Analyzing-the-harmonic-wave-propagation-in-the-la_2011_applied-mathematical-.pdf

Applied Mathematical Modelling 35 (2011) 4091–4102

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Analyzing the harmonic wave propagation in the layered thick tubes Hakan Erol ⇑, Hasan Gönen 1, Hasan Selim Sß engel 2 Department of Civil Engineering, Eskisßehir Osmangazi University, Eskisßehir, Turkey

a r t i c l e

i n f o

Article history: Received 18 May 2010 Received in revised form 9 February 2011 Accepted 17 February 2011 Available online 26 February 2011 Keywords: Arteries Constitutive equations Dispersion equation

a b s t r a c t In this study, the propagation of time harmonic waves in prestressed, anisotropic elastic tubes filled with viscous fluid is studied. The fluid is assumed to be incompressible and Newtonian. A two layered hyperelastic anisotropic structural model is used for the compliant arterial wall. The tube is subjected to a static inner pressure Pi and an axial stretch k. The governing differential equations of tube are obtained in cylindrical coordinates, utilizing the theory of ‘‘Superposing small deformations on large initial static deformations’’. The analytical solutions of the equations of motion for the fluid have been obtained. Due to variability of the coefficients of the resulting equations for the solid body they are solved numerically. The dispersion relation is obtained as a function of the stretch and material parameters. Ó 2011 Elsevier Inc. All rights reserved.

1. Introduction The theoretical analysis of propagation of pressure waves in an inflated tube has attracted the attention of many researchers since the time of Thomas Young who first determined the speed of pulsatile waves in arteries. As a mathematical theory of blood flow in arteries various models have been adopted by Morgan and Kiely [1], Womersley [2], Mirsky [3], Whirlow and Rouleau [4], Lawton [5], Fenn [6], Klip [7] and they have made significant contributions in this area. The initial deformation of an arterial wall was first considered by Atabek and Lew [8]. They have analyzed the effect of the circumferential and longitudinal initial stresses on wave propagation. Rachev [9] considered the effects of the initial stress, but assumed the artery to be a membrane. Cox [10] and Kizilova [11] took into account the thick-walled viscoelastic tube. They analyzed governing equations of propagation of wave through a Newtonian fluid contained within a thick walled viscoelastic tube. Demiray and Antar [12] sought the effects on the wave propagation of the thickness of the elastic tube. They used a truncated power series in terms of the thickness ratio to obtain the solution of the governing equations. In this study, wave propagation of a prestressed, a two layer hyperelastic anisotropic tube filled with viscous fluid is studied. The arterial wall is composed of layered collagen and assumed to be incompressible. Artery is taken to be thick because of thickness ratio changes between l/6 and l/4. The arteries in a body are inflated with a mean pressure of approximately 13 kPa. Considering this physiological condition, in the analysis the tube is assumed to have a uniform inner pressure Pi. Also, arteries in a body are under the axial stretch k. It is further assumed that the amplitude of the pressure disturbance is sufficiently small so that nonlinear terms in the inertia of the fluid are negligible compared with linear ones. Utilizing the theory of small deformations superimposed on initial static deformations, for a symmetrical perturbed motion, the governing differential equations are obtained in cylindrical polar coordinates. The governing equations of the fluid can be solved by analytical. Thus, closed form solutions can be ob⇑ Corresponding author. Tel.: +90 222 2393750/3228; fax: +90 222 2393613. 1 2

E-mail addresses: [email protected] (H. Erol), [email protected] (H. Gönen), [email protected] (H.S. S ß engel). Tel.: +90 222 2393750/3220; fax: +90 222 2393613. Tel.: +90 222 2393750/3232; fax: +90 222 2393613.

0307-904X/$ - see front matter Ó 2011 Elsevier Inc. All rights reserved. doi:10.1016/j.apm.2011.02.037

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H. Erol et al. / Applied Mathematical Modelling 35 (2011) 4091–4102

tained. But, due to the variable coefficients of the differential equations for the tube wall a closed form solution is not possible. Therefore, the wall solution is obtained numerically by the finite-difference method. The dispersion relation is written by a function of the inner pressure, the axial stretch and parameters of the tube wall. 2. Governing equations and boundary conditions The phenomenon which is investigated here is caused by the interactions of the fluid with the tube. Therefore, a mathematical study of the problem should include expressions with regard to the motion of the fluid, motion of the wall, and condition on their interface. 2.1. Equations of the fluid To express the problem, blood will be considered as an incompressible Newtonian fluid subjected to an initial static pressure Pi. In the course of flow small velocity and pressure increments created by the heart are added on this initial field. Assuming flow is axially symmetric, in the absence of initial velocity and body forces, the governing differential equations in the cylindrical polar coordinates are given by

