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Particle Tracking in the Circle of Willis

Pim van Ooij

A report of work carried out at

the Centre of Bioengineering

University of Canterbury

TU Eindhoven

BMTE05.40

June 2005

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Summary

An ischemic stroke is most commonly caused by a blood clot that blocks an artery in the brain, resulting in oxygen deficit in brain tissue and accompanying brain damage. To examine the route a blood clot travels in the Circle of Willis, the ring-like structure of blood vessels that distribute the blood flow to the cerebral mass, the influence of outlet diameter, bifurcation angle and mass flux of five different geometries on particle trajectory is studied, with the intention to predict particle trajectories in the Circle of Willis. The finite volume package Fluent is used, which supports two particle models: the Discrete Phase Model (DPM) and the Macroscopic Particle Model (MPM). The difference between the models is the degree of interaction between the fluid and the particles. The results show a larger effect of outlet diameter and bifurcation angle on the particle trajectory for MPM than for DPM. For DPM, there is a linear relation between mass flux and density of particles. The higher the mass flux, the higher the density of particles in the flow. Since momentum transfer between fluid and particles for MPM is larger, the relation between mass flux and amount of particles is not as obvious. A simulation using DPM of particle tracks in the Circle of Willis produces predicted results. Unfortunately it was not possible to verify the prediction of particle tracks in the Circle of Willis for MPM.

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Contents

1 Introduction 3

2 Theory and Methodology 4

2.1 Emboli . . . 4 2.2 Circle of Willis . . . 4 2.3 Geometry model . . . 6 2.3.1 Geometries . . . 6 2.3.2 Circle of Willis . . . 8 2.4 Fluids model . . . 9 2.4.1 Geometries . . . 9 2.4.2 Circle of Willis . . . 10 2.5 Particle model . . . 10

2.5.1 Discrete Phase Model . . . 11

2.5.2 Macroscopic Particle Model . . . 13

3 Results 16 3.1 Geometries . . . 16 3.1.1 General . . . 16 3.1.2 Outlet diameter . . . 18 3.1.3 Bifurcation angle . . . 23 3.1.4 Mass flux . . . 28 3.2 Circle of Willis . . . 33 3.2.1 General . . . 33 3.2.2 Outlet diameter . . . 33 3.2.3 Bifurcation angle . . . 34 3.2.4 Mass flux . . . 35 4 Discussion 36 4.1 Geometries . . . 36 4.1.1 Outlet diameter . . . 36 4.1.2 Bifurcation angle . . . 37 4.1.3 Mass flux . . . 38

4.1.4 Differences between DPM and MPM . . . 41

4.2 Circle of Willis . . . 42

5 Conclusion 43

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CONTENTS CONTENTS

A Discrete Phase Model Manual 46

A.1 Introduction . . . 46

A.2 Limitations . . . 47

A.3 Overview of Discrete Phase Modeling Procedures . . . 48

A.4 Equations of Motion of Particles . . . 49

A.5 Coupling Between the Discrete and Continuous Phases . . . 52

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Chapter 1

Introduction

The cardiovascular disease called stroke is the third largest cause of death, af-ter heart diseases and all forms of cancer. There are two types of strokes: the ischemic stroke and the hemorrhagic stroke. Although the latter type of stroke has a much higher fatality rate than the former, this project will focus on the former type of stroke which causes 70-80 percent of all strokes. An ischemic stroke occurs when a blood vessel to the brain is blocked by an abnormal blood clot, called embolus. Due to the following deprivation of oxygen, brain tissue dies which causes irreparable brain damage or death.

The embolus most often blocks an artery in the Circle of Willis, a ring-like struc-ture of blood vessels found beneath the hypothalamus at the base of the brain, which main function is to distribute oxygen-rich blood to the cerebral mass. To know more about the route a blood clot travels in the brain, simulations of particle trajectories in the Circle of Willis are needed, which is the main goal of this project.

Five different geometries are used to study the influence of artery diameter, bifurcation angles and mass flux on the particle trajectories. Mass flux is varied by prescribing different boundary conditions on the outlets of the geometries. With this information the particle tracks in the Circle of Willis are predicted. This prediction is then verified by a simulation of particle tracks on the Circle of Willis.

The null hypothesis of this project is that there is no relationship between the geometry of the Circle of Willis and the distribution of the particles. How-ever, it is expected that more mass flux through an artery causes more particle flow through this artery. Also, for equal pressure gradients, a large diameter of an artery is supposed to cause more mass flux through this particular artery, bringing more particles with it, then through an artery with a smaller diameter. Furthermore, it is assumed that a smaller bifurcation angle causes more mass flux through the adjacent artery, causing more particle flow. By comparing the resulting particle distribution on the outlets with the particle distribution on the inlet of the geometries, it is possible to predict particle tracks in the Circle of Willis, so that more knowledge is obtained about the trajectory of blood clots in the brain.

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Chapter 2

Theory and Methodology

2.1

Emboli

Bloods clots are created in so-called thromboembolic conditions, which are usu-ally caused twofold [1]. First, a roughened surface, caused by arteriosclerosis, infection or trauma, is likely to initiate the clotting process. Secondly, clotting occurs in blood that flows very slowly through blood vessels because small quan-tities of thrombin and other procoagulants are always being formed. These are generally removed from the blood by the macrophage system, mainly the Kupf-fer cells of the liver. The concentration of procoagulants rises often high enough to initiate clotting, but when blood flows rapidly, they are mixed with large quantities of blood and are removed during passage through the liver. When an abnormal blood clot develops in a blood vessel, it is called a thrombus. Con-tinued flow of blood past the clot is likely to break it away from its attachment resulting in a clot that flows along with the blood. These free flowing clots are known as emboli, which do not stop flowing until they come to a narrow point in the circulatory system. Thus, emboli originated in the venous system and in the right side of the heart flow into the vessels of the lung and cause pulmonary arterial embolism. Emboli originated in large arteries or in the left side of the heart will plug smaller arteries of arterioles in the brain, kidneys or elsewhere.

2.2

Circle of Willis

The Circle of Willis is a ring-like structure of blood vessels in the brain, displayed in figure 2.1 and distributes oxygenated blood throughout the cerebral mass.

It is estimated that among the general population, 50% have a complete Circle of Willis [2]. Variations include underdeveloped of even absent blood vessels. This can present a health risk, mostly for ischaemic stroke, while an individual with one of these variations may suffer no ill effects. Examples of these variations are the fetal P1, where the P1 section of the Posterior Cerebral Artery (PCA) is underdeveloped, and the missing A1, where the A1 section of the Anterior Cerebral Artery is missing, as displayed in figure 2.2. In these figures are displayed the abbreviations of the names of the arteries in the Circle of Willis. These will be briefly discussed.

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Theory and Methodology 2.2 Circle of Willis

Figure 2.1: The Circle of Willis, frontal

(a)Fetal P1 (b)Missing A1

Figure 2.2: Variations in the Circle of Willis

(ICA), the Vertebral Arteries (VA) and the Basilar Artery (BA) are the vessels that transport blood into the Circle of Willis and are called afferent arteries. The Middle Cerebral Artery (MCA), Posterior Cerebral Artery (PCA) and the Anterior Cerebral Artery (ACA) are the vessels that transport blood from the Circle of Willis and are called efferent arteries. The ICA is connected with the PCA via the Posterior Communicating Artery (PCoA) and the Left ACA is connected with the Right ACA via the Anterior Communicating Artery (ACoA).

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Theory and Methodology 2.3 Geometry model

2.3

Geometry model

2.3.1

Geometries

To study the influence of outlet diameter, bifurcation angle and mass flux on particle trajectories, five different geometries are used. The geometry meshes are created by the meshing software package GAMBIT and comprise of approx-imately 300,000 tetrahedral volumes. The different geometries all consist of one inlet and two outlets. The diameter of the inlet is 5 millimeters. From inlet to outlet the geometries measure approximately 45 millimeters and from outlet to outlet the geometries measure approximately 30 millimeters, see figure 2.3.1.

