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R. Ben-Mansour, H.M. Badr, A. Qaiyum Shaik, and N. Maalej

October 2008 The Arabian Journal for Science and Engineering, Volume 33, Number 2B 529

MODELING OF PULSATILE BLOOD FLOW IN AN

AXISYMMETRIC TUBE WITH A MOVING INDENTATION

R. Ben-Mansour, H.M. Badr, and A. Qaiyum Shaik

Mechanical Engineering Department, College of Engineering

and N. Maalej

Physics Department, College of Sciences

King Fahd University of Petroleum and Minerals, Dhahran, Saudi Arabia, 31261

:

--. .

.Hz1 : .

)200Re=( .

.

* Address for correspondence:

KFUPM P. O. Box 1724

King Fahd University Of Petroleum & Minerals

Dhahran 31261, Saudi Arabia

Paper Received 3 June 2006; Revised 1 March 2008; Accepted 4 June 2008

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The Arabian Journal for Science and Engineering, Volume 33, Number 2B October 2008530

ABSTRACT

The time-dependent flow in an axisymmetric tube with a moving indentation is

numerically simulated using a dynamic mesh model. The model was first validated

for a two-dimensional planar channel with a moving indentation. The results

exhibited very good agreement with the published experimental results. The model

was then used to simulate the blood flow with steady and pulsatile inflows in anaxisymmetric tube with an indentation moving at a frequency of Hz1 . For the same

value of Reynolds number of 200, vortex doubling downstream of the moving

indentation was more enhanced in the case of pulsatile flow inlet conditions. Higher

wall shear stresses and pressure drops were obtained for the pulsatile inflow as

compared with the steady inflow.

Key words: blood, unsteady flow, moving indentation, pulsatile flow, stenosis

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MODELING OF PULSATILE BLOOD FLOW IN AN AXISYMMETRIC TUBE WITH A

MOVING INDENTATION

1. INTRODUCTION

Flows in domains with moving boundaries are encountered in many practical situations. Applications, in which

considerable research interest has been shown in recent years, include flow in blood vessels, in-cylinder flows in internal

combustion engines, free surface flows, etc. The main feature of these flows is their unsteadiness, both with respect to

flow patterns and to the shape of the boundaries. Flow inside moving-wall channels results in transient and complex flow

phenomena. The complexity of these flows is mainly due to the moving boundaries of the domain and the interaction

between the moving wall and the flowing fluid. Occurrence of flow detachment from the wall may result in oscillatory

flow motion downstream of the moving wall.

The pressure and shear patterns across a narrowing mimic the clinical situation of arterial narrowing due to

atherosclerotic plaque. Hemodynamic variables are known to have significant clinical applications. For example, clot

formation in narrowed arteries has been observed to occur in both areas of high shear and flow stagnation. Qualifying

and quantifying these hemodynamic variables and correlating them with clinical observations may prove to be very

valuable for clinical diagnosis and prevention of thrombosis. The importance of this study arises from the fact that heart

disease is one of the leading causes of death in the west. The American Heart Association (AHA) reports that thenumber one killer in the US is cardiovascular disease which claimed 871 500 lives in 2004 (36.3 percent of all deaths).

Coronary heart disease caused about 452 300 deaths in 2004 and is the single leading cause of death in America, today.

Coronary heart disease is caused by atherosclerosis, the narrowing of the coronary arteries due to fatty build ups of

plaque. It is likely to produce angina pectoris (chest pain), heart attack, or both. The narrowed arteries create very

irregular flow conditions that can exacerbate acute coronary thrombosis.

In relation to the development of computational models for flow in diseased human carotid arteries, Younis et al. [1]

have recently reported a simulation of flow in an exact replica of a diseased human carotid artery. Their three-

dimensional transient simulation has revealed the presence of complex flow structures. They have observed an unsteady

flow behavior inside the artery even though they maintained a steady inlet condition.

Experimental investigation of 2-D flow in a closed channel with an asymmetric oscillating constriction was carried

out by Stephanoffet al.[1] who observed a train of waves appearing in the core flow downstream of the constriction.

Pedley and Stephanoff [2] conducted an experimental study for 2-D flow in a channel with moving indentation in one

wall. They considered a steady inflow while moving the indentation in and out periodically. They found a vortex wave

for both viscous and inviscid flows, but with a complex pattern in the viscous case. In recent years, increased

computational power has facilitated many studies of unsteady incompressible flow with substantial flow complexity.

