o u ð1 þ U Þ ðR k þ Uz Þ ; 0 z ot2 k2z
ð15Þ
where q0 is the mass density of the membrane and G is the fluid reaction force density defined by oVr oVr oVz ou þ : ð16Þ G ¼ P 2lv þ lv /þ or oz or oz r¼r0 þ/z þu Here, P is the fluid pressure function, Vr and Vz are the fluid velocity components in the radial and axial directions, respectively, and lv is the fluid viscosity. Let lR be the strain energy density function of the membrane, where l is the shear modulus of the membrane material. The membrane forces may be expressed in terms of the stretch ratios as T1 ¼
lH oR ; k2 ok1
T2 ¼
lH oR : k1 ok2
ð17Þ
Introducing (17) into Eq. (15), the equation of motion in the radial direction takes the following form 8 9 > > l l o < ðkz R0 þ Uz Þ / þ ou oR = 2 1=2 oR oz Hð1 þ U Þ þ H þ G ðr0 þ /z þ u Þ h i1=2 kz ok2 kz oz > ok1 > ; : 1 þ / þ ou 2 oz ¼
q0 o2 u H ð1 þ U2 Þ1=2 ðkz R0 þ Uz Þ 2 : 2 ot kz
ð18Þ
2.2. Equations of fluid Blood is known to be an incompressible non-Newtonian fluid. However, in the course of flow in arteries, the red cells in the vicinity of arteries move to the central region of the artery so that the hematocrit ratio 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 viscosity of blood is further reduced. Therefore, for flow problems in large blood vessels, the blood may be treated as a Newtonian fluid. For an axisymmetric motion of blood, the equations of motion in the cylindrical polar coordinates may be given as follows:
1902
H. Demiray / International Journal of Engineering Science 40 (2002) 1897–1918 oVr 1 oP lv oVr oVr þ V ¼ þ V þ r z ot or oz qf or qf oVz 1 oP lv oVz oVz þ V þ ¼ þ V r z ot or oz qf oz qf
oVr Vr oVz þ þ ¼0 or r oz
o2 Vr 1 oVr Vr o2 Vr 2 þ 2 ; þ r or or2 r oz
ð19Þ
o2 Vz 1 oVz o2 Vz þ þ 2 ; r or or2 oz
ð20Þ
ðincompressibility conditionÞ;
ð21Þ
where qf is the mass density of the fluid body. These field equations must satisfy the following boundary conditions ou ou þ V j ¼ Vr jr¼r þ/z þu : 0 ot oz z r¼r0 þ/z þu
ð22Þ
At this stage it is convenient to introduce the following nondimensionalized quantities
t ¼
R0 t; z ¼ R0 z; r ¼ R0 x; u ¼ R0 u; Vr ¼ c0 v; Vz ¼ c0 w; G ¼ qf c20 g; c0
P ¼ qf c20 p; m ¼
q0 H lH ; r0 ¼ kh R0 ; lv ¼ qf R0 c0 m; c20 ¼ : qf R0 qf R0
ð23Þ
Introducing (23) into equations (18)–(22) we obtain the following nondimensionalized equations: 2 ov ov ov op o v 1 ov v o2 v þ ¼ 0; þv þw þ m þ ot ox oz ox ox2 x ox x2 oz2
ð24Þ
2 ow ow ow op o w 1 ow o2 w þ ¼ 0; þv þw þ m þ ot ox oz oz ox2 x ox oz2
ð25Þ
ov v ow þ þ ¼ 0; ox x oz
ð26Þ
with boundary conditions
ov ov ow ou ðp0 þ pÞ 2m þ m g ¼ 0; þ /þ ox oz ox oz x¼kh þ/zþu
ou ou þ wj ¼ vjx¼kh þ/zþu ; ot oz x¼kh þ/zþu
ð27Þ
H. Demiray / International Journal of Engineering Science 40 (2002) 1897–1918
1903
where the function g is defined by g¼
m ðkz þ UzÞ o2 u ð1 þ U2 Þ1=2 oR 2 1=2 ð1 þ U Þ þ ðkh þ /z þ uÞ ot2 kz ðkh þ /z þ uÞ ok2 k2z 8 9 > > < ou / þ oz ðkz þ UzÞ oR = 1 o : h i1=2 kz ðkh þ /z þ uÞ oz > ok1 > : 1 þ / þ ou 2 ; oz
ð28Þ
These equations give sufficient relations to determine the field quantities u, v, w and p completely. 3. Nonlinear wave modulation In this section we shall examine the amplitude modulation of weakly nonlinear waves in a fluidfilled thin elastic tube whose dimensionless governing equations are given in Eqs. (24)–(28). Considering the dispersion relation of the linearized field equations and the nature of the problem of concern, which is a boundary value type, the following stretched coordinates may be introduced n ¼ ðz ktÞ;
s ¼ 2 z;
ð29Þ
where is a small parameter measuring the weakness of nonlinearity and k is the 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 4 . Hence, the tapering angles can be expressed as U ¼ A4 ;
/ ¼ a4 :
ð30Þ
Here A and a are some finite constants describing the tapering angles and will be specified later. Noting the differential relations o o o o ! þ þ 2 ; oz oz on os
o o o ! k ot ot on
ð31Þ
and expanding the field variables into asymptotic series as u ¼ u1 þ 2 u2 þ 3 u3 þ ; v ¼ v1 þ 2 v2 þ 3 v3 þ ; w ¼ w1 þ 2 w2 þ 3 w3 þ ; 2
ð32Þ
3
g ¼ g0 þ g1 þ g2 þ g3 þ ; p ¼ p0 þ p1 þ 2 p2 þ 3 p3 þ ; where u1 ; . . . ; p3 ; . . . are functions of the fast variables ðt; zÞ and slow variables ðn; sÞ, and introducing (31) and (32) into the field equations (24)–(27), the following sets of differential equations are obtained:
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H. Demiray / International Journal of Engineering Science 40 (2002) 1897–1918
OðÞ order equations: ov1 op1 þ ¼ 0; ot ox
ow1 op1 þ ¼ 0; ot oz
ov1 v1 ow1 þ þ ¼ 0; ox x oz
ð33Þ
and the boundary conditions p1 jx¼kh ¼ g1 ;
ou1 ¼ v1 jx¼kh : ot
ð34Þ
Oð2 Þ order equations: ov2 ov1 ov1 ov1 op2 k þ v1 þ w1 þ ¼ 0; ot on ox oz ox ow2 ow1 ow1 ow1 op2 op1 k þ v1 þ w1 þ þ ¼ 0; ot on ox oz oz on ov2 v2 ow2 ow1 þ þ þ ¼ 0; ox x oz on and the boundary conditions op1 p2 þ u1 ¼ g2 ; ox x¼kh
ð35Þ
ov1 ou2 ou1 ou1 v 2 þ u1 ¼ : k þ w1 ox x¼kh ot on oz x¼kh
ð36Þ
Oð3 Þ order equations: 2 ov3 ov2 o ov1 ov2 ov1 op3 o v1 1 ov1 v1 o2 v1 k þ ðv1 v2 Þ þ w2 þ w1 þ w1 þ m þ þ ¼ 0; ot on ox oz oz on ox ox2 x ox x2 oz2 ow3 ow2 ow2 ow1 o ow1 op3 op2 op1 k þ v1 þ v2 þ ðw1 w2 Þ þ w1 þ þ þ oz ot on ox ox on oz on os 2 2 o w1 1 ow1 o w1 þ ¼ 0; m þ 2 x ox ox2 oz ov3 v3 ow3 ow2 ow1 þ þ þ þ ¼ 0; ð37Þ ox x oz on os and the boundary conditions
1 2 o2 p1 op2 op1 ov1 u þ ðas þ u2 Þ þ p3 2m ¼ g3 ; þ u1 2 1 ox2 ox ox ox x¼kh 1 2 o2 v 1 ov1 ov2 ou3 ou2 ¼ u1 2 þ ðas þ u2 Þ þ u1 þ v3 k 2 ox ox ox ot on x¼kh ou2 ou1 ou1 ow1 þ w1 jx¼kh : þ þ w2 þ u1 oz on oz ox x¼kh
ð38Þ
H. Demiray / International Journal of Engineering Science 40 (2002) 1897–1918
1905
In order to make the equations self-sufficient, or complete, we need the explicit expressions of g1 , g2 and g3 , which should be obtained from Eq. (28). Under the coordinate transformations (30), the tapering terms in these expressions become ðAsÞ2 and ðasÞ2 , in this approximation. Furthermore, we shall assume that the generators before and after the static deformation remain parallel to each other. As pointed out before, for this case the relation between a and A becomes a ¼ Akh =kz . Hence the following expansions are permissible: 2 1 ou1 ou1 ou2 ou1 3 2 k1 ¼ kz 1 þ þ þ þ
; 2 oz oz oz on ! As 2 k2 ¼ kh þ u1 þ u2 þ u3 u1 3 þ ; kz ( ! 1 1 u1 u21 u2 A 2 ½kh þ as þ u ¼ 1 þ s 2 kh kh k2h kh kz " # ) u31 2 2 u3 3 þ
; Asu2 þ 3 þ 2 u1 u2 þ kh kh kz kh kh !2 # " " 1 oR ou 1 ¼ b0 þ b1 u1 þ b2 u21 þ b1 u2 þ a1 kz þ 2 þ b3 u31 þ 2b2 u1 u2 þ b1 u3 kh kz okz oz ! # 2 ou1 ou2 ou1 ou1 A þ þ a2 kz u1 þ 2a1 kz b1 su1 3 þ ; kz oz oz on oz " # 2 1 oR ou1 2 2 þ ; ¼ a0 þ 2a1 u1 þ a2 u1 þ 2a1 u2 þ c1 kz kh kz ok1 oz
ð39Þ
where the coefficients a0 , a1 , a2 , b0 ; . . . ; b3 and c1 are defined by 1 o2 R ; 2kh kz okh okz
a0 ¼
1 oR ; kh kz okz
a1 ¼
b1 ¼
1 o2 R ; kh kz ok2h
b2 ¼
1 o3 R ; 2kh kz ok3h
1 o3 R 1 oR ; b0 ¼ ; 2 2kh kz okh kz kh kz okh 1 o4 R 1 o2 R b3 ¼ ; c ¼ : 1 6 ok4h 2kh kz ok2z a2 ¼
ð40Þ
Introducing Eq. (39) into Eq. (28) the fluid reaction force terms g1 , g2 and g3 may be given by m o2 u1 o2 u1 b0 g1 ¼ a0 kz 2 þ b1 ð41Þ u1 ; kh kz ot2 oz kh ! 2 m o u2 o2 u1 m o2 u1 b1 b0 2 b0 2l u2 g2 ¼ 2 u1 2 þ b2 þ 2 u1 þ b1 kh kz ot2 oton ot kh kh kh kh kz 2 o2 u2 o2 u1 a0 o2 u1 ou1 b 2a1 u1 2 a1 kz a0 kz 2 2a0 kz 0 As; ð42Þ þ kz oz ozon kh oz oz kz
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H. Demiray / International Journal of Engineering Science 40 (2002) 1897–1918
! m o2 u3 o2 u3 b0 m o2 u2 o2 u2 u g3 ¼ a k þ b 2k k 2a 0 z 3 0 z 1 kh kz ot2 kh kz oton ot2 kh ozon 2 2 2 2 m 2 o u1 m o u1 o u2 o u1 m 2 o2 u1 k a0 kz þ 2ku u u1 2 þ þ u 1 1 2 kh kz oton ot2 ot2 ot k3h kz on2 k2h kz ! ! b b b b b o2 u1 þ b3 2 þ 21 30 u31 þ 2 b2 1 þ 20 u1 u2 2a0 kz kh kh kh kh kh ozos ! ! 2 2 3 ou1 o u1 a0 o2 u2 o2 u1 a0 3c1 kz 2a1 u1 2 þ u2 2 þ kz þ kz 2 oz oz2 kh oz oz ! ou1 ou2 ou1 a0 o2 u1 2a1 a0 2 o2 u1 þ þ kz a2 þ 2a1 u1 2 u1 2 2a1 kz þ 2kz oz oz on kh ozon kh oz kh 2 a1 ou1 2 b o2 u1 a2 u1 þ kz b1 0 Asu1 þ a0 As 2 : kz kh oz kh oz
