Internship Report: Interaction of two particles in a pipe flow

INTERACTION OF TWO PARTICLES
IN A PIPE FLOW
Single-phase flow study of hydrodynamic interaction forces
Internship Report
by
Pau Molas Roca
Degree: BSc in Aerospace Technology Engineering
Course: BSc Final Project - Erasmus Internship
Laboratory: Laboratoire des Écoulements Géophysiques et Industriels
Team: MOST Research Group - Turbulence Modeling and Simulation
Dr. Giovanni Ghigliotti, Thesis Adviser
INP Grenoble
Grenoble, France
Spring 2017

c Copyright 2017
by
LEGI Laboratory
All Rights Reserved
ii

CONTENTS
LIST OF FIGURES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . v
ABSTRACT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vii
1. INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1 PREFACE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 STATE OF THE ART . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.2.1 Which is the purpose of the research? . . . . . . . . . . . . . . . . . 7
1.3 ORGANIZATION OF THE WORK . . . . . . . . . . . . . . . . . . . . . . 9
2. PROBLEM DEFINITION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
2.1 DETAILED PROBLEM DESCRIPTION . . . . . . . . . . . . . . . . . . . 10
2.1.1 Red Blood Cells . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
2.1.2 Geometries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
2.1.2.1 A - Trapezoid . . . . . . . . . . . . . . . . . . . . . . . . . 11
2.1.2.2 B - Trapezoid with wings . . . . . . . . . . . . . . . . . . . 12
2.2 PHYSIC SETTING . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
2.2.1 Stokes Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
2.2.2 Hydrodynamic force . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
2.2.3 Finite Element Method . . . . . . . . . . . . . . . . . . . . . . . . . 17
2.2.3.1 Discretization of the domain . . . . . . . . . . . . . . . . . 17
2.2.3.2 Mesh . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
2.3 TOOLS USED . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
2.3.1 Software . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
2.3.2 Post-processing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
2.3.3 Servers’ structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
2.3.3.1 Calculations . . . . . . . . . . . . . . . . . . . . . . . . . . 20
3. ANALYSIS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
3.1 PARAMETERS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
3.1.1 Force . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
3.1.2 Convergence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
3.2 GEOMETRY A - TRAPEZOID . . . . . . . . . . . . . . . . . . . . . . . . 30
3.2.1 Sinusoidal velocity ﬁeld . . . . . . . . . . . . . . . . . . . . . . . . . 31
3.2.2 Sinusoidal-1-Sinusoidal velocity ﬁeld . . . . . . . . . . . . . . . . . . 35
3.3 GEOMETRY B - TRAPEZOID WITH WINGS . . . . . . . . . . . . . . . 37
iii

3.3.1 Code verification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
3.3.2 Other studies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
3.4 CONCLUSIONS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
APPENDICES
A. GEOMETRY A - TRAPEZOID . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
A.1 Code - Sinusoidal velocity field . . . . . . . . . . . . . . . . . . . . . . . . . 53
A.2 Code - Sinusoidal-1-Sinusoidal velocity field . . . . . . . . . . . . . . . . . . 56
B. GEOMETRY B - TRAPEZOID WITH WINGS . . . . . . . . . . . . . . . . . . . 59
B.1 Code . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
REFERENCES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
iv

LIST OF FIGURES
1.1 Snapshopt of two vesicles in deluite suspension.[3] . . . . . . . . . . . . . . . . 2
1.2 Representation of alike parachute particles.[6] . . . . . . . . . . . . . . . . . . 3
1.3 Modeled particles considered to carry out the study. . . . . . . . . . . . . . . 3
1.4 Cluster evolution for different number of vesicles N in an unbounded parabolic
flow.[6] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.5 Scheme of the problem studied. . . . . . . . . . . . . . . . . . . . . . . . . . . 6
1.6 Perturbation to the velocity field created by a single vesicle in an unbounded
Poiseuille flow.[6] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
1.7 Velocity field in the frame moving with the vesicles, the two vortices between
the two leading RBCs merge on the centreline.[6] . . . . . . . . . . . . . . . . 8
2.1 Colored geometries considered. . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
2.2 First geometry analyzed. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
2.3 Second geometry analyzed. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
2.4 s - Voigt’s notation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
2.5 Schem of Kareline cluster. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
3.1 Geometry A. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
3.2 Geometry B. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
3.3 Blue colored border 2 amounts for FLeftx . Red colored border 4 amounts for
FRightx . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
3.4 Blue colored borders 3 and 4 amount for FLeftx . Red colored borders 6 and
7 amount for FRightx . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
3.5 Convergence-Force study. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
3.6 Convergence-Time study. Amount of time spent for the code to compute
the reference geometry (Geometry B - L = 1, 0 - d = 0, 25 - D = 0, 25 -
r1 = r2 = 0, 4) for difference mesh sizes. . . . . . . . . . . . . . . . . . . . . . 29
3.7 Geometry A. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
3.8 Results with rectangular geometry, L = [0, 1−2], r1 = r1 = 0, 4 and δx = 1/100. 31
3.9 Results of r1 and r2 variation with L = 1 and δx = 1/100. . . . . . . . . . . . 32
v

3.10 Results with Sinusoidal velocity profile, r1 = r1 = 0, 4. . . . . . . . . . . . . . 33
3.11 Mesh Geometry A Sinusoidal - L = 1, 5, r1 = r2 = 0, 4 and δx = 1/100. . . . . 33
3.12 Velocity Field Geometry A Sinusoidal - L = 1, 5, r1 = r2 = 0, 4 and δx = 1/100. 34
3.13 Results with Sinusoidal-1-Sinusoidal velocity profile, r1 = r2 = 0, 4. . . . . . . 35
3.14 Mesh Geometry A Sinusoidal-1-Sinusoidal - L = 1, 5, r1 = r2 = 0, 4 and
δx = 1/100. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
3.15 Velocity Field Geometry A Sinusoidal-1-Sinusoidal - L = 1, 5, r1 = r2 = 0, 4
and δx = 1/100. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
3.16 Geometry B. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
3.17 Linear velocity profile. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
3.18 Results of r1 and r2 variation with L = [0, 5 − 2], d = 0, 5, D = 0, 5 and
δx = 1/100. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
3.19 Results of r1 variations with r2 = 0, 4, L = [0, 5 − 2], d = 0, 5, D = 0, 5 and
δx = 1/100. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
3.20 Results of r2 variations with r1 = 0, 4, L = [0, 5 − 2], d = 0, 5, D = 0, 5 and
δx = 1/100. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
3.21 Results of L = [0, 1 − 2] with r1 = r2 = 0, 4, d = [0, 05 − 0, 25], D = 0, 25 and
δx = 1/100. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
3.22 Results of L = [0, 1 − 2] with r1 = r2 = 0, 4, d = [0, 25 − 1], D = 1 and
δx = 1/100. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
3.23 Results of D = [0, 05 − 0, 5] with r1 = r2 = 0, 4, d = 0, 25, L = [0, 1 − 2] and
δx = 1/100. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
3.24 Geometry B considering larger RBCs, D = 1, L = 1 and d = 0, 25. . . . . . . 45
3.25 Results of D = [0, 05 − 2] with r1 = r2 = 0, 4, d = 0, 5, L = [0, 5 − 2] and
δx = 1/100. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
3.26 Results of d = [0, 05 − 0, 25] with r1 = r2 = 0, 4, D = 1, L = [0, 1 − 2] and
δx = 1/100. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
3.27 Results of d = [0, 5 − 1, 5] with r1 = r2 = 0, 4, D = 1, L = [0, 1 − 2] and
δx = 1/100. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
3.28 Mesh Geometry B - L = 1, 5, r1 = r2 = 0, 4, d = 0, 5, D = 0, 5 and δx = 1/100. 48
3.29 Velocity Field Geometry B - L = 1, 5, r1 = r2 = 0, 4, d = 0, 5, D = 0, 5 and
δx = 1/100. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
3.30 Geometry C. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
vi

ABSTRACT
The present document sums up the development and results of the research internship car-
ried out at LEGI Laboratory. The study aimed to understand the hydrodynamic forces
involvement in the interaction between two red blood cells located in a capillary (pipe
ﬂow). The problem regarding Red Blood Cells (RBCs) moving through a capillary has
been tackled from a two-dimensional point of view and has been both analytically and
numerically outlined. Finite elements have been used to discretize the geometries consid-
ered. Several boundary conditions and geometries were simulated and deeply examined
aiming to understand the mechanism governing hydrodynamic attraction and repulsion
between red blood cells. The consequent results are analyzed in this report.
vii

