Effect of the Thickness of Tube on Wave

5—10 July 2004, Antalya, Turkey – Dynamical Systems and Applications, Proceedings, pp. 553—562

Effect of the Thickness of Tube on Wave Propagation in Elastic Tubes Filled with Viscous Fluid Saadet Özer, Ali Ercengiz Istanbul Technical University, Faculty of Science and Letters, Department of Engineering Science, 34469 Maslak, Istanbul, Turkey E-mail: [email protected], [email protected] Abstract In order to understand the effect of initial stress and the thickness of the tube on flow in elastic tubes, the propagation of time harmonic waves in a prestressed elastic, isotropic tube filled with a viscous fluid is studied. The thick walled tubes are important to understand the blood flow through the arteries. Although the blood is known to be a non-Newtonian fluid, for simplicity in the mathematical analysis it is assumed to be a viscous fluid. After obtaining the field equations, for the sake of better understanding of the effect of thickness, a finite difference method is used to solve the equations. The variations of primary and secondary wave speeds and the transmission coefficients are investigated with respect to the thickness ratio and Womersley parameter by considering long wave approximation. Also a cut-off frequency which could not have been obtained in the previous works that used a truncated power series method is obtained.

Key words: Harmonic waves, Thick tube, Initial stress.

1

Introduction

The importance of fluid mechanics and mathematical models in understanding the blood flow process in human and animal bodies has been reviewed by Rudinger [1]. Witzig [2] is the first one who took the viscosity into account but ignored the effects of Poisson’s ratio and obtained the propagation constants as a function of the frequency and viscosity. Morgan and Kiely [3] studied the same problem by assuming the artery as an elastic tube and the blood as a viscous but incompressible fluid and obtained the dispersion relation in which the effects of Poisson’s ratio is also included. Womersley [4] in his pioneering work treated the artery as a thick walled shell and blood as incompressible viscous fluid. In reality, the artery is subject to average pressure which is about 100 mm Hg and axial stretch which is 553

554

S. Özer and A. Ercengiz

1.5 under physiological conditions. The initial stress (or deformation) of arterial wall had been first taken into consideration by Atabek and Lew [5], and the effects of initial stresses and tethering were numerically analyzed. The effects of initial deformation were properly taken into account by Rachev [6], but he simply treated artery as a membrane. However, physiological studies on arteries show that the ratio of thickness to mean radius of the artery (large blood vessels in general) changes from 16 to 14 . This means that the arterial wall is not thin enough to use the membrane theory and to take the constant initial stresses distributions. Demiray and Antar [7] have taken into account the initial stresses and the thickness of artery, but to obtain solution of equations governing the solid body, they have used a truncated power series in terms of thickness ratio. In the present work the propagation of harmonic waves in an initially stressed isotropic, elastic cylindrical tube filled with an incompressible viscous fluid is studied. Considering the arterial wall as an elastic, isotropic and incompressible material subjected to a large initial static deformation, the governing differential equations in cylindrical coordinates are obtained for fluid flow and solid body. Although a closed form solution can be obtained for equations governing the fluid body, due to the variability of coefficients of differential equations of solid body, such a closed form solution is not possible. Therefore the solution is obtained numerically by using the finite difference method.

2 2.1

Equations of motion Equations of fluid

Although the blood is known to be a non-Newtonian fluid, the incremental behavior of blood in arteries can be treated as Newtonian fluid. For an axially symmetric motion, the equations of fluid are given by µ 2 ¶ ∂ u ¯ u ¯ ¯ 1 ∂u ¯ ∂2u ∂u ¯ ∂ p¯ +µ − 2 + 2 − ρ¯ = 0, + − 2 ∂r ∂r r ∂r r ∂z ∂t µ 2 ¶ ∂ p¯ ∂ w ¯ ∂2w ¯ 1 ∂w ¯ ∂w ¯ − +µ + = 0, (1) − ρ¯ + 2 2 ∂z ∂r r ∂r ∂z ∂t where ρ¯ is the mass density of the fluid, µ is the viscosity, p¯ is the incremental inner pressure, the velocity field is given as u = (¯ u, 0, w) ¯ and the equation of incompressibility is ¯ ∂w ¯ ∂u ¯ u + + = 0. ∂r r ∂z

(2)