Figure 2.3: Measures of geometry 1

The diameters of the outlets vary, as well as the bifurcation angles of the outlets. The geometries are displayed in figure 2.4. The different features of the geometries are summarized in table 2.1. For bifurcations of vessels in the human body, a physiological relation exist between outlet diameter and bifurcation angle and is given by Zamir [3]:

cosθ1= (1 +α3)4/3+ 1α4 2(1 +α3)2/3 (2.1) cosθ2= (1 +α3)4/3+α41 2α2(1 +α3)2/3 (2.2) with α=a2 a1 (2.3) where θ1 and θ2 are the bifurcation angles, a1 is the diameter of outlet 1

anda2 the diameter of outlet 2. The diameters of the geometries are randomly

chosen. These equations are used to calculate the bifurcation angles of geometry 5. Then geometry 4 is taken as the opposite of geometry 5. The properties of geometry 5 are calculated by equations 2.1 and 2.2 and can therefore actually be found in the human body, while geometries 1, 2, 3 and 4 are hypothetical geometries.

In figure 2.4 the right outlets of the geometries are outlets 1, the left ones outlets 2.

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Theory and Methodology 2.3 Geometry model

(a)Geometry 1 (b)Geometry 2

(c)Geometry 3 (d)Geometry 4

(e)Geometry 5

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Theory and Methodology 2.3 Geometry model

Table 2.1: Properties of the five different geometries Geometry D outlet 1 D outlet 2 Angle outlet 1 Angle outlet 2

mm mm o o 1 3 3 40 40 2 1.5 3 40 40 3 2.5 2.5 30 50 4 2 3 21.26 54.68 5 3 2 21.26 54.68

2.3.2

Circle of Willis

The 3D mesh of the Circle of Willis comprises of approximately 1 million tetra-hedral volumes. The mesh is displayed in figure 2.5. The diameters of the various arterial segments of the Circle of Willis were obtained from a popula-tion study of retrospective MRA scans [2]. The measurements and standard deviations are displayed in table 2.2.

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Theory and Methodology 2.4 Fluids model

Table 2.2: Circle of Willis Measurements

Artery Diameter Std Dev

mm mm

ACA - A1 Anterior Cerebral Artery - A1 2.33 0.22

ACA - A2 Anterior Cerebral Artery - A2 2.40 0.31

MCA Middle Cerebral Artery 2.86 0.17

PCA - P1 Posterior Cerebral Artery - P1 2.13 0.25

PCA - P2 Posterior Cerebral Artery - P2 2.10 0.21

ACoA Anterior Communicating Artery 1.47 0.17

PCoA Posterior Communicating Artery 1.45 0.31

BA - B1 Basilar Artery - B1 3.17 0.51

BA - B2 Basilar Artery - B2 3.29 0.44

ICA Internal Carotid Artery 4.72 0.26

Since the bifurcation angles in the Circle of Willis are not measured, a rough estimation of 90o is made for the bifurcation between the ICA and the PCoA, and between the ICA and the ACA. There is no angle between ICA and MCA. This estimation is done by rotating the Circle of Willis in Fluent and study its features by eye.

2.4

Fluids model

2.4.1

Geometries

The blood flow through the geometries is modeled as unsteady, incompressible and viscous. This means that the governing transport equations to be solved by Fluent are the continuity equation and the momentum equation:

Z A ρu·dA= 0 (2.4) Z V δu δt + Z A ρuu·dA=− Z A pIdA+ Z A η(¯)¯·dA+ Z V FdV (2.5)

whereu is the velocity vector,ρis the blood density, with a value of 1410 kg/m3,pis the pressure,ηis the fluid viscosity andFis the body forces vector on the fluid. V is a closed volume andAis the edge of a closed volume. ¯is the strain rate tensor and is represented as:

¯ = 1

2(∇u+∇u

T) (2.6)

In this project blood is simulated as a non-Newtonian fluid and therefore the Carreau-Yasuda model for the viscosity is implemented:

η−η∞

η0−η∞

= (1 + (λγ˙)a)n−a1 (2.7) where η∞ is the infinite shear viscosity, set at 0.0022 P a·s and η0 is the

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Theory and Methodology 2.5 Particle model

as 0.392 [2]. ˙γ is the strain rate magnitude, derived from the second invariant of the strain rate tensor, which for an incompressible fluid becomes:

˙ γ= 2p

¯

ijij¯ (2.8)

These equation properties are taken from Moore [2]. To study the influence of different mass fluxes on the particle trajectories, for each geometry, three different pressure profiles are prescribed on the outlets. In the first situation, the pressure on outlet 1 is the same as the pressure on outlet 2: 98mm Hg. In the second situation, a pressure of 95 mm Hgis prescribed on outlet 1, while the pressure on outlet 2 remains 98mm Hg. Finally, the pressure on outlet 1 returns to 98mm Hg, while the pressure on outlet 2 is set to 95mm Hg. The pressure on the inlet remains constant at 100mm Hgat all times. These values are an estimation of pressure differences in arteries of equal size in the human body. The boundary conditions on the outlets of the geometries are summarized in tabel 2.3. In this report, they will be referred to as boundary condition 1, boundary condition 2 and boundary condition 3.

Table 2.3: Boundary conditions on the geometries Boundary outlet Pressure

condition mm Hg 1 1 98 2 98 2 1 95 2 98 3 1 98 2 95

A time step of 0.01 seconds is used in all the simulations. The Reynolds number of the flows are defined on the inlets and are calculated by:

Re= 4 ˙m πdη∞

(2.9) with ˙mthe mass flux through the inlet anddthe diameter of the inlet.

2.4.2

Circle of Willis

All the equations mentioned in subsection 2.4.1 apply to the Circle of Willis as well. The boundary conditions on the inlet and outlets of the geometries are chosen as described in the previous subsection, thus they differ from the boundary conditions on the inlets and outlets of the Circle of Willis, since these are chosen as the mean values of the pressures in systole and diastole of arteries (afferent arteries) and veins (efferent arteries). These are summarized in table 2.4 and are equal for left and right arteries.

2.5

Particle model

Two models to simulate particle tracks are implemented in Fluent: the Discrete Phase Model (DPM) and the Macroscopic Particle Model (MPM). In this section

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Theory and Methodology 2.5 Particle model

Table 2.4: Boundary conditions on the Circle of Willis Artery Pressure mm Hg VA 100 ICA 100 ACA 4 MCA 4 PCA 4

both models will be described. Unfortunately, it is not yet possible to apply MPM to the Circle of Willis, since the MPM algorithms are not fully developed yet.

To study particle tracks in geometries, first of all, particles need to be injected in a certain prescribed distribution. The distribution of the particles in the injection is the same for DPM and MPM at the inlet of the geometries as for DPM in the Circle of Willis. In the latter, the particles are injected in the RICA, see figure 2.2. The injection is displayed in figure 2.6. The distribution of the injection is chosen after the fact that emboli develop attached to vessels and thus are transported at the sides of a flow. No initial velocity is chosen for the particles. The particles used both in DPM and MPM, and in the geometries and the Circle of Willis, are of the material anthracite, with a density of 1550 kg/m3 and a diameter of 0.5 mm. The value of the density is chosen after

the assumption that a particle has a higher density than a liquid of the same material. The value of the diameter chosen so that multiple particles in an injection can be simulated.

Figure 2.6: Injected particle distribution in all geometries

2.5.1

Discrete Phase Model

In addition to solving the transport equations for the continuous phase, Fluent is able to simulate a discrete second phase that consists of spherical particles dispersed in the continuous phase. To calculate the discrete phase trajectories, Fluent uses a Lagrangian formulation that includes the discrete phase inertia and hydrodynamic drag [4]. This formulation is only suited for a continuous phase flow with a well-defined entrance and exit. It contains the assumption that the second phase is sufficiently dilute, so that particle-particle interactions are negligible. This implies that the volume fraction of the discrete phase must be sufficiently low, usually less than 10-12%. At a total particle volume of approximately 5mm3 and a fluid volume around 500mm3, this is the case for

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Theory and Methodology 2.5 Particle model

volume of a geometry, the volume fraction of the discrete phase of the Circle of Willis is lower than the volume fraction of a geometry.