! ^ 1 @u ^ @2u ^ ^ ^ ^ u @u @p @2u ^ þ þ ¼ þl  ; @r 2 r @r r2 @z2 @t @r ! ^ ^ ^ @2w ^ 1 @w ^ @w @p @2w ^ ; ¼ þl þ q^ þ @t @z r @r @r2 @z2

q^

ð1Þ ð2Þ

^ Þ are the velocity components of fluid, along radial and axial directions, respectively, p ^; x ^ is the pressure, l is the where ðu ^ is the density of the fluid. The equation of continuity of an incompressible fluid is viscosity, q

^ @w ^ u ^ @u þ þ ¼ 0: @r r @z

ð3Þ

The stress components that will be used in boundary conditions are given as

^ @u ^trr ¼ p ^ þ 2l ^ ; @r

  ^ @w ^ @u ^t rz ¼ l ^ : þ @z @r

ð4Þ

2.2. Equations of the elastic tube The tube material is assumed to be incompressible, elastic and a composite reinforced with two layered collagen fibers. The strain energy function R of the elastic tube is used as proposed by Holzapfel et al. [13]

h i h i k3 k1 ðexp k2 ðI4  1Þ2 þ exp k2 ðI6  1Þ2  2Þ; ðI1  3Þ þ 2 2k2 x2 1 1 2 2 I1 ¼ 2 þ 2 þ k ; I4 ¼ I6 ¼ eI ¼ 2 cos2 b þ k2 sin b; x x k

R¼

ð5Þ

where k1, k2 and k3 are material constants, I1, I4 and I6 are invariants of the Green deformation tensor C, b is angle between two families of (collagen) fibers of the artery shown in Fig. 1. When a cylindrical tube is subjected to a static inner pressure Pi and the axial stretch k, an initial stress field t0kl will be developed in the body. The stress components and the inner pressure Pi as a function of the axial stretch k are given in cylindrical coordinates by Erol [14]:

  k f þ df; f xo h h ii 1 t 0hh ¼ P0 þ 2 k3 þ 4k1 cos2 bðeI  1Þ exp k2 ðeI  1Þ2 ; x h i 2 t 0zz ¼ P0 þ k2 ½k3 þ 4k1 sin bðeI  1Þ exp k2 ðeI  1Þ2 ;  Z xi  x2 k f þ df; t 0hz ¼ t 0rz ¼ t 0rh ¼ 0; P0 ¼ t0rr  k3 2 ; Pi ¼ k3 f k xo

t 0rr ¼ k3

Z

x

ð6Þ

x

R ; r

where subscripts (o) and (i) stand for the values of a quantity evaluated on the outer and inner surface of the elastic tube, and R and r are the radial coordinates of a materials point before and after deformation, respectively.

H. Erol et al. / Applied Mathematical Modelling 35 (2011) 4091–4102

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Fig. 1. Geometrical properties of arteries.

When a small incremental deformation is superimposed on this initial field, the governing equations can be taken as proposed by Eringen and Suhubi [15] 2

@ ul T kl;k þ qf l ¼ q 2 ; @t

ð7Þ

where q is the mass density of the elastic tube, T kl ; f l and ul are the incremental Piola–Kirchhoff stress tensor, body force and incremental displacement vectors, respectively. The incremental Piola–Kirchhoff stress tensor may be given as Eringen and Suhubi [15]

T kl ¼ t 0km ul;m þ ~t kl ;

ð8Þ

where ~tkl can taken as Erol [14]

" # @2R @2R ~tkl ¼ p dkl  2P0 ekl þ 4 d d þ d d emn ; 1kl 1mn 2kl 2mn ~ @I24 @I26

ð9Þ

 incremental hydrostatic pressure, ekl strain tensor, ~emn is the infinitesimal strain tensor, d1 and d2 are deformation where p gradient vectors of collagen fiber directions which are defined by d1kl = a1  a1dKLFkKFlL, d2kl = a2  a2dKLFkKFlL where a1 and a2 are introduced to the families of collagenous fibers of direction vectors, and described by a1 = [0, cosb, sinb]T, a2 = [0, cosb, sinb]T. The incompressibility assumption of the elastic tube imposes

uk;k ¼ 0:

ð10Þ

In order to be able to determine the incremental field completely Eqs. (11) and (12) are to be supplemented with the boundary conditions

T kl nk ¼ t l  ~eðnÞ t0l

on S;

eðnÞ ¼ eij ni nj

ð11Þ

where ni is the unit exterior normal vector of the surface S, t0l and t l are the initial and the incremental surface tractions, respectively. For the axially symmetric incremental motions of the prestressed cylindrical tube, the displacement field is set

u1 ¼ uðr; zÞ;

u2 ¼ 0;

u3 ¼ wðr; zÞ;