ð43Þ
4. Solution of the field equations 4.1. The solution of OðÞ order equations The solution of the differential equations given in (33) and (41) suggests us to seek the following type of solution to these differential equations u1 ¼ U1 exp½iðxt kzÞ þ c:c:;
g1 ¼ G1 exp½iðxt kzÞ þ c:c:;
v1 ¼ V1 exp½iðxt kzÞ þ c:c:;
w1 ¼ W1 exp½iðxt kzÞ þ c:c:;
p1 ¼ P1 exp½iðxt kzÞ þ c:c:;
ð44Þ
where x is the angular frequency, k is the wave number, U1 and G1 are amplitude functions which depend on the slow variables n and s, and V1 , W1 and P1 are amplitude functions which depend on x as well as the slow variables ðn; sÞ and c.c. stands for the complex conjugate of the corresponding quantities. The solution satisfying the field equations (33) and the boundary conditions (34) are given as follows: U1 ¼ U ;
G1 ¼
x2 f0 U ; k
V1 ¼ ixf1 ðkxÞU ;
W1 ¼ xf0 ðkxÞU;
P1 ¼
x2 f0 ðkxÞU ; k
provided that the following dispersion relation holds true x2 mx2 b0 f0 þ b1 a0 kz k 2 ¼ 0; k kh kz kh
ð45Þ
ð46Þ
where, for brevity, we have defined the following functions fn ðkxÞ ¼ In ðkxÞ=I1 ðkh kÞ; ðn ¼ 0; 1; 2; . . .Þ;
f0 ¼ f0 ðkh kÞ;
u ¼ xt kz:
ð47Þ
H. Demiray / International Journal of Engineering Science 40 (2002) 1897–1918
1907
Here In ðxÞ is the modified BesselÕs function of order n and U ðn; sÞ is an unknown function of the slow variables, whose governing equation will be obtained later. The group velocity vg is defined by vg ¼
kh x2 ðf02 1Þ þ 2a0 kz k 2 : 2xðf0 þ mk=kh kz Þ
ð48Þ
4.2. The solution of Oð2 Þ order equations Introducing the solution given in (45) into Eq. (35) we obtain # " 2 2 2 ov2 op2 oU x f ðkxÞ f ðkxÞ 1 1 þ ixk U 2 e2iu þ 2x2 2kf0 ðkxÞf1 ðkxÞ jU j2 þ c:c: ¼ 0; f1 ðkxÞeiu þ ot ox x x on ow2 op2 x oU iu k f0 ðkxÞ þ þx e þ ix2 k½f1 ðkxÞ2 f0 ðkxÞ2 U 2 e2iu þ c:c: ¼ 0; ot oz k on ov2 v2 ow2 oU iu e þ c:c: ¼ 0; þ þ þ xf0 ðkxÞ ð49Þ on ox x oz
and the boundary conditions g2 p2 jx¼kh ¼ 2x2 jUj2 þ x2 U 2 e2iu þ c:c:; ou2 oU i/ 1 e þ ix 2kf0 v2 jx¼kh ¼ k U 2 e2iu þ c:c: on kh ot
ð50Þ
Here, for convenience, we have defined jUj2 ¼ UU , where U is the complex conjugate of U, and noticed that f1 ðkh kÞ ¼ 1. The form of the differential equations in (49) suggests us to seek the following type of solution
v2 ¼
ð0Þ V2
þ
!
2 X
ðlÞ V2 ðx; n; sÞ expðiluÞ
þ c:c: ;
l¼1 ð0Þ
w 2 ¼ W2 þ
2 X
! ðlÞ
W2 ðx; n; sÞ expðiluÞ þ c:c: ;
l¼1
p2 ¼
ð0Þ P2
þ
2 X
!
ðlÞ P2 ðx; n; sÞ expðiluÞ
l¼1
u2 ¼
ð0Þ U2
þ
2 X
! ðlÞ U2 ðn; sÞ expðiluÞ
l¼1 ð0Þ
g2 ¼ G2 þ
2 X l¼1
þ c:c: ;
þ c:c: ; !
ðlÞ
G2 ðn; sÞ expðiluÞ þ c:c: ;
ð51Þ
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H. Demiray / International Journal of Engineering Science 40 (2002) 1897–1918
where, from (42) we have b0 x2 b0 kh ð0Þ ð0Þ 2 2 G2 ¼ b 1 U2 þ 2 a Ab1 s; f0 þ b2 þ a1 k jU j þ b1 kh kh k kh kz x2 mx oU ð1Þ ð1Þ G2 ¼ k ; f0 U2 þ 2i a0 kz k kh kz on k 2 x b0 x2 ð2Þ ð2Þ 2 U2 þ G2 ¼ 4 f 0 3 b 1 f0 þ b2 þ 3a1 kz k U 2 : k kh kh k
ð52Þ ð53Þ ð54Þ
Introducing (51) into (49) and (50) we obtain the following set of differential equations and the boundary conditions " # ð0Þ ð0Þ ð0Þ 2 oP2 f ðkxÞ oV2 V2 1 2 2 þ 2x 2kf0 ðkxÞf1 ðkxÞ jU j ¼ 0; þ ¼ 0; ð55Þ ox x ox x and the boundary conditions ð0Þ
ð0Þ
ð0Þ
2
G2 P2 jx¼kh ¼ 2x2 jU j ;
V2 jx¼kh ¼ 0:
ð56Þ
The equations and the boundary conditions resulting from the coefficients of expðiuÞ becomes ð1Þ
oP2 oU f1 ðkxÞ ¼ 0; ixk on ox x oU ð1Þ ð1Þ k f0 ðkxÞ ¼ 0; ixW2 ikP2 þ x k on ð1Þ
ixV2 þ
ð1Þ
ð57Þ
ð1Þ
oV2 V oU ð1Þ f0 ðkxÞ ¼ 0; þ 2 ikW2 þ x on ox x and the boundary conditions ð1Þ
ð1Þ
G2 P2 jx¼kh ¼ 0;
ð1Þ
ð1Þ
ixU2 V2 jx¼kh ¼ k
oU : on
ð58Þ
The equations and boundary conditions resulting from the coefficients of expð2iuÞ become, ð2Þ
oP2 f 2 ðkxÞ 2 U ¼ 0; þ x2 1 x ox ð2Þ ð2Þ 2xW2 2kP2 þ x2 k f12 ðkxÞ f02 ðkxÞ U 2 ¼ 0; ð2Þ
2ixV2 þ ð2Þ
ð2Þ
oV2 V ð2Þ þ 2 2ikW2 ¼ 0; ox x
ð59Þ
H. Demiray / International Journal of Engineering Science 40 (2002) 1897–1918
1909
and the boundary conditions ð2Þ
ð2Þ
G2 P2 jx¼kh ¼ x2 U 2 ð2Þ 2ixU2
ð2Þ V2 jx¼kh
1 ¼ ix 2kf0 U 2: kh
ð60Þ
The solution of Eqs. (55) and (56) yields ð0Þ
V2
ð0Þ
U2
ð0Þ
2
P2 ¼ H ðn; sÞ x2 ½f12 ðkxÞ þ f02 ðkxÞ jUj ; " ! # jU j2 2f0 b0 A Hðn; sÞ 2 2 2 x 1þ ; f0 2b2 2a1 kz k þ ¼ sþ ðb1 b0 =kh Þ ðb1 b0 =kh Þ kh k ðb1 b0 =kh Þ kz
¼ 0;
ð61Þ provided that b1 b0 =kh 6¼ 0. The special case where b1 b0 =kh ¼ 0 will be discussed later. From the solution of Eqs. (57) and (58) it is readily seen that, in order to have a nonzero solution for U ðn; sÞ the scale parameter k must be equal to the group velocity given in Eq. (48). Furthermore, in this work, since we shall be concerned with the governing equation of the lowest order term in the perturbation expansion, without loosing the generality of the problem we can set the homogeneous solution equal to zero. Hence, the nonhomogeneous solution of Eqs. (57) and (58) may be given by oU k oU ð1Þ ; U2 ¼ i k h f 0 2 ; ¼ ½kf1 ðkxÞ xxf0 ðkxÞ on x on x oU x2 oU ð1Þ ð1Þ W2 ¼ i xxf1 ðkxÞ þ k f0 ðkxÞ ; P2 ¼ i xf1 ðkxÞ : k on on k ð1Þ V2
ð62Þ
The solution of Eq. (59) gives ð2Þ
V2
ð2Þ
¼ ixCF1 ð2kxÞ;
P2 ¼
2
ð2Þ
W2
¼ xCF0 ð2kxÞ;
2
x x CF0 ð2kxÞ þ ½f12 ðkxÞ f02 ðkxÞ U 2 ; k 2
ð63Þ
where C is another unknown function of slow variables, F0 and the functions Fi ð2kxÞ are defined by F0 ¼ F0 ð2kÞ;
Fi ð2kxÞ ¼
Ii ð2kxÞ ; I1 ð2kÞ
ði ¼ 0; 1; 2; 3; . . .Þ:
ð64Þ
1910
H. Demiray / International Journal of Engineering Science 40 (2002) 1897–1918
The use of the boundary conditions (60) results in 1 ð2Þ C ¼ 2U2 þ 2kf0 U 2 ; kh 2 1 x b0 2x2 1 2 f0 x2 1 ð2Þ 2 3 x 2kf0 F0 f þ F0 þ U2 ¼ 4 f 0 3 b 1 2 2 0 kh k k kh k k kh ! 2 b2 3a1 kz k U 2 :
ð65Þ
4.3. The solution of Oð3 Þ order equations For the solution of this order of differential equations and the boundary conditions given in Eqs. (37) and (38) the field quantities may be expressed as follows: ! 3 X ð0Þ ðlÞ u3 ¼ U3 þ U3 ðn; sÞ expðiluÞ þ c:c: ; l¼1
v3 ¼
ð0Þ V3
þ
3 X
! ðlÞ V3 ðx; n; sÞ expðiluÞ
þ c:c: ;
l¼1 ð0Þ
w2 ¼ W3 þ
3 X
! ðlÞ
W3 ðx; n; sÞ expðiluÞ þ c:c: ;
l¼1 ð0Þ
p3 ¼ P3 þ
3 X
! ðlÞ
P3 ðx; n; sÞ expðiluÞ þ c:c: ;
l¼1
g3 ¼
ð0Þ G3
þ
3 X
ð66Þ
! ðlÞ G3 ðn; sÞ expðiluÞ
þ c:c: ;
l¼1
where ð1Þ G3
! 2 2 ð1Þ x2 oU mk oU mkx oU2 ð1Þ ¼ þ 2i a0 kz k a0 kz þ f0 U3 þ 2ia0 kz os kh kz k kh kz on on2 # " " x2 b b x2 ð0Þ f0 þ 2a1 kz k 2 þ 2b2 1 þ 20 U2 U þ 5 f0 þ 6a1 kz k 2 þ kh k kh kh kh k !# " b1 b0 x2 b2 a1 ð2Þ þ 2a2 5 kz k 2 þ 2b2 þ 3 U2 U 3 2 f0 þ 3 b3 kh k2h kh kh kh k ! # " ! # 3 b A 2 kz sU: a0 3c1 kz k 4 jU j U 2 b1 0 þ a0 kz k 2 kh 2 kz
ð67Þ
Introducing the expansion (66) into Eqs. (37) and (38), we obtain the following sets of differential equations and the boundary conditions
H. Demiray / International Journal of Engineering Science 40 (2002) 1897–1918 ð0Þ
k
ð0Þ
oW2 oP o þ 2 þ x2 ðf02 ðkxÞ þ f12 ðkxÞÞ jU j2 ¼ 0 on on on
ð0Þ
1911
ð0Þ
ð0Þ
oV3 V oW þ 3 þ 2 ¼ 0; ox x on with the boundary condition ð0Þ oU2 k ojU j2 ð0Þ V3 jx¼kh ¼ k þ 2xf0 : kh on on
ð68Þ
ð69Þ
4.4. l ¼ 1 equations ð1Þ
ð1Þ
oP3 oV o ð2Þ ð0Þ ð2Þ k 2 ixU ½f1 ðkxÞV2 þ kxf1 ðkxÞðW2 U þ W2 U Þ ox ox on 2 o V1 1 oV1 V1 ð2Þ 2 2ixkf0 ðkxÞV2 U m þ ¼ 0; k V 1 x ox x2 ox2 ! ð1Þ ð0Þ ð2Þ oW2 x2 oU oW2 oW ð1Þ ð1Þ ixW3 ikP3 k þ ixf1 ðkxÞ þ f0 ðkxÞ U U 2 os on k ox ox ð1Þ oP2 o2 W1 1 oW1 ð2Þ ð0Þ ð2Þ 2 þ xkf1 ðkxÞV2 U ixkf0 ðkxÞ½W2 U þ W2 U þ m k W1 ¼ 0; þ x ox on ox2 ð1Þ
ixV3 þ
ð1Þ
ð1Þ
ð1Þ
oV3 V oW oU ð1Þ ¼ 0; þ 3 ikW3 þ 2 þ xf0 ðkxÞ os ox x on
ð70Þ
and the boundary conditions ð1Þ G3
¼
ð1Þ P3 jx¼kh
2
ð0Þ U2 U
2
ð2Þ U2 U
x2 þ 2
3 13kf0 jU j2 U kh
þx þ 5x 1 A U þ kh x2 sU ; 2imx kf0 kh kz ð1Þ oU2 3 2 3 k ð1Þ ð1Þ 2 kf0 þ F0 2k f0 F0 jUj2 U k þ þ ix ixU3 V3 jx¼kh ¼ k 2 2kh kh on 1 1 1 A ð0Þ ð0Þ ð2Þ þ ikW2 U þ ix kf0 U2 U þ ix 2kF0 þ kf0 U2 U þ ikh x kf0 sU : kh kh kh kz ð71Þ The solution of (68) under the boundary conditions (69) is given as follows: ð0Þ W2
H ðn; sÞ ¼ ; k
ð0Þ V3