Effect of the thickness of tube on wave propagation

555

The components of stress tensor which we need in using the boundary conditions are given by ¶ µ ¯ ∂u ¯ ∂u ¯ ∂w σ ¯rz = µ + . (3) p + 2µ , σ ¯rr = −¯ ∂r ∂z ∂r

2.2

Equations of solid body

In this work the arterial wall material is assumed to be incompressible, isotropic and elastic. The cylindrical tube is subject to a large static deformation under the effects of inner pressure Pi and the axial force N . The governing field equations for such a deformation may be given by (see Eringen and Suhubi [8]) ∂ 2 uk , (4) ∂t2 where ρ is the mass density of solid body, u is the incremental displacement field and the incremental stress tensor σkl is defined by σkl,l = ρ

σkl = tkl + mkl ,

(5)

mkl = uk,m t◦ml

(6)

here and tkl is the incremental stress tensor referred to the final deformed area. The material is assumed to be incompressible neo-Hookean. So the incompressibility condition and the constitutive relations become uk,k = 0, α t◦kl = P ◦ gkl + αckl , W = (I1 − 3), (7) 2 where P ◦ is the hydrostatic pressure, gkl is the metric tensor of the spatial frame, ckl the Finger deformation tensor, I1 is the first invariant of ckl , and α is a material constant to be determined from experimental measurements. Assuming the geometry of the tube is a cylindrical thick shell, the stress distribution in cylindrical coordinates may be given as follows (Demiray and Dost [9]) µ ¶ x α x2 − x2o + λ ln , t◦rr = 2 λ 2 xo µ ¶ x α α x2 + x2o ◦ tθθ = 2 − 2 − λ ln , x λ 2 xo µ ¶ α x2 + x2o x R ◦ 2 tzz = αλ − 2 − λ ln , x≡ , λ 2 xo r µ 2 ¶ µ 2 ¶ 2 x xo α x + xo α xo − x2i ◦ − λ ln + λ ln , Pi = 2 , (8) P =− 2 λ 2 xo λ 2 xi

556


where Pi is the inner pressure, λ is the axial stretch. R and r are the radial coordinates of a material point before and after deformation and the subscripts (i) and (o) stand for the values of a quantity evaluated on the inner and outer surfaces, respectively. On the other hand, the incremental stress tensor is given by [8] tkl = pgkl − 2P ◦ ekl ,

(9)

1 2 (uk,l

where ekl is defined as ekl = + ul,k ). In order to obtain the complete field equations, one must also know the initial stress distribution t◦kl through the elastic wall. The tube that we shall study here is subject to the inner pressure Pi and axial stretch λ. After writing the explicit form of the incremental stress tensor from (9) and the components of mkl from (6) by considering (7), the governing equations can be written in the following form: ∂p ¯ ∂2u ∂u ¯ ∂2u ∂2u + β1 (r) 2 + β¯2 (r) − β3 (r)u + β¯4 (r) 2 = ρ 2 , ∂r ∂r ∂r ∂z ∂t ∂p ¯ ∂2w ¯ ∂w ¯ ∂2w + β1 (r) 2 + β5 (r) + β4 (r) 2 ∂z ∂r ∂r ∂z 2w ∂ ∂u =ρ 2, +(β¯2 (r) − β¯5 (r)) ∂z ∂t ¯ where the coefficients βi (r) (i = 1, 2, . . . , 5) are defined by

(10)

t◦ t◦ dP ◦ P ◦ P◦ − , β¯3 (r) = θθ2 − 2 , β¯2 (r) = θθ − 2 r dr r r r ◦ ◦ ◦ t dP P − . (11) β¯4 (r) = t◦zz − P ◦ , β¯5 (r) = θθ − r dr r These differential equations are to be supplemented by the boundary conditions given by β¯1 (r) = t◦rr − P ◦ ,

trr (ri ) = σ ¯rr (ri ), trz (ri ) = σ ¯rz (ri ), trr (ro ) = trz (ro ) = 0, ∂u ∂w (ri ) = u (ri ) = w, ¯, ¯ (12) ∂t ∂t where ri and r0 represent the inner and outer radius of the tube, respectively. On the other hand, the effects of tethering on the outer boundary have been neglected here.