ð12Þ

where u and w are the incremental displacement components in the radial and axial directions, respectively. Introducing (12) into (9) the incremental stress components become

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H. Erol et al. / Applied Mathematical Modelling 35 (2011) 4091–4102

@u ~t rr ¼ p   2P0 ; @r " !# cos4 b @ 2 R @ 2 R u k2 2 ~t hh ¼ p  þ 2P0 þ 4 þ 4 2 sin b cos2 b þ 2 2 2 x r x @I4 @I6 ! " k2 @2R @2R u 4 ~t zz ¼ p  þ 4 sin2 b cos2 b þ 2P0 þ 4k4 sin b þ x2 @I24 @I26 r !  2 2 ~t hz ¼ 4 k cos b sin b @ R þ @ R 1 cos2 b u þ k2 sin2 b @w ; x x2 r @z @I24 @I26   ~t rz ¼ ~t zr ¼ P 0 @w þ @u ; ~t rh ¼ ~t hr ¼ 0: @r @z

! @ 2 R @w ; @I24 @I26 @z !# @2R @2R @w ; þ @z @I24 @I26 @2R

þ

ð13Þ

Thus, the components of the incremental Piola–Kirchhoff stress tensor (8) are written

    @u u @w u 2 1 þ a  þ t0rr  2P0 þ2 a þ t0hh ; ; T hh ¼ p T rr ¼ p @r r @z r   u @w @w 4 3 þ a þ2 a þ t0zz ; T rh ¼ T hr ¼ 0; T zz ¼ p r @z @z     @w @u @w @w @u @u T rz ¼ P0 þ þ t0rr ; T zr ¼ P0 þ þ t 0zz ; @r @z @r @r @z @z

ð14Þ

T hz ¼ T zh ¼ 0;  i ði ¼ 1; 2; . . . 4Þ and F(x) are defined as where the coefficients a

a 1 ¼ P0 þ 8

cos4 b k1 k2 ðeI  1Þ2 FðxÞ; x4

k2 2 sin b cos2 bk1 k2 ðeI  1Þ2 FðxÞ; x2 k2 a 3 ¼ 8 2 sin2 b cos2 bk1 k2 ðeI  1Þ2 FðxÞ; x a 4 ¼ P0 þ 8k4 sin4 bk1 k2 ðeI  1Þ2 FðxÞ; FðxÞ ¼ exp½k2 ðeI  1Þ2 :

a 2 ¼ 8

ð15Þ

If body force is vanished and introducing (14) and (6) into the governing differential Eq. (7) 2 2 2  @p 1 ðrÞ @ u þ 1 b 2 ðrÞ @u þ b 3 ðrÞ u þ b 4 ðrÞ @ u ¼ q @ u ; þb @r @r2 r @r r2 @z2 @t 2 2 2  @p @ w 1 @w @u @ w @2w 1 ðrÞ 5 ðrÞ 6 ðrÞ 7 ðrÞ þ ¼ q ; þb b þ b þ b @z @r2 r @r @z @z2 @t 2 @u u @w þ þ ¼ 0; @r r @z

ð16Þ

where the coefficients bi (i = 1, 2, . . . , 7) are defined by

  2 ¼ r d t 0  2P0 þ t 0  P0 þ 2a 2; b rr rr dr 3 ¼ 2ða 4 ¼ t0  P0 ; 2  a  1 Þ  P0  t 0hh ; b b zz

1 ¼ t0  P0 ; b rr 0

0

5 ¼ r dP þ r dt rr  P0 þ t 0 ; b rr dr dr 0 0  4 þ t zz þ P : b7 ¼ 2a

0

6 ¼ r dP þ 2a 3; b dr

ð17Þ

These differential field equations have to be supplemented with the boundary conditions on the inner ri and outer ro surfaces of the tube:

  @u  T rr jr¼ri ¼ ^t rr  t0rr ; @r r¼ri

T rz jr¼ri ¼ ^trz jr¼ri ;

  @u ^  ; jr¼ri ¼ u @t r¼ri   A M T rz  ¼ T rz  :

T rr jr¼ro ¼ T rz jr¼ro ¼ 0;   T Arr 

r¼r

  ¼ TM rr 

r¼r

;

r¼r

r¼r

 @w ^ r¼r ; ¼ wj i @t r¼ri

ð18Þ

H. Erol et al. / Applied Mathematical Modelling 35 (2011) 4091–4102

4095

3. Solution of the governing equations Considering the pulsatile motion of the heart, harmonic wave type of solution to the field equation (16) will be sought under the boundary conditions (18). For that purpose we set

b c ðrÞ; PðrÞÞ b ^ ; w; ^ p ^Þ ¼ ð UðrÞ; ðu W exp½iðxt  kzÞ;

