Variable coefficient modified KdV equation in fluid-filled elastic tubes with stenosis: Solitary waves

Chaos, Solitons and Fractals 42 (2009) 358–364

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Variable coefficient modified KdV equation in fluid-filled elastic tubes with stenosis: Solitary waves Hilmi Demiray Department of Mathematics, Faculty of Arts and Sciences, Isik University, 34980 Sile-Istanbul, Turkey

a r t i c l e

i n f o

Article history: Accepted 29 December 2008

a b s t r a c t In the present work, treating the arteries as a thin walled prestressed elastic tube with variable radius, and using the longwave approximation, we have studied the propagation of weakly nonlinear waves in such a fluid-filled elastic tube, by employing the reductive perturbation method. By considering the blood as an incompressible non-viscous fluid, the evolution equation is obtained as variable coefficients Korteweg–de Vries equation. Noticing that for a set of initial deformations, the coefficient characterizing the nonlinearity vanish, by re-scaling the stretched coordinates we obtained the variable coefficient modified KdV equation. Progressive wave solution is sought for this evolution equation and it is found that the speed of the wave is variable along the tube axis. Ó 2009 Elsevier Ltd. All rights reserved.

1. Introduction Due to its applications in arterial mechanics, the propagation of pressure pulses in fluid-filled distensible tubes has been studied by several researchers (Pedley [1] and Fung [2]). Most of the works on wave propagation in compliant tubes have considered small amplitude waves ignoring the nonlinear effects and focused on the dispersive character of waves (see Atabek and Lew [3], Rachev [4] and Demiray [5]). However, when the nonlinear terms arising from the constitutive equations and the 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 [6], Ling and Atabek [7], Anliker et al. [8] and Tait and Moodie [9] by using the method of characteristics, in studying the shock formation. On the other hand, the propagation of small-but-finite amplitude waves in distensible tubes has been investigated by Johnson [10], Hashizume [11], Yomosa [12] and Demiray [13,14] by employing various asymptotic methods. In all these works [10–14], depending on the balance between the nonlinearity, dispersion and dissipation, the Korteweg–de Vries (KdV), Burgers’ or KdV–Burgers’ equations are obtained as the evolution equations. In obtaining such evolution equations, they treated the arteries as circularly cylindrical long thin tubes with a constant cross-section. In essence, the arteries have variable radius along the axis of the tube. In the present work, treating the arteries as a thin walled prestressed elastic tube with variable radius, and using the longwave approximation, we have studied the propagation of weakly nonlinear waves in such a fluid-filled elastic tube, by employing the reductive perturbation method. By considering the blood as an incompressible non-viscous fluid, the evolution equation is obtained as variable coefficients Korteweg–de Vries equation. Noticing that for a set of initial deformations, the coefficient characterizing the nonlinearity vanish, by re-scaling the stretched coordinates we obtained the variable coefficient modified KdV equation. Progressive wave solution is sought for this evolution equation and it is found that the speed of the wave is variable along the tube axis.

E-mail address: [email protected] 0960-0779/$ - see front matter Ó 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.chaos.2008.12.014

H. Demiray / Chaos, Solitons and Fractals 42 (2009) 358–364

359

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 elastic tube, with an axially symmetric stenosis, and filled with a non-viscous fluid. For that purpose, we consider a circularly cylindrical tube of radius R0 . It is assumed that such a tube is subjected to an axial stretch kz and the static inner pressure P 0 ðZ  Þ. Under the effect of such a variable pressure the position vector of a generic point on the tube is assumed to be described by

z ¼ kz Z  ;

r0 ¼ ½r 0  f  ðz Þer þ z ez ;

ð1Þ

where er ; eh and ez are the unit base vectors in the cylindrical polar coordinates, r 0 is the deformed radius at the origin of the coordinate system, Z  is axial coordinate before the deformation, z is the axial coordinate after static deformation and f  ðz Þ is a function characterizing the axially symmetric stenosis on the surface of the arterial wall and will be specified later. Upon this initial static deformation, we shall superimpose a dynamical radial displacement u ðz ; t Þ, where t  is the time parameter, but, in view of the external tethering, the axial displacement is assumed to be negligible. Then, the position vector r of a generic point on the tube may be described by

r ¼ ½r 0  f  ðz Þ þ u er þ z ez :