1 oH ¼ x; 2k on
k2 ð0Þ k 1 jU j2 : 2xf0 H ¼ 2 U2 2 kh kh kh
ð72Þ
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H. Demiray / International Journal of Engineering Science 40 (2002) 1897–1918
Introducing the expression of H into Eq. (61) we have ð0Þ U2
2
k2 þ 2b2 kh þ 2a1 kh kz k 2 kh ! 2f0 A 2 2 2 kh x jU j b0 s : þ kh f0 kz k 1
¼ ½2k ðkh b1 b0 Þ
4kxf0 2
ð73Þ
Finally the solution of Eq. (70) under the boundary conditions (71) is given by ð1Þ P3
¼
2 x2 x 2 x2 x oU x2 ½F0 ð2kxÞf0 ðkxÞ þ F1 ð2kxÞf1 ðkxÞ CU f0 ðkxÞ þ 2 f1 ðkxÞ 2k 2k on2
þi ð1Þ V3
ð1Þ W3
ð1Þ
2 ikHU x x 2 k2 oU oU f1 ðkxÞ þ i k x þ ; ¼ xxf0 ðkxÞ xf0 ðkxÞ f1 ðkxÞ 2 k 2k 2 os x on 2 kHU x x 2 k2 k oU ¼ f0 ðkxÞ þ k xf1 ðkxÞ þ x þ f0 ðkxÞ k 2k 2 x k on2 x oU þ i xxf1 ðkxÞ þ f0 ðkxÞ ; k os
G3 ¼
ð1Þ U3
x2 oU ; xf1 ðkxÞ os k
x2 kh o2 U ð0Þ ð2Þ ð1 kk f Þ x2 ð1 þ f0 F0 ÞCU þ x2 U2 U þ 5x2 U2 U h 0 2 2k 2 on 2 x 3 1 A 13kf0 jU j2 U 2imx kf0 U þ kh x2 sU ; þ kh kz 2 kh
2kFU 2i oU k 1 k2 k2h o2 U ¼ þ þ 2 kh f0 3 2 xk x os x 2k x 2 on2 3 2 3 k 1 ð0Þ 2 2 U2 U þ kf0 þ F0 2k f0 F0 jU j U þ kf0 k þ 2 2kh kh kh 1 1 A ð2Þ þ 2kF0 þ kf0 sU : U2 U þ kh kf0 kh kh kz ð1Þ
ð74Þ
ð1Þ
Eliminating G3 and U3 between the Eqs. (71) and (74) the following dissipative nonlinear Schr€ odinger equation with variable coefficient is obtained i
oU o2 U 2 þ l1 2 þ l2 jU j U þ il3 U þ l4 sU ¼ 0; os on
ð75Þ
H. Demiray / International Journal of Engineering Science 40 (2002) 1897–1918
1913
1 x2 2 m f0 x2 kh kk 2 2 2 f kh f 0 kh ¼ 2a0 kz k þ kh ðf0 1Þ 3k þ þ 2 4 kh kz k x 0 k 2k mx k f0 þ 2a0 kz a0 kz ð1 þ 2kh kf0 Þ ; þ 2k kz x ! ( " 1 x2 2 3 13 k 3 3 ¼ 2a0 kz k þ kh ðf0 1Þ x2 þ k þ 4 2 þ k þ 2 f0 2kh 2 k kh 2 kh k x 3 1 x b f2 þ 2k f0 F0 þ 4 þ 3 b3 2 þ 8 þ kh 2kh 0 kh kh kh ! " ! a1 3 2 þ 2a2 5 kz k þ 2 kz a0 3c1 kz k 4 þ x2 f02 f0 1 2 kh k kh # b1 b0 x ð0Þ þ 2b2 þ 2 þ 2a1 kz k 2 þ 4k f0 U2 U kh kh kh " ! # ) 6 b1 b0 ð2Þ 2 2 2 þ 6a1 kz k U2 U þ x 4f0 F0 þ f0 f0 3 þ 2b2 þ 3 kh k kh k2h 2mx kf0 k1h ¼ 2 2a0 kz k þ kh xk ðf02 1Þ ! ! 1 ( 2 x2 2 f f b 0 ¼ 2a0 kz k þ kh ðf0 1Þ x2 0 2 kh 2 b1 0 a0 kz k 2 k kh kh k kh h i9 =A b0 x2 f02 k2h k f0 1 þ 2b2 bkh1 þ bk20 þ 2a1 kz k 2 þ 4 kx f kh 0 h : ; kz 2k2 ðkh b1 b0 Þ
ð76Þ
where
l1
l2
l3 l4
When the tapering angle parameter A is set equal to zero the coefficient l4 vanishes and the resulting evolution equation becomes the dissipative nonlinear Schr€ odinger equation i
oU o2 U þ l1 2 þ l2 jUj2 U þ il3 U ¼ 0: os on
ð77Þ
Therefore, the coefficient l4 stands for the contribution of tapering to the evolution equation. 4.5. Progressive wave solution In this sub-section we shall try to present a progressive wave solution to the evolution equation (75). For that purpose, following Demiray [21], we shall propose a solution of the form U ðn; sÞ ¼ aðsÞV ðfÞ expfi½XðsÞ Kn g;
f ¼ aðsÞ½n 2l1 Ks ;
ð78Þ
1914
H. Demiray / International Journal of Engineering Science 40 (2002) 1897–1918
where K is a constant aðsÞ, aðsÞ, and V ðfÞ are some real functions to be determined from the solution of (75). Introducing (78) into Eq. (75) and setting the real and imaginary parts of the resulting equation equal to zero we obtain the following sets of ordinary differential equations
a0 ðsÞ a0 ðsÞ 0 fV ¼ 0; þ l3 V þ aðsÞ aðsÞ
ð79Þ
½X0 ðsÞ l1 K 2 þ l4 s V þ l1 aðsÞ2 V 00 þ l2 a2 ðsÞV 3 ¼ 0;
ð80Þ
where primes denote the derivative of the corresponding quantity with respect to its argument. In the present work we shall be concerned with the localized travelling wave solution to the field equation, i.e., V and its various order derivatives vanish as f ! 1. Under such assumptions one can show that V is square integrable over the interval ð1; 1Þ. Thus, multiplying Eq. (79) by V and integrating the resulting equation with respect to f from 1 to þ1 we obtain
a0 ðsÞ 1 a0 ðsÞ þ l3 hV 2 i ¼ 0; aðsÞ 2 aðsÞ
2
hV i ¼
Z
1
V 2 df:
ð81Þ
1
Since hV 2 i is bounded and nonzero, we obtain the following ordinary differential equation a0 ðsÞ 1 a0 ðsÞ þ l3 ¼ 0: aðsÞ 2 aðsÞ
ð82Þ
Now, in the first place, we shall propose a solution to Eq. (80) of the following form V ðfÞ ¼ sech f:
ð83Þ
Introducing (83) into Eq. (80), the following differential equations are obtained 2
aðsÞ ¼
l2 2 aðsÞ ; 2l1
X0 ðsÞ ¼
l2 2 aðsÞ þ l4 s l1 K 2 : 2
ð84Þ
It is seen from the first term of Eq. (84) that, in order to have a real solution for aðsÞ and aðsÞ the coefficient l1 and l2 must satisfy the inequality l1 l2 > 0, which is the case for the classical NLS equation. Now, let us return to the investigation of Eq. (79). It can be shown from the first term of Eq. (84) that a0 ðsÞ=aðsÞ ¼ a0 ðsÞ=aðsÞ. Introducing this relation into Eq. (79) we obtain a0 ðsÞ þ 2l3 ¼ 0: aðsÞ
ð85Þ
The solution of this equation may be given as follows: aðsÞ ¼ a0 e2l3 s ;
ð86Þ
H. Demiray / International Journal of Engineering Science 40 (2002) 1897–1918
1915
where a0 is a constant. Introducing (86) into equation (84) we obtain aðsÞ ¼
l2 2l1
1=2
a0 e2l3 s ;
XðsÞ ¼ l1 K 2 s þ
l4 2 l2 a20 ð1 e4l3 s Þ: s þ 2 8l3
ð87Þ
Here, in obtaining the function XðsÞ we have utilized the regularity condition that XðsÞ ¼ 0 at s ¼ 0. Thus, the final solution may be given as follows: U ðn; sÞ ¼ a0 e
2l3 s
sech f expfi½XðsÞ Kn g;
f¼
l2 2l1
1=2 aðsÞðn 2l1 KsÞ:
ð88Þ
It should be noted that the Eq. (88) is not a solution in the classical sense, it is rather a solution in the sense of distribution. If the coefficient l3 is vanishingly small then one can approximate a ¼ a0 ;
a¼
l2 2l1
1=2 a0 ;
XðsÞ ¼ ðl1 K 2 þ l2 a20 =2Þs þ
l4 2 s: 2
ð89Þ
This result is exactly the same with that of obtained by Demiray [21] for nondissipative NLS equation. If the tapering parameter is set equal to zero, i.e., l4 ¼ 0; the result will be that of the progressive wave solution of the classical NLS equation. Now, secondly, we shall propose a solution to Eq. (80) of the following form V ¼ tanh f:
ð90Þ
Since this function is not square integrable we cannot follow the above procedure to handle this problem. As is seen from Eq. (79), for this type of solution fV 0 ðfÞ approaches to zero as f ! þ1. Therefore, if we investigate the far field behavior of Eq. (79) and consider that fV 0 ðfÞ vanishes as f ! þ1 and obtain the following differential equation a0 ðsÞ þ l3 ¼ 0: aðsÞ
ð91Þ
The solution of this may be given by aðsÞ ¼ a0 el3 s ;
ð92Þ
where a0 is a constant. Now, let us introduce the proposed solution (90) into Eq. (80). In order to satisfy this equation the following relations must be satisfied 2l1 aðsÞ2 þ l2 aðsÞ2 ¼ 0;
ð93Þ
X0 ðsÞ l1 K 2 þ l4 s þ l2 aðsÞ2 ¼ 0:
ð94Þ
1916
H. Demiray / International Journal of Engineering Science 40 (2002) 1897–1918
The relation (93) holds true if and only if l1 and l2 have different signs, which is the case for classical nonlinear Schr€ odinger equation. Introducing (92) into (93) we obtain aðsÞ as aðsÞ ¼
l 2 2l1
1=2
a0 el3 s :
ð95Þ
Finally, integrating (94) and utilizing the regularity condition that XðsÞ at s ¼ 0, we obtain XðsÞ ¼ l1 K 2 s þ
l4 2 a20 l2 ð1 e2l3 s Þ: s þ 2 2l3
ð96Þ
Thus, the final solution may be given by l3 s
U ðn; sÞ ¼ a0 e
tanh f expfi½XðsÞ Kn g;
f¼
l 2 2l1
1=2
a0 el3 s 2l1 Ks:
ð97Þ
If the coefficient l3 is vanishingly small, then one can approximate a ¼ a0 ;
a¼
l2 2l1
1=2 a0 ;
XðsÞ ¼ l1 K 2 s þ l2 a20 s þ
l4 2 s: 2
ð98Þ
As is seen from these solutions, the existence of dissipation causes the decay in wave amplitude with the distance parameter s. Similar interpretation applies to the phase velocity of the enveloping wave, which might be given by vp ¼ l1 K þ
l4 l s þ 2 a20 e4l3 s : K 2K
ð99Þ
Here, again, the dissipative effect causes to decay in phase speed with distance parameter s. In order to see the contribution of the tapering to the phase velocity of the enveloping wave, one has to know the sign of the coefficient l4 . For convenience we shall re-write l4 as l4 ¼ v
A ; kz
ð100Þ