CHAPTER 1
INTRODUCTION
1.1 PREFACE
The internship has been carried out under the overseeing of Dr. Giovanni Ghigliotti
from February until July of 2017 at LEGI. The 5-month research has been focused on un-
derstanding the behavior of hydrodynamic interaction forces between two particles placed
on a flow inside a cylindrical pipe. The present investigation is also a Final Project of the
intern’s Bachelor’s Degree in Aerospace Technology Engineering.
The subject is treated using blood and capillaries as the fluid and pipe of the prob-
lem. Blood is a non homogeneous material, formed mainly by a fluid and red blood cells
(white cells are neglected in this study due to its low presence). It is classified on the
category of complexfluids 1. In contrast with simplefluids, whose properties have been
studied since long and the motion equation, the Navier-Stokes equation, is well-known; for
nearly any complexfluid the governing equation is still unknown. The complex interplay
between the macroscopic scale of the flow and the mesoscale structures that does not allow
for a simple continuum description is the main reason of this scarcity.
At present, although many features of blood flow have been studied and understood,
many questions remain unanswered. The arising questions are linked, on one side, to the
fundamental understanding of the underlying physics and, on the other, to the biological
relevance of physiological conditions. This thesis deals with the first set of problems, more
related to physics and specially regarding fluid mechanics and hydrodynamic interactions
between to consecutive particles in a pipe flow. The long-term goal of this research is
to contribute to the knowledge of blood flow and the particles composing this complex
fluid. The short-term goal is the understanding of the interaction mechanism between two
consecutive particles, in this case RBCs. Numerical simulations were carried out in two
dimensions in order to solve the problem outlined. This remarkable choice is motivated
to greatly simplify geometries and to explore a wide range of parameters thought to have
a certain involvement on the matter. In addition, computational costs are reduced. The
research focused only on the interaction between two particles.
1
Material composed by a fluid phase (simplefluid) and some mesoscopic structures dispersed in the
fluid.
1

When choosing the type of particle to consider, several were taken into account and
compared by their properties. The focus is on vesicles and red blood cells.
Figure 1.1: Snapshopt of two vesicles in deluite suspension.[3]
Red blood cells are biconcave shape particules composed by a membrane that keeps a
dense suspension of haemoglobin (high-viscosity Newtonian fluid) inside it. The membrane
has an elastic cyoskeleton fixed below and it is composed by a lipid bilayer in liquid
state that strongly resist surface dilatation and opposes surface bending. The cyoskeleton
allows deformations but helps to recover the original shape of the membrane by exerting a
restoring force towards it. So, it has the function of preserving the cell integrity. However,
since RBCs do not have a nucleus, they are highly deformable. Consequently, RBCs under
flow exhibits a highly non-trivial behavior.
Vesicles are liquid drops surrounded by a phospholipidic bilayer and, contrary to
RBCs, they do not have the elasticity provided by the cytoskeleton. However, its mem-
brane has similar properties to the membrane of RBCs: strong resistance of surface dilata-
tion and bending contention. A vesicle has a constant volume due to the incompressibility
of the enclosed fluid and it membrane is inextensible as well as impermeable.
2

After having described the main properties of a vesicle and a RBC, the particle
modelled is defined by a closed and inextensible membrane to separate its inside from
the outer fluid. Therefore, none deformation of the membrane caused by the action of
the velocity field of the surrounding fluid is considered. The former implies a fixed ge-
ometry and a single-phase flow. Despite the option of modelling particles with a concave
parachute-like shape as shown in Figure 1.2, a simplified model by using straight borders
was chosen. This election is supported by the fact that more parameters can be easily
changed and thus analyzed to the better understanding of the problem, see Figure 1.3
Figure 1.2: Representation of alike parachute
particles.[6]
Figure 1.3: Modeled particles considered to
carry out the study.
The hydrodynamic interaction of deformable objects is a challenging topic, encoun-
tered in biological situations - RBCs in capillaries, cell motility - as well as in technological
applications in microfluidics - drops. A model resulting on a single-phase flow problem,
was proposed in order to identify the origin of this complex behavior.
Despite its extremely simple formulation, this problem reveals a very complex dy-
namics with non-trivial dependence between the main parameters such as confinement,
distance between particles, viscosity, flow’s velocity profile and particles’ shape. By vary-
ing these parameters, the resultant hydrodynamic force acting can be either attractive or
repulsive, and a steady behavior can show up or not. Following the analytical modelling
of the problem, numerical calculations are carried with the aim of obtaining precise results
and thus be able to comprehend its behavior taking into account several geometric con-
figurations. This study might help to answer further questions considering more complex
and realistic configurations of drops’ dynamics and deeply analyzing the effect of flow
inertia.
3

1.2 STATE OF THE ART
Through the years, several studies have been published considering multiple ap-
proaches of RBCs’ spatial organization. How they organize themselves inside arterioles
and capillaries, where they are submitted to a Poiseuille flow profile (parabolic) [1], may
impact flow efficiency and oxygen transport due to its function as gas exchangers in the
human body. Therefore, as it is of great relevance to biology and medicine, and following
many attempts to model the dynamics of an ensemble of RBCs in a capillary; the research
held claimed to investigate a single, but relevant, part of the whole, which has not been
studied yet.
Early attempts have shown that RBCs’ deformability plays a fundamental role in
the collective behavior, which turns out to a tendency to aggregate and form clusters2.
Since the distances between neighbouring cells are too large to be explained by depletion
forces or even by chemical bonds between membranes; clusters are formed then due to
hydrodynamic interactions between RBCs.
As under a parabolic flow, a single vesicle3 assumes a stationary centered shape [2],
which may enjoy the mirror symmetry with respect to the centreline (so-called parachute),
the studied geometries in the present research try to simulate this shape by defining the
RBCs using straight borders.
Cluster formation has been numerically observed in the case of vesicles [3], [4] and
also experimentally observed for RBCs in vitro [5]. Since the only forces implemented in
the simulations were hydrodynamic forces, they are meant to be responsible of clustering.
In addition, a direct relationship between hydrodynamic interaction and cluster formation-
destruction has been recently shown [6]. By considering a two-dimensional RBCs cluster
problem in a parabolic flow, a self-regulating mechanism showed up. Hence, it provides
the existence of an intrinsic maximal cluster size (a number N∗ of maximum RBCs in the
cluster). Under N∗, RBCs are captured in the front by the converging vortices (see Figure
1.4). But, when N > N∗, the extra vesicles are expelled from the front of the cluster [6].
2
Clusters: RBCs are close one to another in a single-file configuration.
3
liquid drops delimited by a lipid bilayer
4

Figure 1.4: Cluster evolution for diﬀerent number of vesicles N in an unbounded parabolic
ﬂow.[6]
5

To better understand the behavior of RBCs escaping or being captured by the clus-
ter, the present research has been focused on calculating the interaction forces between
two consecutive RBCs on a geometrically static configuration with periodic boundary con-
ditions defined along the flow direction, considered from left to right. This approach is
expected to lead to the comprehension of the hydrodynamic interactions that show up on
a 2D set up. Multiple rigid and non-deformable RBCs’ geometries are considered and,
by comparing its resultant forces, discussed to finally explain the most probable behav-
iors depending on the case’s parameters. The Reynolds number is, in the physiological
conditions and in the available experiments, of the order of 10−3 ≤ Re ≤ 10−2 inside a
capillary. Having a Reynolds number less than 1, implies that viscosity dominates iner-
tia allowing the use of Stokes equation to describe the hydrodynamics (see Chapter 2).
Higher Reynolds number implies inertia being the flow’s dominant and thus dealing with
an uncertain and complex turbulent flow. This procedure may look like as an overly sim-
plified case, which is far from false. Nonetheless, the initial intention was to find valuable
answers of the hydrodynamic interaction in between RBCs without getting a lot into bi-
ological detail and to provide serviceable statements for further related studies. It is as
well, a way to adapt the research to a feasible project regarding the intern’s capabilities
and the internship duration.
Figure 1.5: Scheme of the problem studied.
6

1.2.1 Which is the purpose of the research?
The root of choosing this configuration arises from the statement of a convergent-
divergent flow pattern around a RBC - convergent in the front and divergent in the back,
see Figure 1.6. Hence, for a cluster of two vesicles, since the velocity of the fluid between
them is smaller than that of the surrounding fluid, vortices due to viscous drag appear.
The size of these votices is given by the interdistance between the two vesicles.[6]
The purpose of this research then is to simulate the free space between two consec-
utive RBCs and, by considering a Stokes flow, compute the forces generated due to their
hydrodynamic interaction.
Figure 1.6: Perturbation to the velocity field created by a single vesicle in an unbounded
Poiseuille flow.[6]
7

Figure 1.7: Velocity ﬁeld in the frame moving with the vesicles, the two vortices between
the two leading RBCs merge on the centreline.[6]
8

1.3 ORGANIZATION OF THE WORK
The contents of this report are organized as follows.
Chapter 2 contains an extensive description of the physics applied to perform the
studies. All the fundamental equations and hypothesis are presented together with the
numerical method applied and a brief explanation of the tools used. The simulated ge-
ometries are also outlined.
Chapter 3 is dedicated to the deep scientific analysis of the results obtained as
well as presenting as series of perspectives. It has been arranged respecting the scientific
conclusions that gradually lead to the last stage of the research.
Appendix A shows the complete code used to compute the first geometry considered
with short explanations for the better understanding of it (section 2.1.2.1).
Appendix B has the same content but with an extended code used for the second
geometry (section 2.1.2.2).
9