Methods using moving grids for simulating unsteady incompressible flows with moving boundaries have been reported.

Ralph and Pedley [3, 4] studied numerically the problem of flow through a channel with moving indentation. A time-

dependent coordinate transformation was applied, in order to resolve the difficulties of specifying boundary conditions

arising from the moving wall. Rosenfeld and Kwak [5] used a finite volume fraction step method on moving grids to

compute a channel flow with moving indentation. Peric and Demirdzic [6] developed a finite volume method for

prediction of fluid flow in arbitrarily shaped domains with moving boundaries.

Luo and Pedley [79] performed a time-dependent simulation of a coupled flowmembrane problem, using the

Arbitrary Lagrangian Eulerian (ALE) and spine method to treat the moving boundary. A moving mesh method for the

computation of compressible viscous flow past deforming and moving aerofoils was developed by Gaitonde [10, 11]. A

sequence of body conforming grids and the corresponding grid speeds were required, where inner and outer boundariesof the grid moved independently. The interaction between fluid and rigid body motions was analyzed by Mendes and

Branco [12] using a finite element procedure. They incorporated the ALE method into a two-step projection scheme, and

assumed a 2-D incompressible viscous flow. Also, Anju et al. [13] presented a finite element analysis of a fluidstructure

interaction problem by the ALE method and a fractional step NavierStokes solver. The method was applied to analyze

flow around an oscillating rectangular cylinder. A three-dimensional steady Stokes flow in an elastic tube was studied by

Heil [14] using non-linear shell equations. Since the flow was steady, only the final equilibrium state was presented.

Lefrancois et al. [15] developed a finite-element model for studying fluidstructure interaction. An ALE formulation was

used to model the compressible inviscid flow with moving boundaries with large deformation. A new in vivo method

was designed by Maalej [16, 17] to study the blood hemodynamic effects on platelet kinetics in canine stenosed carotid

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arteries and the wall shear stress was calculated using a finite-difference scheme. Recently, Yong and Ahmed [17]

developed a general method for simulating fluid flow with moving and compliant boundaries on unstructured grids using

ALE approach. They adopted a new dynamic mesh method to handle the large deformation of the flow field.

Pulsation of blood flow is also an important factor dominating the unsteady flow phenomena in a cardiovascular

system, because it complicates the vortical flow under time-varying inflow and pressure conditions. Pulsatile blood flow

may show very different features between normal physiological and pathological situations, among different parts ofarterial system, or even at different tiniest, e.g., still or exercising under normal physiological conditions [19]. With

regard to the pulsation effect on the vortical blood flow many points still remained unclear. The velocity distribution

resulting from its oscillatory blood flow has been extensively studied. The classical works of Womersley, Uchida,

Atabek, and Chang gave the fundamentals of oscillatory flow field theory.Mirsa and Sing [20] investigated the pulsatile

flow of blood through arteries by treating the blood vessel as a thin walled anisotropic, non linear viscoelastic,

incompressible material and blood as an incompressible Newtonian fluid whose motion is non linear. A numerical study

on pulsatile non-Newtonian flow characteristics in a three-dimensional Human Carotid Bifurcation model was carried

out by Perkfold et al. [21]. They considered both Newtonian and non-Newtonian behavior of the blood. The comparison

between Newtonian and non-Newtonian fluid models showed no change in the essential flow characteristics; however a

minor difference was found in the secondary flow. Xu et al. [22] predicted the three-dimensional flows through canine

femoral bifurcation models by numerically solving the time-dependant three-dimensional NavierStokes equations. They

considered both Newtonian fluid and non-Newtonian fluid obeying the power law. They found that the non-Newtonian

characteristics might not be an important factor in determining the general flow patterns for these bifurcations, but could

have logical significance. He and Ku [23] studied the unsteady entrance flow development in a straight tube. They

observed the variations in the entrance length during the pulsatile cycle. The effect of blood velocity pulsations on

bioheat transfer in 2-D straight rigid blood vessel was numerically studied by Oana and Scott [24]. Their results showed

that the pulsating axial velocity produces a pulsating temperature distribution. Stroud et al. [25] carried out a numerical

analysis of flow through a severely stenotic carotid artery bifurcation. They considered both steady and pulsatile flow

conditions for different Reynolds numbers. They found that both dynamic pressure and wall shear stress were very high,

proximal to the stenosis throat in the internal carotial theory. They also observed vortex shedding downstream of the

most severe occlusion. Pulsatile turbulent flow in stenotic vessels was numerically modeled by Sonu and Steven [26]

using the Reynolds-averaged NavierStokes equation approach.