ð2Þ

The arc-lengths along the meridional and circumferential curves are, respectively, given by

"

 2 #1=2 @u  0 dsz ¼ 1 þ f þ  dz ; @z

dsh ¼ ðr 0  f  þ u Þdh:

ð3Þ

Then, the stretch ratios along the meridional and circumferential curves are given by 0

k1 ¼ kz ½1 þ ðf  þ @u =@z Þ2 1=2 ;

k2 ¼

1 ðr0  f  þ u Þ: R0

ð4Þ

where the prime denotes the differentiation of the corresponding quantity with respect to z . The unit tangent vector t along the deformed meridional curve and the unit exterior normal vector n to the deformed tube are given by 0

t¼

0

ðf  þ @u =@z Þer þ ez ½1 þ ðf 0 þ @u =@z Þ2 

n¼

; 1=2

er  ðf  þ @u =@z Þez ½1 þ ðf 0 þ @u =@z Þ2 1=2

:

ð5Þ

Let T 1 and T 2 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 can be given by

" ( )  2 #1=2 0 @u @ ðr0  f  þ u Þðf  þ @u =@z Þ HR0 @ 2 u 0 þ P ðr 0  f  þ u Þ ¼ q0 T 2 1 þ f  þ  þ  T ; 1 2 1=2 0    @z @z kz @t2 ½1 þ ðf þ @u =@z Þ 

ð6Þ

where q0 is the mass density of the tube, H is the thickness of the tube in the undeformed configuration and P is inner pressure applied by the fluid. Let lR be the strain energy density function of the membrane, where l is the shear modulus of the tube material. Then, the membrane forces may be expressed in terms of the stretch ratios as

T1 ¼

lH @ R k2 @k1

T2 ¼

;

lH @ R k1 @k2

ð7Þ

:

Introducing (7) into Eq. (6), the equation of motion of the tube in the radial direction takes the following form



l @R kz @k2

þ lR0

@ @z

( ½1 þ

0

ðf  þ @u =@z Þ ðf 0

þ

@u =@z Þ2 1=2

@R @k1

) þ

P R0 @ 2 u : ðr 0  f  þ u Þ ¼ q0 H kz @t2

ð8Þ

2.2. Equations of fluid In general, blood is known to be an incompressible non-Newtonian fluid. The main factor for blood to behave like a nonNewtonian fluid is the level of cell concentration (hematocrit ratio) and the deformability of red blood cells. In the course of blood flow in arteries, the red cells move to the central region of the artery and, thus, the hematocrit ratio is reduced near the arterial wall, where the shear rate is quite high, as can be seen from Poiseuille flow. Experimental studies indicate that when the hematocrit ratio is low and the shear rate is high, blood behaves like a Newtonian fluid (see Ref. [2]). Moreover, as pointed out by Rudinger [6], for flows in large blood vessels, the viscosity of blood may be neglected as a first approximation. For this case, the variation of the field quantities with the radial coordinate may be neglected. Thus, the averaged equations of motion of an incompressible fluid may be given by

360

H. Demiray / Chaos, Solitons and Fractals 42 (2009) 358–364

@A @ þ ðA w Þ ¼ 0; @t  @z

ð9Þ

@w @w 1 @P þ w  þ ¼ 0; @t  @z qf @z

ð10Þ

where A is the cross-sectional area of the tube, w is the averaged axial fluid velocity and P  is the averaged fluid pressure. Noting the relation between the cross-sectional area and the final radius, i.e. A ¼ pðr 0  f  þ u Þ2 ; the Eq. (9) reads