where 8 ! 1 < 2 x f f b0 0 2 2 0 x þ 2 þ kh þ 2 b1 þ a0 kz k 2 v ¼ 2a0 kz k þ kh ðf0 1Þ : k kh kh k kh h i9 = b0 x2 f02 k2h k f0 1 þ 2b2 bkh1 þ bk20 þ 2a1 kz k 2 þ 4 kx f kh 0 h þ : ; 2k2 ðkh b1 b0 Þ
2
ð101Þ
H. Demiray / International Journal of Engineering Science 40 (2002) 1897–1918
1917
In order to get information about the sign of v one has to know the constitutive relation of the tube material. In this work we shall utilize the constitutive relation proposed by Demiray [22] for soft biological tissues. Following Demiray [22], the strain energy density function may be expressed as R¼
1 fexp½aðI1 3Þ 1g; 2a
ð102Þ
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 (102) into Eq. (44), the coefficients a0 , b0 , b1 and b2 are obtained as ! 1 1 F ðkh ; kz Þ; a0 ¼ kh k3h k4z " ! !# 1 1 1 a1 ¼ þa 1 4 2 1 2 4 F ðkh; kz Þ; k4h k4z kh kx kh kz ! 1 1 F ðkh ; kz Þ; b0 ¼ kz k4h k3z 2 ! !2 3 1 3 a 1 þ 5 3 þ2 kh 3 2 5F ðkh ; kz Þ; b1 ¼ 4 kh kz kh kz kh kz kh kz 2 ! ! !3 3 2 6 a 3 1 a 1 b2 ¼ 4 4 3 þ 3 kh 3 2 þ 2 1þ 4 2 kh 3 2 5F ðkh ; kz Þ; k k k k kh kz kh kz kh kz kh kz h z h z
ð103Þ
where the function F ðkh ; kz Þ is defined by "
!# 1 F ðkh ; kz Þ ¼ exp a k2h þ k2z þ 2 2 3 : kh kz
ð104Þ
In order to study the variation of the coefficient l4 with the initial deformation, we need the value of material constant a. For the static case, the present model was compared by Demiray [23] with the experimental measurements by Simon et al. [24] on canine abdominal artery with the characteristics Ri ¼ 0:31 cm, R0 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. The numerical analysis of the coefficient v as a function of kh , kz and the wave number k indicates that the coefficient v is always positive. This means that for descending tube (A < 0) the coefficient l4 is positive and the wave speed given in (99) increases with the distance parameter s. On the other hand for ascending tube (A > 0) the coefficient l4 is negative and the wave speed in (99) decreases with distance from the origin. Such a result is to be expected from physical considerations.
1918
H. Demiray / International Journal of Engineering Science 40 (2002) 1897–1918
Acknowledgement This work was supported by the Turkish Academy of Sciences.
References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24]
S.C. Ling, H.B. Atabek, A nonlinear analysis of pulsatile blood flow in arteries, J. Fluid Mech. 55 (1972) 492–511. T.J. Pedley, Fluid Mechanics of Large Blood Vessels, Cambridge University Press, Cambridge, 1980. Y.C. Fung, Biodynamics: Circulation, Springer-Verlag, New York, 1981. 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. A.J. Rachev, Effects of transmural pressure and muscular activity on pulse waves in arteries, J. Biomech. Engng., ASME 102 (1980) 119–123. H. Demiray, Wave propagation through a viscous fluid contained in a prestressed thin elastic tube, Int. J. Eng. Sci. 30 (1992) 1607–1620. G. Rudinger, Shock waves in a mathematical model of aorta, J. Appl. Mech. 37 (1970) 34–37. 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. R.J. Tait, T.B. Moodie, Waves in nonlinear fluid filled tubes, Waves Motion 6 (1984) 197–203. T.B. Moodie, J.B. Haddow, Int. J. Non-linear Mech. 12 (1977) 223–231. T.B. Moodie, J.B. Haddow, J. Acoustic. Soc. Am. 67 (1980) 446–452. T.B. Moodie, G.E. Swaters, Quart. Appl. Math. 47 (1989) 705–722. R.S. Johnson, J. Fluid Mech. 42 (1970) 49–60. Y. Hazhizume, J. Phys. Soc. Jpn. 54 (1985) 3305–3312. Y. Yomosa, J. Phys. Soc. Jpn. 56 (1987) 506–520. H.A. Erbay, S. Erbay, S. Dost, Acta Mech. 95 (1992) 87–102. H. Demiray, Bull. Math. Biol. 58 (1996) 939–955. A.E. Ravindran, P. Prasad, Acta Mech. 31 (1979) 253–280. H.A. Erbay, S. Erbay, Acta Mech. 104 (1994) 201–214. H. Demiray, Int. J. Nonlinear Mech. 36 (2001) 649–661. H. Demiray, Applied Mathematics and Computation, in press. H. Demiray, J. Biomech. 5 (1972) 309–311. H. Demiray, Bull. Math. Biol. 38 (1976) 701–711. B.R. Simon, A.S. Kobayashi, D.E. Stradness, C.A. Wiederhielm, Circulation Res. 30 (1972) 491–500.
Modulation of nonlinear waves in a viscous fluid contained in a tapered elastic tube Hilmi Demiray
*
Department of Mathematics, Faculty of Arts and Sciences, Isik University, B€ uy€ ukdere Caddesi Maslak, 80670 Istanbul, Turkey Received 15 May 2002; accepted 10 June 2002
Abstract In the present work, treating the arteries as a tapered, thin walled, long and circularly conical prestressed elastic tube and the blood as a Newtonian fluid, we have studied the amplitude modulation of nonlinear waves in such a fluid-filled elastic tube, by use of the reductive perturbation method. The governing evolution equation is obtained as the dissipative nonlinear Schr€ odinger equation with variable coefficients. It is shown that this type of equations admit solitary wave solutions with variable wave amplitude and speed. It is observed that, the wave speed increases with distance for tubes of descending radius while it decreases for tubes of ascending radius. The dissipative effects cause a decay in wave amplitude and wave speed. Ó 2002 Elsevier Science Ltd. All rights reserved. Keywords: Solitary waves; Tapered tubes; Wave modulation
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 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 their applications in arterial mechanics, the propagation of pressure pulses in fluid-filled distensible tubes has been studied by several researchers (Pedley [2] and Fung [3]). Most of the
*
Tel.: +90-212-286-29-60; fax: +90-212-286-57-96. E-mail address: [email protected] (H. Demiray).
0020-7225/02/$ - see front matter Ó 2002 Elsevier Science Ltd. All rights reserved. PII: S 0 0 2 0 - 7 2 2 5 ( 0 2 ) 0 0 1 1 3 - 1
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H. Demiray / International Journal of Engineering Science 40 (2002) 1897–1918
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 terms 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 Moodie and his co-workers [9–12] 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 [13], Hashizume [14], Yomasa [15], Erbay et al. [16] and Demiray [17] by employing various asymptotic methods. In all these works [13–17], 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 phenomena admit harmonic wave solutions 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. When 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 selfmodulation 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-butfinite amplitude waves in fluid-filled distensible tubes was considered by Ravindran and Prasad [18], in which they showed that for a linear elastic tube wall model, the nonlinear self-modulation of pressure waves is governed by the NLS equation. Nonlinear self-modulation in fluid-filled distensible tubes had been investigated by Erbay and Erbay [19], by employing the nonlinear equations of a viscoelastic thin tube and the approximate fluid equations and they showed that the nonlinear pressure waves are governed by the dissipative NLS equation. Demiray [20], 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. In the present work, treating the arteries as a tapered elastic thin walled long circularly conical tube and the blood as a Newtonian fluid, we have studied the amplitude modulation of weakly nonlinear waves in such a fluid-filled elastic tube by use of the reductive perturbation method. The governing evolution equation is found to be the dissipative nonlinear Schr€ odinger equation with variable coefficients. It is shown that this type of equations admit solitary wave solutions with variable wave speed and amplitude. It is observed that, the wave speed increases with distance for tubes of descending radius while it decreases for tubes of ascending radius. It is further observed that the dissipation causes to decay in wave amplitude and the speed of enveloping wave with the distance parameter s.