3

Solution of the governing equations

In the present work we shall seek a harmonic wave type of solution to the field equations. For that purpose we set ¯ (r), W ¯ (r), P¯ (r)] exp[i(ωt − kz)], (¯ u, w, ¯ p¯) = [U (13)


557

¯ (r), W ¯ (r), P¯ (r) are where ω is the angular frequency, k is the wave number and U the complex amplitudes to be determined from the solution of the field equations and the boundary conditions. Substituting (13) into (1) and (2), the solution of the ¯ (r), W ¯ (r) and P¯ (r) can be obtained: resulting equations for U ¯ ¯ = k[I1 (kr)A¯ + J1 (sr)B], U ¯ ¯ = −i[kI0 (kr)A¯ + sJ0 (sr)B], W ¯ (14) P¯ = −i¯ ρωI0 (kr)A, ´ ³ where we defined s2 = − k2 + i¯ρµω and Jn (sr), In (kr) are the first kind and ¯ are integration constants modified Bessel functions of order n, respectively, A¯ and B to be determined by using the boundary conditions. In order to obtain the solution of the field equations of solid body, we shall seek again a harmonic type solution which can be written as ˆ (r), W ˆ (r), Pˆ (r)] exp[i(ωt − kz)]. (u, w, p) = [U (15) Considering the incompressibility condition and substituting this solution into the eqs. (10), we have ˆ ˆ d2 U dU dPˆ ˆ = 0, + β¯1 2 + β¯2 + (ρω2 − β¯3 − k 2 β¯4 )U dr dr dr ˆ ˆ d2 W dW ˆ − ik(β¯2 − β¯5 )U ˆ = 0, + (ρω2 − k 2 β¯4 )W −ikPˆ + β¯1 2 + β¯5 dr dr ˆ 1ˆ dU ˆ = 0. + U − ikW dr r By defining a function φ(r) as Z r ˆ (ξ) dξ, ξW φ(r) =

(16)

(17)

0

from the last equation of (16) the following can be obtained:

ˆ (r) = 1 dφ . ˆ = ik φ(r), W (18) U r r dr By using this function, the first two equations of (16) can be written in the following form: ¶ µ ¶ µ β1 d2 φ¯ iη 2β1 dφ¯ iη 2β1 β2 dP 2 2 + iη + + − +Ω −7β3 −η β4 φ¯ = 0, β2 − dξ ξ dξ 2 ξ ξ dξ ξ ξ2 ξ µ ¶ 2¯ 3 ¯ β1 d φ 2β1 β5 d φ −iηP + + − 2 + 3 ξ dξ ξ ξ dξ 2 · ¸ ¯ 2β1 β5 1 2 dφ η 2 2 (Ω + (β2 − β5 )φ¯ = 0, + − + − η β ) (19) 4 ξ3 ξ2 ξ dξ ξ

558


where we used the following non-dimensionalized quantities β¯1 = αβ1 , Pˆ = αP, Ω=

ω¯ r , c0

α α α β¯2 = β2 , β¯3 = β3 , β¯4 = αβ4 , β¯5 = β5 , r¯ r¯ r¯ α 3¯ 2 r, c0 = , γ = r¯s, r = r¯ξ, φ = r¯ φ, η = k¯ ρ ηI1 (η) γJ0 (γ) ρ i¯ ρω¯ r2 , f= , g= , q= . α0 = 2µ I0 (η) J1 (γ) ρ¯

(20)

On the other hand, we obtain boundary conditions in the following form µ ¯ ¶¸ ¶ µ ¶ µ · dφ φ¯ 1 f ◦ iη 2 − B, = η − + α0 A + η g − P − 2P ξ dξ ξ ξ=ξi ξi ξi · ◦µ ¶¸ P d2 φ¯ 1 dφ¯ 2¯ −η φ − 2 + = −iηfA − i(α0 + η 2 )B, ξ dξ ξ dξ ξ=ξi µ ¯ ¶¸ · dφ φ¯ ◦ iη − = 0, P − 2P ξ dξ ξ ξ=ξo ¸ · d2 φ¯ 1 dφ¯ = 0, −η 2 φ¯ − 2 + dξ ξ dξ ξ=ξo ¶ µ iΩ2 ¯ f A + B = 0, φ(ξi ) + qα0 ξi η ¶ µ iΩ2 dφ¯ + qα0 (ηA + gB) = 0, (21) ξi dξ ξ=ξi where