2

 @u @w  @u þ ðr 0  f  þ u Þ  ¼ 0:  þ 2w  @t @z @z

ð11Þ

At this stage it is convenient to introduce the following non-dimensional quantities

t ¼

  R0 t; c0

z ¼ R0 z;

u ¼ R0 u;

m¼

q0 H lH ; w ¼ c0 w; f  ¼ R0 f ; r 0 ¼ R0 kh ; P  ¼ qf c20 p; c20 ¼ : q f R0 q f R0

ð12Þ

Introducing (12) into Eqs. (8)–(10) we obtain the following non-dimensional equations:

2

  @u @u @w þ 2 f 0 þ w þ ðkh  f þ uÞ ¼ 0; @t @z @z

ð13Þ

@w @w @p þw þ ¼ 0; @t @z @z ( " #)  1 @2u @R @ @u k2z @ R 0 : p¼ m 2þ  f þ kz k2 @z k1 @k1 @k2 @z @t

ð14Þ ð15Þ

 ¼ u  f and its temporal and spatial derivaFor our future purposes we need the power series expansion of p in terms of u  , up to and including the cubic terms, we obtain tives. If the expression (15) is expanded into a power series in u

 Þ þ L 2 ðu  Þ þ L 3 ðu Þ þ . . . ; p ¼ p0 þ L1 ðu

ð16Þ

 Þ; L2 ðu  Þ and L3 ðu  Þ are defined by where p0 is the constant ambient pressure, L1 ðu

  m @2u @2u ; p0 ¼ b0 ;  a0 2 þ b1 u 2 @z kh kz @t  2     m @2u @u a0  @ 2 u Þ ¼  2 u  2  a1 2 ;  2a1  L2 ðu u 2 þ b2 u @z kh @z kh kz @t !          m 2 @2u a1  @ u 2 2a1 a0 2 @ 2 u a0  @ u 2 @ 2 u   3 ;  L3 ðuÞ ¼ 3 u  a2   a2  þ 2 u  3 c1  þ b3 u u 2 2 @z @z @z2 kh kh 2 @z kh kh kz @t Þ ¼ L1 ðu

ð17Þ ð18Þ

ð19Þ

where the coefficients a0 ; a1 ; . . . ; b0 ; . . . ; b3 and c1 are defined by

1 @2R ; 2kh @kh kz

a0 ¼

1 @R ; kh @kz

b2 ¼

1 @ 3 R b1  ; 2kh kz @k2h kh

a1 ¼

b3 ¼

a2 ¼

1 @3R ; 2kh @k2h @kz

1 @ 4 R b2  ; 6kh kz @k4h kh

c1 ¼

b0 ¼

1 @R ; kh kz @kh

kz @ 2 R : 2kh @k2z

b1 ¼

1 @ 2 R b0  ; kh kz @k2h kh ð20Þ

Eqs. (13), (14) and (17)–(19) give sufficient relations to determine the field variables u; w and p completely. In our subsequent analysis we need the explicit expressions of the coefficients a0 ; b1 ; . . . ; b3 . For that purpose we shall utilize the constitutive equation proposed by Demiray [17] for soft biological tissues as

( " # ) 1 1 2 2 R¼ exp aðkh þ kz þ 2 2  3  1 ; 2a kh kz

ð21Þ

where a is a material constant. Introducing (21) into (20), the explicit expressions of the coefficients a0 ; a1 ; b1 ; b2 ; b3 may be given as follows

! 1 1 a0 ¼ kz  2 3 F; kh kh kz 2 !2 3 4 a 1 4 b1 ¼ 5 3 þ 2 kh  3 2 5 F; kh kz kh kz kh kz

H. Demiray / Chaos, Solitons and Fractals 42 (2009) 358–364

2

!3 3 b2 ¼ 4 6 3 þ ðkh  3 2 Þ 1 þ 4 2 þ 2 kh  3 2 5 F; kh kz kh kz kh kz kh kz kh kz kh kz 2 ! ! !2 !4 3 20 36 20 1 7 1 4 a3 1 2 4 5F; b3 ¼ 7 3 þ a 9 5  5 3 þ 2a kh  3 2 þ þ kh  3 2 kh kz k5h k3z 3 kh kz kh kz kh kz kh kz kh kz kh kz 

a

10

1

11

!