CHAPTER 2
PROBLEM DEFINITION
2.1 DETAILED PROBLEM DESCRIPTION
The problem is approached from a 2D static point of view. To summarize, attention
is focused on only two particles and a uniform flow is generated to simulate the RBCs’ real
movement. In this particular case (see Figure 1.5), if we were to consider the real motion,
particles would move from left to right. Instead, to exclude the time variable, RBCs’
position is fixed and it is the flux around them that flows from right to left simulating
their real movement.
2.1.1 Red Blood Cells
The particles are considered to be rigid and non-deformable as the goal is to identify
the origin of the complex behavior. RBCs are separated or attached each other for one or
several reasons not yet known and it is our purpose to gather if hydrodynamic interaction
induces this not yet interpreted behavior.
2.1.2 Geometries
The geometries considered are the spaces between the two particles:.
Figure 2.1: Colored geometries considered.
Due to evident horizontal symmetry, to perform numeric calculations and to reduce
significantly the computational cost, only the upper half is considered. The results corre-
sponding to the entire space between the two RBCs are obtained by doubling the initial
result calculated when considering only Geometries A and B.
The origin of coordinates for both geometries is considered to be located on the same
exact place, which facilitates the formulation of the borders’ equations. All the geometries
considered during the study in Chapter 3, are presented in sections 2.1.2.1 and 2.1.2.2.
10

2.1.2.1 A - Trapezoid
Distances:
Figure 2.2: First geometry analyzed.
• L distance between two consecutive RBCs.
• h half of the RBC width.
• r1 parameter used to ﬁx the front shape of the pursuer RBC.
• r2 parameter used to ﬁx the back shape of the leading RBC.
11

2.1.2.2 B - Trapezoid with wings
Distances:
Figure 2.3: Second geometry analyzed.
• L distance between two consecutive RBCs.
• h half of the RBC width.
• r1 parameter used to ﬁx the front shape of the pursuer RBC.
• r2 parameter used to ﬁx the back shape of the leading RBC.
• d distance between the side of the RBC and the capillarys wall.
• D length of the RBC’s sides considered on the force calculations.
12

2.2 PHYSIC SETTING
2.2.1 Stokes Equation
The case of study could be defined in the realm of microhydrodynamics. Then, it is
legitimate to reduce the Navier-Stokes equation to the Stokes equation: inertia in the flow
is negligible - low Reynolds number (Re → 0) - relative to viscous effects. This reduction
provides a simplification of the fluid-mechanical description, as Stokes equations are linear.
Consequently, the mathematical solutions are analytically derivable for a number of basic
but important solutions. The former means that the principle of superposition of solutions
may be applied, by which adding different solutions of the Stokes equations one obtains
also a solution of the Stokes equations. On these equations, bodies are assumed to be
solid, idealized as non-deformable (rigid).
The first step to take is to solve the Stokes equations. To do so, the most general
case is considered and simplified regarding the determined initial conditions.
Governing equations of a general fluid-mechanics problem (Newtonian Fluid):
Conservation of mass
∂ρ
∂t
+ ρ · u = 0 (2.1)
Balance of linear momentum
· σ + f = ρ
du
dt
(2.2)
Constitutive equation
σ = −pl + λ · ul + 2µ s
u (2.3)
where
• ρ constant density.
• u velocity.
• l lenght.
• µ dynamic viscosity.
• f external body force per unit volume.
• s Voigt’s notation, see Figure 2.4.
• λ Lamé modulus.
13

Figure 2.4: s - Voigt’s notation.
The superscript a indicates an absolute pressure and a corresponding absolute stress
tensor. The term absolute stress is used to indicate the actual stress (with the absolute
pressure being the true pressure) rather than a modified stress to be defined below, in which
the hydrostatic stress field is removed. In the last equality, the constitutive equation for
a Newtonian fluid is assumed, which implies the symmetric stress tensor σa:
σa
ij = σa
ji = −pa
δij + 2µeij (2.4)
where pa is the absolute pressure, defined as:
pa
= −
1
3
σa
ii (2.5)
and the rate-of-strain tensor e:
eij = eji =
1
2
(
∂ui
∂xj
+
∂uj
∂xi
) (2.6)
Additional hypothesis:
• Steady state du
dt = 0
• Incompressibility · u = 0
· u = 0 (2.7)
· σa
+ f = 0 (2.8)
σa
= −pa
l + 2µ s
u (2.9)
Note that the nonlinear convective acceleration and the time-dependent term are
lost.
14

Shall be assumed that the boundary Γ admits the decompositions:
Γ = Γi
u ∪ Γi
σ (2.10)
where
Γi
u ∩ Γi
σ = (2.11)
For simplicity, homogeneous Dirichlet Boundary conditions are considered (velocities
are prescribed on boundaries) and viscosity is ﬁxed to a constant value µ = 1.
To solve the system, the weak formulation of Stokes equations is used. It is obtained
by integrating over the domain the product of the balance and incompressibility equations
with the corresponding test functions, and by employing the Divergence Theorem to shift
the derivatives from the unknowns to the test functions. As Re4 and St (Stokes Number)5
are small, the inertial and acceleration terms in the equation can be neglected, which gives
the so called Stokes equations:
· u = 0 (2.12)
· σa
= − pa
+ 2µ ·( s
u) = −f (2.13)
also called the creeping ﬂow equations.
Finally, it leads to
− pa
+ µ( 2
u) + f = 0 (2.14)
4
Re = UρL
µ
5
St = a2
T ν
∼ |∂u/∂t|
|ν 2u|
15

2.2.2 Hydrodynamic force
Once the velocity field is obtained, the hydrodynamic force can be calculated on
each border. To do so, previous analytical development is required.
The total fluid force on a particle in a fluid is given by the integral of the traction
vector σa · n over the surface:
F =
Sp
σa
· ndS (2.15)
where
• σa fluid stress tensor
• n outward unit normal from the particle surface
The external body force per unit volume can be expressed as:
−→
f =
−→−→σ · −→n (2.16)
Introducing equation 2.6, σa
ij results in:
σa
ij = −pa
δij + µ(
∂ui
∂xj
+
∂uj
∂xi
) (2.17)
Which, expressed in 2D:
−→−→σ = µ
2∂u
∂x − p
µ
∂u
∂y + ∂v
∂x
∂u
∂y + ∂v
∂x 2∂v
∂y − p
µ
(2.18)
Considering the normal vector for a 2D surface k:
−→nk = Xk Yk =
X
Y
k
(2.19)
The external body force per unit volume expression results in:
−→
f k
= µ
2∂u
∂x Xk − p
µXk + ∂u
∂y Yk + ∂v
∂x Yk
∂u
∂y Xk + ∂v
∂x Xk + 2∂v
∂y Yk − p
µYk
=
fk
x
fk
y
(2.20)
When integrating
−→
f over the surface along a border formed by several elements,
the mathematical procedure can be expressed as:
Fx =
m
k=1
−→
fx
k
· Sk (2.21)
Fy =
m
k=1
−→
fy
k
· Sk (2.22)
16

2.2.3 Finite Element Method
The name Finite Element Method summarizes a numerical technique applied to
solve Partial Differential Equations (PDE)6. That is, looking at the geometry, the shape
of a region, and immediately imagine it broken down into smaller subregions. The idea
is to use a simple approximation method, but the errors in this approximation method
become unnoticeable as the size of the subregion gets small. So if small enough subregions
are used, approximate over each one, and then stitch all the answers back together, a
smooth and believable answer is obtained to the original full size problem.
2.2.3.1 Discretization of the domain
In two dimensions the element domains might be simply triangles and quadrilaterals.
Nodal points may exist anywhere on the domain but most frequently appear at the element
vertices and interelement boundaries and less often in the interiors.
Numerical calculations are used to solve Stokes equations on the already designed
domain. To properly do so, a mesh of small enough elements is created in the area inside
the borders. This mesh is likely to change depending on how refined it is desired it to
be. More accurate results will be obtained as smaller are the elements generated since
the Stokes equation will be solved at each of them. Nonetheless, computational cost rises
with refinement. The adequate convergence value has to be precisely decided.
The set of prescribed velocity nodes pertaining to Dirichlet Boundary conditions are
denoted by vector ri = (i = 1, nsd). The set of remaining velocity nodes is denoted by
li = (i = 1, nsd). Together; ri ∪ li = 1, 2, .., npt. Where npt is the total number of velocity
nodes.
2.2.3.2 Mesh
Even though the hard part of the Finite Element Method involves considering ab-
stract approximation spaces, sequences of approximating functions, the issue of boundary
conditions, weak forms and so on, it all starts with a very simple idea:
Take a geometric shape, and break it into smaller, simpler shapes, in such a way
that they can all be put back together.
6
PDE: relation between a function of several variables and its (partial) derivatives.
17

2.3 TOOLS USED
2.3.1 Software
Following the analytical approach, to develop and then compute the problem stud-
ied, an open-source French-developed software has been used: FreeFEM++. As its name
implies, it is a free software based on the Finite Element Method which uses C++-like
programming language. It is a high-level Integrated Development Environment (IDE)
for numerically solving Partial Differential Equations in dimension 2 and 3. Moreover,
FreeFEM++ is highly adaptive. With its all-included interface and an advanced auto-
matic mesh generator, FreeFEM++ allows to quickly test new ideas and multi-physics
problems.
Its solving method relies on a general-purpose elliptic solver interfaced with fast
algorithms. It has several triangular finite elements, including discontinuous elements.
Finally, everything is there in FreeFEM++ to prepare research quality reports: color
display plots with zooming and other features and postscript printouts.
The characteristics of FreeFEM++ that made it suitable for the research are:
• Easy geometric input by analytic description of boundaries by pieces using paramet-
ric equations for x and for y coordinates. They are referred either by name or by
label. Not a CAD system.
• Automatic mesh generator under input parameters to control its density.
• Problem description by its variational formulation.
• Instantaneous output plots. Enabling the user to navigate in situ on them with
zoom commands.
• Fast execution speed.
• Generation of: .dat, .eps and mesh files for further manipulations of the output data.
2.3.2 Post-processing
Once the data had been obtained using FreeFEM++, several graph generator soft-
ware were used. To quickly build experimental graphs from the data generated, Microsoft
Excel and Gnuplot were used depending on the size of the data that was waiting to be
treated. When some progression was made and the present document started to shape,
XMGrace was introduced in order to create stylish and scientifically recognized graphs.
This software stand out because of its simplicity of use and its high quality outcome.
18