The trajectory of a discrete phase particle is calculated by use of integration of the force balance on the particle, which is written in a Lagrangian reference frame. This force balance equates the particle inertia with the forces acting on the particle, and can be written as:

dup

dt =FD(u−up) +

gx(ρp−ρ)

ρp +Fx (2.10)

Since gravity is omitted, and the additional forces,Fxonly play a role when ρ > ρp[4], equation A.1 reduces to:

dup

dt =FD(u−up) (2.11)

withFD(u−up) the drag force per unit particle mass andFDdefined as FD= 18µ

ρpd2

p CDRe

24 (2.12)

with Reynolds numberRe defined as

Re=ρdp|up−u|

µ (2.13)

and drag coefficientCD as

CD=a1+

a2

Re + a3

Re2 (2.14)

whereu is the continuous phase (fluid) velocity, up the particle velocity, µ the kinematic viscosity of the fluid,ρthe density of the fluid,ρpthe density of the particle, dp the particle diameter and a1, a2 and a3 constants that apply

for smooth spherical particles over several ranges of Re given by Morsi and Alexander [5].

The trajectory equations are solved by stepwise integration over discrete time steps. Integration in time of equation 2.11 yields the velocity of the particle at each point along the trajectory, with the trajectory itself predicted by:

dx

dt =up (2.15)

Withyandzin place ofxfor every coordinate direction. Equation 2.11 can be rewritten in simplified form as:

dup dt =

1

τp(u−up) (2.16)

whereτp is the particle relaxation time. Then a trapezoidal scheme is used for integrating 2.16: un+1 p −unp ∆t = 1 τp(u ∗un+1 p ) +... (2.17)

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Theory and Methodology 2.5 Particle model u∗= 1 2(u n +un+1) (2.18) unp+1=unp+ ∆tunp· ∇unp (2.19) These final two equations are solved simultaneously to determine the veloc-ity and position of the particle at any given time.

In these simulations the coupled approach is used, which means that the con-tinuous flow pattern is impacted by the discrete phase and vice versa. The calculations of the continuous and discrete phase are alternated until a con-verged solution is achieved. The momentum transfer from the continuous phase to the discrete phase is computed in Fluent by examining the change in mo-mentum of a particle as it passes through each control volume in the model, as illustrated in figure 2.7.

Figure 2.7: Momentum transfer between the discrete and the continuous phases

The momentum transfer from the continuous phase to the discrete phase is computed by examining the change in momentum of a particle as it passes through each control volume in the Fluent model:

F =X(18µCDRe ρpd2

p24

(up−u) +Fother) ˙mp∆t (2.20) whereFother are other interaction forces, in this case 0, ˙mp is the mass flow rate of the particles and ∆t is the time step.

When a particle strikes a wall, it slides along the wall depending on the particle properties and the impact angle [4].

2.5.2

Macroscopic Particle Model

The Macroscopic Particle Model takes into account:

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Theory and Methodology 2.5 Particle model

• Evaluation of the drag force and the torque experienced by the particles.

• Particle-particle as well as particle-wall collision.

In MPM, the particles are treated in a Lagrangian frame of reference as well. Rigid body velocity of a particle is imposed on the fluid cells that are touched by the particle, as displayed in figure 2.8 [6].

Figure 2.8: Particle velocity of a particle imposed on touched cells

Firstly, this means that the particles add momentum to the fluid. The momentum is integrated and the particle drag and particle torque vectors are calculated for each particle. From these vectors the new position, velocity and angular velocity of the particles are calculated. Secondly, this means that mo-mentum can also diffuse from touched cells to the particles. This momo-mentum represents the hydrodynamic forces on the particles.

Particle-wall collision works in the following manner. First, the boundary faces that are intersected by the particle are identified. Secondly, the particle velocity onto the normal and tangential vector of the wall is projected. Finally, the resti-tution coefficient is applied to calculate the outgoing velocity of the particle. Particle-particle collision is determined in a similar manner. At first, the parti-cles which are going to collide are detected. The line-of-action of the collision is found next, which identifies the normal direction. Then, incoming particle velocities are projected onto the line-of-action to get the normal and tangential components. Finally, the coefficient of restitution and conservation of momen-tum to the normal components of the incoming velocities are applied to obtain the final velocities of the particles. The coefficients of restitution in the sim-ulations are set to 0.8. For MPM, it is necessary to define the mass and the moment of inertia of the particles. The mass of the particles is set to 1e-5 kg, the moment of inertia to 1e-3kg/m2, sufficiently small to neglect. Furthermore, particle-particle attraction force and particle-wall attraction force are set to 1. For simulations of flow and pressure on the Circle of Willis, Fluent needs a library containing user-defined functions that describes the behavior of these properties. However, the MPM code for Fluent is also written in a library of user-defined functions and unfortunately, for this version of Fluent, it was not

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Theory and Methodology 2.5 Particle model

possible to use multiple libraries for the same simulation. Nor was it possible to combine both the libraries, since the user-defined functions of MPM are not known, as it is a part of the Fluent program. In future versions of Fluent and MPM, it will be possible to use multiple libraries and the simulation using MPM on the Circle of Willis can be done.

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Chapter 3

Results

3.1

Geometries

3.1.1

General

On each geometry, three different boundary condition situations are set. Since 5 geometries are used, 15 simulations are done for DPM and 15 simulations for MPM. A total of 30 simulations.

The DPM results are obtained using Fluent. It is possible in DPM to trap the particles on the outlets, so that the positions of the particles on the outlets are easy to determine. For MPM, trapping particles on outlets is not possible. The positions of the particles in MPM are saved in files, the final positions are isolated, and displayed using Matlab. Therefore, the results of DPM and MPM look slightly different. Unfortunately, it was not possible in Matlab to display the particles at real size. An example of a result is given in figure 3.1.

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Results 3.1 Geometries

(a) DPM, Geometry 1, equal boundary conditions

(b)MPM, Geometry 1, equal boundary conditions

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Results 3.1 Geometries

From these results, it is easy to determine the amount of particles on outlet 1 and how many on outlet 2. This data in combination with the geometry and boundary conditions is used to evaluate particle trajectories in various circum-stances.

In tables 3.1 and 3.2 duration of the simulations is summarized.

Table 3.1: Time for all particles to be trapped on the outlets for DPM in seconds

Geometry 1 2 3 4 5 boundary conditions 98 outlet 1 98 outlet 2 0.64 0.52 1.11 0.49 0.49 95 outlet 1 98 outlet 2 0.41 0.49 0.42 0.49 0.40 98 outlet 1 95 outlet 2 0.40 0.27 0.43 0.41 0.69

Table 3.2: Time for all particles to exit the outlets for MPM in seconds

Geometry 1 2 3 4 5 boundary conditions 98 outlet 1 98 outlet 2 0.60 0.71 0.68 0.69 0.64 95 outlet 1 98 outlet 2 0.54 0.67 0.56 0.55 0.59 98 outlet 1 95 outlet 2 0.50 0.85 0.53 0.58 0.57

First, the influence of diameter of the outlets of the geometries and bifurca-tion angle on the particle trajectories is evaluated. Then, the influence of mass flux on the particle trajectories is evaluated.

3.1.2

Outlet diameter

To evaluate the influence of outlet diameter on the distribution of the particles on the outlets, first, a comparison is made between geometry 1 and geometry 2, since the diameter of outlet 1 of geometry 2 is half the size of outlet 1 of geometry 1, while outlets 2 of both geometries remain the same size. Secondly, a comparison is made between geometries 4 and 5, where geometry 5 is the inverse is of geometry 4, see figure 2.4 and table 2.1. The results for DPM are displayed in figure 3.3, the results for MPM are displayed in figure 3.5

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Results 3.1 Geometries

(a) Percentage of particles against different mass fluxes on geometry 1 where the diameters of outlets 1 and 2 both are 3mm.