a2

361

1

ð22Þ

where the function F is defined by

" F ¼ exp a k2h þ k2z þ

!#

1

3 k2h k2z

ð23Þ

:

3. Longwave approximation In this section, we shall examine the propagation of small-but-finite amplitude waves in a fluid-filled thin elastic tube with stenosis, whose dimensionless governing equations are given in (13), (14), and (17)–(19). For this, we adopt the longwave approximation and employ the reductive perturbation method [15,16]. The nature of the problem suggests us to consider it as a boundary value problem. For such problems, the frequency is specified and the wave number is calculated through the use of dispersion relation. Thus, it is convenient to introduce the following type of stretched coordinates

n ¼ 1=2 ðz  gtÞ;

s ¼ 3=2 z;

ð24Þ

where  is a small parameter measuring the weakness of nonlinearity and dispersion, g is the scale parameter to be determined from the solution. Solving z in terms of s we get z ¼ 3=2 s. Introducing this expression of z into the expression of the function f ðzÞ we have f ðzÞ ¼ f ð3=2 sÞ. We shall assume that the function f ðzÞ has the following form

f ðzÞ ¼ hðsÞ:

ð25Þ

We shall further assume that the field variables u; w and p may be expressed as asymptotic series of the form

u ¼ u1 ðn; sÞ þ 2 u2 ðn; sÞ þ . . . w ¼ w1 ðn; sÞ þ 2 w2 ðn; sÞ þ . . .

ð26Þ

p ¼ p0 þ p1 ðn; sÞ þ 2 p2 ðn; sÞ þ . . . Noting the differential relations

@ @ ! 1=2 g ; @t @n

@ @ @ ! 1=2 þ 3=2 @z @n @s

ð27Þ

and introducing the expansion (26) into Eqs. (13), (14), and (17)–(19) the following sets of differential equations are obtained: O ðÞ order equations :

@u1 @w1 þ kh ¼ 0; @n @n

@w1 @p1 þ ¼ 0; @n @n

p1 ¼ b1 ½u1  hðsÞ:

ð28Þ

@u2 @w2 @w1 @u1 @w1 @w1 þ 2w1 þ kh þ kh þ u1  hðsÞ ¼ 0; @n @n @s @n @n @n @w2 @p2 @p1 @w1 þ w1 þ þ ¼ 0; g @n @n @s @n  2  2 mg @ u1  a0 þ b1 u2 þ b2 ðu1  hÞ2 : p2 ¼ kh kz @n2

ð29Þ

2g

g

O ð2 Þ order equations :

 2g

3.1. Solution of the field equations From the solution of the set (28), we obtain

u1 ¼ Uðn; sÞ;

w1 ¼

2g  1 ðsÞ; ½U þ w kh

p1 ¼

2g 2 ½U  hðsÞ; kh

ð30Þ

provided that g satisfies the following relation:

g2 ¼

kh b 1 ; 2

ð31Þ

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H. Demiray / Chaos, Solitons and Fractals 42 (2009) 358–364

 1 ðsÞ corresponds to the steady where Uðn; sÞ is an unknown function whose governing equation will be obtained later and w flow of order of . Introducing the solution given in (30) into Eq. (29), we have

  1 @u2 @w2 @U dw 4g @U 2g @U 2g @U  1Þ þ ðU þ w U hðsÞ þ  ¼0 þ kh þ 2g þ @s kh @n kh @n kh @n @n @n ds   @w2 @p2 2g 2 @U dh 4g 2 @U  1 ÞÞ þ 2 ðU þ w  ¼ 0; þ þ g @n @n @n kh @ s ds kh  2  2 mg @ U  a0 þ b2 ðU  hÞ2 þ b1 u2 : p2 ¼ kh kz @n2  2g

ð32Þ

Eliminating w2 between Eq. (32) we have



2g 2 @u2 @p2 4g 2 @U 10g 2 @U 2g 2 @U 2g 2 d  1  hÞ  1  hÞ ¼ 0: þ 2 U ðw þ 2 ð4w þ þ þ @n @n kh @n @n kh @ s kh ds kh kh