H. Demiray / International Journal of Engineering Science 40 (2002) 1897–1918
1899
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 prestressed 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 Þ1=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 k01
1=2 ds0z 1 þ /2 ¼ ¼ kz ; dSZ 1 þ U2
k02 ¼
ds0h ðr0 þ /z Þ ¼ kz : ðkz R0 þ Uz Þ dSh
ð5Þ
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 "
ou dsz ¼ 1 þ / þ oz
2 #1=2
dz ;
dsh ¼ ðr0 þ /z þ u Þ dh:
ð7Þ
1900
H. Demiray / International Journal of Engineering Science 40 (2002) 1897–1918
Then, the stretch ratios in the final configurations read " 2 #1=2 ou 1þ /þ oz ðr0 þ /z þ u Þ k1 ¼ kz ; k ¼ k : 2 z kz R0 þ Uz ð1 þ U2 Þ1=2 Thengent vector t along the deformed meridional curve may be given by ! ou / þ er þ ez oz t¼" 2 #1=2 : ou 1þ /þ oz Similarly, the unit exterior normal vector n to the deformed membrane is given as ! ou er / þ ez oz n¼2 !2 31=2 : 41 þ / þ ou 5 oz
ð8Þ
ð9Þ
ð10Þ
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
H ð1 þ U2 Þ1=2 kz R0 þ Uz ; k2z ð1 þ /2 Þ1=2 r0 þ /z
ð11Þ
H ð1 þ U2 Þ1=2 ðkz R0 þ Uz Þ 2 3 !2 1=2 ½r0 þ /z þ u : ou k2z 41 þ / þ 5 oz
h0 ¼
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 h0 ¼
H ð1 þ U2 Þ1=2 ; kh kz ð1 þ /2 Þ1=2
h1 ¼
H ð1 þ U2 Þ1=2 U 1=2
k2z ð1 þ /2 Þ
/
;
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
Hð1 þ U2 Þ1=2 kh kz ð1 þ /2 Þ1=2
:
ð14Þ
H. Demiray / International Journal of Engineering Science 40 (2002) 1897–1918
1901
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 8 9 #1=2 " > > 2 = ou o < ðr0 þ /z þ u Þ / þ ou oz þ T1 h T2 1 þ / þ þ G ðr0 þ /z þ u Þ i 1=2 > oz > oz ou 2 : ; 1 þ / þ oz ¼
2 q0 H 2 1=2 o u ð1 þ U Þ ðR k þ Uz Þ ; 0 z ot2 k2z
ð15Þ
where q0 is the mass density of the membrane and G is the fluid reaction force density defined by oVr oVr oVz ou þ : ð16Þ G ¼ P 2lv þ lv /þ or oz or oz r¼r0 þ/z þu Here, P is the fluid pressure function, Vr and Vz are the fluid velocity components in the radial and axial directions, respectively, and lv is the fluid viscosity. Let lR be the strain energy density function of the membrane, where l is the shear modulus of the membrane material. The membrane forces may be expressed in terms of the stretch ratios as T1 ¼
lH oR ; k2 ok1
T2 ¼
lH oR : k1 ok2
ð17Þ
Introducing (17) into Eq. (15), the equation of motion in the radial direction takes the following form 8 9 > > l l o < ðkz R0 þ Uz Þ / þ ou oR = 2 1=2 oR oz Hð1 þ U Þ þ H þ G ðr0 þ /z þ u Þ h i1=2 kz ok2 kz oz > ok1 > ; : 1 þ / þ ou 2 oz ¼
q0 o2 u H ð1 þ U2 Þ1=2 ðkz R0 þ Uz Þ 2 : 2 ot kz
ð18Þ
2.2. Equations of fluid Blood is known to be an incompressible non-Newtonian fluid. However, in the course of flow in arteries, the red cells in the vicinity of arteries move to the central region of the artery so that the hematocrit ratio 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 viscosity of blood is further reduced. Therefore, for flow problems in large blood vessels, the blood may be treated as a Newtonian fluid. For an axisymmetric motion of blood, the equations of motion in the cylindrical polar coordinates may be given as follows:
1902
H. Demiray / International Journal of Engineering Science 40 (2002) 1897–1918 oVr 1 oP lv oVr oVr þ V ¼ þ V þ r z ot or oz qf or qf oVz 1 oP lv oVz oVz þ V þ ¼ þ V r z ot or oz qf oz qf
oVr Vr oVz þ þ ¼0 or r oz
o2 Vr 1 oVr Vr o2 Vr 2 þ 2 ; þ r or or2 r oz
ð19Þ
o2 Vz 1 oVz o2 Vz þ þ 2 ; r or or2 oz
ð20Þ
ðincompressibility conditionÞ;
ð21Þ
where qf is the mass density of the fluid body. These field equations must satisfy the following boundary conditions ou ou þ V j ¼ Vr jr¼r þ/z þu : 0 ot oz z r¼r0 þ/z þu
ð22Þ
At this stage it is convenient to introduce the following nondimensionalized quantities
t ¼
R0 t; z ¼ R0 z; r ¼ R0 x; u ¼ R0 u; Vr ¼ c0 v; Vz ¼ c0 w; G ¼ qf c20 g; c0
P ¼ qf c20 p; m ¼
q0 H lH ; r0 ¼ kh R0 ; lv ¼ qf R0 c0 m; c20 ¼ : qf R0 qf R0
ð23Þ
Introducing (23) into equations (18)–(22) we obtain the following nondimensionalized equations: 2 ov ov ov op o v 1 ov v o2 v þ ¼ 0; þv þw þ m þ ot ox oz ox ox2 x ox x2 oz2
ð24Þ
2 ow ow ow op o w 1 ow o2 w þ ¼ 0; þv þw þ m þ ot ox oz oz ox2 x ox oz2
ð25Þ
ov v ow þ þ ¼ 0; ox x oz
ð26Þ
with boundary conditions
ov ov ow ou ðp0 þ pÞ 2m þ m g ¼ 0; þ /þ ox oz ox oz x¼kh þ/zþu
ou ou þ wj ¼ vjx¼kh þ/zþu ; ot oz x¼kh þ/zþu
ð27Þ
H. Demiray / International Journal of Engineering Science 40 (2002) 1897–1918
1903
where the function g is defined by g¼
m ðkz þ UzÞ o2 u ð1 þ U2 Þ1=2 oR 2 1=2 ð1 þ U Þ þ ðkh þ /z þ uÞ ot2 kz ðkh þ /z þ uÞ ok2 k2z 8 9 > > < ou / þ oz ðkz þ UzÞ oR = 1 o : h i1=2 kz ðkh þ /z þ uÞ oz > ok1 > : 1 þ / þ ou 2 ; oz
ð28Þ
These equations give sufficient relations to determine the field quantities u, v, w and p completely. 3. Nonlinear wave modulation In this section we shall examine the amplitude modulation of weakly nonlinear waves in a fluidfilled thin elastic tube whose dimensionless governing equations are given in Eqs. (24)–(28). Considering the dispersion relation of the linearized field equations and the nature of the problem of concern, which is a boundary value type, the following stretched coordinates may be introduced n ¼ ðz ktÞ;
s ¼ 2 z;
ð29Þ
where is a small parameter measuring the weakness of nonlinearity and k is the 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 4 . Hence, the tapering angles can be expressed as U ¼ A4 ;
/ ¼ a4 :
ð30Þ
Here A and a are some finite constants describing the tapering angles and will be specified later. Noting the differential relations o o o o ! þ þ 2 ; oz oz on os
o o o ! k ot ot on
ð31Þ
and expanding the field variables into asymptotic series as u ¼ u1 þ 2 u2 þ 3 u3 þ ; v ¼ v1 þ 2 v2 þ 3 v3 þ ; w ¼ w1 þ 2 w2 þ 3 w3 þ ; 2
ð32Þ
3
g ¼ g0 þ g1 þ g2 þ g3 þ ; p ¼ p0 þ p1 þ 2 p2 þ 3 p3 þ ; where u1 ; . . . ; p3 ; . . . are functions of the fast variables ðt; zÞ and slow variables ðn; sÞ, and introducing (31) and (32) into the field equations (24)–(27), the following sets of differential equations are obtained:
1904
H. Demiray / International Journal of Engineering Science 40 (2002) 1897–1918
OðÞ order equations: ov1 op1 þ ¼ 0; ot ox
ow1 op1 þ ¼ 0; ot oz
ov1 v1 ow1 þ þ ¼ 0; ox x oz
ð33Þ
and the boundary conditions p1 jx¼kh ¼ g1 ;
ou1 ¼ v1 jx¼kh : ot
ð34Þ
Oð2 Þ order equations: ov2 ov1 ov1 ov1 op2 k þ v1 þ w1 þ ¼ 0; ot on ox oz ox ow2 ow1 ow1 ow1 op2 op1 k þ v1 þ w1 þ þ ¼ 0; ot on ox oz oz on ov2 v2 ow2 ow1 þ þ þ ¼ 0; ox x oz on and the boundary conditions op1 p2 þ u1 ¼ g2 ; ox x¼kh
ð35Þ
ov1 ou2 ou1 ou1 v 2 þ u1 ¼ : k þ w1 ox x¼kh ot on oz x¼kh
ð36Þ
Oð3 Þ order equations: 2 ov3 ov2 o ov1 ov2 ov1 op3 o v1 1 ov1 v1 o2 v1 k þ ðv1 v2 Þ þ w2 þ w1 þ w1 þ m þ þ ¼ 0; ot on ox oz oz on ox ox2 x ox x2 oz2 ow3 ow2 ow2 ow1 o ow1 op3 op2 op1 k þ v1 þ v2 þ ðw1 w2 Þ þ w1 þ þ þ oz ot on ox ox on oz on os 2 2 o w1 1 ow1 o w1 þ ¼ 0; m þ 2 x ox ox2 oz ov3 v3 ow3 ow2 ow1 þ þ þ þ ¼ 0; ð37Þ ox x oz on os and the boundary conditions
1 2 o2 p1 op2 op1 ov1 u þ ðas þ u2 Þ þ p3 2m ¼ g3 ; þ u1 2 1 ox2 ox ox ox x¼kh 1 2 o2 v 1 ov1 ov2 ou3 ou2 ¼ u1 2 þ ðas þ u2 Þ þ u1 þ v3 k 2 ox ox ox ot on x¼kh ou2 ou1 ou1 ow1 þ w1 jx¼kh : þ þ w2 þ u1 oz on oz ox x¼kh
ð38Þ
H. Demiray / International Journal of Engineering Science 40 (2002) 1897–1918
1905