Based on the cited literature, it can be concluded that no work has been published on the subject of pulsatile blood

flow in an axisymmetric tube with a moving indentation. In the present study, a 2-D numerical model is used to simulate

the time-dependent flow in a wall deforming channels. The model is first used to simulate the flow in a 2-D planar

channel with a moving indentation; for validation against the published experimental and numerical results by Pedley etal. [2, 3]. The model is then extended to simulate the steady and pulsatile blood flow in an axisymmetric tube with an

indentation moving at a frequency of Hz1=wf . This frequency is chosen close to typical heart beat frequencies. The

model will be used to investigate and compare the flow behavior between a pulsatile blood inflow and a steady inflow

conditions.

2. PROBLEM DESCRIPTION AND MATHEMATICAL MODEL

2.1. Two-Dimensional Channel Flow with a Moving Indentation

2.1.1. Description of the Problem

The first problem considered is that of 2-D flow in a channel with a moving indentation as shown in Figure 1. The

shape of the indentation is the same as that considered by Pedley and Stephanoff [2]. The time varying height of the

indentation is given by:

3

31

1

2

0

0

)]}(tanh[1){(5.0

)(

),(

xxfor

xxxfor

xxfor

xxath

th

txyw

>

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first quarter of the period and goes on advancing but decelerates in the second quarter of the period. It accelerates

towards the wall for the third quarter of the period, and then decelerates back to its flush position in the final quarter of

the period. The flow downstream in the channel is therefore accelerated for the first and third quarter of the period and

decelerated for the second and final quarter of the period.

Figure 1. Geometry of planar collapsible channel (not to scale) b=1cm, l1 = 10cm, l2 = 18 cm

2.1.2. Governing Equations

In order to write the governing equations in dimensionless form, we first introduce the following dimensionlessvariables,

02

000

,,,,,t

t

b

yY

b

xX

U

pP

U

vV

U

uU ======

(3)

where u and v are the dimensional velocity components in x andy directions respectively,p is the pressure, tis time, U0

is the average velocity at inlet, is the density of the fluid, and t0 is the indentation motion period. The governingequations can be expressed as:

Conservation of Mass:

0=

+

y

V

x

U

(4)

Conservation of X-momentum:

+

+

=

+

+

2

2

2

22 1

Y

U

X

U

ReX

P

Y

UV

X

UU

U

Re

(5)

Conservation of Y-momentum:

+

+

=

+

+

2

2

2

22 1

Y

V

X

V

ReY

P

Y

VV

X

VU

V

Re

(6)

where Re is the Reynolds number based on the channel height, St is the Strouhal number and is the frequencyparameter respectively defined as

StRetU

bSt

bURe ===

,,

00

0 (7)

where is the kinematic viscosity of the fluid.

At the start of fluid motion ( = 0), the flow is assumed to be steady and fully developed in the entire flow domainand the velocity components are given by

( , ) 6 (1 ); ( , ) 0 0U X Y Y Y V X Y at = = = (8)

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The boundary conditions are expressed in terms of the conditions at the inlet, at the walls, and at the exit of the flow

domain. The velocity profile at inlet (X= l1/b) is assumed to be parabolic and invariant with time, thus

0),,();1(6),,( 11 ==== YblXVYYYblXU (9)

At the stationary wall (Y = 1), the no slip condition is applied while the velocity components at the moving

indentation are derived from Equations (1) and (2). These are expressed as:

0;0 == VU at Y=1 (10 a)

))2cos(1)](/([sec095.0 22

=

bxXabhab

VU ;

(10 b)

[ ]20.597

1 [ ( / )] (2 )St

V tanh ab X x b sinb

= at Y=yw/b and 0

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0002

000

,,,,,t

t

r

rR

r

xX

U

pP

U

vV

U

uU ======

(14)

where u and v are the dimensional velocity components inx and rdirection respectively,p is the pressure, tis the time,

U0 is the average velocity at the inlet, r0is the radius of the cylinder, is the density of the fluid, and 0t is the moving

indentation period. The governing equations are the continuity and NavierStokes equations that can be expressed as:

0)(1

=

+

R

RV

rX

U,

(15)

+

+

+

=

+

+

R

U

RR

U

X

U

ReX

P

R

UV

X

UU

U

Re

1212

2

2

22

,

(16)

+

+

+

=

+

+

22

2

2

22 121

R

V

R

V

RR

V

X

V

ReR

P

R

VV

X

VU

V

Re

(17)

where Re is the Reynolds number based on the channel height, is the Wormersley, parameter, which are defined as

0

0 ,2

rUr

Re == (18)

where is the kinematic viscosity of the fluid.

