An analytical estimate of the flow-field in a porous stenotic artery subject to body acceleration

PERGAMON

International Journal of Engineering Science 36 (1998) 1083±1102

An analytical estimate of the ¯ow-®eld in a porous stenotic artery subject to body acceleration Santabrata Chakravarty *, Ashis Kr. Sannigrahi Department of Mathematics, Visva-Bharati Univ., Santiniketan 731235, India Received 18 August 1997; accepted 8 October 1997

Abstract The paper deals with a theoretical study of the ¯ow-®eld in a porous stenotic artery when it is subjected to a single cycle of body acceleration using an appropriate mathematical model. The simulated artery is taken as an isotropic elastic tube containing a viscous incompressible ¯uid representing the ¯owing blood. The shape of the stenosis in the arterial lumen is chosen to be irregular in order to improve resemblance to the in-vivo situation. Instead of having a constant seepage rate along the axis of the artery, the wall deformability has been accounted for so that the hydraulic membrane permeability is treated to be contained in the wall velocity. The equations governing the motion of the system are solved analytically in both the steady and the unsteady states with the use of the appropriate boundary conditions. Numerical computations have ®nally been performed in order to have a thorough quantitative measure of the e€ects of body acceleration and the hydraulic membrane permeability on the ¯ow velocity, the ¯ux, the resistive impedances and the wall shear stress just to validate the applicability of the present mathematical model. # 1998 Elsevier Science Ltd. All rights reserved.

1. Introduction Partial occlusion of blood vessels due to abnormal growth of tissues or the deposition of cholesterol as pearly substances on the arterial wall, that is, the coarctation of the aorta in the arterial lumen, commonly referred to as a stenosis, is one of the most frequently occurring abnormalities in the cardiovascular system of humans. Such constriction of the arterial lumen grows inward and restricts the normal movement of blood where the transport of blood to the region beyond the narrowing is reduced drastically depending upon the severity of the stenosis. The problem becomes more grave in some exceptional situations where humans experience * Corresponding author. Tel.: +91-(03463)-52751; fax: +91-(03463)-52672; E-mail: [email protected] 0020-7225/98/$19.00 # 1998 Elsevier Science Ltd. All rights reserved. PII: S 0 0 2 0 - 7 2 2 5 ( 9 8 ) 0 0 0 0 9 - 3

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whole-body accelerations. While riding in a vehicle or while ¯ying in a space craft, the external accelerations of considerable magnitudes imparted to the human body can cause serious problems in the cardiovascular system leading to the impairment of certain physiological functions. The root cause for the development of such unnatural growth (stenosis) is not completely clear to the theoreticians, but its e€ect on the cardiovascular system has been determined by studying the ¯ow characteristics of blood in the stenotic region with an accurate knowledge of the mechanical properties of the vascular wall together with that of the ¯owing blood in the arterial system. To understand the e€ects of stenosis in the arterial lumen, several theoretical studies (cf. Young [1]; Forrester and Young [2]; Lee and Fung [3]; Young and Tsai [4]; Misra and Chakravarty [5]) related to ¯ow through stenosed arteries have been performed. All these studies are based on the assumption that blood behaves like a Newtonian ¯uid and that the geometry of the stenosis could be represented by a smooth mathematical function. Subsequently Chakravarty et al. [6±8] improved the geometry of the stenosis by introducing a time parameter and a shape parameter so that it becomes ¯exible with respect to time and shape in their analytical investigations. They also explored the possibility of forming the multiple stenoses and overlapping stenoses in an artery and the e€ects of such constrictions on the ¯ow characteristics of blood were quanti®ed thereby. The ¯owing blood was however treated to be non-Newtonian in concert with the experimental observations that blood behaves like a non-Newtonian ¯uid at low shear rates in smaller arteries. But Taylor [9] pointed out that at high shear rates, commonly found in larger arteries (above 1 mm in diameter) blood behaves like a Newtonian ¯uid. According to modern conceptions the shape of the stenosis is quite irregular and it contains many small valleys and ridges like a mountain range instead of having the general trend of a smooth curve. The studies mentioned above have totally disregarded the e€ect of whole-body accelerations and hence attention has only been centered on the stenotic ¯ow behavior of blood under normal physiological conditions. The body acceleration bears the potential to in¯uence signi®cantly the ¯ow phenomenon in the vicinity of an irregular stenosis developed in an artery as indicated by the present authors in their recent investigation [10] where the ¯ow mechanism was subjected to a pulsatile pressure gradient owing to the normal functioning of the heart together with a single cycle of body acceleration. All the aforesaid studies treat the vessel wall to be impervious. But according to recent literatures, the endothelial walls are permeable having ultra-microscopic pores through which ®ltration does occur. The permeability of the endothelial walls increases with the deposition of cholesterol and it is deemed to be believed that such increase in permeability also results from damaged or in¯amed arterial walls. Seepage from the arterial ends of the capillary beds and corresponding back ¯ow at the venous end are very important to the mass transfer of various vital elements between blood and tissue, and are also critical to the functioning of kidneys. With the above discussion in mind we propose to study the ¯ow behavior of blood in a porous stenotic artery in both steady and unsteady states analytically by considering blood to be a Newtonian ¯uid and by properly accounting for vessel wall deformability. The stenosed artery is subjected to whole-body acceleration which is expressed in terms of unit functions such that it builds up from zero to a maximum value at a uniform rate, remains constant at its maximum value for some time and thereafter drops to zero also at a uniform rate. The published data from the experimental investigation of Back et al. [11] were used to de®ne the