ð33Þ

Substituting the expression of p2 into Eq. (33) we obtain

! !   3 4g 2 @U 10g 2 @U @U 8g 2 2g 2 mg 2 @ U 2g 2 d  1  hÞ ¼ 0:  U þ þ 2b  h  2b h þ  a þ ðw þ w 1 0 2 2 2 2 2 @n @n kh kh @ s kh kz kh ds kh kh @n3

ð34Þ

The Eq. (34) should even be valid when U ¼ 0, which corresponds to steady flow. In order to satisfy the Eq. (34) for this spe 1 ðsÞ ¼ hðsÞ. Thus, the Eq. (33) reduces to the following Korteweg–de Vries equation with variable cial case we must have w coefficient

@U @U @3U @U þ l1 U þ l2 3 þ l3 hðsÞ ¼ 0; @s @n @n @n where the coefficients

l1 ¼

ð35Þ

l1 ; l2 and l3 are defined by

5 b þ 2; 2kh b1



 m a0 ;  4kz 2b1

l2 ¼

l3 ¼

3 b  2: 2kh b1

ð36Þ

This evolution equation has the following type of progressive wave solution (Demiray [18]) 2

U ¼ a sech f;



l1 a 12l2

f¼

1=2   Z s la n  l3 hðsÞds  1 s ; 3 0

ð37Þ

where a is the wave amplitude. The detail of this solution is given in Ref. [18]. As can be seen from the expression of the phase function (37)1, the trajectory of the wave is not a straight line anymore it is rather a curve in the (n; s) plane. The equation of the trajectory may be given by

n ¼ n0 þ

l1 as 3

þ l3

Z s

hðsÞds;

ð38Þ

0

where n0 is a constant. Considering that s is a space and n is the temporal variables, the speed of the propagation is given by

vp ¼

hl a i1=2 1 : þ l3 hðsÞ 3

ð39Þ

As can be seen from the expression (39) the wave speed is also variable along the tube axis. In general, the coefficient li ði ¼ 1; 2; 3Þ depend on the initial deformation kh and kz . For some set of values of ðkh ; kz Þ it may happen that l1 becomes zero. In this case, due to imbalance between the nonlinearity and dispersion, the variable coefficient KdV equation degenerates into the linear equation. Setting the coefficient l1 equal to zero, from (36) we obtain

kh 

!"

1 k3h k2z

3þ

3 k4h k2z

þ aðkh 

1 k3h k2z

#

Þ

2

¼ 0:

ð40Þ

Since the term in the bracket is positive, in order to satisfy the Eq. (40) we must have

kh 

1 k3h k2z

¼ 0;

or kh ¼ kz1=2 :

ð41Þ

The case of vanishing l1 . If the initial deformation satisfies the relation given in (41), in order to balance the nonlinearity and dispersion, it might be convenient to introduce the following coordinate stretching

n ¼ ðz  gtÞ;

s ¼ 3 z:

ð42Þ

H. Demiray / Chaos, Solitons and Fractals 42 (2009) 358–364

363

For this case, the function f ðzÞ will be of the form f ðzÞ ¼ 2 hðsÞ. Introducing the expansions (26) and (42) into the field Eqs. (13),(14), and (17)–(19), the following sets of differential equations are obtained: O ðÞ order equations :

2g

@u1 @w1 þ kh ¼ 0; @n @n

g

@w1 @p1 þ ¼ 0; @n @n

p1 ¼ b1 u1 :

ð43Þ

O ð2 Þ order equations :

@u2 @w2 @u1 @w1 þ kh þ 2w1 þ u1 ¼ 0; @n @n @n @n @w2 @p2 @w1 þ þ w1 ¼0 g @n @n @n 2 p2 ¼ b1 ðu2  hÞ þ b2 u1 :  2g

ð44Þ

O ð3 Þ order equations :