At the start of motion (t = 0), a steady Poiseuille flow with an average velocity U0 is assumed at the entrance of the

flow domain, thus

2( , ) 2(1 ); ( , ) 0 0U X R R V X R at = = = (19)

Boundary Conditions

Inlet conditions:

At the inlet, a parabolic velocity profile is introduced. i.e., atX = l1/b

0),,();1(2),,( 12

1 ==== RblXVRRblXU (20)

Wall

At the non-moving wall, the velocity components are given atR = 1 andX>X3

0;0 == VU (21 a)

At the moving indentation, the velocity components are derived using Equations (1) and (2) and are given by:

))2(1)](/([095.0 0202

0 CosrxXarhSecar

VU

=

(21 b)

0 2 00

0.5971 [ ( / )] (2 )

StV Tanh ar X x r Sin

r

=

(21 c)

atR = rw/b and 0

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3. METHOD OF SOLUTION

The computational fluid dynamics software package Fluent 6.1 with dynamic mesh model (released in 2005 by

Fluent Inc.) was used to solve the NavierStokes equation for the two-dimensional unsteady flow. A non-uniform

unstructured grid (shown in Figure 3) is used for numerical simulation. The code is based on the Finite Volume Methodthat is well documented in the literature. In this method, the flow domain is divided into sub-domains or control volumes

with one control volume around every grid point. Each differential equation is integrated over this control volume toyield the discretized equation. Thus, the discretized equation represents the same conservation principle over a finite

region as the differential equation does over an infinitesimal region [28]. This direct interpretation of the discretized

equation makes the method easy to understand in physical terms; the coefficients in the equation can be identified, even

when they appear in a computer program, as familiar quantities such as flow rates, pressure forces, areas, volumes,diffusivities, etc. The coupling of pressure and velocity is achieved using the Semi-Implicit Method for Pressured-Linked

Equation (SIMPLE) algorithm [29]. Second order upwind descritization was used for the momentum equation.

Figure 3. Section of the planar channel grid (55 637 nodes) at two time instants (a) = 0 and (b) = 0.5

The motion of the moving indented wall was described by means of a user-defined functions (UDF) using the

dynamic mesh model. The UDF was written in C programming language. Spring-based smoothing method is used to

update the volume mesh in the deforming regions subjected to the wall motion. In the spring-based smoothing method,

the edges between any two-mesh nodes are idealized as a network of interconnected springs. The initial spacing of the

edges before any boundary motion constitutes the equilibrium state of the mesh. A displacement at a given boundarynode will generate a force proportional to the displacement along all the springs connected to the node [30].

4. RESULTS AND DISCUSSION4.1. Flow in a Two-Dimensional Channel with a Moving IndentationBenchmark Comparison

4.1.1. Grid Independence Test

A number of grid independence tests were carried out in the present study. Three meshes with different sizes were

tested: Mesh 1 with 10 173 nodes; Mesh 2 with 38 310 nodes and Mesh 3 with 55 637 nodes. A section of the fine grid

(Mesh 3) is shown in Figure 3 for the time instants = 0 and = 0.5. The grid is refined downstream of the indentationwhere the flow is affected by its movement. Figure 4 shows the comparison of longitudinal velocity calculated on

different grids at the centre of the channel at = 0.5 and = 0.7 (both obtained with time step, t = t0/1000). Thesecomparisons show that there is no significant difference in the longitudinal velocity between the meshes with 38 310 and

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55 637 nodes. Figure 5 shows the comparison of the wall shear stress calculated on different grids at = 0.5 and = 0.7(both obtained with t = t0/1000). With the coarse grid (10 173 nodes), the wall shear stresses were underestimated andsmooth streamlines could not be obtained. Hence the finer Mesh 3 is adopted.