ð19Þ

ðu; w; pÞ ¼ ðUðrÞ; WðrÞ; PðrÞÞ exp½iðxt  kzÞ;

ð20Þ

b where x is the angular frequency, k is the wave number and UðrÞ . . . PðrÞ are the complex amplitudes to be determined from the solution of the field equations with the boundary conditions. If (19) is substituted into (1, 2), the solution of the resulting differential equations for the fluid variables are obtained as

b UðrÞ ¼

k ik I1 ðkrÞA0 þ J 1 ðsrÞB0 ; 2 2 s ^ lðk þ s Þ

c ðrÞ ¼  W

k I1 ðkrÞA0 þ J 0 ðsrÞB0 ; 2 ^ lðk þ s2 Þ

ð21Þ

b PðrÞ ¼ I0 ðkrÞA0 : here 2

s2 ¼ k 

ix

m

;

l^ m¼^ q

ð22Þ

and Ji(sr), Ii(kr) are the Bessel and modified Bessel functions of the first kind of order i, respectively, A0 and B0 are two integration constants to be determined from boundary conditions. If, (21) is substituted into (4) the stress components of the fluid are obtained as

^t rr ¼

8h > < I0 ðkrÞ þ

2k ðI1 ðkrÞ rðk2 þs2 Þ

i 9 = þ krI0 ðkrÞÞ A0 >

exp½iðxt  kzÞ; > > : þ 2il^ k ðj ðsrÞ þ srJ ðsrÞÞB ; 0 1 0 sr ( ) 2 l^ 2 2 ^t rz ¼  2ik I1 ðkrÞA0  ðs  k ÞJ 1 ðsrÞB0 exp½iðxt  kzÞ: 2 s ðk þ s2 Þ

ð23Þ

In order to obtain the solution of the field equations of solid body, a harmonic type solution shown in (20) is sought again. For this purpose (20) is substituted into (16) ordinary differential equations for the elastic tube are given by

  2 dPA;M A;M d U A;M 1 A;M dU A;M A;M k2 þ qA;M x2 U A;M ¼ 0; A;M 1  b þ b2 þ b1 þ b 3 4 2 2 r r dr dr dr  A;M 1 A;M dW A;M ik A;M A;M  2 A;M d W A;M k2 W A;M ¼ 0;  ikPA;M þ b b b þ U þ qx  b  5 7 1 6 2 r r dr dr 2

ð24Þ

dU A;M U A;M þ  ikW A;M ¼ 0; dr r where superscripts (A,M) represent Adventitia and Media, respectively and for convenience, the following dimensionless quantities are defined,

U A;M ¼ r U A;M ; A;M ¼ kA;M bA;M b 3 i i

W A;M ¼ rW A;M ; ði ¼ 1; . . . 7Þ;

A;M

PA;M ¼ k3 PA;M ; r2 x2 qA;M g ¼ kr; X2 ¼ ; A;M k3

where r is the midradius of the tube after deformation.

r n¼ ; r

ð25Þ

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H. Erol et al. / Applied Mathematical Modelling 35 (2011) 4091–4102

If (23) is introduced into (18) the boundary conditions written as

"

M

M

P  "

2bM 1

#

 f   þ

M

" 0A

"

þ



bA1

P

0A

#

 dU A

A

bA1

g

n¼ni

dW þ igP0M U M dn

bM 1 P

dU dn

dW þ igP 0A U A dn

dn #

þ n¼ni

 2g 2ig f  1Þ Aþ ð1  ni gÞB ¼ 0; þ ðn i ni ðg2 þ c2 Þ ni c

2ig2 ðc2  g2 Þ Aþ B ¼ 0; 2 ðg þ c Þ c 2

#

¼ 0; n¼no

¼ 0; ð26Þ

n¼no

ga2 q qiga2 A B ¼ 0; ½iX U n¼ni  2 2 ðg þ c Þ c ia2 fq gqa2 ½iX2 W M n¼ni þ 2 A B ¼ 0; 2 ðg þ c Þ c 2

M

"

P

0A

" bA1

þ



bA1

P

0A

A

 dU A

dW þ igP 0A U A dn

dn #

#

"