In order to make the equations self-sufficient, or complete, we need the explicit expressions of g1 , g2 and g3 , which should be obtained from Eq. (28). Under the coordinate transformations (30), the tapering terms in these expressions become ðAsÞ2 and ðasÞ2 , in this approximation. Furthermore, we shall assume that the generators before and after the static deformation remain parallel to each other. As pointed out before, for this case the relation between a and A becomes a ¼ Akh =kz . Hence the following expansions are permissible: 2 1 ou1 ou1 ou2 ou1 3 2 k1 ¼ kz 1 þ þ þ þ
; 2 oz oz oz on ! As 2 k2 ¼ kh þ u1 þ u2 þ u3 u1 3 þ ; kz ( ! 1 1 u1 u21 u2 A 2 ½kh þ as þ u ¼ 1 þ s 2 kh kh k2h kh kz " # ) u31 2 2 u3 3 þ
; Asu2 þ 3 þ 2 u1 u2 þ kh kh kz kh kh !2 # " " 1 oR ou 1 ¼ b0 þ b1 u1 þ b2 u21 þ b1 u2 þ a1 kz þ 2 þ b3 u31 þ 2b2 u1 u2 þ b1 u3 kh kz okz oz ! # 2 ou1 ou2 ou1 ou1 A þ þ a2 kz u1 þ 2a1 kz b1 su1 3 þ ; kz oz oz on oz " # 2 1 oR ou1 2 2 þ ; ¼ a0 þ 2a1 u1 þ a2 u1 þ 2a1 u2 þ c1 kz kh kz ok1 oz
ð39Þ
where the coefficients a0 , a1 , a2 , b0 ; . . . ; b3 and c1 are defined by 1 o2 R ; 2kh kz okh okz
a0 ¼
1 oR ; kh kz okz
a1 ¼
b1 ¼
1 o2 R ; kh kz ok2h
b2 ¼
1 o3 R ; 2kh kz ok3h
1 o3 R 1 oR ; b0 ¼ ; 2 2kh kz okh kz kh kz okh 1 o4 R 1 o2 R b3 ¼ ; c ¼ : 1 6 ok4h 2kh kz ok2z a2 ¼
ð40Þ
Introducing Eq. (39) into Eq. (28) the fluid reaction force terms g1 , g2 and g3 may be given by m o2 u1 o2 u1 b0 g1 ¼ a0 kz 2 þ b1 ð41Þ u1 ; kh kz ot2 oz kh ! 2 m o u2 o2 u1 m o2 u1 b1 b0 2 b0 2l u2 g2 ¼ 2 u1 2 þ b2 þ 2 u1 þ b1 kh kz ot2 oton ot kh kh kh kh kz 2 o2 u2 o2 u1 a0 o2 u1 ou1 b 2a1 u1 2 a1 kz a0 kz 2 2a0 kz 0 As; ð42Þ þ kz oz ozon kh oz oz kz
1906
H. Demiray / International Journal of Engineering Science 40 (2002) 1897–1918
! m o2 u3 o2 u3 b0 m o2 u2 o2 u2 u g3 ¼ a k þ b 2k k 2a 0 z 3 0 z 1 kh kz ot2 kh kz oton ot2 kh ozon 2 2 2 2 m 2 o u1 m o u1 o u2 o u1 m 2 o2 u1 k a0 kz þ 2ku u u1 2 þ þ u 1 1 2 kh kz oton ot2 ot2 ot k3h kz on2 k2h kz ! ! b b b b b o2 u1 þ b3 2 þ 21 30 u31 þ 2 b2 1 þ 20 u1 u2 2a0 kz kh kh kh kh kh ozos ! ! 2 2 3 ou1 o u1 a0 o2 u2 o2 u1 a0 3c1 kz 2a1 u1 2 þ u2 2 þ kz þ kz 2 oz oz2 kh oz oz ! ou1 ou2 ou1 a0 o2 u1 2a1 a0 2 o2 u1 þ þ kz a2 þ 2a1 u1 2 u1 2 2a1 kz þ 2kz oz oz on kh ozon kh oz kh 2 a1 ou1 2 b o2 u1 a2 u1 þ kz b1 0 Asu1 þ a0 As 2 : kz kh oz kh oz
ð43Þ
4. Solution of the field equations 4.1. The solution of OðÞ order equations The solution of the differential equations given in (33) and (41) suggests us to seek the following type of solution to these differential equations u1 ¼ U1 exp½iðxt kzÞ þ c:c:;
g1 ¼ G1 exp½iðxt kzÞ þ c:c:;
v1 ¼ V1 exp½iðxt kzÞ þ c:c:;
w1 ¼ W1 exp½iðxt kzÞ þ c:c:;
p1 ¼ P1 exp½iðxt kzÞ þ c:c:;
ð44Þ
where x is the angular frequency, k is the wave number, U1 and G1 are amplitude functions which depend on the slow variables n and s, and V1 , W1 and P1 are amplitude functions which depend on x as well as the slow variables ðn; sÞ and c.c. stands for the complex conjugate of the corresponding quantities. The solution satisfying the field equations (33) and the boundary conditions (34) are given as follows: U1 ¼ U ;
G1 ¼
x2 f0 U ; k
V1 ¼ ixf1 ðkxÞU ;
W1 ¼ xf0 ðkxÞU;
P1 ¼
x2 f0 ðkxÞU ; k
provided that the following dispersion relation holds true x2 mx2 b0 f0 þ b1 a0 kz k 2 ¼ 0; k kh kz kh
ð45Þ
ð46Þ
where, for brevity, we have defined the following functions fn ðkxÞ ¼ In ðkxÞ=I1 ðkh kÞ; ðn ¼ 0; 1; 2; . . .Þ;
f0 ¼ f0 ðkh kÞ;
u ¼ xt kz:
ð47Þ
H. Demiray / International Journal of Engineering Science 40 (2002) 1897–1918
1907
Here In ðxÞ is the modified BesselÕs function of order n and U ðn; sÞ is an unknown function of the slow variables, whose governing equation will be obtained later. The group velocity vg is defined by vg ¼
kh x2 ðf02 1Þ þ 2a0 kz k 2 : 2xðf0 þ mk=kh kz Þ
ð48Þ
4.2. The solution of Oð2 Þ order equations Introducing the solution given in (45) into Eq. (35) we obtain # " 2 2 2 ov2 op2 oU x f ðkxÞ f ðkxÞ 1 1 þ ixk U 2 e2iu þ 2x2 2kf0 ðkxÞf1 ðkxÞ jU j2 þ c:c: ¼ 0; f1 ðkxÞeiu þ ot ox x x on ow2 op2 x oU iu k f0 ðkxÞ þ þx e þ ix2 k½f1 ðkxÞ2 f0 ðkxÞ2 U 2 e2iu þ c:c: ¼ 0; ot oz k on ov2 v2 ow2 oU iu e þ c:c: ¼ 0; þ þ þ xf0 ðkxÞ ð49Þ on ox x oz
and the boundary conditions g2 p2 jx¼kh ¼ 2x2 jUj2 þ x2 U 2 e2iu þ c:c:; ou2 oU i/ 1 e þ ix 2kf0 v2 jx¼kh ¼ k U 2 e2iu þ c:c: on kh ot
ð50Þ
Here, for convenience, we have defined jUj2 ¼ UU , where U is the complex conjugate of U, and noticed that f1 ðkh kÞ ¼ 1. The form of the differential equations in (49) suggests us to seek the following type of solution
v2 ¼
ð0Þ V2
þ
!
2 X
ðlÞ V2 ðx; n; sÞ expðiluÞ
þ c:c: ;
l¼1 ð0Þ
w 2 ¼ W2 þ
2 X
! ðlÞ
W2 ðx; n; sÞ expðiluÞ þ c:c: ;
l¼1
p2 ¼
ð0Þ P2
þ
2 X
!
ðlÞ P2 ðx; n; sÞ expðiluÞ
l¼1
u2 ¼
ð0Þ U2
þ
2 X
! ðlÞ U2 ðn; sÞ expðiluÞ
l¼1 ð0Þ
g2 ¼ G2 þ
2 X l¼1
þ c:c: ;
þ c:c: ; !
ðlÞ
G2 ðn; sÞ expðiluÞ þ c:c: ;
ð51Þ
1908
H. Demiray / International Journal of Engineering Science 40 (2002) 1897–1918
where, from (42) we have b0 x2 b0 kh ð0Þ ð0Þ 2 2 G2 ¼ b 1 U2 þ 2 a Ab1 s; f0 þ b2 þ a1 k jU j þ b1 kh kh k kh kz x2 mx oU ð1Þ ð1Þ G2 ¼ k ; f0 U2 þ 2i a0 kz k kh kz on k 2 x b0 x2 ð2Þ ð2Þ 2 U2 þ G2 ¼ 4 f 0 3 b 1 f0 þ b2 þ 3a1 kz k U 2 : k kh kh k
ð52Þ ð53Þ ð54Þ
Introducing (51) into (49) and (50) we obtain the following set of differential equations and the boundary conditions " # ð0Þ ð0Þ ð0Þ 2 oP2 f ðkxÞ oV2 V2 1 2 2 þ 2x 2kf0 ðkxÞf1 ðkxÞ jU j ¼ 0; þ ¼ 0; ð55Þ ox x ox x and the boundary conditions ð0Þ
ð0Þ
ð0Þ
2
G2 P2 jx¼kh ¼ 2x2 jU j ;
V2 jx¼kh ¼ 0:
ð56Þ
The equations and the boundary conditions resulting from the coefficients of expðiuÞ becomes ð1Þ
oP2 oU f1 ðkxÞ ¼ 0; ixk on ox x oU ð1Þ ð1Þ k f0 ðkxÞ ¼ 0; ixW2 ikP2 þ x k on ð1Þ
ixV2 þ
ð1Þ
ð57Þ
ð1Þ
oV2 V oU ð1Þ f0 ðkxÞ ¼ 0; þ 2 ikW2 þ x on ox x and the boundary conditions ð1Þ
ð1Þ
G2 P2 jx¼kh ¼ 0;
ð1Þ
ð1Þ
ixU2 V2 jx¼kh ¼ k
oU : on
ð58Þ
The equations and boundary conditions resulting from the coefficients of expð2iuÞ become, ð2Þ
oP2 f 2 ðkxÞ 2 U ¼ 0; þ x2 1 x ox ð2Þ ð2Þ 2xW2 2kP2 þ x2 k f12 ðkxÞ f02 ðkxÞ U 2 ¼ 0; ð2Þ
2ixV2 þ ð2Þ
ð2Þ
oV2 V ð2Þ þ 2 2ikW2 ¼ 0; ox x
ð59Þ
H. Demiray / International Journal of Engineering Science 40 (2002) 1897–1918
1909
and the boundary conditions ð2Þ
ð2Þ
G2 P2 jx¼kh ¼ x2 U 2 ð2Þ 2ixU2
ð2Þ V2 jx¼kh
1 ¼ ix 2kf0 U 2: kh
ð60Þ
The solution of Eqs. (55) and (56) yields ð0Þ
V2
ð0Þ
U2
ð0Þ
2
P2 ¼ H ðn; sÞ x2 ½f12 ðkxÞ þ f02 ðkxÞ jUj ; " ! # jU j2 2f0 b0 A Hðn; sÞ 2 2 2 x 1þ ; f0 2b2 2a1 kz k þ ¼ sþ ðb1 b0 =kh Þ ðb1 b0 =kh Þ kh k ðb1 b0 =kh Þ kz
¼ 0;
ð61Þ provided that b1 b0 =kh 6¼ 0. The special case where b1 b0 =kh ¼ 0 will be discussed later. From the solution of Eqs. (57) and (58) it is readily seen that, in order to have a nonzero solution for U ðn; sÞ the scale parameter k must be equal to the group velocity given in Eq. (48). Furthermore, in this work, since we shall be concerned with the governing equation of the lowest order term in the perturbation expansion, without loosing the generality of the problem we can set the homogeneous solution equal to zero. Hence, the nonhomogeneous solution of Eqs. (57) and (58) may be given by oU k oU ð1Þ ; U2 ¼ i k h f 0 2 ; ¼ ½kf1 ðkxÞ xxf0 ðkxÞ on x on x oU x2 oU ð1Þ ð1Þ W2 ¼ i xxf1 ðkxÞ þ k f0 ðkxÞ ; P2 ¼ i xf1 ðkxÞ : k on on k ð1Þ V2
ð62Þ
The solution of Eq. (59) gives ð2Þ
V2
ð2Þ
¼ ixCF1 ð2kxÞ;
P2 ¼
2
ð2Þ
W2