(b) Percentage of particles against different mass fluxes on geometry 2 where the diameter of outlet 1 is 1.5mmand the diameter of outlet 2 is 3mm.

Figure 3.2: Percentage of particles against the diameters of geometries 1 and 2 for DPM

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Results 3.1 Geometries

(a) Percentage of particles against different mass fluxes on geometry 4 where the diameter of outlet 1 is 2mmand the diameter of outlet 2 is 3mm.

(b) Percentage of particles against different mass fluxes on geometry 5 where the diameter of outlet 1 is 3mmand the diameter of outlet 2 is 2mm.

Figure 3.3: Percentage of particles against the diameters of geometries 4 and 5 for DPM.

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Results 3.1 Geometries

(a) Percentage of particles against different mass fluxes on geometry 1 where the diameters of outlets 1 and 2 both are 3mm.

(b) Percentage of particles against different mass fluxes on geometry 2 where the diameter of outlet 1 is 1.5mmand the diameter of outlet 2 is 3mm.

Figure 3.4: Percentage of particles against the diameters of geometries 1 and 2 for MPM.

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Results 3.1 Geometries

(a) Percentage of particles against different mass fluxes on geometry 4 where the diameter of outlet 1 is 2mmand the diameter of outlet 2 is 3mm.

(b) Percentage of particles against different mass fluxes on geometry 5 where the diameter of outlet 1 is 3mmand the diameter of outlet 2 is 2mm.

Figure 3.5: Percentage of particles against the diameters of geometries 4 and 5 for MPM.

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Results 3.1 Geometries

3.1.3

Bifurcation angle

To evaluate the influence of the bifurcation angles of the geometries, the results of geometry 1 are compared with the results of geometry 3. This is because both geometries have equal diameters on the outlets, while the bifurcation angles differ. There is, however, a difference between the diameters of the outlets of geometry 1 and geometry 3, but this is not considered as of influence on the particle distribution, since equal diameter is thought to cause equal distribution of particles. Furthermore, a comparison is made between the outlets with the same diameter of geometry 4 and 5. This means that outlet 1 of geometry 4 is compared with outlet 2 of geometry 5 and the opposite. See figure 2.4 and table 2.1 for details.

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Results 3.1 Geometries

(a) Percentage of particles against different mass fluxes on geometry 1 where the bifurcation angles of outlets 1 and 2 both are 40o.

(b) Percentage of particles against different mass fluxes on geometry 3 where the bifurcation angle of outlet 1 is 30oand where the bifurcation angle of outlet 2 is 50o.

Figure 3.6: Percentage of particles against the bifurcation angles of geometries 1 and 3 for DPM.

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Results 3.1 Geometries

(a) Percentage of particles against different mass fluxes on geometry 4 where the bifurcation angles of outlet 1 is 21.26oand where the bifurcation angle of outlet 2 is 54.68o.

(b) Percentage of particles against different mass fluxes on geometry 4 where the bifurcation angles of outlet 1 is 21.26oand where the bifurcation angle of outlet 2 is 54.68o.

Figure 3.7: Percentage of particles against the bifurcation angles of geometries 4 and 5 for DPM.

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Results 3.1 Geometries

(a) Percentage of particles against different mass fluxes on geometry 1 where the bifurcation angles of outlets 1 and 2 both are 40o.

(b) Percentage of particles against different mass fluxes on geometry 3 where the bifurcation angles of outlet 1 is 30oand where the bifurcation angle of outlet 2 is 50o.

Figure 3.8: Percentage of particles against the bifurcation angles of geometries 1 and 3 for MPM.

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Results 3.1 Geometries

(a) Percentage of particles against different mass fluxes on geometry 4 where the bifurcation angles of outlet 1 is 21.26oand where the bifurcation angle of outlet 2 is 54.68o.

(b) Percentage of particles against different mass fluxes on geometry 5 where the bifurcation angles of outlet 1 is 21.26oand where the bifurcation angle of outlet 2 is 54.68o.

Figure 3.9: Percentage of particles against the bifurcation angles of geometries 4 and 5 for MPM.

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Results 3.1 Geometries

3.1.4

Mass flux

Different boundary conditions on the outlets will cause different mass fluxes through the artery. For every geometry, three different pressure situations is applied, see table 2.3. To examine the influence of mass flux on particle trajec-tory, the percentage of particles on the outlets is plotted against the mass flux for every geometry. This means that each plot contains six points: the percent-age of particles on outlet 1 for three different mass fluxes, and the percentpercent-age of particles on outlet 2 for three different mass fluxes. First, the results for DPM are displayed. The results of DPM can be found in figure 3.10, the results of MPM in figure 3.11. In figure 3.12(a) and figure 3.12(b) all percentages of the particles on the outlets of all simulations are plotted against every mass flux on the outlets. The mass fluxes on the inlets are used to calculate the Reynolds numbers of the different flows. These are displayed in table 3.3 for DPM and in table 3.4 for MPM.

(a)Geometry 1 (b)Geometry 2

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Results 3.1 Geometries

(e)Geometry 5

Figure 3.10: Percentage of particles on the outlets as a function of mass flux in DPM

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Results 3.1 Geometries

(a)Geometry 1 (b)Geometry 2

(c)Geometry 3 (d)Geometry 4

(e)Geometry 5

Figure 3.11: Percentage of particles on the outlets as a function of mass flux in MPM

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Results 3.1 Geometries

(a) Percentage of particles as a function of mass flux of every simulation in DPM

(b) Percentage of particles as a function of mass flux of every simulation in MPM

Figure 3.12: The percentage of particles as a function of mass flux for every simulation in DPM and MPM

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Results 3.1 Geometries

Table 3.3: Reynolds numbers for different boundary conditions of the geometries based on the inlet for DPM

Geometry 1 2 3 4 5 boundary conditions 98 outlet 1 98 outlet 2 407 242 284 272 282 95 outlet 1 98 outlet 2 574 274 384 345 493 98 outlet 1 95 outlet 2 572 443 380 476 368

Table 3.4: Reynolds numbers for different boundary conditions of the geometries based on the inlet for MPM

Geometry 1 2 3 4 5 boundary conditions 98 outlet 1 98 outlet 2 442 258 278 307 305 95 outlet 1 98 outlet 2 699 305 423 396 560 98 outlet 1 95 outlet 2 678 526 427 566 397

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Results 3.2 Circle of Willis

3.2

Circle of Willis

3.2.1

General

In figure 3.13 the end state of the the results of DPM on the Circle of Willis is displayed. The whole simulation can be found in .mpeg file called sequence-1 or ParticlesCoW. The development of each particle from injection to end state can be studied in this movie. From the figure only the final distribution on the outlets of the Circle of Willis can be derived. Since some particles are trapped in the same positions on the MCA-outlet, it’s not clear to see that there are 6 particles trapped on the the MCA-outlet. Two particles are trapped on the ACA, and one on the PCA.

Figure 3.13: Result of particle trajectory on the Circle of Willis using DPM

3.2.2

Outlet diameter

In figure 3.14 the percentage of the particles on the outlets is plotted. The blood flow through the PCA is an addition of the blood flow through the BA and the

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Results 3.2 Circle of Willis

blood flow through the PCoA. Since the particle trajectory of the particle found on the PCA contains the PCoA and not the BA, the contribution of blood flow from the BA is not considered. Therefore, instead of displaying the percentage of particles on the PCA, the percentage of particles through the PCoA is displayed. The diameters of the displayed outlets can be found in table 2.2 in subsection 2.1.2.

Figure 3.14: Percentage of particles as a function of diameter outlet of the Circle of Willis using DPM

3.2.3

Bifurcation angle

For the results of bifurcation angle, figure 3.14 can be used. The angles between the arteries are mentioned in subsection 2.1.2.

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Results 3.2 Circle of Willis

3.2.4

Mass flux

Figure 3.15: Percentage of particles as a function of mass flux through the outlets of the Circle of Willis using DPM

The Reynolds number defined on the inlet of the right ICA is 769. The simulation time is 1.09 seconds.