 P

0M

n¼nm

"

 bM 1 n¼nm

  M 0M dU þ bM 1 P dn #

# ¼ 0; n¼nm

M

dW þ igP 0M U M dn

¼ 0; n¼nm

here

xr2 q^ q^ gI0 ðgnÞ ; q¼ ; f ¼ ; q I1 ðgnÞ l^ cJ ðcnÞ A0 I1 ðgnÞ lB0 J1 ðcnÞ ;B ¼ ; g¼ 0 ; A¼ r k3 J 1 ðcnÞ k3

a 2 ¼

r

r

ð27Þ

r

c ¼ rs; ni ¼ i ; nm ¼ m ; no ¼ o : r r r By defining a function /(n) as

/ðnÞ ¼

Z

n

ð28Þ

fWðfÞdf: 0

From the last equation of (24) the following can be obtained:

WðnÞ ¼

1 d/ðnÞ ; n dn

UðnÞ ¼

ig /ðnÞ: n

ð29Þ

By using this function, the first two equations of (24) can be written in the form: A;M 2  d/A;M ig   bA;M ig d /A;M ig  A;M dP 2 2 2 2 /A;M ¼ 0; þ 2 b2  2bA;M  bA;M þ bA;M  bA;M þ 1 þ 3 2bA;M 1 1 2 3 4 g n þX n 2 dn n dn dn n n 3  d2 /A;M 1   A;M bA;M g2 bA;M d /A;M 1  A;M 2 2 d/ 2 2 þ 2 b5  2bA;M þ 3 2bA;M  bA;M  bA;M þ 6 2 /A;M ¼ 0:  igPA;M þ 1 5 7 g n þX n 1 1 3 2 n dn dn dn n n n

ð30Þ On the other hand, boundary conditions may be expressed as

     ig f 2g 2ig PM þ 2bM  2 /M   þ f  1Þ Aþ ð1  ni gÞB ¼ 0; þ ðn i 1 2 þ c2 Þ g n ð g ni c n i n¼ni " ! # 2 1 d/M 1 d /M g2 P0M M 2ig2 ðc2  g2 Þ   þ 2 Aþ B ¼ 0; bM þ / 1 2 dn 2 2 n dn n ðg þ c Þ c n "

n¼ni

!#   ig2 ig d/A A 0A A A P þ b1  P / þ ¼ 0; n dn n2 n¼no " ! # 2 1 d/A 1 d /A g2 P0A A A  ¼ 0; þ / b1  2 n dn2 n n dn n¼no

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H. Erol et al. / Applied Mathematical Modelling 35 (2011) 4091–4102

X2

ga2 q qiga2 A B ¼ 0; 2 2 ðg þ c Þ ni c   1 d/ ia2 fq gqa2 þ 2 A  B ¼ 0; iX2 n dn ðg þ c2 Þ c n¼ni 

/

"

!#

ð31Þ "

!#

  ig2   ig2 ig d/A ig d/M 0M /A þ  P M þ bM /M þ ¼ 0; PA þ bA1  P0A 1 P 2 2 n dn n dn n n n¼nm n¼nm " ! # " ! # 2 2 1 d/A 1 d /A g2 P0A A 1 d/M 1 d /M g2 P0M M A M   b1  2  b1  2 ¼ 0: þ / þ / n dn2 n dn2 n n n dn n dn n¼nm

n¼nm

Because of the complex structure of the coefficient appearing in (30), it is almost impossible to give a closed form of analytical solutions to the field equations of the tube. Therefore, the numerical solution of the equations with the finite-difference  is divided into n equal intervals, thus method will be sought. To use the finite-difference method the thickness of the tube h some quantities are defined as

 ¼r r ¼r r ; h m i o m n0 ¼ ni ¼ 1 

nh ; 2

nj ¼ n0 þ jh ðj ¼ 0; 1; 2 . . . nÞ;  h nn ¼ no ; h ¼ : nr

ð32Þ

The value of function at the point nj = n0 + jh is characterized by a subscript j. Introducing appropriate finite-difference expressions for various derivatives, the following difference equations are obtained for the Eq. (30),

" # 2 ig A;M h 2 A;M A;M A;M 2 /jþ2 þ igzA;M / þ i g z þ ð X  b g Þ /A;M ¼ 0; 1j jþ1 2j 4j j nj nj " # " # 2 2   b1j A;M h  2 h  2 3 A;M A;M A;M A;M 2 A;M A;M A;M 2  igh PA;M / /A;M þ / þ z / þ z þ X  b g þ z  X  b g ¼0 j 3j jþ2 4j 7j jþ1 5j 7j j nj jþ3 nj nj A;M

A;M

hP jþ1  hP j

þ bA;M 1j

ð33Þ

and for the Eq. (31),

PM 0

þ

2bM 10

!   ig M ig ig f 2g 2ig M /1  2b10 2 þ ðn0 f  1Þ A þ þ ðn g  1ÞB ¼ 0; /M 0   2 2 cn0 0 n0 h g n0 ðg þ c Þ n0 hn0