2µJ1 (γ) ¯ 2µI0 (η) ¯ A, B= B (22) 2 α¯ r α¯ r2 and r¯ is the midradius of the tube after deformation. It is almost impossible to give a closed form analytical solutions to (19). To see the effects of thickness of the tube on wave propagation more clearly, we will investigate numerical solutions of the equations by using finite difference method. It is obvious that the results will be obtained by using this method, will be more realistic than the power series method in which only first order terms have been taken into account. To use the finite difference method we divide the thickness of ¯ = r0 − ri , into n equal intervals, thus we define the tube, h A=

ξj = ξ0 + jh, where ξ0 =

nh ri =1− , r¯ 2

j = 0, 1, . . . , n,

ξn =

ro , r¯

h=

(23) ¯ h . n¯ r


559

We characterize the value of a function at the point ξj = ξ0 + jh by a subscript j. By introducing appropriate finite-difference expressions for various derivatives, we obtain the following difference equations for the equations (21) and (19): iη [α1j + h2 (Ω2 − η 2 β4j )]φ¯j ξj iη iη + α2j φ¯j+1 + β1j φ¯j+2 = 0, ξj ξj h i 2 h h3 η 2 −iηh3 Pj + α3j − (Ω2 − η 2 β4j ) + (β2j − β5j ) φ¯j ξj ξj h i 2 β1j ¯ h + α4j + (Ω2 − η 2 β4j ) φ¯j+1 + α5j φ¯j+2 + φj+3 = 0, ξj ξj µ ¶ h ¯ 2P0◦ iη 2P ◦ iη 1+ φ0 − 0 φ¯1 hP0 + ξ0 ξ0 ξ0 µ ¶ µ ¶ f 1 2 −h η − + α0 A − hη g − B = 0, ξ0 ξ0 µ ¶ µ ¶ P0◦ h ¯ P◦ P◦ h ¯ 2 2 − η h +1+ φ0 + 2+ φ1 − 0 φ¯2 ξ0 ξ0 ξ0 ξ0 ξ0

−hPj + hPj+1 +

+iηf h2 A + ih2 (α0 + η 2 )B = 0, ¶ µ iΩ2 ¯ f A + B = 0, φ0 + qα0 ξ0 η 2 iΩ ¯ (φ1 − φ¯0 ) + qα0 h(ηA + gB) = 0, ξ0 µ ¶ h ¯ ◦ hξn Pn + 2Pn iη 1 + φn − 2Pn◦ iη φ¯n+1 = 0, ξn µ µ ¶ ¶ h ¯ h ¯ 2 2 − η h +1+ φn + 2 + φn+1 − φ¯n+2 = 0, ξn ξn

where α1j α2j α3j

Ã ! ¶ µ 2β1j β 2β 1j 2j + h2 = β1j − h β2j − − − β3j , ξj ξj ξj2 ¶ µ 2β1j , = −2β1j + h β2j − ξj Ã ! Ã ! β1j 2β1j β 2β 1j 5j = − + h β5j ξj − 2 − 2 , − h2 ξj ξj ξj3 ξj

(24)

560


α4j = α5j = β1j = β5j =

Ã

! Ã ! β5j 2β1j β 2β 1j 5j − 2 − 2 , + h2 ξj ξj ξj3 ξj Ã ! 3β1j 2β1j − + h β5j ξj − 2 , ξj ξj " µ ¶ # x2j 1 1 2 λ 2 1 , β2j = − 2 xj − , β3j = 2 2 , β4j = λ2 , 2 2 λ ξj xj λ xj ξj xj " " # µ ¶2 # xj 1 1 1 λ 1 x2j + x2o ◦ − λ ln − 2 xj − Pj = − 2 , ξj x2j λ xj λ 2 xo 3β1j − 2h ξj

1/2

[λ(¯ r2 ξj2 − ri2 ) + Ri2 ] xj = r¯ξj

Ro . ro In order to have a non-zero solution for the coefficients Pj , φ¯j (j = 0, 1, . . . , n), A and B, the determinant of the coefficients matrix must vanish.