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outline of the irregular stenosis under consideration. The cylindrical coordinates system has been taken for analytical formulation. Unlike the previous analytical studies relevant to the present one, particular emphasis has been put on both the radial and axial ¯ow through the artery under stenotic conditions. The equations governing the motion of the system are solved analytically in both the steady and the unsteady states using the appropriate boundary conditions. Final attempt is made quantitatively to address the question whether the body acceleration and the hydraulic membrane permeability bear the potential to in¯uence signi®cantly the velocity pro®le, the ¯ux, the resistive impedances and the wall shear stress throughout the stenosed arterial segment under consideration. 2. Formulation of the problem The stenosed arterial segment under consideration is modelled as a cylindrical tube whose wall material is being treated as isotropic, linear and elastic containing an incompressible Newtonian ¯uid representing blood. Let (r, y, z) be the coordinates of a material point in the cylindrical polar coordinate system in which the z-axis is taken along the axis of the artery while r, y are taken along the radial and the circumferential directions, respectively. The irregularly stenosed arterial segment (cf. Fig. 1) includes the severity of the stenosis as 48% areal occlusion. Considering the stenotic blood ¯ow to be axisymmetric, laminar, two dimensional and fully developed, the basic equations of motion governing such ¯ow subject to whole body acceleration may be written as   @u 1 @p mf @ 2 u 1 @u ‡ ˆÿ ‡ ‡ G…t†, …1† @t r @z r @r2 r @r @v 1 @p mf ˆÿ ‡ @t r @r r



 @ 2 v 1 @v v ‡ , ÿ @r2 r @r r2

Fig. 1. Geometry of the irregular stenosed arterial segment.

…2†

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together with the equation of continuity given by @u 1 @…rv† ‡ ˆ 0: @z r @r

…3†

The absence of the convection and axial di€usion terms in Eqs. (1) and (2) is due to smallness of the arterial radius with respect to its length under consideration. The variables (u, v) involved in the equations represent the axial and radial velocity components of the ¯owing blood, p the pressure, mf the viscosity and G(t) designates the body acceleration. It may be mentioned here that only the axial component of the body acceleration present in Eq. (1) would be e€ective in accelerating or decelerating blood ¯ow in the stenosed artery while the component of body acceleration normal to the arterial axis would be ine€ective. For pulsatile nature of the ¯ow we assume G(t) = F(t)eiot where F(t) denotes the acceleration pro®le. Suppose at time t>0, the stenosed arterial ¯ow is subjected to a single cycle of body acceleration (cf. Fig. 2) expressible in terms of unit functions which may be represented mathematically as F…t† ˆ a0 …t ÿ t1 †d…t ÿ t1 † ÿ a0 …t ÿ t2 †d…t ÿ t2 † ÿ a1 …t ÿ t3 †d…t ÿ t3 † ‡ a1 …t ÿ t4 †d…t ÿ t4 †,