" #   2 2 bM bM h 1 bM 2ig2 h h ðc2  g2 Þ M 2 2 0 10 M 10 10 h /2  þ 2 /1 þ þ b10  g h P0 /M A þ B ¼ 0; 0 þ n0 n0 n0 n0 n0 ðg2 þ c2 Þ c

ð34Þ

  ig   ig  1 1 /A ¼ 0; PAn þ bA1n  P 0A /Anþ1  bA1n  PAn þ n hnn nn nn h n !   bA1n A bA h 1 hbA1n 2 /nþ2  1n þ 2 /Anþ1 þ þ bA1n  g2 h P An /An ¼ 0; nn nn nn nn nn

X2 g n0

/M 0 þ

ga2 q qiga2 Aþ B ¼ 0; 2 2 ðg þ c Þ c

iX2 M ia2 fq g a2 q A B ¼ 0; /1  /M 0 þ 2 2 ðg þ c Þ n0 h c PA0 

! ! " # M M   ig P A0  bA10 PA0  bA10 A ig  A ig PM PM ig  M M M M 2  b12 2  b12 /0  / ¼ 0; þ P0  bA10 /A1  P M  þ  P  b / 2 12 3 2 n0 n0 h hn0 n2 n2 h hn2 2

! " ! #     2 2 P 0A bA10 bA10 bA10 1 2 A bA10 A P 0M bM bM bM 1 2 M bM A M M 12 12 12 12 0 g 2 g /  /  þ þ þ / þ /   þ þ þ / þ / ¼ 0;  n0 hn0 h2 n0 0 hn0 n0 h 1 h2 n0 2 n2 hn2 h2 n2 2 hn2 n2 h 2 h2 n2 3

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H. Erol et al. / Applied Mathematical Modelling 35 (2011) 4091–4102

where

zA;M 1j ¼  zA;M 2j ¼

2bA;M 1j

bA;M 1j nj

nj 

þ

 h  A;M b2j  2bA;M ; 1j 2 nj

 h2   h  A;M A;M A;M b2j  2bA;M þ 3 2bA;M ; 1j 1j  b2j þ b3j 2 nj nj

Table 1 Parameters of materials and geometry.

Media

Adventitia

Materials

Geometry

M k3 M k1 M k2

¼ 3:0000 ½kPa

HM = 0.26 [mm]

¼ 2:3632 ½kPa

bM = 29.0°

¼ 0:8393 [–] HA = 0.13 [mm]

A

k3 ¼ 0:3000 ½kPa A k1 A k2

bA = 62.0°

¼ 0:5620 ½kPa ¼ 0:7112 [–]

Ri = 0.71 [mm]

a

12

λ=1.3 λ=1.5 λ=1.7

11 10 9

v1

8 7 6 5 4 3 2 1 0

2

4

6

8

10

6

8

10

α

b

1

λ=1.3 λ=1.5 λ=1.7

χ1

0.75

0.5

0.25

0 0

2

4 α

Fig. 2. (a) The variations of the primary wave speed different values of k.

v1 with a . (b) The variations of the transmission coefficient of the primary wave v1 with a , for and

H. Erol et al. / Applied Mathematical Modelling 35 (2011) 4091–4102

3bA;M 1j

zA;M 3j ¼  zA;M 4j ¼

nj

3bA;M 1j nj

h n2j

 A;M bA;M ; 5j  2b1j

 h   2h  A;M A;M A;M A;M b  2b b  2b  ; 5j 1j 5j 1j n3j n2j 2



 h2   h3 g2 h  A;M A;M A;M A;M b  2b b  2b þ þ 2 bA;M 5j 1j 5j 1j 6j ; nj n3j nj n2j h   i1=2 k r2 n2j  r2i þ R2i ¼ bA;M ðxj Þ; xj ¼ i r nj

zA;M 5j ¼ 

bijA;M

þ

4099

bA;M 1j

þ

ð35Þ

and Ri is the radius of the tube before deformation. In order to have a non-zero solution for the coefficients PA;M ; /A;M ðj ¼ 0; 1; . . . ; nÞ; A and B, the determinant of the coeffij j cients matrix must be vanish. This gives the dispersion relation of the problem. 3.1. Long wave approximation and thin tube Even for large arteries the wavelength is very large as compared to mean radius of the artery Atabek and Lew [8]. Therefore, in this special case the wave number k is very small and g  1; f defined by (27) approaches to 2. Using the definitions ^ in (22) following equations are obtained c^ ¼ sr and a 2 ¼ q^ ixr2 =l

c^  a 2 :

ð36Þ

a

1.2

λ=1.3 λ=1.5 λ=1.7

v2

0.9

0.6

0.3

0 0

2

4

6

8

10

6

8

10

α

b 0.75

λ=1.3 λ=1.5 λ=1.7

χ2

0.5

0.25

0 0

Fig. 3. (a) The variations of the secondary wave speed different values of k.