2.3.3 Servers’ structure
Working on a numerical laboratory implies a complex informatics structure specif-
ically build to proportionate the required resources and to assure the right network and
memory for fast and secure calculations. The whole of MOST has a reliable and up-to-date
system, checked daily by a dedicated engineer whose work remains on the supervision and
improvement of the structure.
The core of the structure is called Kareline. It is a cluster with:
• A DELL poweredge C6100 consisting of four calculation nodes (the sleds) intercon-
nected by a specialized low latency network and very high speed inﬁniband QDR
(40 Gbits per second, red in Figure 2.5). This part of the cluster (48 cores) receives
by default the interactive sessions managed by OAR.
• Five DELL poweredge C6320s each consisting of four compute nodes, interconnected
by the inﬁniband network QDR. This part of the cluster (400 cores) receives by
default the batch sessions managed by OAR.
Figure 2.5: Schem of Kareline cluster.
19

These nodes are frontalized by a powerful Poweredge R720XD server on which the
user connections (Kareline in the strict sense) are made. This is a dual-processor Sandy
Bridge (2 times 6 cores) clocked at 2.5GHz with 32GB of memory. It manages 60TB of fast
storage (up to 1.6 GB/s in writing) and redundant (2 raid6 units, raid6 hardware protected
by battery). This storage space is accessible to all compute nodes either via the 10 GB/s
Ethernet network or through the infiniband network, allowing remote inputs/outputs at
speeds up to 600 MB/s.
The Kareline cluster is associated with the sge visualization station, a two-processor
Westmere-EP processor running at 2.8GHz (2 times 6 cores), equipped with 24GB of
memory and two graphics processors Nvidia Quadro FX 4800 (one per screen). This
station is accessible in H106 in ”free access” mode. These two machines are connected by
a high-speed network (10 GB/s) on optical fiber for a high-performance access to the data
to be processed.
A ”fat node” Charlie completes the solution for problems of large size in memory
and little parallel. Charlie does not have devices directly attached to the server. It is
therefore necessary to use a desktop offset solution on your workstation.
2.3.3.1 Calculations
To perform the required computations a sge virtualization was used. To do so, a
previous connection to sge server was required under the command vncviewer -via user-
name@sge sge:1. Once the virtualization was running, FreeFEM++ was load and ready to
be used. It is interesting to point out the amount of security involved in the process since
a specific password is requested at every step. To create graphs from the data obtained, a
connection to another machine without virtualization was used; in this case: ssh -X thor.
20

CHAPTER 3
ANALYSIS
The results presented in this section show the evolution of the scientific research with
justified decisions. They have been accurately selected from a considerable amount of
data generated to precisely show the findings and for the lector’s proper understanding.
3.1 PARAMETERS
Previous to the results analysis, it is helpful to introduce the most significant aspects
considered to describe the problem numerically. They are summarized and justified below
this lines for the lector understanding. They will be related to the lines of the code
attached on Appendix B, where a full version can be found.
As the writing language is C-like, all the variables used must be firstly defined. On
the code developed, basic integers (int) and real variables (real) are used - depending on
their values. A boolean variable (bool) is also introduced to easily manage the online-
simulation plotting procedure.
Border
Once all the variables are defined, the first step is to define the border of the domain.
As mentioned, straight borders are used. FreeFEM++ has a particular way to construct
the border. The boundary Γ is described analytically by a parametric a equation for x
and for y. A counterclockwise sense has to be used when defining all the segments, it has
no relevance the point of origin. In the code developed, the top surface was decided to be
the first segment to define, see Figures 3.1 and 3.2 below.
Figure 3.1: Geometry A. Figure 3.2: Geometry B.
Each border is named (i.e. w1, w2,...) and by the use of an external parameter t,
the segment is defined allways considering the same origin of coordinates.
21

Geometry A borders’ equations:
1. x = L − t ; y = h with t = (0, L)
2. x = r1 × t ; y = −h
r1 × x + h with t = (0, 1)
3. x = t ; y = 0 with t = (r1, L + r2)
4. x = (L + r2) − r2 × t ; y = h × t with t = (0, 1)
Geometry B borders’ equations:
1. x = t ; y = h + d with t = (L + D, −D)
2. x = −D ; y = t with t = (h + d, h)
3. x = t ; y = h with t = (−D, 0)
4. x = r1 × t ; y = −h
r1 × x + h with t = (0, 1)
5. x = t ; y = 0 with t = (r1, L + r2)
6. x = (L + r2) − r2 × t ; y = h × t with t = (0, 1)
7. x = t ; y = h with t = (L, L + D)
8. x = L + D ; y = t with t = (h, h + d)
The keyword label is introduced so the border can be referred further on the code
(i.e. boundary conditions). To make it simple, each segment has been labelled depending
on its position starting from the first one defined and so on.
An interesting improvement was made in order to facilitate the calculations. On
the most relevant segments for further calculations, a multiborder technique was applied
by making a loop on the segment generation. Consequently, the border is directly divided
in n small borders of infinitesimal size. This procedure was introduced in order to refine
the upcoming calculations. The number of multiborders created on each firstly defined
border, depends on the number of elements initially given. The elements of each border
have the same exact size: δx = 1/100. Nonethless, the multiborder technique allows to
chose how many elements are created at each smaller border. As an example, on Figures
3.11 and 3.14, two elements were generated at each multiborder of borders 2 and 4.
See lines 35-63 - Code B
22

Mesh
Triangular elements have been used to build the mesh. The syntax used consists on
defining the mesh (Th in this case) followed by the command buildmesh with the name of
each border created associated with the desired number of elements. As mentioned, each
border has been thought to have the same δx and thus create a uniform mesh.
The triangulation Th of the domain Ω is automatically generated by buildmesh.
The domain is assumed to be on the left side of the boundary which is implicitly oriented
by the parametrization. The automatic mesh generation is based on the Delaunay-Voronoi
algorithm and mesh refinement is done by increasing the number of points of Γ.
Spaces and elements
A finite element space is, usually, a space of polynomial functions on elements (tri-
angles in FreeFEM++) with certain matching properties at edges, vertices etc. By using
fespace command, a finite element space is created (i.e. Uh and Ph). Then, Uh and Ph
are defined as the space of continuous functions. A hand-picked type of element is associ-
ated too each finite element space. FreeFem++ implements a large variety of elements in
2D. The chosen ones to solve the present problem are:
P1 continuous piecewise linear. The degrees of freedom are the vertices values. It
is used to define the pressure variables.
P2 continuous piecewise quadratic. It is the set of polynomials of R2 of degrees ≤
2. Used to define the velocity variables of the problem.
23

Boundary conditions
They are introduced in the Stokes Problem definition. Velocity boundary conditions
are set for each border. By using the label of each, border boundary conditions can be
fixed precisely for both velocity components u and v.
Velocity Profile
Several velocity profiles have been considered to carry out the simulations and get
close to a realistic configuration. A sinusoidal velocity was implemented for Geometry A
and a linear velocity profile for Geometry B. Both had a maximum value of U = −17
that was placed at the upper border. More detailed information can be found at sections
3.2 and 3.3.
As the real focus of the research is made on the force tendency rather that on its
value, µ is defined with unit value as it acts only as a multiplying factor. Stokes flow
are linear and, thus, increasing or decreasing the velocity simply equals multiplying the
results by a constant.
Stokes Problem
The way of writing the expression 2.14, so FreeFEM++ understands it, is:
1 solve stokes ( [ u , v , p ] , [ uu , vv , pp ] ) =
int2d (Th) (dx(u) ∗dx(uu)+dy(u) ∗dy(uu) + dx(v) ∗dx( vv )+ dy(v) ∗dy( vv )
3 + dx(p) ∗uu + dy(p) ∗vv + pp∗(dx(u)+dy(v) )
−1e−10∗p∗pp)
5 + on ( border , boundary conditions ) ;
3 Codes/stokes equations.edp
Plots
To control the simulations’ output, both mesh and velocity field plots were displayed.
To do so, plot command was used introducing additionally some fancy and useful aspects
such as name, values and generation of an output file (ps).
7
Maximum absolute value for the x component of the velocity field (u, v).
24