¼ xCF0 ð2kxÞ;
2
x x CF0 ð2kxÞ þ ½f12 ðkxÞ f02 ðkxÞ U 2 ; k 2
ð63Þ
where C is another unknown function of slow variables, F0 and the functions Fi ð2kxÞ are defined by F0 ¼ F0 ð2kÞ;
Fi ð2kxÞ ¼
Ii ð2kxÞ ; I1 ð2kÞ
ði ¼ 0; 1; 2; 3; . . .Þ:
ð64Þ
1910
H. Demiray / International Journal of Engineering Science 40 (2002) 1897–1918
The use of the boundary conditions (60) results in 1 ð2Þ C ¼ 2U2 þ 2kf0 U 2 ; kh 2 1 x b0 2x2 1 2 f0 x2 1 ð2Þ 2 3 x 2kf0 F0 f þ F0 þ U2 ¼ 4 f 0 3 b 1 2 2 0 kh k k kh k k kh ! 2 b2 3a1 kz k U 2 :
ð65Þ
4.3. The solution of Oð3 Þ order equations For the solution of this order of differential equations and the boundary conditions given in Eqs. (37) and (38) the field quantities may be expressed as follows: ! 3 X ð0Þ ðlÞ u3 ¼ U3 þ U3 ðn; sÞ expðiluÞ þ c:c: ; l¼1
v3 ¼
ð0Þ V3
þ
3 X
! ðlÞ V3 ðx; n; sÞ expðiluÞ
þ c:c: ;
l¼1 ð0Þ
w2 ¼ W3 þ
3 X
! ðlÞ
W3 ðx; n; sÞ expðiluÞ þ c:c: ;
l¼1 ð0Þ
p3 ¼ P3 þ
3 X
! ðlÞ
P3 ðx; n; sÞ expðiluÞ þ c:c: ;
l¼1
g3 ¼
ð0Þ G3
þ
3 X
ð66Þ
! ðlÞ G3 ðn; sÞ expðiluÞ
þ c:c: ;
l¼1
where ð1Þ G3
! 2 2 ð1Þ x2 oU mk oU mkx oU2 ð1Þ ¼ þ 2i a0 kz k a0 kz þ f0 U3 þ 2ia0 kz os kh kz k kh kz on on2 # " " x2 b b x2 ð0Þ f0 þ 2a1 kz k 2 þ 2b2 1 þ 20 U2 U þ 5 f0 þ 6a1 kz k 2 þ kh k kh kh kh k !# " b1 b0 x2 b2 a1 ð2Þ þ 2a2 5 kz k 2 þ 2b2 þ 3 U2 U 3 2 f0 þ 3 b3 kh k2h kh kh kh k ! # " ! # 3 b A 2 kz sU: a0 3c1 kz k 4 jU j U 2 b1 0 þ a0 kz k 2 kh 2 kz
ð67Þ
Introducing the expansion (66) into Eqs. (37) and (38), we obtain the following sets of differential equations and the boundary conditions
H. Demiray / International Journal of Engineering Science 40 (2002) 1897–1918 ð0Þ
k
ð0Þ
oW2 oP o þ 2 þ x2 ðf02 ðkxÞ þ f12 ðkxÞÞ jU j2 ¼ 0 on on on
ð0Þ
1911
ð0Þ
ð0Þ
oV3 V oW þ 3 þ 2 ¼ 0; ox x on with the boundary condition ð0Þ oU2 k ojU j2 ð0Þ V3 jx¼kh ¼ k þ 2xf0 : kh on on
ð68Þ
ð69Þ
4.4. l ¼ 1 equations ð1Þ
ð1Þ
oP3 oV o ð2Þ ð0Þ ð2Þ k 2 ixU ½f1 ðkxÞV2 þ kxf1 ðkxÞðW2 U þ W2 U Þ ox ox on 2 o V1 1 oV1 V1 ð2Þ 2 2ixkf0 ðkxÞV2 U m þ ¼ 0; k V 1 x ox x2 ox2 ! ð1Þ ð0Þ ð2Þ oW2 x2 oU oW2 oW ð1Þ ð1Þ ixW3 ikP3 k þ ixf1 ðkxÞ þ f0 ðkxÞ U U 2 os on k ox ox ð1Þ oP2 o2 W1 1 oW1 ð2Þ ð0Þ ð2Þ 2 þ xkf1 ðkxÞV2 U ixkf0 ðkxÞ½W2 U þ W2 U þ m k W1 ¼ 0; þ x ox on ox2 ð1Þ
ixV3 þ
ð1Þ
ð1Þ
ð1Þ
oV3 V oW oU ð1Þ ¼ 0; þ 3 ikW3 þ 2 þ xf0 ðkxÞ os ox x on
ð70Þ
and the boundary conditions ð1Þ G3
¼
ð1Þ P3 jx¼kh
2
ð0Þ U2 U
2
ð2Þ U2 U
x2 þ 2
3 13kf0 jU j2 U kh
þx þ 5x 1 A U þ kh x2 sU ; 2imx kf0 kh kz ð1Þ oU2 3 2 3 k ð1Þ ð1Þ 2 kf0 þ F0 2k f0 F0 jUj2 U k þ þ ix ixU3 V3 jx¼kh ¼ k 2 2kh kh on 1 1 1 A ð0Þ ð0Þ ð2Þ þ ikW2 U þ ix kf0 U2 U þ ix 2kF0 þ kf0 U2 U þ ikh x kf0 sU : kh kh kh kz ð71Þ The solution of (68) under the boundary conditions (69) is given as follows: ð0Þ W2
H ðn; sÞ ¼ ; k
ð0Þ V3
1 oH ¼ x; 2k on
k2 ð0Þ k 1 jU j2 : 2xf0 H ¼ 2 U2 2 kh kh kh
ð72Þ
1912
H. Demiray / International Journal of Engineering Science 40 (2002) 1897–1918
Introducing the expression of H into Eq. (61) we have ð0Þ U2
2
k2 þ 2b2 kh þ 2a1 kh kz k 2 kh ! 2f0 A 2 2 2 kh x jU j b0 s : þ kh f0 kz k 1
¼ ½2k ðkh b1 b0 Þ
4kxf0 2
ð73Þ
Finally the solution of Eq. (70) under the boundary conditions (71) is given by ð1Þ P3
¼
2 x2 x 2 x2 x oU x2 ½F0 ð2kxÞf0 ðkxÞ þ F1 ð2kxÞf1 ðkxÞ CU f0 ðkxÞ þ 2 f1 ðkxÞ 2k 2k on2
þi ð1Þ V3
ð1Þ W3
ð1Þ
2 ikHU x x 2 k2 oU oU f1 ðkxÞ þ i k x þ ; ¼ xxf0 ðkxÞ xf0 ðkxÞ f1 ðkxÞ 2 k 2k 2 os x on 2 kHU x x 2 k2 k oU ¼ f0 ðkxÞ þ k xf1 ðkxÞ þ x þ f0 ðkxÞ k 2k 2 x k on2 x oU þ i xxf1 ðkxÞ þ f0 ðkxÞ ; k os
G3 ¼
ð1Þ U3
x2 oU ; xf1 ðkxÞ os k
x2 kh o2 U ð0Þ ð2Þ ð1 kk f Þ x2 ð1 þ f0 F0 ÞCU þ x2 U2 U þ 5x2 U2 U h 0 2 2k 2 on 2 x 3 1 A 13kf0 jU j2 U 2imx kf0 U þ kh x2 sU ; þ kh kz 2 kh
2kFU 2i oU k 1 k2 k2h o2 U ¼ þ þ 2 kh f0 3 2 xk x os x 2k x 2 on2 3 2 3 k 1 ð0Þ 2 2 U2 U þ kf0 þ F0 2k f0 F0 jU j U þ kf0 k þ 2 2kh kh kh 1 1 A ð2Þ þ 2kF0 þ kf0 sU : U2 U þ kh kf0 kh kh kz ð1Þ
ð74Þ
ð1Þ
Eliminating G3 and U3 between the Eqs. (71) and (74) the following dissipative nonlinear Schr€ odinger equation with variable coefficient is obtained i
oU o2 U 2 þ l1 2 þ l2 jU j U þ il3 U þ l4 sU ¼ 0; os on
ð75Þ
H. Demiray / International Journal of Engineering Science 40 (2002) 1897–1918
1913
1 x2 2 m f0 x2 kh kk 2 2 2 f kh f 0 kh ¼ 2a0 kz k þ kh ðf0 1Þ 3k þ þ 2 4 kh kz k x 0 k 2k mx k f0 þ 2a0 kz a0 kz ð1 þ 2kh kf0 Þ ; þ 2k kz x ! ( " 1 x2 2 3 13 k 3 3 ¼ 2a0 kz k þ kh ðf0 1Þ x2 þ k þ 4 2 þ k þ 2 f0 2kh 2 k kh 2 kh k x 3 1 x b f2 þ 2k f0 F0 þ 4 þ 3 b3 2 þ 8 þ kh 2kh 0 kh kh kh ! " ! a1 3 2 þ 2a2 5 kz k þ 2 kz a0 3c1 kz k 4 þ x2 f02 f0 1 2 kh k kh # b1 b0 x ð0Þ þ 2b2 þ 2 þ 2a1 kz k 2 þ 4k f0 U2 U kh kh kh " ! # ) 6 b1 b0 ð2Þ 2 2 2 þ 6a1 kz k U2 U þ x 4f0 F0 þ f0 f0 3 þ 2b2 þ 3 kh k kh k2h 2mx kf0 k1h ¼ 2 2a0 kz k þ kh xk ðf02 1Þ ! ! 1 ( 2 x2 2 f f b 0 ¼ 2a0 kz k þ kh ðf0 1Þ x2 0 2 kh 2 b1 0 a0 kz k 2 k kh kh k kh h i9 =A b0 x2 f02 k2h k f0 1 þ 2b2 bkh1 þ bk20 þ 2a1 kz k 2 þ 4 kx f kh 0 h : ; kz 2k2 ðkh b1 b0 Þ
ð76Þ
where
l1
l2
l3 l4
When the tapering angle parameter A is set equal to zero the coefficient l4 vanishes and the resulting evolution equation becomes the dissipative nonlinear Schr€ odinger equation i
oU o2 U þ l1 2 þ l2 jUj2 U þ il3 U ¼ 0: os on
ð77Þ
Therefore, the coefficient l4 stands for the contribution of tapering to the evolution equation. 4.5. Progressive wave solution In this sub-section we shall try to present a progressive wave solution to the evolution equation (75). For that purpose, following Demiray [21], we shall propose a solution of the form U ðn; sÞ ¼ aðsÞV ðfÞ expfi½XðsÞ Kn g;
f ¼ aðsÞ½n 2l1 Ks ;
ð78Þ
1914
H. Demiray / International Journal of Engineering Science 40 (2002) 1897–1918
where K is a constant aðsÞ, aðsÞ, and V ðfÞ are some real functions to be determined from the solution of (75). Introducing (78) into Eq. (75) and setting the real and imaginary parts of the resulting equation equal to zero we obtain the following sets of ordinary differential equations
a0 ðsÞ a0 ðsÞ 0 fV ¼ 0; þ l3 V þ aðsÞ aðsÞ
ð79Þ
½X0 ðsÞ l1 K 2 þ l4 s V þ l1 aðsÞ2 V 00 þ l2 a2 ðsÞV 3 ¼ 0;
ð80Þ
where primes denote the derivative of the corresponding quantity with respect to its argument. In the present work we shall be concerned with the localized travelling wave solution to the field equation, i.e., V and its various order derivatives vanish as f ! 1. Under such assumptions one can show that V is square integrable over the interval ð1; 1Þ. Thus, multiplying Eq. (79) by V and integrating the resulting equation with respect to f from 1 to þ1 we obtain
a0 ðsÞ 1 a0 ðsÞ þ l3 hV 2 i ¼ 0; aðsÞ 2 aðsÞ
2
hV i ¼
Z
1
V 2 df:
ð81Þ
1
Since hV 2 i is bounded and nonzero, we obtain the following ordinary differential equation a0 ðsÞ 1 a0 ðsÞ þ l3 ¼ 0: aðsÞ 2 aðsÞ
ð82Þ
Now, in the first place, we shall propose a solution to Eq. (80) of the following form V ðfÞ ¼ sech f:
ð83Þ
Introducing (83) into Eq. (80), the following differential equations are obtained 2
aðsÞ ¼
l2 2 aðsÞ ; 2l1
X0 ðsÞ ¼
l2 2 aðsÞ þ l4 s l1 K 2 : 2
ð84Þ
It is seen from the first term of Eq. (84) that, in order to have a real solution for aðsÞ and aðsÞ the coefficient l1 and l2 must satisfy the inequality l1 l2 > 0, which is the case for the classical NLS equation. Now, let us return to the investigation of Eq. (79). It can be shown from the first term of Eq. (84) that a0 ðsÞ=aðsÞ ¼ a0 ðsÞ=aðsÞ. Introducing this relation into Eq. (79) we obtain a0 ðsÞ þ 2l3 ¼ 0: aðsÞ