Figure 4. Comparison of X-Velocity component using three different grids at two different times: (a) = 0.5 and (b) = 0.7;

using a time step t = t0 /1000; for the planar channel

Figure 5. Comparison of predicted wall shear stress using three different grids at two different times: (a) = 0.5

and (b) = 0.7; using a time step t = t0 /2000; for the planar channel

4.1.2. Flow Development

The volume flow rate Q0 per unit depth upstream of the moving indentation is assumed invariant. The first case

considered is that ofRe=507 that corresponds to Q0 = 0.0306 m2/min. Figure 6(i)(al) shows the predicted velocity

streamlines at times ranging from = 0.1 to = 1. Only the section downstream of the indentation is shown since theflow upstream is not affected much by its movement. The flow development is found to be the same as that of Pedley

[11,12] and a brief summary is presented here. A train of vorticity waves is generated downstream of the indentation,

every cycle. At some time between = 0.2 and = 0.25, separation occurs in the lee of the indentation, and the resultingvortex grows rapidly. A second vortex of opposite sign forms on the upper wall at some distance downstream of the first

and a third appears still further downstream on the lower wall and so on until there is a sequence of such vortices of

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alternating sign, bounded by a wavy core flow. At time = 0.6, five vortices (marked A, B, C, D, and E) appeareddownstream of the indentation as shown in Figure 6(i)(g). As time increases not only the number of vortices increases

but also the subsequent extent of existing vortices and the amplitude of the core waviness also increases. As the

indentation recedes late in the cycle, the vortices shrink in size and strength and are swept downstream of the indentation,

but at = 1, there is again vorticity of uniform sign at each wall. Figure 6(ii) shows the instantaneous streamlinesobtained by the Pedley at different instants. It can be observed that a qualitative agreement exists between the present

numerical results (Figure 6(ii)) and Pedleys [2]. It should be noted that the two sets of figures are plotted on differentscales, thus only visual comparison is possible.

Figure 6. Predicted velocity streamlines downstream of indentation at various nondimensional times for the planar channel

The velocity profiles at inlet are parabolic throughout the cycle. As the indentation grows, the profiles becomedistorted due to the upward movement of the indentation. Separation occurs at the sections where both positive and

negative velocity vectors exist. When the indentation moves back, the profiles tend to go back to their original shapes.

This trend ultimately creates a wavy core flow along the channel.

Figure 7 shows the longitudinal velocity profiles at different cross sections of the channel at various time intervals.Figures 8(a, b) and Figures 9(a, b) show the variation of the wall shear stress on the lower (indented) and upper

(unindented) wall respectively at various time intervals during the cycle. They indicate the strength of the vortices, the

positions of separation and reattachment (change of sign in wall shear stress) and thus the movement of vortices along

the wall. The first four vortices marked A, B, C, and D in Figure 6(i)(f), are the strongest; vortices E and F are relatively

weak. Vortex A (marked in Figure 6(i)(f)) is strongest at = 0.5 (maximum shear stress occurs then); peaks in othervortices occur later. The shear stress variation along the indentation and the opposite wall also indicates an acceleration

of the flow in this region when the indentation is moving inwards ( < 0.5) and a deceleration when it is retracting ( >0.5 > 0). Figures 8 and 9 also show that the steadiness of the flow upstream of the indentations is hardly affected by the

indentation motion.

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A quantitative comparison between the present results and the experimental and numerical result by Pedley [2, 3] isshown graphically in Figure 10. This comparison is shown as the time evaluation of the positions of crests/troughs of the

wave after the indentation. Wave-crest positions as functions of time are obtained from the turning points in the axial

direction of stream function at the centre of the channel. According to the Pedley and Stephanoff, the abscissa is definedas

bStxxx /)10)((*3/1

1=

Figure 7. Velocity profiles at different cross sections of the collapsible tube at various time intervals (a) = 0.25, (b) = 0.5,

(c) = 0.75, (d) = 1 for a 2D planar channel

It can be observed from the figure, that there is a good agreement between the present numerical result andexperimental result, with discrepancies within the range of the experimental scatter.

4.2. Flow in an Axisymmetric Cylindrical Tube with a Moving Indentation

4.2.1. Grid Independence Test

A number of grid independence studies were carried out for the axisymmetric case. An unstructured grid has been

used for the current computation. Three meshes with different sizes were tested: Mesh 1 with 9 194 nodes; Mesh 2 with

10 828 nodes and Mesh 3 with 55 425 nodes. A section of the Mesh 3 at = 0 and = 0.5 is shown in Figure 11. Themain features of the flow were captured with the coarse grid but the details of the flow could not be captured using this

grid. Hence Mesh 3 is used in the computational scheme.