Output files
To save the data generated, as well as the mesh files, specific commands were imple-
mented. By using the command ofstream, the data calculated was written in an output
file whose name and variables printed were conveniently chosen. By using the command
append the new data generated was added to the existent file or printed to a new file, in
case it did not exist previously.
Only when a deep analysis of the mesh was needed, the mesh information was saved
using savemesh command.
3.1.1 Force
After numerically solving the Stokes Problem, Equation 2.20 is introduced to cal-
culate the forces acting on the borders. To do so, the keyword int1d(Th, border), which
represents an integral along the indicated border, is used.
Γk
f(x, y)ds = int1d(Th, k)(f) (3.1)
As the borders where the force is calculated had been previously divided in multi-
borders, a loop over the integral of each border is required. The total force acting on the
k border is the final output of the calculation.
To integrate, the normal vector (N.x, N.y) of each border is extracted directly from
FreeFEM++ (internal function). It always points outward to the border. A negative sign
then must be added to respect the procedure presented in section 2.2.2 as the force made
by the fluid to the wall is the one searched.
FFluid−Wall = −
Sp
σa
· ndS (3.2)
Special emphasis is made on the abscissa component of FFluid−Wall since the study
gathers the horizontal interaction between two RBCs. The total vertical component of the
force, FFluid−Wally , is zero due to domain’s symmetry. The vertical force calculated for
the geometries presented is compensated by the one that appears on the non computed
lower half of the gap: same module but opposite sign.
25

Attraction - Repulsion
The point of interest, nevertheless, remains on the global resultant force that the
fluid makes towards both red blood cells. The force acting on the left side and on the
right side of the geometries considered, are calculated separately. Finally, the total force
FFluid−Wallx
8 is calculated as the difference between FRightx
9 and FLeftx
10. As a conse-
quence; when the final result is negative, attractive behavior takes place and thus, when
positive, repulsive force is induced, see Table 3.1.
Table 3.1: Attractive - Repulsive behavior.
FLeftx FRightx FFluid−Wallx Behavior
|FRightx | < |FLeftx |
+ + - Attractive
+ - - Attractive
- + + Repulsive
- - + Repulsive
|FRightx | > |FLeftx |
+ + + Repulsive
+ - - Attractive
- + + Repulsive
- - - Attractive
|FRightx | = |FLeftx |
+ + 0 Attractive
+ - - Steady
- + + Repulsive
- - 0 Steady
Figure 3.3: Blue colored border 2 amounts for FLeftx . Red colored border 4 amounts for
FRightx
8
FF luid−W allx = FRightx − FLeftx
9
Sum of all the forces acting on the front RBC.
10
Sum of all the forces acting on the trailing RBC.
26

Figure 3.4: Blue colored borders 3 and 4 amount for FLeftx . Red colored borders 6 and 7
amount for FRightx
27

3.1.2 Convergence
In order to obtain accurate results, a convergence test was conducted having in
mind that the purpose of the research is to understand the behavior of RBCs rather than
focusing to decimal values of the force calculated. Therefore, the variables considered on
this test were: computation time, element’s size and force value.
Having several input variables with multiple values each means performing a con-
siderable amount of simulations. To scientiﬁcally adjust the tests, a ﬁxed geometry was
taken as a reference11 to carry out a convergence study. Four meshes were tested for values
of δx
12: 1/50, 1/100, 1/200 and 1/300. The force values obtained were normalized with
the value obtained with the biggest elements (δx=1/50).
0 50 100 150 200 250 300 350
1/ δx
1
1,2
1,4
1,6
1,8
2
fx/fx(δ1/50)
Convergence
Figure 3.5: Convergence-Force study.
11
Geometry B - L = 1, 0 - d = 0, 25 - D = 0, 25 - r1 = r2 = 0, 4
12
Element size
28

0 50 100 150 200 250 300 350
1 / δx
0
20
40
60
80
100Time(s)
Convergence Time
Figure 3.6: Convergence-Time study. Amount of time spent for the code to compute
the reference geometry (Geometry B - L = 1, 0 - d = 0, 25 - D = 0, 25 -
r1 = r2 = 0, 4) for difference mesh sizes.
From Figure 3.5, a huge gap between δx=1/50 and δx=1/100 allows to neglect
the possibility of using bigger elements due to the low refined mesh generated. From
then on, the decision was more complex and required to take a glance at Figure 3.6.
Taking into account that the time showed in the graph is only for one shape of the
geometry simulated, the most refined mesh (δx=1/300) was dismissed too due to its high
time demand. Comparing the two remaining meshes, an optimal decision was taken and
δx=1/100 was chosen to be the elements’ size because of its lower time invested and a
slight difference in the force value calculated.
29

3.2 GEOMETRY A - TRAPEZOID
The first geometry considered, already presented in section 2.1.2.1, is the initial
approach to the problem.
Figure 3.7: Geometry A.
For this geometry, the shifting values are:
• L distance between two consecutive RBCs. L = [0, 1 − 2]
• r1 parameter used to fix the front shape of the pursuer RBC. r1 = [0, 1 − 0, 5]
• r2 parameter used to fix the back shape of the leading RBC. r2 = [0, 1 − 0, 5]
*The RBC’s width parameter is fixed at h = 0, 5 to avoid repetitive results; to do not
repeat the computations for similar proportional gemoetries.
The boundary conditions summerized below can be found at Code A:
• Border 1: two options explained next, sections 3.2.1 and 3.2.2.
• Border 2: u = 0 and v = 0 as the wall of the RBC cannot move.
• Border 3: v = 0 due to symmetry.
30

3.2.1 Sinusoidal velocity field
The first configuration tested was with a sinusoidal velocity profile at Border 1
reaching the maximum speed U = −1 at the center of the border and thus, U = 0 at both
tips. Several simulations were made varying all the shifting parameters.
Owing to the linearity of Stokes equation, the case with a rectangular Geometry A
should give FFluid−Wallx = 0. This particular case was computed for several values of L
and the data confirmed the theory. Following this lines a graph of the former is presented.
0 0,5 1 1,5 2
L
-1
-0,75
-0,5
-0,25
0
0,25
0,5
0,75
1
fx
Figure 3.8: Results with rectangular geometry, L = [0, 1 − 2], r1 = r1 = 0, 4 and δx =
1/100.
The r1 and r2 independent variation was analyzed as well, see Figure 3.9. The x
axis corresponds to r1 for the red and blue lines and to r2 for the green line. The red line
represents the simultaneous variation of r1 and r2. For the blue and the green line, r2
and r1 are fixed respectively while the oder variable changes its value.
The results show an always repulsive behavior for the three cases presented. The
outcome do not fulfil the expectations of finding an attractive-repulsive behavior in func-
tion of the computed parameters. The former is used to justify the approach of the next
simulations with an arbitrary fixed value of r1 = r2 = 0, 4 until a change of the force’s
behavior is noticed when varying other parameters of the geometry.
Note that a relevant assumption is made here since two consecutive RBCs do not
show the same shape as it has been explained in previous studies[6].
31

0,1 0,2 0,3 0,4 0,5
r1
0
2
4
6
8
fx
r1=r2
r2=0,4
r1=0,4
r1 and r2 study
Figure 3.9: Results of r1 and r2 variation with L = 1 and δx = 1/100.
As a consequence of the previous graph, their value was ﬁxed at r1 = r2 = 0, 4 and
the outcome is presented in Figure 3.10.
32

0 0,5 1 1,5 2
L
6
6,2
6,4
6,6
fx
Sinusoidal Velocity Profile
Figure 3.10: Results with Sinusoidal velocity proﬁle, r1 = r1 = 0, 4.
Having in mind the force explanations made on section 3.1.1, the graph shows that
RBCs behave only repulsively. Although the value of the FFluid−Wallx shifts with L’s
variation, it remains all the time in a positive value.
The mesh and velocity ﬁeld output plots are presented next.
Figure 3.11: Mesh Geometry A Sinusoidal - L = 1, 5, r1 = r2 = 0, 4 and δx = 1/100.
33

Figure 3.12: Velocity Field Geometry A Sinusoidal - L = 1, 5, r1 = r2 = 0, 4 and δx =
1/100.
The vortex mentioned in section 1.2.1, appears and converges on the centreline as
expected. From the results yielded with the sinusoidal velocity ﬁeld, an always repulsive
behavior is obtained which does not match with the statements announced by previous
studies (Chapter 1). A more complex model should be considered. Therefore, another
velocity ﬁeld on this geometry is analyzed next.
34

3.2.2 Sinusoidal-1-Sinusoidal velocity field
As the previous boundary condition showed unexpected results, a little modification
was made. Instead of having a full sinusoidal velocity profile at Border 1, a combination of
sinusoidal and uniform velocity value was applied. A sinusoidal velocity is considered on
both tips of Border 1 with a maximum value of U = −1 at a 0,1 distance from the corner
and U = 0 on the vertex. These two fractions are linked by a U = −1 uniform velocity
profile. This new set up considers a wider velocity field rather than with the previous one
in which a full sinusoidal profile was applied.
Keeping in track of the decisions made in the preceding section, the simulations
made with the present velocity field considered r1 = r2 = 0, 4 to carry out the next
calculations. In Figure 3.13, the outcome of the L variation is introduced.
0 0,5 1 1,5 2
L
6
8
10
12
14
16
18
20
fx
Sinusoidal-1-Sinusoidal Velocity Profile
Figure 3.13: Results with Sinusoidal-1-Sinusoidal velocity profile, r1 = r2 = 0, 4.
The graph shows a gradual change when distance between RBCs grows. These
results prove definitely that the geometry and the boundary conditions considered are not
valid (over-simplified configuration) taking into account the previous expectations.
35