…4†

in which a0, a1 are the gradients, t1 is the moment of application of the body acceleration, (t2ÿt1) is the build up time, (t4ÿt3) is the climb-down time, t4 is the moment of withdrawal of the external acceleration, d(t) is a unit step function and t indicates the generic time. The governing equations of motion of the vessel wall when it is subjected to the inertial forces and the surface forces may be put in the form

Fig. 2. Schematic representation of body acceleration.

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@srr srr ÿ syy @srz @ 2Z ‡ ‡ ˆ rv 2 , @t @r r @z

…5†

@srz srz @szz @ 2x ‡ ‡ ˆ rv 2 , @r r @z @t

…6†

in which srr, syy, szz and srz represent the elastic stress components, rv the mass density of the vascular wall material and x, Z are the respective displacement components in the longitudinal and radial directions. 3. Boundary conditions Considering that the blood particles in the stream adhere to the inner surface of the vessel wall, the axial velocity of the blood particles on the wall surface may be taken to be equal to the velocity of the vascular wall material points along the same sense. This may be stated mathematically as uˆ

@x , @t

on r ˆ R…z†,

…7†

while the radial component of the ¯owing blood may be matched with the ®ltration velocity of the wall surface, that means, vˆ

@Z ˆ Lp p, on r ˆ R…z†, @t

…8†

Lp being the hydraulic membrane permeability. Also, the stresses are assumed to be continuous at the ¯uid±solid interface, that is, sfrr ˆ svrr ,

sfrz ˆ svrz

on r ˆ R…z†

…9†

where the superscripts f and v stand for the ¯uid and the vessel wall, respectively, and R(z) represents the radius of the irregular stenosed artery under consideration. Moreover the axial velocity gradient of the ¯uid representing blood may be assumed to be equal to zero throughout the axis and there is no radial ¯ow along the axis which may be written as @u ˆ0 @r

on r ˆ 0

…10†

and vˆ0

on r ˆ 0

…11†

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4. Method of solution We assume that the ¯ow ®eld, the pressure exerted by blood and the displacement components of the arterial wall are composed of a steady part and an unsteady part as follows: u ˆ us ‡ uf ˆ us ‡ u0 eio t ,

…12†

v ˆ vs ‡ vf ˆ vs ‡ v0 eio t ,

…13†

p ˆ ps ‡ pf ˆ … ps0 ÿ ks z† ‡ … pf 0 ÿ f …z, r††eio t ,

…14†

Z ˆ Zs ‡ Zf ˆ Zs ‡ Z0 eio t ,

…15†

x ˆ xs ‡ xf ˆ xs ‡ x0 eio t ,

…16†

and

in which the subscripts s and f designate the steady and unsteady states, respectively, while the 0 subscript indicates the quantities independent of time and spatial coordinates. Since pressure damps out due to energy dissipation through the membrane and function in the pressure term, we need to consider f(z, r) as a damping function in pressure term with ks as a friction parameter and o the angular frequency. 4.1. Case I: steady state solutions The Eqs. (1) and (2) are sought into steady state which are reduced to @ 2 us 1 @us 1 @p ‡ ˆ , 2 @r r @r mf @z

…17†

@ 2 vs 1 @vs vs ‡ ÿ ˆ 0, @r2 r @r r2

…18†

while the steady state conditions are us ˆ 0, vs ˆ Lp … ps0 ÿ Ks z† dus ˆ 0, vs ˆ 0 …sfrr †s ˆ …svrr †s , …sfrz †s ˆ …svrz †s , dr

on r ˆ R…z†, on r ˆ 0:

…19†

Using these boundary conditions one may obtain the steady state solutions of the stenosed ¯ow problem as us ˆ and

ks …R2 …z† ÿ r2 †, 4mf

…20†

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vs ˆ

Lp … ps0 ÿ ks z†r , R…z†

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…21†

Also, the general solutions of the vessel wall displacements from Eqs. (5) and (6) in the steady state may be obtained by incorporating the constitutive relations [10] and following the same method as mentioned there, as p k2 z 0 0 …22† Zs ˆ k1 fAs I0 …k1 r† ‡ Bs K0 …k1 r†ge , p p k2 z xs ˆ k k2 fAs I0 …k1 r† ‡ Bs K0 …k1 r†ge ,

…23†

where As and Bs are two arbitrary constants to be determined from the boundary conditions and k21=c/(l + 2m), k2= ÿ c/(m + k(l + m)) in which l, m are the elastic constants, k and c are two non-zero constants. On the application of a pair of conditions of Eq. (19), the constants As and Bs are determined whose expression should be read as As ˆ

f1 f22 ÿ f2 f12 , f11 f22 ÿ f12 f21

…24†

Bs ˆ

f2 f11 ÿ f1 f21 , f11 f22 ÿ f12 f21

…25†

and

where the expressions for the coecients fij and fk are included in Appendix A.

4.2. Case II: unsteady state solutions The unsteady or the time-dependent part of Eqs. (1) and (2) are given by   @uf 1 @pf mf @ 2 uf 1 @uf ‡ ˆÿ ‡ ‡ G…t† @t r @r2 r @z r @r @vf 1 @pf mf ˆÿ ‡ @t r r @r



@ 2 vf 1 @vf vf ÿ ‡ @r2 r @r r2

…26†

 …27†

The equation of continuity (Eq. (3)) can also be expressed as a composition of a steady part and an unsteady part of which the steady state solutions are already made known so that Eq. (3) can now be expressed in terms of the time-dependent variables only. In order to solve the unsteady part of the radial velocity, the damping function f(z, r) needs to be determined ®rst. Further @u/@z term in Eq. (3) can not be disregarded and we assume this as v which when introduced into the composite relation corresponding to Eq. (3) and integrated with respect to r, one ®nds

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r r vf ˆ ÿv ÿ Lp … ps0 ÿ ks z† : 2 R…z†

…28†

Using Eq. (8) corresponding to unsteady state, one asserts that vˆÿ

2Lp ‰… ps0 ÿ ks z† ‡ f pf 0 ÿ f …z, R…z††geio t Š, R…z†

…29†

and hence vf ˆ

rLp ‰ pf ÿ f …z, R…z††Šeio t : R…z† 0

…30†

Introducing this expression of vf and the unsteady part of the pressure into Eq. (27) and then integrating with respect to r, one obtains the damping function as f …z, r† ˆ qz ‡

iorLp r2 ‰ pf 0 ÿ f …z, R…z††Š, 2R…z†

…31†

where q being the constant of integration and f …z, R…z†† ˆ

qz ‡ pf 0 iorLp R…z†=2 : 1 ‡ iorLp R…z†=2

…32†

Since oLpR(z)/2<<1, one can disregard the higher order terms, retaining up to the second order terms only, which helps ®nally obtaining the damping function and consequently the unsteady part of the radial velocity given by f …z, r† ˆ qz ‡

iorLp r2 … pf 0 ÿ qz†…1 ÿ iorLp R…z†=2†, 2R…z†

…33†

and vf ˆ

rLp … pf ÿ qz†…1 ÿ iorLp R…z†=2†eio t : R…z† 0

…34†

From Eqs. (12) and (26) we also have, d2 u0 1 du0 ior ÿ ‡ u0 ˆ g…r†, r dr mf dr2

…35†

where g…r† ˆ ÿ

q iorLp r2 F…t†r ÿ …1 ÿ iorLp R…z†=2† : mf mf 2R…z†

…36†

Solving Eq. (35) and making use of Eq. (10) corresponding to the unsteady state one obtains u0 ˆ ÿ

g…r† 2qiorLp …1 ÿ iorLp R…z†=2† ‡ V1 I0 …m1 r†, ÿ m21 mf m41 R…z†

…37†

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p in which m1= …ior†=mf and V1 is an arbitrary constant to be determined from the boundary conditions. The displacement components of the vessel wall in the unsteady state are obtained as p p p p …38† Z0 ˆ c‰Af I0 0 … cr† ‡ Bf K0 0 … cr†Še cÿc1 z , and x0 ˆ