ð85Þ
The solution of this equation may be given as follows: aðsÞ ¼ a0 e2l3 s ;
ð86Þ
H. Demiray / International Journal of Engineering Science 40 (2002) 1897–1918
1915
where a0 is a constant. Introducing (86) into equation (84) we obtain aðsÞ ¼
l2 2l1
1=2
a0 e2l3 s ;
XðsÞ ¼ l1 K 2 s þ
l4 2 l2 a20 ð1 e4l3 s Þ: s þ 2 8l3
ð87Þ
Here, in obtaining the function XðsÞ we have utilized the regularity condition that XðsÞ ¼ 0 at s ¼ 0. Thus, the final solution may be given as follows: U ðn; sÞ ¼ a0 e
2l3 s
sech f expfi½XðsÞ Kn g;
f¼
l2 2l1
1=2 aðsÞðn 2l1 KsÞ:
ð88Þ
It should be noted that the Eq. (88) is not a solution in the classical sense, it is rather a solution in the sense of distribution. If the coefficient l3 is vanishingly small then one can approximate a ¼ a0 ;
a¼
l2 2l1
1=2 a0 ;
XðsÞ ¼ ðl1 K 2 þ l2 a20 =2Þs þ
l4 2 s: 2
ð89Þ
This result is exactly the same with that of obtained by Demiray [21] for nondissipative NLS equation. If the tapering parameter is set equal to zero, i.e., l4 ¼ 0; the result will be that of the progressive wave solution of the classical NLS equation. Now, secondly, we shall propose a solution to Eq. (80) of the following form V ¼ tanh f:
ð90Þ
Since this function is not square integrable we cannot follow the above procedure to handle this problem. As is seen from Eq. (79), for this type of solution fV 0 ðfÞ approaches to zero as f ! þ1. Therefore, if we investigate the far field behavior of Eq. (79) and consider that fV 0 ðfÞ vanishes as f ! þ1 and obtain the following differential equation a0 ðsÞ þ l3 ¼ 0: aðsÞ
ð91Þ
The solution of this may be given by aðsÞ ¼ a0 el3 s ;
ð92Þ
where a0 is a constant. Now, let us introduce the proposed solution (90) into Eq. (80). In order to satisfy this equation the following relations must be satisfied 2l1 aðsÞ2 þ l2 aðsÞ2 ¼ 0;
ð93Þ
X0 ðsÞ l1 K 2 þ l4 s þ l2 aðsÞ2 ¼ 0:
ð94Þ
1916
H. Demiray / International Journal of Engineering Science 40 (2002) 1897–1918
The relation (93) holds true if and only if l1 and l2 have different signs, which is the case for classical nonlinear Schr€ odinger equation. Introducing (92) into (93) we obtain aðsÞ as aðsÞ ¼
l 2 2l1
1=2
a0 el3 s :
ð95Þ
Finally, integrating (94) and utilizing the regularity condition that XðsÞ at s ¼ 0, we obtain XðsÞ ¼ l1 K 2 s þ
l4 2 a20 l2 ð1 e2l3 s Þ: s þ 2 2l3
ð96Þ
Thus, the final solution may be given by l3 s
U ðn; sÞ ¼ a0 e
tanh f expfi½XðsÞ Kn g;
f¼
l 2 2l1
1=2
a0 el3 s 2l1 Ks:
ð97Þ
If the coefficient l3 is vanishingly small, then one can approximate a ¼ a0 ;
a¼
l2 2l1
1=2 a0 ;
XðsÞ ¼ l1 K 2 s þ l2 a20 s þ
l4 2 s: 2
ð98Þ
As is seen from these solutions, the existence of dissipation causes the decay in wave amplitude with the distance parameter s. Similar interpretation applies to the phase velocity of the enveloping wave, which might be given by vp ¼ l1 K þ
l4 l s þ 2 a20 e4l3 s : K 2K
ð99Þ
Here, again, the dissipative effect causes to decay in phase speed with distance parameter s. In order to see the contribution of the tapering to the phase velocity of the enveloping wave, one has to know the sign of the coefficient l4 . For convenience we shall re-write l4 as l4 ¼ v
A ; kz
ð100Þ
where 8 ! 1 < 2 x f f b0 0 2 2 0 x þ 2 þ kh þ 2 b1 þ a0 kz k 2 v ¼ 2a0 kz k þ kh ðf0 1Þ : k kh kh k kh h i9 = b0 x2 f02 k2h k f0 1 þ 2b2 bkh1 þ bk20 þ 2a1 kz k 2 þ 4 kx f kh 0 h þ : ; 2k2 ðkh b1 b0 Þ
2
ð101Þ
H. Demiray / International Journal of Engineering Science 40 (2002) 1897–1918
1917
In order to get information about the sign of v one has to know the constitutive relation of the tube material. In this work we shall utilize the constitutive relation proposed by Demiray [22] for soft biological tissues. Following Demiray [22], the strain energy density function may be expressed as R¼
1 fexp½aðI1 3Þ 1g; 2a
ð102Þ
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 (102) into Eq. (44), the coefficients a0 , b0 , b1 and b2 are obtained as ! 1 1 F ðkh ; kz Þ; a0 ¼ kh k3h k4z " ! !# 1 1 1 a1 ¼ þa 1 4 2 1 2 4 F ðkh; kz Þ; k4h k4z kh kx kh kz ! 1 1 F ðkh ; kz Þ; b0 ¼ kz k4h k3z 2 ! !2 3 1 3 a 1 þ 5 3 þ2 kh 3 2 5F ðkh ; kz Þ; b1 ¼ 4 kh kz kh kz kh kz kh kz 2 ! ! !3 3 2 6 a 3 1 a 1 b2 ¼ 4 4 3 þ 3 kh 3 2 þ 2 1þ 4 2 kh 3 2 5F ðkh ; kz Þ; k k k k kh kz kh kz kh kz kh kz h z h z
ð103Þ
where the function F ðkh ; kz Þ is defined by "
!# 1 F ðkh ; kz Þ ¼ exp a k2h þ k2z þ 2 2 3 : kh kz
ð104Þ
In order to study the variation of the coefficient l4 with the initial deformation, we need the value of material constant a. For the static case, the present model was compared by Demiray [23] with the experimental measurements by Simon et al. [24] on canine abdominal artery with the characteristics Ri ¼ 0:31 cm, R0 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. The numerical analysis of the coefficient v as a function of kh , kz and the wave number k indicates that the coefficient v is always positive. This means that for descending tube (A < 0) the coefficient l4 is positive and the wave speed given in (99) increases with the distance parameter s. On the other hand for ascending tube (A > 0) the coefficient l4 is negative and the wave speed in (99) decreases with distance from the origin. Such a result is to be expected from physical considerations.
1918
H. Demiray / International Journal of Engineering Science 40 (2002) 1897–1918
Acknowledgement This work was supported by the Turkish Academy of Sciences.
References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24]
S.C. Ling, H.B. Atabek, A nonlinear analysis of pulsatile blood flow in arteries, J. Fluid Mech. 55 (1972) 492–511. T.J. Pedley, Fluid Mechanics of Large Blood Vessels, Cambridge University Press, Cambridge, 1980. Y.C. Fung, Biodynamics: Circulation, Springer-Verlag, New York, 1981. 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. A.J. Rachev, Effects of transmural pressure and muscular activity on pulse waves in arteries, J. Biomech. Engng., ASME 102 (1980) 119–123. H. Demiray, Wave propagation through a viscous fluid contained in a prestressed thin elastic tube, Int. J. Eng. Sci. 30 (1992) 1607–1620. G. Rudinger, Shock waves in a mathematical model of aorta, J. Appl. Mech. 37 (1970) 34–37. 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. R.J. Tait, T.B. Moodie, Waves in nonlinear fluid filled tubes, Waves Motion 6 (1984) 197–203. T.B. Moodie, J.B. Haddow, Int. J. Non-linear Mech. 12 (1977) 223–231. T.B. Moodie, J.B. Haddow, J. Acoustic. Soc. Am. 67 (1980) 446–452. T.B. Moodie, G.E. Swaters, Quart. Appl. Math. 47 (1989) 705–722. R.S. Johnson, J. Fluid Mech. 42 (1970) 49–60. Y. Hazhizume, J. Phys. Soc. Jpn. 54 (1985) 3305–3312. Y. Yomosa, J. Phys. Soc. Jpn. 56 (1987) 506–520. H.A. Erbay, S. Erbay, S. Dost, Acta Mech. 95 (1992) 87–102. H. Demiray, Bull. Math. Biol. 58 (1996) 939–955. A.E. Ravindran, P. Prasad, Acta Mech. 31 (1979) 253–280. H.A. Erbay, S. Erbay, Acta Mech. 104 (1994) 201–214. H. Demiray, Int. J. Nonlinear Mech. 36 (2001) 649–661. H. Demiray, Applied Mathematics and Computation, in press. H. Demiray, J. Biomech. 5 (1972) 309–311. H. Demiray, Bull. Math. Biol. 38 (1976) 701–711. B.R. Simon, A.S. Kobayashi, D.E. Stradness, C.A. Wiederhielm, Circulation Res. 30 (1972) 491–500.