The mesh and velocity field output plots are presented below:
Figure 3.14: Mesh Geometry A Sinusoidal-1-Sinusoidal - L = 1, 5, r1 = r2 = 0, 4 and
δx = 1/100.
Figure 3.15: Velocity Field Geometry A Sinusoidal-1-Sinusoidal - L = 1, 5, r1 = r2 = 0, 4
and δx = 1/100.
The velocity field shows a substantial change than the one from Figure 3.12, which
responds to the wider velocity field introduced.
From the information collected from previous studies, an evolution from repulsive
to attractive of the force interaction between the RBCs expected. However, the data still
reveals a solely repulsive behavior, which implies to think over the problem. The lack
of expected results may be caused by a wrong choice of the geometry selected to study.
Thus, it was found convenient to enlarge the shape to compute. To do so, ”wings” were
added to the original Geometry A and by this way, the action of the flow at the RBCs’
lateral walls can be adhered to force calculations. Then, after revaluation, Geometry B is
introduced as the step forward on this research since it considers partially or entirely the
side border of the RBCs.
36

3.3 GEOMETRY B - TRAPEZOID WITH WINGS
The new geometry considered has some more parameters to shift, which provide the
opportunity to study more configurations and allow, mainly, to consider the FFluid−Wall
on borders 3 and 7 (lower part of the wings, see Figure 3.16 below). As explained in
the previous section, the total force on the vertical direction equals to zero due to the
problem’s symmetry.
Figure 3.16: Geometry B.
For this geometry, the shifting values are:
• L distance between two consecutive RBCs. L = [0, 1 − 2]
• r1 parameter used to fix the front shape of the pursuer RBC. r1 = [0, 1 − 0, 5]
• r2 parameter used to fix the back shape of the leading RBC. r2 = [0, 1 − 0, 5]
• d distance between the side of the RBC and the capillarys wall. d = [0, 05 − 1, 5]
• D length of the RBC’s sides considered on the force calculations. D = [0, 05 − 2]
*The RBC’s width parameter is fixed at h = 0, 5 to avoid repetitive results; to do not
repeat the computations for similar proportional gemoetries.
The range of values considered for all the parameters was established after prove-
error simulations to finally show the proper results for the lector’s understanding. They
are linked one to another: L has been studied from [0, 1 − 2] since the total width of the
37

RBCs is 1 (2 times h) and thus, the maximum length of the RBCs varies from [0, 05 − 2].
The reason of introducing values of D smaller than h relies on the desire to know the
influence of the RBCs’ lateral walls in the total hydrodynamic force. The range of d
focuses on the influence of confinement13 so, it is liked to the RBCs’ overall size with the
aim of simulationg several configurations with different proportions. Here it can be seen
the utility of fixing h value.
The boundary conditions summarized below can be found at Appendix B:
• Border 1: u = −1 and v = 0.
• Border 2: u =linear profile and v = 0.
• Border 5: v = 0 due to symmetry.
• Border 8: u =linear profile and v = 0.
The linear velocity profile u(y) mentioned is presented next (Figure 3.17). The
maximum value of u(y) is umax = U = −1 and has been considered to be placed on the
capillary’s wall (Border1). As written in the above boundary conditions, u(y = 0) = 0
and u(y) grows gradually with a slope of α = −1/2, being u(y = d/2) = U/2 = −1/2.
In section 3.1.1, Figure 3.4 was introduced to explain the borders considered to
perform the force calculations. To do so, multiborder technique has been applied as well
in borders 3 and 7.
It is important to remark that on the results presented next, when variation of d
parameter is involved, the maximum velocity U = −1 is always considered at the maximum
d value: u(y = dmax) = −1. So, for values of di < dmax, the velocity u(y = di) = −1 and
it will always follow the relation: u(y) = U y−h
dmax
, where y = di + h.
13
Relation between the RBCs’ and the capillary’s width. In short, how wide is the gap between them
both.
38

Figure 3.17: Linear velocity proﬁle.
39

r1 and r2 study
As with the previous geometry, a study of r1 and r2 parameters was conducted to
determine their involvement in the hydrodynamic attraction and repulsion mechanism.
Three cases are presented: one with both parameters varying at the same time from
[0, 1 − 0, 5] and two considering one of them fixed and the other one shifting its value
inside the range [0, 1 − 0, 5].
0,1 0,2 0,3 0,4 0,5
r1
-0,8
-0,6
-0,4
-0,2
0
0,2
fx
L=0,5
L=1,0
L=1,5
L=2,0
Figure 3.18: Results of r1 and r2 variation with L = [0, 5 − 2], d = 0, 5, D = 0, 5 and
δx = 1/100.
From the graph above, Figure 3.18, several behaviors can be detected depending
on the distance between the RBCs. Firstly, for the smallest plotted distance between the
RBCs, a negative slope of the curve is observed. This means that for this separation, as
higher is the slope defining the RBCs’ shape, more attractive becomes their interaction.
Nonetheless, this result loses its relevance due to the already explained aspect of non-
identical shape between two consecutive RBCs. As L grows, the tendency to an attractive
behavior disappears remaining on a repulsive conduct.
The same behavior is experienced when keeping r2 value fixed at 0, 4 and varying
r1, see Figure 3.19. In this case, the result is more interesting since the shape is not the
same for the two RBCs. The front one keeps its back slope constant while the trailing
one is tested for several slope values. The significant aspect to note is that with a fixed
distance between RBCs, as more pronounced is the front slope of the back particle, more
attractive (or less repulsive) is its behavior.
For the third case presented, Figure 3.20, the former is not true: it is the opposite.
The force value does not vary much when shifting the back slope shape of the leading
40

RBC. This induces the conclusion of little involvement of the back slope shape of the
leading RBC on the hydrodynamic interaction mechanism.
0,1 0,2 0,3 0,4 0,5
r1
-0,8
-0,6
-0,4
-0,2
0
0,2
0,4
fc
L=0,5 - r2=0,4
L=1,0 - r2=0,4
L=1,5 - r2=0,4
L=2,0 - r2=0,4
Figure 3.19: Results of r1 variations with r2 = 0, 4, L = [0, 5 − 2], d = 0, 5, D = 0, 5 and
δx = 1/100.
0,1 0,2 0,3 0,4 0,5
r2
-0,8
-0,6
-0,4
-0,2
0
0,2
0,4
fx
L=0,5 - r1=0,4
L=1,0 - r1=0,4
L=1,5 - r1=0,4
L=2,0 - r1=0,4
Figure 3.20: Results of r2 variations with r1 = 0, 4, L = [0, 5 − 2], d = 0, 5, D = 0, 5 and
δx = 1/100.
41

L study
The next step was to see if for this new geometry and linear velocity field considered,
the distance between RBCs had a relevant involvement on the interaction mechanism. To
do so, the problem was simulated for several configurations varying L = [0, 1 − 2] as well
as the newly introduced parameters: d = [0, 05 − 0, 25] (with u(di = 0, 25) = −1 for all
the calculations) and D = 0, 25. r1 and r2 were kept fixed at 0, 4 waiting for a relevant
result to arise.
0 0,5 1 1,5 2
L
-2
-1,5
-1
-0,5
0
0,5
1
fx
d=0,05
d=0,10
d=0,15
d=0,20
d=0,25
Figure 3.21: Results of L = [0, 1 − 2] with r1 = r2 = 0, 4, d = [0, 05 − 0, 25], D = 0, 25
and δx = 1/100.
The above graph does not show the expected result and keeps showing an unstable
behavior since the force turns from attractive to repulsive when L grows. It can be observed
that as the capillary widens (d growth), the interaction force experienced becomes smaller
(moving down in the graph). These results mean that, from a certain distance between
RBCs (depending on the confinement), if L is lower than the critical value L∗14 the RBCs
will experience a growing attraction force until they collide. But, if L > L∗ the RBCs with
repulse and separate. In short, an unstable behavior emerged, the opposite of the expected
14
Value of L at which force changes from attractive to repulsive.
42

behavior stated by previous studies: a soft change between repulsion and attraction at L∗
(neutral point).
To gather if a wider pipe changes the behavior of the RBCs, the problem was
simulated again for several configurations varying L = [0, 5 − 2] with D = 1 and r1 =
r2 = 0, 4. The parameter controlling the confinement was computed for d = [0, 25 − 1].
See Figure 3.22. In this case the maximum velocity U = −1 was considered each time at
the corresponding d value, either 0, 25, 0, 5 or 1. The decision was taken since d starts to
considerably vary from one value to the other and so does the velocity if the maximum is
not relocated.
0 0,5 1 1,5 2
L
-3
-2,5
-2
-1,5
-1
-0,5
0
0,5
fx
D=1 - d=0,25
D=1 - d=0,5
D=1 - d=1,0
Figure 3.22: Results of L = [0, 1 − 2] with r1 = r2 = 0, 4, d = [0, 25 − 1], D = 1 and
δx = 1/100.
From the graph above it can be deduced that, even after enlarging the capillary’s
width, with a fixed RBCs length D = 1, the bahavior does not change from the one seen
in Figure 3.21. The main difference relies on the fact that for a wider pipe, the whole
curve moves up in the graph, contrary to the tendency experienced before. The former
opens the new path of the research: shift the newly introduced parameters of Geometry B
(d and D) to gather if they have any direct influence to the interaction mechanism, more
relevant than the little ones already commented. Several configurations are presented next
in order to explain the involvement of the mentioned parameters.
43