2

4

α

v2 with a . (b) The variations of the transmission coefficient of the secondary wave v2 with a , for

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H. Erol et al. / Applied Mathematical Modelling 35 (2011) 4091–4102

In order to compare the present work with previous works, thin tube filled with a fluid of zero viscosity is taken. Therefore, ^ ! 1 the dispersion relation for this case may be obtained in term of the complex phase velocity c = X/g by setting n = 1, c and g ! 1 become as follows:

B1ðcÞ4 þ B2ðcÞ2 þ B3 ¼ 0;

ð37Þ

where B1, B2 and B3 coefficients are obtained from the system given by Eqs. (30) and (34), under the above assumptions. If it is assumed that there is no initial deformation, tube material is considered not to include fibers (c1 = 0), these coefficients can be obtained,

B1 ¼ 2hqi; 2

B2 ¼ 8hqi þ 4ih q2 ;

ð38Þ

2

B3 ¼ 12ih q2 : For thin tube the parameter h is quite, therefore, the roots (38) may be approximated as

c21 ¼ 4 þ OðhÞ;

c22 ¼

3qh 2 þ Oðh Þ: 2

ð39Þ

These roots are decomposed into real and imaginary parts

c ¼ X þ iY;

ð40Þ

a

2.6

γ1=1.0 γ1=0.5 γ1=0.0

2.4 2.2 2

v1

1.8 1.6 1.4 1.2 1 0.8 0

b

2

4

1

α

6

8

10

6

8

10

γ1=1.0 γ1=0.5 γ1=0.0

χ1

0.75

0.5

0.25 0

Fig. 4. (a) The variations of the primary wave speed and different values of c1.

2

4

α

v1 with a . (b) The variations of the transmission coefficient of the primary wave v1 with a, for k = 1.3

H. Erol et al. / Applied Mathematical Modelling 35 (2011) 4091–4102

a

0.75

4101

γ1=1.0 γ1=0.5 γ1=0.0

v2

0.5

0.25

0 0

b

2

4

0.6

α

6

8

10

6

8

10

γ1=1.0 γ1=0.5 γ1=0.0

χ2

0.4

0.2

0 0

Fig. 5. (a) The variations of the secondary wave speed k = 1.3 and different values of c1.

2

4

v2 with a. (b) The variations of the transmission coefficient of the secondary wave v2 with a, for

as proposed by Atabek and Lew [8], the speed of propagation

v¼

X2 þ Y 2 ; X



v ¼ exp 2p

α

v and transmission coefficients v are defined by



Y : X

ð41Þ

Using the real physical quantities and setting k3 = E/3, where E is the Young modulus of the tube, the speeds of the propagation may be expressed as

v 21 ¼

4E ; 3q

Eh

v1 ¼ 1; v 22 ¼ ^ ; v2 ¼ 1: 2qr

ð42Þ

These wave speeds correspond to the Lamb mode and Moens–Korteweg speed, respectively. 3.2. Numerical example For the numerical analysis of the system given by Eqs. (33) and (34), the geometrical and mechanical values shown in Table 1 are used as Holzapfel et al. [13] From the values is given Table 1, the system given by (33) and (34) provides fourteen algebraic equations for P A;M ; /kA;M ; A j and B. In order to have a non-zero solution for these coefficients, the determinant of the coefficients matrix must be vanish. This gives the dispersion relation in term of the complex phase velocity. Finally, the wave speeds and the transmission coefficients are obtained by using (41). These quantities depend on the influence of the parameters, stretch ratio k, Womersley A;M  , and fibers ratio of tube c1 ¼ kA;M parameter a 1 =k3 .  for different values of axial Figs. 2 and 3 represent variation of velocity of propagation and transmission of the waves vs. a stretch k. Fig. 2(a) shows that for all values of Womersley parameters primary wave speed v1 increases with the increases of

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H. Erol et al. / Applied Mathematical Modelling 35 (2011) 4091–4102