p p p p c ÿ c1 ‰Af I0 … cr† ‡ Bf K0 … cr†Še cÿc1 z ,

…39†

in which c1= ÿ (rvo2)/(l + 2m). After having obtained the general solutions for the dependent variables involved in the stenotic ¯ow in the unsteady state in terms of four unknown constants, one needs to evaluate them by properly using the boundary conditions Eqs. (7)±(9) in the unsteady state. On application of these conditions one arrives at the following equations involving the unknowns. a11 Af ‡ a12 Bf ‡ a13 q ˆ b1 ,

…40†

a21 Af ‡ a22 Bf ‡ a23 q ˆ b2 ,

…41†

a31 Af ‡ a32 Bf ‡ a33 q ˆ b3 ,

…42†

a41 Af ‡ a42 Bf ‡ a43 q ‡ a44 V1 ˆ b4 ,

…43†

where the coecients aij and the constants bk have got their expressions included in Appendix B. These equations are nonhomogeneous and hence one can solve them for the unknowns Af, Bf, q and V1. After having determined these unknowns, the ¯ow velocity components and the arterial wall displacements are completely determined in the unsteady state.

5. Volumetric ¯ow rate, resistive impedance and wall shear stress The volumetric ¯ow rate (Q) can be determined with the knowledge of the axial ¯ow velocity in both the steady and the unsteady states, as … R…z† ru dr: …44† Q ˆ 2p 0

The resistive impedance (L) experienced by the ¯owing blood through the stenosed arterial system under consideration may be calculated from the following relation, Lˆ

Pi ÿ Po , Q

…45†

where Pi and Po are, respectively, the incoming and the outgoing pressures exerted by blood.

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Finally, the shear stress (t) developed at the stenosed wall may be obtained as,    @u @v R…z†ks ‡ t ˆ mf ˆ ÿ m ‡ Lp k s f @r @z rˆR…z† 2mf  0   G …R…z†† ‡ ÿ m V I …m R…z†† ‡ qL …1 ÿ iorL R…z†=2† eio t 1 1 1 1 p p m21

6. Numerical results and discussion In order to illustrate the applicability of the present mathematical model and to have a thorough quantitative measure of the combined e€ects of body acceleration and the vessel wall permeability on the ¯ow-®eld, the ¯ux, the resistance to ¯ow and the wall shear stress, the following relevant data have been made use of (cf. Atabek [12], McDonald [13], Milnor [14]) m ˆ 4  105 N=m2 , mf ˆ 0:004 kg mÿ1 sÿ1 , l ˆ 0:7  105 N=m2 , r ˆ 1:024  103 kg mÿ3 , rv ˆ 1:05  103 kg mÿ3 , a0 ˆ 0:981 msÿ3 ˆ a1 , ps0 ˆ 20 ˆ pf 0 , o ˆ 7 rad=sec, t1 ˆ 0:1 s, t2 ˆ 0:2 s, t3 ˆ 0:3 s, t4 ˆ 0:4 s: Numerical results obtained by using the above mentioned data as also the numerical values for di€erent arterial radii along its length corresponding to an irregular stenosis obtained from Back et al. [11] are presented in the Figs. 3±12. The results of Fig. 3 illustrate the behavior of the axial velocity pro®le of the ¯owing blood at a speci®c location of z = 62.2 mm corresponding to the maximum arterial constriction for two di€erent time periods viz. t = 0.25 and 0.35 s. It appears that the axial ¯ow velocity diminishes gradually from its maximum value at the centre and approaches a very small nonzero value towards the wall surface resulting from the deformability of the vessel wall. One may notice that there is a reduction of the ¯ow velocity as time progresses and the decreasing rate is higher towards the centre line than near the wall surface. The bottom most dotted line curve represents the corresponding results in the absence of any body acceleration at an instant of t = 0.25 s. Unlike the behavior of the other curves this is found to have a little deviation as one moves away from the axis of the arterial segment under consideration. The e€ect of body acceleration on the axial velocity of the streaming blood can thus be quanti®ed by a considerable amount through a comparison of the topmost and the bottommost curves corresponding to the same instant of t = 0.25 s. It is noteworthy that the whole-body acceleration enhances the maximum ¯ow velocity on the axis by nearly 70% to that under normal physiological condition. The axial velocity distributions of the streaming blood over the stenosed arterial segment under study are exhibited in Fig. 4 for two di€erent radial positions Ð one at the stenotic region and the other at the nonstenotic region at a particular instant of t = 0.25 s. The curves are noted to follow the outline of the irregular stenosis and the velocity distribution becomes minimum at two critical locations quite close to each other where the arterial constrictions are maximum. Thus the irregular stenosis present in the arterial lumen a€ects signi®cantly the axial