D study
The first approach, with the aim of knowing if the RBCs’ length has a major involve-
ment in the interaction mechanism, is presented in Figure 3.23. For longitudinal length
values from [0, 05 − 0, 5], the total force value shows that as longer is the RBC, more
attractive the behavior is. Also, as shorter is the distance between RBCs, more negative
(less repulsive) is their behavior, which matches with the previous analyzis made varying
L parameter. The calculations were made considering umax = U = −1 at d = 0, 25.
0 0,1 0,2 0,3 0,4 0,5
D
-0,4
-0,2
0
0,2
fx
L=0,1
L=0,2
L=0,3
L=0,4
L=0,5
L=0,6
L=0,7
L=0,8
L=0,9
L=1,0
L=1,1
L=1,2
L=1,3
L=1,4
L=1,5
L=1,6
L=1,7
L=1,8
L=1,9
L=2,0
Figure 3.23: Results of D = [0, 05 − 0, 5] with r1 = r2 = 0, 4, d = 0, 25, L = [0, 1 − 2] and
δx = 1/100.
To better understand and confirm the influence of D parameter, larger RBCs were
computed, D = [0, 05−2]. In this case, calculations were made considering umax = U = −1
at d = 0, 5, owing to a change in the configuration analyzed. The results in Figure 3.25
show a more pronounced slope for values of D from 0,05 to 1. From then (D = [1 − 2]),
the variation of the force value does not change much; probably due to long enough RBCs.
For this reason the following study of d parameter will consider the length of RBCs to be
D = 1, see Figure 3.24.
44

Figure 3.24: Geometry B considering larger RBCs, D = 1, L = 1 and d = 0, 25.
0 0,5 1 1,5 2
D
-0,8
-0,6
-0,4
-0,2
0
0,2
fx
L=0,5
L=1,0
L=1,5
L=2,0
Figure 3.25: Results of D = [0, 05 − 2] with r1 = r2 = 0, 4, d = 0, 5, L = [0, 5 − 2] and
δx = 1/100.
45

d study
The last parameter to study is the one used to control the conﬁnement of the RBCs
in the capillary. To perform this simulation, d was considered from being really small so
the RBCs are almost as wide as the capillary where they ﬂow through, to be twice and
three times larger than the RBCs - d = [0, 05 − 1, 5]. This simulation was carried out
considering several distances between RBCs L = [0, 1 − 2] and a RBC’s lateral length of
D = 1. It was divided in two simulations in order to adapt the location of the maximum
velocity umax = U = −1 placed at d = 0, 25 for smaller d values (Figure 3.26) and placed
at d = 0, 5, d = 1, 0 and d = 1, 5 when the capillary computed was wider (Figure 3.27).
The outcome was:
0,05 0,1 0,15 0,2 0,25
d
-3
-2
-1
0
1
fx
L=0,1
L=0,2
L=0,3
L=0,4
L=0,5
L=0,6
L=0,7
L=0,8
L=0,9
L=1,0
L=1,1
L=1,2
L=1,3
L=1,4
L=1,5
L=1,6
L=1,7
L=1,8
L=1,9
L=2,0
Figure 3.26: Results of d = [0, 05 − 0, 25] with r1 = r2 = 0, 4, D = 1, L = [0, 1 − 2] and
δx = 1/100.
46

0,6 0,8 1 1,2 1,4
d
-1
-0,5
0
fx
L=0,1
L=0,2
L=0,3
L=0,4
L=0,5
L=0,6
L=0,7
L=0,8
L=0,9
L=1,0
L=1,1
L=1,2
L=1,3
L=1,4
L=1,5
L=1,6
L=1,7
L=1,8
L=1,9
L=2,0
Figure 3.27: Results of d = [0, 5 − 1, 5] with r1 = r2 = 0, 4, D = 1, L = [0, 1 − 2] and
δx = 1/100.
When the free distance between the RBCs and the capillary’s wall is small, Figure
3.26, the force becomes more attractive as they are closer one to another. In terms of d
affecting directily the interaction mechanism, little implication is observed since the curves
representing its growth show a flat shape.
The second graph, Figure 3.27 shows, for a wide range of distances between RBCs,
three cases of wider separation with the capillary’s wall. The higher reaction when varying
d is experienced when the RBCS are closer (little L values). Nonetheless, as wider is the
distance with the pipe, less attractive is the resultant force. The results stay always
attractive except for the biggest L values where a low repulsive force is observed for all d
values.
To sum up, confinement has some implication in the force value with short distances
between RBCs. Nevertheless, it does not change the behavior of the mechanism since the
force stays attractive for the mentioned L values.
47

Mesh and Velocity Field Plots
An example of the mesh and the velocity ﬁeld which clearly shows the linear velocity
imposed is presented next. The plots were obtained simulating the geometry for L = 1, 5,
r1 = r2 = 0, 4, d = 0, 5, D = 0, 5 and δx = 1/100.
Figure 3.28: Mesh Geometry B - L = 1, 5, r1 = r2 = 0, 4, d = 0, 5, D = 0, 5 and
δx = 1/100.
Figure 3.29: Velocity Field Geometry B - L = 1, 5, r1 = r2 = 0, 4, d = 0, 5, D = 0, 5 and
δx = 1/100.
48

3.3.1 Code verification
After having performed and analyzed several configurations of the designed geome-
tries, the results did not show a reasonable interaction mechanism according to previous
studies made on this field. At the beging of the research, the finding of an estable mech-
anism ruled by hydrodynamic forces was expected to be ecountered. However, due to the
again unexpected results obtained for the second geometry, the code was brought into
question. What if the problem’s code had been wrongly written?
To erase any doubt, several test cases were conducted in order to confirm the code’s
validity. All the test problems outlined had a known physic answer so it was easy to verify
the outcome. They were done either using a velocity field or a pressure field. Although
the finding of a code mistake was expected, the test cases proved that the code had been
written correctly. The sign of the force was an all-time doubt on whether a negative sign
had to be added or not, as explained previously in section 3.1.1. All the results confirmed
the mathematical development done and the negative sign was properly used in the force
calculations.
49

3.3.2 Other studies
When the results obtained were not the expected ones regarding the precedent
studies named in Chapter 1, several new configurations were shortly introduced to gather
if another configuration could show what the research aimed to find. Since the time of the
internship is limited, any of the new setups could be deeply analyzed but still, with little
simulations done, they did not show any relevant change related to the hydrodynamic
interaction mechanism.
Configurations
• Simulate a much larger back RBC in order to resemble a cluster of several RBCs,
Dtrailing = [0, 5 − 3].
• Widen the pipe to gather the behavior in a completely different confinement, d =
[0, 5 − 5].
• Evolution of the geometry considered. The new configuration, Geometry C considers
the two entire RBCs, see Figure 3.30. Several parameters more were added to do
so.
Figure 3.30: Geometry C.
The third new configuration mentioned might be the one to consider in further
studies. Although the first simulations made showed the same comportment as Geometry
B, a deeper study of the multiple parameters used to define the geometry may show up a
reasonable hydrodynamic interaction mechanism.
50

3.4 CONCLUSIONS
After a deep and dedicated research during the 5-month internship, the problem
outlined showed to be more complex than initialy anticipated due to subtle hydrodynamic
interactions. All the results summarized in this report show an unstable behavior in
which two RBCs experience an attractive-repulsive mechanism completely differing from
the stable behavior exposed by previous studies.
Nevertheless, the overall result obtained remains as a scientific result, which shows
the problem to be more complex than expected. This report presents a study where proba-
bly too much simplifications were made and the outcome exemplifies it. Maybe, to obtain
relevant results about the hydrodynamic interaction mechanism, more variables should
be taken into account or the ones used in this research should be considered differently.
What is sure, is that the way followed to solve the problem outlined proves that matters
regarding RBCs and blood are more complex than expected.
The question about the governing mechanism of hydrodynamic interaction between
two consecutive RBCs remains to be answered. It is sure though, that the path has been
already established and, from now on, new doors await to be opened. The simplifications
made, all summed together, are meant to be responsible of the result gardened: none in
concrete, but all the same way.
From the second configuration presented, Geometry B, short statements can be
written:
- r1 and r2: For small distances between RBCs and a fixed back slope of the leading
RBC (r2 = 0, 4), when the front slope of the trailing RBC is more pronounced more
attractive is the behavior of the set.
- L: It has a relevant influence on the dynamics of the problem. When smaller is the
distance between RBCs, the interaction force becomes more attractive and as separation
grows, from a unstable neutral point, it turns more repulsive.
- D: An important contribution to the total force calculated is observed when con-
sidering reasonable large enough RBCs.
- d: Although having some implication in the force values when L is little, it does
not imply any change in the interaction mechanism.
- Linear velocity field: It was introduced as an improvment of the velocity fields
considered in Geometry A. Although the linear flow considered between the RBCs and
the capillary is very close to reality, future studies with Geometry C should better apply
a parabolic flow profile on the front and the back of the fluid boundaries, which is the one
encountered in blood.
51