 < 3 with the axial stretch k. Fig. 2(b) reveals that the transmission coefficient v1 of the primary wave speed decreases for a  and increases of the k. In the Fig. 3(a) secondary wave speed v2 decreases for all values of a  with increases of the increase of a k. Fig. 3(b) displays that the transmission coefficient v2 of the secondary wave speed increases with the increases of k and the . increases a In Fig. 4 and the effect of the variation of velocity of propagation and transmission of the waves vs. fiber parameter c1 for axial stretch k = 1.3 are shown. c1 shows effect of the collagen fibers. c1 approaches zero, the effect of the fibers is reduced. When c1 is zero that means there are not fibers and tube behaves as rubber-like materials. c1 is same value, for both of layer of the tube. Fig. 4(a) shows that v1 increases with the increases of the c1. In the Fig. 4(b) v1 increases with the increase of the c1. From Fig. 5(a), it can be seen that, v2 increases with the increases of the c1. Fig. 5(b) reveals that v2 decreases with the increases of the c1. 4. Conclusion In this study, the propagation of time harmonic waves in, prestressed, anisotropic, combined of two layered elastic tubes filled with viscous fluid is studied. The effects of the axial stretch, fibers of the material on the wave speeds and transmission coefficients have been analyzed. The governing differential equations of the layered tube’s material are obtained also in the cylindrical coordinates utilizing the theory of small deformations superimposed on large initial static deformations. For the axially symmetric motion the field equations are solved by assuming harmonic wave solutions. The field equations are solved by assuming harmonic wave solutions in case of the axially symmetric motion. Dispersion relation is obtained by the finite-difference method and, the wave velocities and the transmission coefficients are computed. It is observed that axial stretch has strong effect on the all wave characteristics. The obtained results in the Figs. 2 and 3 are in agreement with the results of Demiray and Antar [12] and Demiray and Akgün [16]. In previous studies in which wave propagation is analyzed, arteries are not considered as fibered and layered. However, the obtained results for the fibered artery shown in Figs. 4 and 5, put the importance of the fibers. New studies should be realized with the fibered arteries. Additionally, by the help of experimental studies, the behavior of fibers should be modeled more effectively. Special cases of the long wave and thin tube obtained after some simplifications are made in field equations in the most general form, agree with the results given in literature. Wave velocities and transmission coefficients in Eq. (42) are the same of that of Demiray and Akgün [16]. This may be accepted as the verification of presented study. The results obtained in this study yield that in order to offer a more realistic approach for the problem of wave propagation; arteries material should be taken as fibered and layered. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16]

G.W. Morgan, J.P. Kiely, Wave propagation in a viscous liquid contained in a flexible tube, J. Acoust. Soc. Am. 26 (1954) 323–328. J.R. Womersley, An elastic tube theory of pulse transmission and oscillatory flow in mammalian arteries, WADC, Technical Report, TR., 56-614, 1957. I. Mirsky, Wave propagation in a viscous fluid contained in an orthotropic elastic tube, Biophys. J. 7 (1967) 165–186. D.K. Whirlow, W.T. Roulwau, Bull. Math. Biophys. 27 (1965) 355–370. R.W. Lawton, Circ. Res. 3 (1955) 403–408. W.O. Fenn, in: J.W. Remington (Ed.), Tissue Elasticity, American Physiology Society, Washington, DC, 1957, p. 154. W. Klip, Velocity and damping of the pulse wave, Martinus Nyhoff, The Hague, 1962. H.B. Atabek, H.S. Lew, Wave propagation through a viscous incompressible fluid contained in an initially stressed elastic tube, Biophys. J. 6 (1966) 481– 503. A.I. Rachev, Effect of transmural pressure and muscular activity on pulse waves in arteries, J. Biomech. Eng. ASME 102 (1980) 119–123. R.H. Cox, Wave propagation through a Newtonian fluid contained within a thick walled viscoelastic tube, Biophys. J. 8 (1968) 691–709. N.N. Kizilova, Pressure wave propagation in liquid-filled tubes of viscoelastic material, Fluid Dynam. 41 (2006) 434–446. H. Demiray, N. Antar, Effects of initial stresses and wall thickness on wave characteristics in elastic tubes, ZAMM 76 (1996) 521–530. G.A. Holzapfel, T.C. Gasser, R.W. Ogden, A new constitutive framework for arterial wall mechanics and a comparative study of material models, J. Elast. 61 (2000) 1–48. _ H. Erol, Içerisinde Parçacıklı Akısßkan Bulunan Öngerilmeli, Lifli, Tabakalı, Elastik Kalın Tüplerde Harmonik Dalga Yayılımı, Doctoral Dissertation, Eskisßehir, 2008 (In Turkish). A.C. Eringen, E.S. Suhubi, Elastodynamics, vol. I, Pergamon Press, New York, 1974. pp. 246–259. H. Demiray, G. Akgün, Wave propagation in a viscous fluid contained in a prestressed viscoelastic thin tube, Int. J. Eng. Sci. 35 (1997) 1065–1079.