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Figure 8. Predicted wall shearstress distributions during one cycle along the indented (collapsing) wall (a) = 0.5 and (b)

= 0.6 to 1, for the 2D-planar channel

Figure 9. Predicted wall shearstress distribution during one cycle along the unindented wall (a) = 0.1 to 0.5 and (b) = 0.6

to 1, for the 2D-planar channel

4.2.2. Time Dependence Test

Two time increments were used for the coarse grid calculations: t = t0/1000and t = t0/2000. Figure 13 showscomparison of pressure drop at various locations and for different time increments. The pressure drops along the

indentation are plotted against time at different points and are shown in Figure 12. The figure shows insignificant change

between the two time increments. Hence, the fine grid calculations were carried out only with t = t0/1000.

In the present work, two different cases considering the two different fluid properties have been simulated.

4.2.3. Case I: Steady Flow at Inlet

In many studies, blood is assumed to be a Newtonian fluid and the same assumption is adopted in this work. Non-

Newtonian flow models arepresently under investigation. For isothermal conditions, the density and dynamic viscosity

of blood are taken as = 1060kg/m3 and = 3.7110

3N.s/m2, respectively. The frequency of wall oscillation was set atvalue 1Hz representative of biomedical flows. The average velocity was chosen to give a Reynolds number ofRe = 200.

For thisRe, the volumetric flow rate upstream of the moving indentation, Q0is 6.6104. The other parameters for this

case are St= 0.57 and = 13.4

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Figure 10. Comparison of predicted and experimentally observed position of wave crests and troughs corresponding to eddies

B, C and D as functions of time for the planar collapsible channel

Figure 11. Section of the axisymmetric collapsible tube grid (55 425 nodes) at (a) = 0 and (b) = 0.5

Figure 12. Points at which the pressure is monitored against time for the axisymmetric collapsible tube

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Figure 13. Comparison of pressure drop histories (a) P1-P2 and; (b) P2-P3; computed with two different time increments for

the axisymmetric collapsible tube

Figure 14(ar) show the velocity streamlines at = 0.1 to 1. In the first quarter of the period, as the indentationaccelerates into the tube, the fluid is squeezed towards the axis of the tube (Figure 14 (a)(II)). At some time between, =0.2 and 0.25 (Figure 14(b)(II) and 14(c)(II)) separation occurs in the lee of the indentation and a vortex (labeled A)

appears. The resulting vortex A grows in size and strength. As a result, a permanent region of flow reversal exists distal

to vortex A, downstream of the indentation.

Figure 14. Velocity streamlines

at different time instants inside

the axisymmetric collapsible

tube for steady-inlet blood flow

(case I)

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At = 0.45 a second vortex (labeled B), of opposite sign, forms near vortex A (Figure 14(g)(II) and the vortex Agrows further in size and moves towards the axis of the tube. At = 0.55 (Figure 14(i)(II), vortex A almost blocks thecore flow and the thickness of the reversal flow region increases, downstream of the indentation. At = 0.65 (Figure14(k)(II), the reverse flow from the downstream of the indentation sweeps out vortex B. As the indentation recedes back

to its original position, the reverse flow region at the downstream moves back into the indentation and collides with the

upstream coming flow. Hence, at = 1 (Figure 14(r)(II)), the flow is fully disturbed in the indentation.

Figure 15(a) and 15(b) show the variation of the wall shear stress on the upper (indented) wall at various time

intervals during the cycle. They indicate the strength of the vortex; the position of separation and the position of

reattachment (change of sign in wall shear stress) and thus the movement of vortex along the wall. Vortex separation can

be observed from these figures. The wall shear stress upstream of the indentation is dominated mostly by the forwardflow but downstream of the indentation it is strongly affected by the vortices. The shear stress variation along the

indentation also indicates an acceleration of the flow in this region when the indentation is moving inwards ( < 0.5) anda deceleration when it is retracting ( > 0.5). High wall shear stresses are observed downstream of just downstream of theindentation. The vortex forming downstream of the indentation gives high velocity gradients, which results in high wall

shear stress.

Figure 15. Wall shear stress distributions during one complete cycle along the indented wall of the axisymmetric collapsible

tube: (a) = 0.1 to 0.5 and (b) = 0.6 to 1, for a steady-inlet blood flow (case I)

Figure 16 shows the history of pressure drops (P1P2 and P1P3) during one complete flow cycle. Both pressuredrop histories are similar in shape but different in magnitude. For the first quarter of the cycle, the pressure drops

increase with the time, while for the second quarter; the pressure drops decrease with time. As the indentation starts

receding, the pressure differences increase again with time resulting in a sinusoidal history similar to the imposedmoving wall condition. The pressure differences P1P2 and P1P3 are positive in the first and last thirds of the cycle;

indicating a pressure drop. During the second third of the cycle, the pressure drops P1P2 and P1P3 are both negative

indicating a pressure rise across the collapsing part of the tube.