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Chapter 4

Discussion

4.1

Geometries

4.1.1

Outlet diameter

DPM

The diameter of outlet 1 of geometry 2 is two times as small as the diameter of outlet 1 of geometry 1. In comparison of figure 3.2(a) and 3.2(b) it shows that, for all three boundary conditions, the percentage of particles on outlet 1 of geometry 2 is significantly less than the percentage of particles on outlet 1 of geometry 1. Since there is no difference in bifurcation angle, this indicates that a smaller diameter of an outlet is of significant influence on the particle distribution on the outlets for DPM. This is supported by comparing figure 3.3(a) with figure 3.3(b) where outlet 1 of geometry 4 traps a significant smaller percentage of particles than outlet 1 of geometry 5 for all pressures submitted on the outlets.

It is obvious that if a smaller diameter of an outlet causes a smaller percentage of particles on this outlet, a larger diameter of an outlet causes a larger percentage of particles on this outlet. This can also be found by observing figure 3.3.

MPM

When comparing figure 3.4(a) with 3.4(b), it is seen that for geometry 2, the diameter outlet of outlet 1 has a large effect on the particle distribution, since the percentages of particles are significantly lower than the percentages on outlet 1 of geometry 1. Furthermore, when figure 3.5(a) is compared with 3.5(b), it can be seen that for the three boundary conditions, the percentage of particles on outlet 1 of geometry 5 is significantly higher than the percentage of particles on outlet 1 of geometry 4. The hypothesis that a smaller diameter of an outlet causes a smaller percentage of particles on the outlet is thus supported by the results of MPM.

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Discussion 4.1 Geometries

4.1.2

Bifurcation angle

DPM

Note that figure 3.6(a) is exactly the same as figure 3.6(b). The same amount of particles is found on the outlets of geometry 3 as on the outlets of geometry 1, while the bifurcation angles of the geometries differ. This result does not support the hypothesis that a larger bifurcation angle causes a smaller percentage of particles through the artery.

The comparison between figure 3.7(a) and figure 3.7(b) is as follows. Outlet 1 of geometry 4 needs to be compared with outlet 2 of geometry 5, since these outlets both have a diameter of 2 mm, while the bifurcation angles differ. Outlet 2 of geometry 4 needs to be compared with outlet 1 of geometry 5, since these outlets have a diameter of 3 mm. This means that boundary condition 2 of geometry 4 needs to be compared with the boundary condition 3 of geometry 5 and the opposite. Now, it can be seen that the results for boundary condition 1 for both geometries and boundary condition 3 for geometry 4 and boundary condition 2 for geometry 5 are equal. The only indication that a different bifurcation angle might be of influence on the distribution of particles on the outlets can be found in the comparison between boundary condition 2 of geometry 4 and boundary condition 3 of geometry 5. Here it can be seen that more particles are found on outlet 1 of geometry 5, where the bifurcation angle is 21.26o, than on outlet 2 of geometry 4, where the bifurcation angle is 54.68o, and less particles on outlet 2 of geometry 5, where the bifurcation angle is 54.68o, than on outlet 1 of geometry 4, where the bifurcation angle is 21.26o. It is from these results, however, not clear that, for DPM, a smaller bifurcation angle causes a larger percentage of particles on the outlet.

MPM

From figure 3.8(a) and figure 3.8(b) it is observed that a higher percentage of particles, for all three boundary conditions, is found on outlet 1 of geometry 3, where the bifurcation angle is 30o, than on outlet 1 of geometry 1, where the bifurcation angle is 40o. Consequently, a lower percentage of particles is found on outlet 2 of geometry 3, where the bifurcation angle is 50o, than on outlet 2 of geometry 1, where the bifurcation angle is 40o. Furthermore, figures 3.9(a) and 3.9(b) show that a difference in bifurcation angle, for MPM, does cause a difference in particle distribution. There are significantly more particles found on outlet 1 of geometry 5 than on outlet 2 of geometry 4, when a pressure of 98 mm Hg is applied on both outlets. This is also the case for the other two boundary condition situations, when outlet 1 of geometry 4 is compared with outlet 2 of geometry 5 and vice-versa. It can be seen that when boundary condition 2 is applied on geometry 4, a larger amount of particles is found than on outlet 2 of geometry 5, when boundary condition 3 is applied. This is also the case for boundary condition 2 on geometry 5 compared with boundary condition 3 on geometry 4. Thus, the smaller the bifurcation angle, the larger the amount of particles. This result indicates that bifurcation angle is of significant influence on particle trajectories in MPM.

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Discussion 4.1 Geometries

4.1.3

Mass flux

DPM

From figure 3.10(a), it can be derived that when boundary condition 1 on geom-etry 1 is applied, this results in a nearly equal mass flux through both outlets. There is a slightly larger mass flux through outlet 2, which causes one particle more on this outlet than on outlet 1. The small difference in mass flux is prob-ably due to the mesh, which is unstructured and thus not totally symmetrical. From this figure, it can also be derived that a pressure of 95 mm Hg on an outlet causes a larger mass flux through this outlet and consequently a smaller mass flux through the other outlet where a pressure of 98 mm Hg is applied. As a result, a larger amount of particles is found on the outlet where the mass flux is higher. This indicates that there exists a linear relation between mass flux and particle percentage.

Since the Reynolds number is a measure for the mass flux through the inlet of the flow, in table 3.3, for geometry 1, it can be seen that the different boundary conditions have an obvious effect on the total mass flux of the flow. When a pressure of 95mm Hg is applied on one on the outlets, and 98mm Hgon the other, this causes a higher total mass flux of the flow than when a pressure of 98mm Hg on both outlets is applied.

When these Reynolds numbers are compared with the time of the simulations found in table 3.1, it can be seen that a higher mass flux through the flow causes a higher particle velocity for geometry 1, since the travel time of the particles is shorter than the travel time of the particles when a lower Reynolds number is found.

In figure 3.10(b) it can be seen that the mass flux through outlet 1 of geometry 2, where the diameter is 1.5mm, is for every boundary condition less than the mass flux through outlet 1 of geometry 1, where the diameter is 3 mm. This results in a small amount of percentages of particles on outlet 1, even 0 when a pressure of 95 mm Hg is applied on outlet 2. The mass flux through outlet 2 of geometry 2 does not differ much from the mass flux through outlet 2 of geometry 1. This means that the total mass flux of the flow through geometry 2 is lower than the total mass flux of the flow through geometry 1. This indicates that there exists a linear relation between mass flux and outlet diameter. The above can be verified by observing the Reynolds numbers in table 3.3. As expected, since total mass flux is lower, the Reynolds number of the flow through geometry 2 is lower than the Reynolds number of the flow through geometry 1. Also, when a pressure of 95mm Hgis applied to outlet 1 of geometry 2, it has a smaller effect on the mass flux through the flow than when a pressure of 95mm Hg is applied to outlet 2. These results support the hypothesis that a smaller diameter of an outlet causes less mass flux, and consequently less particles, to flow through this outlet.

It can be seen in table 3.1 that the simulation times between the first two boundary conditions situations are nearly equal. This is as expected, since the corresponding Reynolds numbers do not differ much. When the Reynolds num-ber increases in boundary condition 3, the time for the particles to be trapped on the outlet decreases.

There is, however, a large difference between simulation times of geometry 1 and geometry 2. The simulation time of equal pressures of geometry 2 is lower

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Discussion 4.1 Geometries

than the simulation time of equal pressures of geometry 1, while the Reynolds number of the flow through geometry 2 is smaller than the Reynolds number of the flow through geometry 1. The opposite is expected. It is not known why this is the case. Apparently, for different geometries, different relations between Reynolds number and duration of simulation apply.