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Fig. 3. E€ect of body acceleration on the axial ¯ow velocity varying radially for di€erent time periods at z = 62.3 mm.

Fig. 4. The axial velocity distribution of the streaming blood over the arterial segment and the e€ect of body acceleration on it at 1 = 0.25 s (Lp=0.01 m2 s kgÿ1).

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Fig. 5. E€ect of permeability on the radial ¯ow velocity at z = 62.3 mm.

velocity distribution of the ¯owing blood. In order to examine the e€ect of body acceleration on the velocity distribution one can compare the relevant curves corresponding to the same radial position. Here also, the presence of body acceleration bears the potential to in¯uence the velocity distribution appreciably keeping the nature of the curve almost unperturbed. Fig. 5 shows the radial velocity component of the ¯owing blood in the stenosed artery varying radially in the constricted region where arterial narrowing is maximum for three di€erent permeability constants. In spite of the axial ¯uctuating velocity pro®le the radial velocity appears to increase linearly from its minimum (zero) at the axis and hence, the velocity gradient is largest near the wall surface. As the permeability increases the radial velocity of the ¯owing blood increases. It may be pointed out that a very small increment of the hydraulic permeability constant can in¯uence the radial velocity signi®cantly. The distributions of the ¯ow rate over the irregular stenosed artery for di€erent time periods have been calculated and plotted in Fig. 6 being in¯uenced by the presence of body acceleration. Following the outline of the irregular stenosis, the ¯ow rate increases a little at the onset of the stenosis, then diminishes downstream excepting at a few speci®c locations forming small valleys until the maximum arterial constrictions near z = 60 mm and thereafter increases upstream barring a particular location of z = 80 mm forming a small ridge. Thus the ¯ow rate enhances or reduces to some extent with the arterial length in the stenotic region depending upon whether the arterial cross-section increases or decreases, respectively. Following the results of the axial velocity pro®le varying with time presented in Fig. 3, the ¯ow

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Fig. 6. E€ect of body acceleration on the ¯ux distribution for di€erent time periods (Lp=0.01 m2 s Kgÿ1).

rate also gets reduced considerably with increasing time. Such behavior of the ¯ow rate with time is solely responsible for the nature of the body acceleration under consideration. The choice of considering the speci®c time of t = 0.25 s corresponds to the time of attaining the maximum amplitude of the body acceleration while t = 0.35 s indicates the time when the acceleration declines from its maximum. The bottommost curve exhibits the corresponding results when there is no body acceleration in the system under consideration at a particular instant of t = 0.25 s. One may observe that when the stenosed arterial ¯ow is allowed only under the pressure gradient arising out of the normal functioning of the heart, the characteristics of the ¯ow rate remains analogous to the bottom most curve, but they di€er in magnitude considerably. The external body acceleration bears the potential to accelerate the ¯ow velocity and consequently the ¯ow rate. Hence one can easily estimate the e€ect of body acceleration quantitatively on the ¯ux distribution through a direct comparison of the two curves Ð one at the top and the other at the bottom computed for the same instant of t = 0.25 s. Fig. 7 displays the e€ect of vessel wall permeability on the ¯ux distribution over the stenosed arterial system at time t = 0.25 s when the body acceleration attains its amplitude maximum. The ¯ow rate is found to diminish signi®cantly with the increase in the hydraulic membrane permeability. Further, if one disregards the permeability, the ¯ow rate increases as shown by the dotted line curve. Such behavior is deemed to be quite feasible for reasons that in case of the porous arterial wall a certain amount of ¯uid representing blood is ®ltrated out instead of

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Fig. 7. E€ect of permeability on the distribution of ¯ux at time 1 = 0.25 s.