@u3 @w3 @u1 @u2 @w1 @w2 @w1 þ u1 þ kh þ2 w2 þ 2 w1 þ kh þ ðu2  hÞ ¼ 0; @n @n @n @n @s @n @n @w3 @p3 @p1 @ þ ðw1 w2 Þ ¼ 0; g þ þ @n @n @ s @n mg 2 @ 2 u1 p3 ¼ ð  a0 Þ 2 þ b1 u3 þ 2b2 u1 ðu2  hÞ þ b3 u31 : kh kz @n  2g

ð45Þ

For our future purposes, we need the solution of the field equations for the steady flow. The governing equations of the steady flow may be obtained from Eqs. (43)–(45) by simply setting u1 ¼ u2 ¼ u3 ¼ :::: ¼ 0. Then the solution of the resulting equations yields

 2 ð sÞ ¼ 2 w

g hðsÞ: kh

ð46Þ

3.2. Solution of the field equations From the solution of the set (43), we have

u1 ¼ Uðn; sÞ; p1 ¼

2g 2 U; kh

2g Uðn; sÞ; kh kh b1 g2 ¼ : 2 w1 ¼

ð47Þ

where Uðn; sÞ is an unknown function whose governing equation will be obtained later. Introducing the solution given in (47) into Eq. (44), we have

2

g 2 @u2 @w2 g 2 @U ¼ 0; þg þ6 2U kh @n @n kh @n

g

@w2 @u2 g 2 @U ¼ 0: þ b1 6 2U @n @n kh @n

ð48Þ

From the solution of the set (48), we obtain

w2 ¼ 2

g g ½u2 þ hðsÞ  3 2 U 2 : kh kh

ð49Þ

Finally, introducing the solutions (47) and (49) into Eq. (45) we obtain



2g 2 @u3 @w3 2g 2 @U g2 @ g2 @U g2 @U @w3 @u3 þ6 2 ðu2 UÞ  12 3 U 2 þ 2 2 hðsÞ ¼ 0; g þg þ þ b1 @n @n @n kh @n @n kh @ s @n @n kh kh kh !  2  3 2 2 2 2 mg @ U g @U g @ g @U g @U þ  a0 þ2 6 2 ðu2 UÞ þ 3b3  18 3 U 2 þ 14 2 hðsÞ ¼ 0: @n @n kh kz kh @ s kh @n kh kh @n3

ð50Þ

Eliminating u3 and w3 between these equations, the following evolution equation is obtained:

@U @U @3U 4 @U þ l4 U 2 þ l2 3 þ hðsÞ ¼ 0; @s @n kh @n @n where the coefficient

l4 is defined by

ð51Þ

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H. Demiray / Chaos, Solitons and Fractals 42 (2009) 358–364

l4 ¼

! 3 b3 5  2 : 2 b1 kh

ð52Þ

This evolution equation, which is known as the modified Korteweg–deVries equation with variable coefficients, assumes the travelling wave solution of the form

U ¼ a sech f;

 f¼

l4 6l2

 1=2  Z l a2 4 s a n 4 s hðsÞds ; kh 0 6

ð53Þ

where a is the wave amplitude. Here, again, it is seen that the trajectory is a curve in the (n; s) plane and the wave is variable along the tube axis. The expressions of them are given by

n ¼ n0 þ

vp ¼

l4 a2



6

l4 a 2 6

þ

sþ

4 kh

4 hðsÞ kh

Z s

hðsÞds;

ð54Þ

0

1=2 :

ð55Þ

4. Results and discussions In the present work, employing the nonlinear equations of a prestressed thin elastic tube with a stenosis and the averaged equations of an inviscid fluid, we have studied the propagation of weakly nonlinear waves in such a medium through the use of reductive perturbation method. The variable KdV equation and the modified variable KdV equation are obtained depending on the initial deformations. The progressive wave solution is sought for these evolution equations. It is observed that the rays are not straight lines anymore, they are rather some curves in the ðn; sÞ plane. It is further observed that the wave speeds are variable in the distant parameter s. Both of these observed phenomenon are resulted from the stenosis in the arterial wall. Acknowledgment 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]

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