4

,

xo =

Long wave approximation

Even for large arteries the wave length is very large as compared to the mean 2 radius of the artery. Therefore, for this special case |η| ¿ 1, f approaches η2 . To understand the effects of the thickness of the arterial wall on the wave characteristics parameter n defining the number of intervals through thickness must be greater than 1. For the values n > 1, the cut-off frequency which could not have been obtained in the previous works have been obtained from the dispersion relation by setting η → 0. For instance, the cut-off frequency for small values of h and n = 2 is obtained as Ω2c = {h[β11 (2β10 − β50 − P00 gq) + β10 (β51 − P10 )gq] + β11 β10 }/(h2 β10 ).

(25)

For the more general case, we will take n = 4, λ = 1.3, Ri = 0.3 cm and Pi = 0.1. Moreover, experimental studies indicate that the density of the arterial wall tissue is quite close to that of blood, therefore, we may approximate q as q ∼ = 1 and decompose the non-dimensional phase velocity c into real and imaginary parts as c = X + iY.

(26)

The speed of propagation v and the transmission coefficient χ are defined by (Atabek and Lew [5]) X2 + Y 2 , χ = exp(−2πY /X). (27) v= X


561

The speeds of propagation and transmission coefficients are evaluated for various values of thickness ratio and Womersley parameter. Figure 1 shows the variation of primary wave speed with Womersley parameter and the thickness ratio. It shows that speed increases with Womersley parameter and thickness ratio. This result indicates the effects of the thickness on the wave speeds. The variation of secondary wave speed is depicted in Figure 2. The variations of transmission coefficients of primary wave are depicted in Fig. 3. Examination of this figure indicates that the coefficient decreases very fast with Womersley parameter at the beginning but stars to increase after a certain value of the parameter and with the thickness ratio. Finally, the variation of transmission coefficient of secondary wave with the same parameters is depicted in Figure 4. As might be seen from the figure, the coefficient increases with Womersley parameter but decreases with the thickness ratio.

References [1] Rudinger G., Reivew of current mathematical methods for the analysis of blood flow, in: Proc. Biomed. Fluid Mechs. Symposium, ASME, New York, 1966. [2] Witzig K., Über erzwungene Wellenbewegungen zäher, inkompressibler Flüssigkeiten in elastischen Rohren, Inaugural-Dissertation, University of Bern, 1914. [3] Morgan G. W. and Kiely J. P., Wave propagation in a viscous liquid contained in a flexible tube, J. Acoust. Soc. Am., 26 (1954), 323—328. [4] Womersley J., An elastic tube theory of pulse transmission and oscillatory flow in mammalian arteries, W.A.D.C., Techinical Report., TR56—614, 1957. [5] Atabek H. B. and Lew H. S., Wave propagation through a viscous incompressible fluid contained in an initially stressed elastic tube, Biophys. J. 7 (1966), 480—503. [6] Rachev A. I., Effect of transmural pressure and muscular activity on pulse waves in arteries, J. Biomechanical Engng. ASME., 102 (1980), 119—123. [7] Demiray H. and Antar N., Effects of initial stresses and wall thickness on wave characteristics, ZAMM, 76 (1996), 521—530. [8] Eringen A. C. and Suhubi E. S., Elastodynamics, Vol.I., New York, Permagon Press, 1974. [9] Demiray H. and Dost S., Pulse wave in prestressed elastic tube, Int. J. Engng. Sci., 28 (1990), 1—9. [10] Demiray H. and Ercengiz A., Wave propagation in a prestressed elastic tube filled with a viscous fluid, Int. J. Engng. Sci., 29 (1991), 575—585.

562


1 .8

1 .0

0 .8 1 .6

0 .6

V1

V

1 .4

2

0 .4

1 .2 0 .2

1 .0

0 .0 0 .2

0 .4

0 .6

0 .8

Figure 1: Variation of the primary wave speed with respect to the thickness ratio

0 .2

0 .6

0 .8

Figure 2: Variation of the secondary wave speed with respect to the thickness ratio

1 .0

0 .5

0 .9

0 .4

0 .8

0 .3

1

0 .4

2

0 .7

0 .2

0 .6

0 .1

0 .5

0 .0 0 .2

0 .4

0 .6

0 .8

Figure 3: Variation of the transmission coefficient of the primary wave speed

0 .2

0 .4

0 .6

0 .8

Figure 4: Variation of the transmission coefficient of the secondary wave speed