Fig. 8. E€ect of body acceleration on the resistive impedance for di€erent time periods (Lp=0.01 m2 s Kgÿ1).

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¯owing through the main stream as a result of which the ¯ow rate is reduced depending upon the degree of permeability while for the impervious wall, the entire blood particles follow the main stream only. The resistances to ¯ow or the impedances experienced by the ¯owing blood through the stenosed arterial segment under consideration for two speci®c time periods are shown through the results presented in Fig. 8. Unlike the characteristics of the ¯ow rate, the resistive impedances get enhanced at the onset of the stenosis from a relatively lower value in the unconstricted part till its maximum constriction near z = 60 mm, except at a speci®c location, and thereafter diminish irregularly as the constrictions assume minimum. It may be noted here that the impedances increase with the increase in time or in other words the ¯owing blood experiences lesser resistance to ¯ow for higher amplitude of the external body acceleration. The dotted line curve represents the corresponding results when there is no body acceleration. Thus the key role played by the whole-body acceleration in¯uencing the resistive impedances as also the ¯ow rate, has got its own importance most in the realm of arterial biomechanics. The results of the ¯ow rate versus time in both the stenotic and the nonstenotic regions have been recorded in Fig. 9. Two continuous curves correspond to the results for stenotic and nonstenotic zone in the presence of whole body acceleration. The ¯ow rate is found to increase from some ®nite initial value the moment the whole body acceleration becomes operative, attains a maximum at an instant of t = 0.2 s followed by a gradual decrease till time t = 0.3 s

Fig. 9. E€ect of body acceleration, permeability and stenosis on the ¯ow rate vs time.

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Fig. 10. Distribution of wall shear stress over the arterial segment.

and thereafter drops sharply until time t = 0.4 s followed by meager ¯uctuations with the rest of time as shown in the ®gure when the body acceleration becomes inoperative. The ¯ow rate is noted to be enhanced more in the nonstenotic region than in the stenotic region. It appears that although the nature of both curves is similar, they di€er considerably in magnitude. Thus one can measure the e€ect of irregular stenosis quantitatively on the time-variant ¯ow rate in the arterial segment under consideration through a numerical comparison of the results represented by these two curves. In the absence of body acceleration, the dotted-line curve shown in the present ®gure represents the ¯ow rate in the same stenotic zone where no peak is observed during the whole period unlike other curves. Here too, one can easily estimate the e€ect of body acceleration on the ¯ow rate versus time in the constricted area during the speci®c period from 0.1 to 0.4 s only when the body acceleration is operative in the arterial system under consideration. However, the vessel wall permeability has got a little in¯uence on the ¯ow rate as evident from the results obtained by disregarding permeability. Fig. 10 includes the wall shear stress distribution over the stenosed arterial segment for two di€erent time periods. Note that the wall shear stress increases or decreases with the arterial length, depending upon the decrease or increase of the arterial cross section, respectively. The curves are found to be compressive in nature for both the times. As time progresses from 0.25 to 0.35 s, the entire wall shear stress distribution gets enhanced considerably. The second curve from the top has been plotted at t = 0.25 s when the presence of the body acceleration has totally been disregarded. Another important observation is that when the body acceleration is

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1099

Fig. 11. Variation of axial wall motion with time (z = 62.3 mm).