A remarkable result is found in figure 3.10(c). For boundary condition 1, the mass flux through outlet 1 is slightly higher than the mass flux through outlet 2, as expected by bifurcation angle difference. On outlet 1, however, a lower amount of particles is found than on outlet 2. This indicates that the interac-tion between continuous phase and discrete phase can cause unexpected results in DPM. This result can be explained by a momentum transfer of fluid to the particles, which causes a decrease in mass flux of the flow, while the velocity of the particles increases. Since the difference in mass flux between the outlets is extremely small, 3.2e-5 kg/s, which is 0.12 % of the mass flux through the inlet, it is assumed that in DPM, this unexpected mass flux difference does not occur often. The results of the other boundary conditions is as expected. When figure 3.10(c) is compared with figure 3.10(a), it is seen that the mass fluxes are smaller for geometry 3 than for geometry 1, in all three boundary conditions. This is not due to the difference in bifurcation angle, as discussed in the previous section, but to the diameter of 2.5mmof the outlets of geometry 3, instead of 3mmfor the outlets of geometry 1.

Although the Reynolds numbers are smaller than the Reynolds numbers of ge-ometry 1, they show a similar pattern. It is possible that the smaller Reynolds number of geometry 3, in boundary condition 3, in comparison with the Reynolds number of boundary condition 2, is caused by the larger bifurcation angle of out-let 2. This can, however, not be known for certain, since the Reynolds number for this simulation in geometry 1 is also smaller.

In table 3.1 a similar result for geometry 3 as for geometry 1 can be found. The simulation time when equal pressures on both outlets are applied, however, is extremely long. This is due to one particle that interacted with the wall of the geometry. It probably lost some of its velocity during this interaction and therefore it took longer to travel to the outlet. After the interaction the velocity of the particle increases, which means that momentum of the fluid is transferred onto the particle, causing less mass flux on the outlet.

In figure 3.10(d) it can be seen that the mass flux through outlet 1, where the diameter is 2mmand the bifurcation angle 21.26o, for the three circumstances (see table 2.3), is smaller than the mass flux through outlet 2, where the diame-ter is 3mmand the bifurcation angle is 54.68o. This indicates that the diameter of the outlet is of larger influence on the mass flux than the bifurcation angle, which means, as seen before, that the outlet with the larger diameter traps more particles than the outlet with the smaller diameter, although the bifurcation an-gle of the latter is smaller than the former.

As a consequence the Reynolds number, in boundary condition 2, is larger than the Reynolds number when both pressures are equal, but is smaller than the Reynolds number of boundary condition 3, as can be seen in table 3.3.

The simulation times for geometry 4 in table 3.1 are not in correspondence with the Reynolds numbers of geometry 4. If it is assumed that the times of bound-ary condition 1 and boundbound-ary condition 3 are correct, than a simulation time of approximately 0.45 seconds is expected for the simulation with boundary con-dition 2. There is no suitable explanation for this.

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Discussion 4.1 Geometries

As expected, the mass flux through outlet 1 of geometry 5, where the diameter is 3mmis, for all three boundary conditions, higher than the mass flux through outlet 2, where the diameter is 2 mm, as can be seen in figure 3.10(e). Since the mass fluxes through geometry 5 are of approximately the same size as the mass fluxes through geometry 4, Reynolds numbers are of same size as well, as can be seen from 3.3. For the simulation time, the same problem arises as for geometry 4. The simulation time of geometry 5 with boundary condition 3, is expected to be lower than the time of boundary condition 1, and higher than the time of boundary condition 3. Here, however, the long simulation time can be explained by a particle losing velocity while interacting with the wall.

MPM

Note in figure 3.11(a) that for boundary condition 1, on outlet 1 a higher per-centage of particles is found, while the mass flux is lower than on outlet 2, where the mass flux is higher, as is seen before in the previous section. This can only be explained by assuming that momentum of the fluid is imposed on the par-ticles that travel towards an outlet, resulting in a lower mass flux through this outlet. The mass flux difference of the outlets is 7.0541e-5 kg/s, which is 1.8 % of the mass flux through the inlet. This particular result, obviously, causes a decrease in linearity in the graph. The results of boundary conditions 2 and 3 are as expected: a higher mass flux causes a higher percentage of particles, a lower mass flux causes a lower percentage.

In table 3.4 a similar pattern for the Reynolds numbers in MPM is found as in DPM. The Reynolds numbers for boundary conditions 2 and 3 are higher than the Reynolds number for boundary condition 1. Furthermore, the former two Reynolds numbers do not differ significantly, as expected.

In table 3.2 it can be seen that the pattern for simulation time in MPM is not as obvious as it is in DPM. This is due to the mutual momentum interaction between the particles and the flow. Since, for a particular flow, the exact momen-tum transfers are not known, it is not possible to verify this effect on simulation time. The results presented in figure 3.11(b) show that mass fluxes through outlets with small diameters are small for the three boundary conditions, and subsequently large through the other outlet, as expected and discussed in the previous section.

The Reynolds numbers show the same pattern as for DPM, where boundary condition 3 causes a larger mass flux through the geometry than boundary con-dition 2.

The time of the simulation does not correspond with the Reynolds numbers. Here, mutual momentum interaction is assumed to be the cause as well. In figure 3.11(c) it is shown that for boundary condition 1, a lower mass flux through outlet 1 causes a higher particle percentage, where the opposite is ex-pected. The momentum of the fluid has a large effect on the particles, as de-scribed earlier. The difference between mass fluxes is 9.5e-5kg/s, which is 3.95 % of the mass flux through the inlet. The other boundary conditions produce the expected results.

There is a slightly higher Reynolds number of geometry 4 when boundary con-dition 3 is applied, than when boundary concon-dition 2 is applied. The opposite is expected. It can be explained by the fluid transferring more momentum to the particles in outlet 2 for boundary condition 3, than fluid transferring momentum

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Discussion 4.1 Geometries

to the particles in outlet 1 for boundary condition 2. This causes the Reynolds number for boundary condition 3 to be lower than the Reynolds number for boundary condition 2.

Here, the simulation times show a correlation with the Reynolds numbers: a lower Reynolds number causes a larger simulation time, and a higher Reynolds number a lower simulation time.

The momentum transfer has a large effect on the particles in geometry 4 for boundary condition 2 as well, as can be seen in figure 3.11(d), where the mass flux through outlet 1 is lower than the mass flux through outlet 2, but causes a higher amount of particles on the outlet. Apparently, a pressure of 95 mm Hg on outlet 1 at first causes a larger mass flux through outlet 1, resulting in more particle flow towards outlet 1. Then momentum transfer causes the mass flux to reduce while the particles continue to travel towards outlet 1. In this simulation the momentum transfer must have been large, since the mass flux difference between the outlets is 72.28 kg/s, which is 21.1 % of the mass flux through the inlet. The other boundary conditions produce the expected results. The Reynolds numbers show the same pattern as for DPM, as expected. The time of the simulations do not corresponds with the Reynolds number. The results presented in 3.11(e) show the expected results, as well as the Reynolds numbers and the simulation times.

4.1.4

Differences between DPM and MPM

As seen in the previous section, there is more interaction between the fluid and the particles in MPM than in DPM. As mentioned in section 2.5, there is a difference in momentum transfer between DPM and MPM. In DPM, the ex-change of momentum between the continuous phase and the discrete phase is calculated as the particle passes through each control volume in the model. In MPM, momentum, supplied by the particles, is added to the fluid and particle velocity is imposed on touched cells. By doing this, momentum is transferred to the particle, which causes a lower particle velocity, while the velocity of the fluid subsequently increases. A higher fluid velocity will cause a higher mass flux through the geometry and consequently a higher Reynolds number, as can be found in tables 3.3 and 3.4. However, since the particles lost velocity to the flow, they will take longer to travel to the geometry. This explains the longer simulation time of MPM, as seen in tables 3.1 and 3.2. Interaction between particles and fluid also means that fluid velocity can be imposed on the parti-cles, so that the velocity of the fluid decreases and the velocity of the particles increases.