4.2.4. Case II:Pulsatile Flow at the Inlet

Eventually we are interested in simulating pulsatile blood flow, which represents a more realistic model ofphysiological blood flows. In this case a sinusoidal time-varying wave is combined with a steady velocity are imposed atinlet of the collapsible axisymmetric tube. Hence the average (over the inlet) velocity imposed at the inlet (Figure 17)

can be written as:

)2()( 0 tsinUUinletU m +=

Note here that the frequency of the wall motion is the same as the frequency of the sinusoidal portion of the inlet

velocity. The average inlet velocity selected, assumed values ofU0 = 0.07m/s and Um = 0.0455m/s, resulting in a time-

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averaged volume flow rate of Q0=6.6x104m3/min and a peak volume flow rate of 1.089103m3/min. In order to take

into account the no-slip flow behavior at the walls, the final form of inlet velocity is given by:

))2()()/(1(),,( 02 tsinUURytyinletU m+=

Figure 16. History of pressure drops (P1-P2 and P1-P3) during one flow cycle inside the axisymmetric collapsible tube with a

steady inlet blood flow (case I)

Figure 17. Average (over area) velocity waveform during one cycle of pulsatile blood flow at the inlet of the axisymmetric

collapsible tube

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Figure 18(ar) show the velocity streamlines at = 0.1 to 1. In the first quarter of the period, as the indentationaccelerates into the tube, the fluid is squeezed towards the axis of the tube (Figure 18 (a)(II). At some time between, =0.2 and 0.25 (Figure 18(b)(II) and 18(c)(II) separation occurs in the lee of the indentation and a vortex (labeled A)

appears. The resulting vortex grows in size and strength. As a result a permanent region of flow reversal exists distal to

vortex A, downstream of the indentation. At = 0.4 a second, vortex (labeled B) of opposite sign forms near vortex A(Figure 18(f)(II)) and the vortex A grows in size and moves towards the axis of the tube.

Figure 18. Velocity streamlines at various times for unsteady (pulsatile) blood flow (case II) at the inlet of the axisymmetric

collapsible tube

At time = 0.5, a small recirculation zone appears upstream of the indentation and exists until time level = 0.75because of the decelerating flow. As the flow accelerates through the end of the pulse, the recirculation zone upstream of

the indentation is pushed into the indentation. Simultaneously, at = 0.5, vortex A splits into a pair of co-rotatingvortices (vortex A and vortex C) downstream of the indentation (Figure 18(h)(II)). As increases these vortices grow intheir size and strength and moves towards the axis of the tube. Furthermore, the extent of the reversal flow region, at the

downstream of the indentation increases with the time.

At = 0.55 (Figure 18(i)(II)), vortices A and C almost block the core flow in the tube. At = 0.65 (Figure 18(k)(II)),the reverse flow from the downstream of the indentation sweeps out vortex B and vortex C. As the indentation recedes

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back to its original position, the reverse flow region at the downstream moves back into the indentation and collides with

the upstream forward flow. Hence, at = 1 (Figure 18(r)(II)), the flow is fully disturbed in the indentation region.

Figure 19(a) and 19(b) show the variation of the wall shear stress on the upper (indented) wall at various timeintervals during the cycle. They indicate the strength of the vortex; the position of separation and the position of

reattachment (change of sign in wall shear stress) and thus the movement of vortices along the wall. Vortex doubling can

be observed from these figures. The wall shear stress upstream of the indentation is dominated mostly by the forwardflow but downstream of the indentation it is strongly affected by the vortices. The shear stress variation along the

indentation also indicates an acceleration of the flow in this region when the indentation is moving inwards ( < 0.5); anda deceleration when it is retracting ( > 0.5). The shear stresses are higher for the pulsatile flow compare to steady flow.