withdrawn from the system under study the wall shear stresses are enhanced largely throughout the constricted arterial segment under consideration. The other curve representing the results of the wall shear stress indicates its behavior when the vessel wall is treated to be impervious. So, in the absence of wall permeability the stresses appear to have an increasing trend throughout the stenosed artery. Thus the permeability of the wall together with the external body acceleration bear the potential to contribute the ¯ow characteristics signi®cantly. The peaks of the shear stresses are believed to cause severe damage to the arterial lumen which help determining the aggregation sites of platelets. Finally, the concluding Figs. 11 and 12 of the present paper illustrate the results for the arterial wall displacements varying radially at a critical location where the arterial constriction is maximum for di€erent time periods. Both the axial and the radial displacements of the vessel wall are found to be ¯uctuating with increasing time where the pulsatile nature of the wall movement is clearly re¯ected. The changing rate of the axial wall displacement is found to be slower than that of the radial displacement as one moves towards the wall surface for all the time periods. Further, since the axial displacements are being lowered considerably with increasing time from t = 0.25 to 0.35 s one may conclude that the axial wall deformability may tend to rigidity with the large passage of time. Moreover, the amount of the axial displacement of the vessel wall appears to be more than that of the radial displacement at a particular

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Fig. 12. Variation of radial wall motion with time at z = 62.3 mm.

instant so far as the magnitudes are concerned, are re¯ected by the ¯ow phenomena since major ¯ow occurs in the axial direction than in the radial direction.

Appendix A The expression for the coecients fij together with the constants fk (k = 1, 2) in Eqs. (24) and (25) should be read as   p l 2 0 f11 ˆ …l ‡ 2m†I0 0…k1 R…z††k1 ‡ I0 …k1 R…z††k1 ‡ lI0 …k1 R…z††kk2 e k2 z , R…z†   p l 0 2 f12 ˆ …l ‡ 2m†K0 0…k1 R…z††k1 ‡ K0 …k1 R…z††k1 ‡ lK0 …k1 R…z††kk2 e k2 z , R…z†   p l I0 0 …k1 R…z††k1 ‡ …l ‡ 2m†I0 …k1 R…z††kk2 e k2 z , f21 ˆ lI0 0…k1 †k21 ‡ R…z†   p l 2 0 K0 …k1 R…z††k1 ‡ …l ‡ 2m†K0 …k1 R…z††kk2 e k2 z , f22 ˆ lK0 0…k1 R…z††k1 ‡ R…z†   2mf Lp ÿ 1 , and f2 ˆ ÿ… ps0 ÿ Ks z†: f1 ˆ… ps0 ÿ Ks z† R…z†

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1101

Appendix B The coecient aij and the independent quantities bk, (k = 1, 2, 3, 4) involved in Eqs. (40)± (43) have got their expressions given by, p p p l p 0 p cI0 … cR…z†† ‡ l c ÿ c1 I0 … cR…z††, a11 ˆc…l ‡ 2m†I0 0… cR…z†† ‡ R…z† p p p l p 0 p cK0 … cR…z†† ‡ l c ÿ c1 K0 … cR…z††, a12 ˆc…l ‡ 2m†K0 0… cR…z†† ‡ R…z†   2mf Lp z iorLp R…z† 1ÿ , a13 ˆ R…z† 2 p p p a21 ˆm‰cI0 0… cR…z†† ‡ I0 … cR…z††…c ÿ c1 †Še …cÿc1 †z , p p p a22 ˆm‰cK0 0… cR…z†† ‡ K0 … cR…z††…c ÿ c1 †Še …cÿc1 †z ,   iorLp R…z† a23 ˆmf Lp 1 ‡ , 2 p p p a24 ˆ ÿ mf I0 0 …m1 r†m1 ,31 ˆ cI1 … cR…z††e …cÿc1 †z , p p p a32 ˆ ÿ cK1 … cR…z††e …cÿc1 †z ,   iorLp R…z† a33 ˆLp 1 ÿ , 2 p p p a41 ˆio …c ÿ c1 †I0 … cR…z††e …cÿc1 †z , p p p a42 ˆio …c ÿ c1 †K0 … cR…z††e …cÿc1 †z , 2iorLp …1 ÿ iorLp R…z†=2† , a43 ˆ m41 mf R…z† a44 ˆ ÿ I0 …m1 R…z††,    iorLp R…z† 2mf Lp ÿ1 , b1 ˆPf 0 1 ÿ R…z† 2 G 0 …R…z††mf , b2 ˆ ÿ m21 b3 ˆLp Pf 0 …1 ÿ iorLp R…z†=2†, G…R…z†† : b4 ˆ m21

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