Since, as seen before, the influence of bifurcation angle is larger in MPM than in DPM, the difference between effect of outlet diameter of the models can not be defined. The only difference in figures 3.2(a) and 3.2(b), and 3.4(a) and 3.4(b), is the smaller percentage of particles on outlet 1 of geometry 2 in boundary condition 2 for MPM than on outlet 1 of geometry 2 in boundary condition 2 for DPM. This indicates a slightly larger influence of outlet diameter on particle distribution for MPM.

The difference between the influence of bifurcation angle on the particle distri-bution between DPM and MPM can be explained by the fact that, for MPM, the velocity vectors of the fluid have a significant influence on the velocity vec-tors of the particles at every touched cell, as explained earlier. The interaction

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Discussion 4.2 Circle of Willis

between the continuous and discrete phase for control volumes in DPM is not as large as the interaction between flow and particles in MPM. This means that when the direction of the fluid velocity is affected by a bifurcation angle, this direction will have a larger influence on the direction of particles in MPM than on the direction of DPM. Since both diameter and bifurcation angle contribute more to the particle tracks in MPM, it is considered more accurate than DPM. The main difference between DPM and MPM is that the particles in MPM in-teract more with the fluid, transferring momentum, which causes an increase or decrease of fluid velocity, a larger or smaller mass flux and a lower or higher particle velocity. This means that under certain boundary conditions, an unex-pected higher mass flux through an outlet is found and thus a higher amount of particles than on the other outlet. This can also mean that a lower mass flux is found, when the flow transfers momentum to the particles, causing a lower amount of particles on the outlet. As a result, the results in DPM are more consistent than the results of MPM, as can be seen in figures 3.12(a) and 3.12(b).

4.2

Circle of Willis

Summarizing these outcomes, it is seen that, for DPM, the diameter of the outlet contributes most to the mass flux found on the outlets and thus to the amount of particles found on the outlets. Bifurcation angle is of small influence on the mass flux and the amount of particles. With these results it is possible to roughly predict the particle distribution on the Circle of Willis. It is expected that in the Circle of Willis, the highest percentage of particles will be found on the MCA, where the outlets is largest, and the bifurcation angle the smallest, causing the largest mass flux. On the ACA a larger percentage of particles will be found than on the PCA, since the diameter of the ACA is larger than the diameter of the PcOA as can be verified by table 2.2, and thus causing a larger mass flux through the ACA, while there is not much difference between the bifurcation angles.

This is exactly what is seen in figures 3.13, 3.14 and 3.15.

For MPM, such a prediction is more difficult to make, since the results of MPM are not as consistent as the results of DPM. In MPM, both outlet diameter and bifurcation angle are of influence on particle distribution. From this result it can be derived that the largest amount of injected particles will travel through the MCA, since the diameter is largest and there is no bifurcation angle between the ICA and MCA. If is assumed that the bifurcation angles between MCA and PCoA, and MCA and ACA is equal, since the influence of outlet diameter for MPM is higher than for DPM, it is most likely that the higher percentage of the particles will be found on the ACA, and the smallest amount of particles on the PCA, as in DPM.

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Chapter 5

Conclusion

To predict particle trajectory in the Circle of Willis, 5 geometries are used to study the effect of outlet diameter, bifurcation angle and mass flux on the particle distribution, Two different models are used to simulate particle tracking: the Discrete Phase Model (DPM) and the Macroscopic Particle Model (MPM). It has been found that outlet diameter effects the particle distribution greatly in DPM, while bifurcation angle barely influences the distribution. Furthermore, for DPM, a larger diameter of outlet causes a higher mass flux through this outlet, while a smaller diameter causes a smaller mass flux. Mass flux is linearly related to the amount of particles. These findings are used for a prediction of particle trajectory in the Circle of Willis, and results are as predicted.

Results of the simulations performed with MPM differ from the the results of DPM. Firstly, it is seen that bifurcation angle is of significantly larger influence on the particle distribution. Secondly, also the outlet diameter has a larger effect on the particle tracks in MPM than in DPM. This difference can be explained by a difference in interaction between fluid and particles for DPM and MPM. In DPM, momentum transfer of the particles to the fluid, or vice-versa, is calculated for every control volume, whereas for MPM, this transfer takes place in every cell that is touched by the particle. As a result, the percentage of particles on the outlets as a function of mass flux does not show as much linearity as DPM. This means that the prediction of particle trajectory in the Circle of Willis is slightly different. It is expected for MPM, that more particles will be found on the outlet where diameter is largest and bifurcation smallest, than for DPM. Consequentially, less particles are found on the other outlets. Unfortunately, it was for MPM not possible to verify this prediction by simulating particle tracks in the Circle of Willis.

Nevertheless, the conclusion can be drawn that, since for DPM most particles are found on the MCA, and that for MPM most particles are expected to be found on the MCA, an embolus, causing a stroke, is most likely to block blood flow through the MCA.

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Chapter 6

Recommendations

Further research includes, obviously, applying MPM to the Circle of Willis, using the same injection and boundary conditions as is used in DPM. Other simulations can be done with the use of boundary conditions that vary in time, simulating pulsatile blood flow. Another interesting simulation is to use MPM to study the particle track of one large particle, simulating the particle trajectory of an emboli, in the Circle of Willis, and the accompanying influence on the flow.

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References

[1] A. C. Guyton and J. E. Hall.Textbook of medical physiology. W.B. Saunders Company, Philadelphia, 9th edition, 1996.

[2] S. Moore. T. David. J. G. Chase. J. Arnold and J. Fink. 3D Models of Blood Flow in the Cerebral Vasculature. Journal of Biomechanics, Article in Press.

[3] M. Zamir. The Physics of Pulsatile Flow. Springer-Verslag, New York, 1st edition, 2000.

[4] FLUENT 6.1 User’s Guide, Appendix A.

[5] S. A. Morsi and A. J. Alexander. An investigation of particle trajectories in two-phase flow systems. J. Fluid Mech, 55.

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Appendix A

Discrete Phase Model

Manual

A.1

Introduction

In addition to solving transport equations for the continuous phase, FLUENT allows you to simulate a discrete second phase in a Lagrangian frame of ref-erence. This second phase consists of spherical particles (which may be taken to represent droplets of bubbles) dispersed in the continuous phase. FLUENT computes the trajectories of these discrete phase entities, as well as heat and mass transfer to/from them. The coupling between the phases and its impact on both the discrete phase trajectories and the continuous phase flow can be included.

FLUENT provides the following discrete phase modeling options:

• Calculation of the discrete phase trajectory using a Lagrangian formula-tion that includes the discrete phase inertia, hydrodynamic drag, and the force of gravity, for both steady and unsteady flows

• Prediction of the effects of turbulence on the dispersion of particles due to turbulent eddies present in the continuous phase

• Heating/cooling of the discrete phase

• Vaporization and boiling of liquid droplets

• Combusting particles, including volatile evolution and char combustion to simulate coal combustion

• Optional coupling of the continuous phase flow field prediction to the discrete phase calculations

• Droplet breakup and coalescence

These modeling capabilities allow FLUENT to simulate a wide range of discrete phase problems including particle separation and classification, spray drying, aerosol dispersion, bubble stirring of liquids, liquid fuel combustion, and coal

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Discrete Phase Model Manual A.2 Limitations

combustion. The physical equations used for discrete phase calculations are described in Sections A.4 and A.5.

A.2

Limitations

Limitation on the Particle Volume Fraction

The discrete phase formulation used by FLUENT contains the assumption that the second phase is sufficiently dilute that particle-particle interactions and the effects of the particle volume fraction on the gas phase are negligible. In practice, these issues imply that the discrete phase must be present at a fairly low volume fraction, usually less than 10-12 %. Note that the mass loading of the discrete phase may greatly exceed 10-12 %: you may solve problems in which the mass flow of the discrete phase equals or exceeds that of the continuous phase.

Limitation on Modeling Continuous Suspension of Particles

The steady-particle Lagrangian discrete phase model described in this chapter is suited for flows in which particle streams are injected into a continuous phase flow with a well-defined entrance and exit condition. The Lagrangian model does not effectively model flows in which particles are suspended indefinitely in the continuum, as occurs in solid suspensions within closed systems such as stirred tanks, mixing vessels, or fluidized beds. The unsteady-particle discrete phase model, however, is capable of modeling continuous suspensions of particles.