Figure 19. Wall shear stress distribution during one complete cycle along the indented wall (a) = 0.1 to 0.5 and (b) = 0.6 to

1, for the pulsatile blood flow at the inlet of the axisymmetric collapsible tube (case II)

Figure 20(ad) shows the comparison of axial velocity profiles of pulsatile flow and steady flow at different cross

sections and time intervals. For the pulsatile flow case a recirculation zone is observed upstream of the indentation

between the time levels = 0.5and = 0.75 because of the decelerating flow between these time levels. The velocitiesare maximum for the both the cases after the indentation (6.5R). The velocity profiles also show the separation and

recirculation zones. From the figure it can also be seen that the reverse flow dominates at the downstream of the

indentation as the increases.

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The main difference that exists between the pulsatile flow and steady flow is vortex doubling. Doubling of vortex is

not observed in the case of steady flow while vortex doubling is observed in the case of pulsatile flow. The wall shear

stresses obtained for the pulsatile flow are greater than that of the steady flow because of the higher shear rate.

Figure 20. Comparison of axial velocity profiles at various time instants (a) = 0.25 (b) = 0.5 (c) = 0.75 and (b) = 1, for

both inlet velocity conditions: pulsatile (- - -) and steady () flow; at different section of the axisymmetric collapsible tube

(cases I & II)

Figure 21 shows the history of pressure drops (P1P2 and P1P3) during one flow cycle. Both pressure drop historiesare similar in shape but different in magnitude. For the first quarter of the cycle, the pressure drops increase with time,

while for the second quarter; the pressure differences decrease with time. As the indentation starts receding, the pressure

drops increase again with time resulting in a history similar to the imposed moving wall condition. The pressure drops

P1P2 and P1P3 are positive in the first and last thirds of the cycle. During the second third of the cycle, the pressure

drops P1P2 and P1P3 are both negative indicating a pressure rise. The pulsatile flow case has higher-pressure drops incomparison to the steady inlet flow case.

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Figure 21. History of pressure drops (P1-P2 and P1-P3) during one flow cycle inside the axisymmetric collapsible tube with a

pulsatile inlet blood flow (case II)

5. CONCLUSIONS

A dynamic-mesh numerical model has been developed to simulate the time dependent flow in a 2-D channel with amoving indentation. A non-uniform unstructured grid was used for the numerical simulation. The above model was first

validated for a 2-D planar channel against Pedleys [2] experimental work. The results showed very good agreement withthe experimental results. The model has been extended to simulate a Newtonian-model blood flow in an axisymmetric

tube with an indentation moving at a frequency of Hz1=wf with two variations of inlet conditions: steady and pulsatileflow inlets. For the case of steady flow at the inlet of the tube, no recirculation zone was observed upstream of theindentation while a small recirculation zone was observed for the pulsatile inlet flow case upstream of the indentation. In

addition, the pulsatile inlet flow condition has enhanced the vortex doubling downstream of the indentation. Reverse

flow was found to be dominating downstream of the indentation for both the cases. The wall shear stresses and the

pressure drops obtained for the pulsatile case were higher than that of the steady inflow case.

6. ACKNOWLEDGMENT

The support of King Fahd University of Petroleum & Minerals during the course of this study is greatly appreciated

and acknowledged.

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[17] N. Maalej and J.D. Folts, Increased Shear Stress Overcomes the Antithrombotic Platelet Inhibitory Effect of Aspirinin Stenosed Dog Coronary Arteries, Circulation, 93(1996), pp. 12011205.[18] Y. Zhao and A. Forhad, A General Method for Simulation of Fluid Flows with Moving and Compliant Boundaries

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[19] D.A. McDonald,Blood Flow in Arteries, 2nd ed. London: Arnold 1974.[20] J.C. Mirsa and S.I. Singh, A Study on the Nonlinear Flow of Blood Through Arteries, Bull. Math. Biol., 49(1987),

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8. NOMENCLATURE

b Depth of unindented channel

maxh Maximum blockage of the channel at 5.0=

p Dimensional pressure

P Non-dimensional pressure

Q0 Volume flow rate

r Radius

0r Radius of unintended cylinder

R Non-dimensional radius

t Dimensional time

t Time Step

0t Indentation motion period

0U Average velocity at the inlet or characteristic velocity

U Non-dimensional velocity component inx -direction

u Dimensional velocity component inx-direction

v Dimensional velocity component iny-direction

maxv Maximum velocity

V Non-dimensional velocity component iny-direction

X Non-dimensionalx co-ordinate

Y Non-dimensionaly co-ordinate

Greek Symbols

Non-dimensional time

w Wall Shear stress

Wormersley parameter

Viscosity

Kinematic viscosity

Density

Radial frequency

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