Limitation on Using the Discrete Phase Model with Other FLUENT Models

The following restrictions on the use of other models with the discrete phase model:

• Streamwise periodic flow (either specified mass flow rate or specified pres-sure drop) cannot be modeled when the discrete phase model is used.

• Adaptive time stepping cannot be used with the discrete phase model.

• Only non-reacting particles can be included when the premixed combus-tion model is used

• Surface injections will not be moved with the grid when a sliding mesh or a moving or deforming mesh is being used.

• When multiple reference frames are used in conjunction with the discrete-phase model, the display of particle tracks will not, by default, be mean-ingful. Similarly, coupled discrete-phase calculations are not meanmean-ingful. An alternative approach for particle tracking and coupled discrete-phase calcula-tions with multiple reference frames is to track particle based on absolute veloc-ity instead of relative velocveloc-ity. To make this change, use the define/models/ dpm/tracking/track-in-absolute-frame text command. Note that the re-sults may be strongly depend on the location of walls inside the multiple ref-erence frame. The particle injection velocities (specified in the Set Injection

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Discrete Phase Model Manual A.3 Overview of Discrete Phase Modeling Procedures

Properties panel) are defined relative to the frame of reference in which the particles are tracked. By default, the injection velocities are specified relative to the local reference frame. If you enable thetrack-in-absolute-frameoption, the injection velocities are specified relative to the absolute frame.

A.3

Overview of Discrete Phase Modeling

Pro-cedures

You can include a discrete phase in your FLUENT model by defining the initial position, velocity, size, and temperature of individual particles. These initial conditions, along with your inputs defining the physical properties of the dis-crete phase, are used to initiate trajectory and heat/mass transfer calculations. The trajectory and heat/mass transfer calculations are based on the force bal-ance on the particle and on the convective/radiative heat and mass transfer from the particle, using the local continuous phase conditions as the particle moves through the flow. The predicted trajectories and the associated heat and mass transfer can be viewed graphically and/or alphanumerically.

You can use FLUENT to predict the discrete phase patterns based on a fixed continuous phase flow field (anuncoupled approach), or you can include the ef-fect of the discrete phase on the continuum (acoupledapproach). In the coupled approach, the continuous phase flow pattern is impacted by the discrete phase (and vice versa), and you can alternate calculations of the continuous phase and discrete phase equations until a converged coupled solution is achieved. See Section A.6 for details.

Outline of Steady-State Problem Setup and Solution Procedure

The general procedure for setting up and solving a steady-state discrete-phase problem is outlined below:

1. Solve the continuous-phase flow. 2. Create the discrete-phase injections. 3. Solve the coupled flow, if desired.

4. Track the discrete-phase injections, using plots or reports.

Outline of Unsteady Problem Setup and Solution Procedure

The general procedure for setting up and solving an unsteady discrete-phase problem is outlined below:

1. Create the discrete-phase injections. 2. Initialize the flow field.

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Discrete Phase Model Manual A.4 Equations of Motion of Particles

Particle positions will be updated as the solution advances in time. If you are solving an uncoupled flow, the particle position will be updated at the end of each time step. For a coupled calculation, the positions are iterated on within each time step.

A.4

Equations of Motion of Particles

Particle Force Balance

FLUENT predicts the trajectory of a discrete phase particle (or droplet or bubble) by integrating the force balance on the particle, which is written in a Lagrangian reference frame. This force balance equates the particle inertia with the forces acting on the particle, and can be written (for the x direction in Cartesian coordinates) as dup dt =FD(u−up) + gx(ρp−ρ) ρp +Fx (A.1)

whereFD(u−up) is the drag force per unit particle mass and FD= 18µ

ρpd2

p CDRe

24 (A.2)

Here,uis the fluid phase velocity,upis the particle velocity,µis the molec-ular velocity of the fluid,ρis the fluid density,ρp is the density of the particle, and dp is the particle diameter. Re is the relative Reynolds number, which is defined as

Re=ρdp|up−u|

µ (A.3)

The drag coefficient,CD, can be taken from either CD=a1+

a2

Re + a3

Re2 (A.4)

wherea1, a2anda3 are constants that apply for smooth spherical particles

over several ranges of Re given by Morsi and Alexander, or CD= 24 Re(1 +b1Re b2) + b3Re b4+Re (A.5) where b1=exp(2.3288−6.4581φ+ 2.4486φ2) b2= 0.0964 + 0.5565φ b3=exp(4.905−13.8944φ+ 18.4222φ2−10.2599φ3) b4=exp(1.4681 + 12.2584φ−20.7322φ2+ 15.8855φ3)

which is taken from Haider and Levenspiel. The shape factor,φ, is defined as

φ= s

S (A.6)

wheresis the surface area of a sphere having the same volume as the par-ticle, andS is the actual surface area of the particle.

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Discrete Phase Model Manual A.4 Equations of Motion of Particles

For sub-micron particles, a form of Stokes’ drag law is available. in this case, FD is defined as FD= 18µ d2 pρpCc (A.7) The factorCcis the Cunningham correction to Stokes’ drag law, which you can compute from

Cc = 1 +2λ dp

(1.257 + 0.4e−(1.1dp/2λ)) (A.8) whereλis the molecular mean free path.

(rest not relevant, omitted)

Including the Gravity Term

While equation 1 includes a force of gravity on the particle, it is important to note that in FLUENT the default gravitational acceleration is zero. If you want to include the gravity force, you must remember to define the magnitude and direction of the gravity vector in theOperating Conditionspanel.

Other Forces

Equation 1 incorporates additional forces (Fx) in the particle force balance that can be important under special circumstances. The first of these is the ”virtual mass” force, the force required to accelerate the fluid surrounding the particle. This force can be written as

Fx=1 2 ρ ρp d dt(u−up) (A.9)

and is important whenρ > ρp. An additional force arises due to the pressure gradient in the fluid:

Fx= (ρ ρp)up

∂u

∂x (A.10)

(rest not relevant, omitted)

Integration of the Trajectory Equations

The trajectory equations, and any auxiliary equations describing heat of mass transfer to/from the particle, are solved by stepwise integration over discrete time steps. Integration in time of Equation 1 yields the velocity of the particle at each point along the trajectory, with the trajectory itself predicted by

dx

dt =up (A.11)

Equations similar to 1 and 11 are solved in each coordinate direction to pre-dict the trajectories of the discrete phase.

(53)

Discrete Phase Model Manual A.4 Equations of Motion of Particles

Assuming that the term containing the body force remains constant over each small time interval, and linearizing any other forces acting on the particles, the trajectory equation can be rewritten in simplified form as

dup dt =

1 τp

(u−up) (A.12)

whereτpis the particle relaxation time. FLUENT uses a trapezoidal scheme for integrating equation 12:

un+1 p −unp ∆t = 1 τp(u ∗un+1 p ) +... (A.13)

where n represents the iteration number and u∗= 1

2(u

n+un+1) (A.14)

un+1=un+ ∆tunp· ∇un (A.15)

Equations 14 and 15 are solved simultaneously to determine the velocity and position of the particle at any given time. For rotating reference frames, the integration is carried out in the rotating frame with the extra terms described above (not relevant) to account for system rotation. In all cases, care must be taken that the time step used for integration is sufficiently small that the trajectory integration is accurate in time.

Discrete Phase Boundary Conditions

When a particle strikes a boundary face, one of several contingencies may arise:

• Reflection via an elastic or inelastic collision.

• Escape through the boundary. (The particle is lost from the calculation at the point where it impacts the boundary.)

• Trap at the wall. non-volatile material is lost from the calculation at the point of impact with the boundary; volatile material present in the particle or droplet is released to the vapor phase at this points.

• Passing through an internal boundary zone, such as radiator or porous jump.

• Slide along the wall depending on particle properties and impact angle. You also have the

References

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