Perspectives
The present report opens a set of perspectives:
• Parabolic flow and Parachute-shape RBCs. To reduce the amount of simplifactions
made on the present report, to consider a parabolic velocity field and more accurate
designed RBCs may be helpful to investigate again the interaction mechanism.
• Consider the elasticity of the cytoskeleton of the RBC. Only with Gemoetry C, which
considers the front part of the RBCs. Nevertheless, the effect of the cytoskeleton is
believed to be minor, since, as exlained in Chapter 1, the difference between vesicles
and red blood cells in two dimensions is really small.
All of the above presented improvments represent a qualitative change in the prob-
lem and may a have a direct implication in the hydrodynamic interaction regulation.
52

APPENDIX A
GEOMETRY A - TRAPEZOID
On the following section, the code implemented to compute the problem regarding geom-
etry A and its multiple conﬁgurations and the data obtained are presented.
A.1 Code - Sinusoidal velocity ﬁeld
1 // Geometry A − Sin Velocity Field
3 f o r ( r e a l L=0.1;L<=2.1;L=L+0.1)
{
5 f o r ( int ns=0; ns <2; ns++)
{
7 f o r ( int np=0; np<5; np++)
{
9 // Create geometry
r e a l r1 = 0.1∗( np+1) ; // Modular Geometry : v a r i e s at each loop cycle .
11 r e a l r2 = 0.1∗( ns+1) ;
// r e a l r1 = 0 . 4 ; // When the values of r1 and r2 were f i x e d .
13 // r e a l r2 = r1 ;
r e a l h = 0 . 5 ;
15 // r e a l L= 1 . 5 ; // When the parameter L was f i x e d
r e a l bc = 0 . 1 ;
17 bool debug = f a l s e ; // To control the online p l o t s
19 // All the borders have 1/100 dx elements (L/ nelements =1/100)
int ne=50;
21 int n=100∗(L) ;
int i ;
23 int [ int ] nne ( ne ) ;
r e a l mu=1; // Viscosity
25 r e a l fx2 , fy2 , fx4 , fy4 , fx , fy , fx2i , fy2i , fx4i , fy4i , fxhalf , f y h a l f
;
int [ int ] r i g h t ( ne ) , l e f t ( ne ) ;
27
// Create Geometry
29 border w1( t =0,L) { x=L−t ; y=h ; l a b e l =1;}; // Top
border w2( t =0 ,1; i )
31 {
x=( r e a l ( i ) ∗ r1 /ne )+(r1 ∗ t /ne ) ;
33 y = (h−( r e a l ( i ) ∗h/ne ) )−(h∗ t /ne ) ;
l a b e l=i +2;
35 }; // Left
border w3( t=r1 , ( L+r2 ) ) { x=t ; y=0; l a b e l=1+ne +1;}; // Base
37 border w4( t =0 ,1; i )
53

{
39 x=(L+r2 )−( r e a l ( i ) ∗ r2 /ne )−(r2 ∗ t /ne ) ;
y=( r e a l ( i ) ∗h/ne )+(h∗ t /ne ) ;
41 l a b e l=i+2+ne+1;
}; // Right
43
// Built Mesh
45 f o r ( i =0; i<ne ; i++)
{
47 nne [ i ]=2;
l e f t [ i ] = i +2;
49 r i g h t [ i ] = i+1+ne+1+1;
}
51
mesh Th = buildmesh (w1(n)+w2( nne )+w3(n)+w4( nne ) ) ;
53
// Define spaces and functions
55 fespace Uh(Th, P2) ;
Uh u , v , uu , vv ;
57 fespace Ph(Th, P1) ;
Ph p , pp ;
59
// Solve Linear Problem
−1e−10∗p∗pp)
65 + on (1 , u=−sin (( x/L) ∗ pi ) ,v=0)
+ on(1+ne+1,v=0)
67 + on ( l e f t , right , u=0,v=0) ;
69 // Plot Mesh and Velocity Field
plot (Th, wait=debug ,cmm= ”Mesh” , ps=” Mesh Geo A sin L ”+L+” r 1 ”+r1+”
e l e 1 0 0 ”+” . eps ” ) ;
71 plot ( [ u , v ] , wait=debug ,cmm= ” Velocity Field ” , value=true , ps=”
VF Geo A sin L ”+L+” r 1 ”+r1+” e l e 1 0 0 ”+” . eps ” ) ;
73 // Calculations of the surface f o r c e s
75 // Force on w2
f o r ( i =4; i<=3+ne ; i++)
77 {
f x 2 i= − int1d (Th, i ) (mu∗((2∗ dx(u)−p/mu) ∗(N. x)+(dy(u)+dx(v) ) ∗(N. y) ) ) ;
79 f y 2 i= − int1d (Th, i ) (mu∗(( dy(u)+dx(v) ) ∗(N. x) +(2∗dy(v)−p/mu) ∗(N. y) ) ) ;
fx2=fx2+f x 2 i ;
81 fy2=fy2+f y 2 i ;
}
83
54

// Force on w4
85 f o r ( i=3+ne+1+1;i<=3+ne+1+ne ; i++)
{
87 f x 4 i= − int1d (Th, i ) (mu∗((2∗ dx(u)−p/mu) ∗N. x+(dy(u)+dx(v) ) ∗N. y) ) ;
f y 4 i= − int1d (Th, i ) (mu∗(( dy(u)+dx(v) ) ∗N. x+(2∗dy(v)−p/mu) ∗N. y) ) ;
89 fx4=fx4+f x 4 i ;
fy4=fy4+f y 4 i ;
91 }
93 // Resultant f o r c e
f x h a l f= fx4−fx2 ;
95 f y h a l f= fy4−fy2 ;
97 fx=2∗ f x h a l f ;
fy=2∗ f y h a l f ;
99
// Print f i n a l values
101 cout<<” fx= ”<<fx<<” ; ”<<endl ;
cout<<” fy= ”<<fy<<” ; ”<<endl ;
103
// Output data f i l e
105 ofstream f p l o t ( ” Geometry A sin loop . dat” , append ) ;
f p l o t . p r e c i s i o n (6) ;
107 fplot <<fx<<” t ”<<r1<<” t ”<<r2<<” t ”<<L<<endl ;
109 }
}
111 }
3 Codes/Geometry A sin loop.edp
55

A.2 Code - Sinusoidal-1-Sinusoidal velocity ﬁeld
// Geometry A − Sin 1 Sin Velocity Field
2
f o r ( r e a l L=0.22;L<=2.1;L=L+0.1)
4 {
f o r ( int ns=0; ns <2; ns++)
6 {
f o r ( int np=0; np<5; np++)
8 {
// Create geometry
10 r e a l r1 = 0.1∗( np+1) ; // Modular Geometry : v a r i e s at each loop cycle .
r e a l r2 = 0.1∗( ns+1) ;
12 // r e a l r1 = 0 . 4 ; // When the values of r1 and r2 were f i x e d .
// r e a l r2 = r1 ;
14 r e a l h = 0 . 5 ;
// r e a l L= 1 . 5 ; // When the parameter L was f i x e d
16 r e a l bc = 0 . 1 ;
bool debug = f a l s e ; // To control the online p l o t s
18
// All the borders have 1/100 dx elements (L/ nelements =1/100)
20 int n1=10;
int n2=100∗(L−(2∗bc ) ) ;
22 int n3=10;
int ne=50;
24 int n=100∗(L) ;
int i ;
26 int [ int ] nne ( ne ) ;
r e a l mu=1; // Viscosity
28 r e a l fx2 , fy2 , fx4 , fy4 , fx , fy , fx2i , fy2i , fx4i , fy4i , fxhalf , f y h a l f ;
int [ int ] r i g h t ( ne ) , l e f t ( ne ) ;
30
// Create Geometry
32 border w11( t=L, L−bc ) { x=t ; y=h ; l a b e l =1;}; // Top 1
border w12( t=L−bc , bc ) { x=t ; y=h ; l a b e l =2;}; // Top 2
34 border w13( t=bc , 0 ) { x=t ; y=h ; l a b e l =3;}; // Top 3
border w2( t =0 ,1; i )
36 {
x=( r e a l ( i ) ∗ r1 /ne )+(r1 ∗ t /ne ) ;
38 y = (h−( r e a l ( i ) ∗h/ne ) )−(h∗ t /ne ) ;
l a b e l=i +4;
40 }; // Left
border w3( t=r1 , ( L+r2 ) ) { x=t ; y=0; l a b e l=3+ne +1;}; // Base
42 border w4( t =0 ,1; i )
{
44 x=(L+r2 )−( r e a l ( i ) ∗ r2 /ne )−(r2 ∗ t /ne ) ;
y=( r e a l ( i ) ∗h/ne )+(h∗ t /ne ) ;
46 l a b e l=i+3+ne+1+1;
56

}; // Right
48
// Built Mesh
50 f o r ( i =0; i<ne ; i++)
{
52 nne [ i ]=2;
l e f t [ i ] = i +4;
54 r i g h t [ i ] = i+3+ne+1+1;
}
56
mesh Th = buildmesh (w11( n1 )+w12( n2 )+w13( n3 )+w2( nne )+w3(n)+w4( nne ) ) ;
58
Ph p , pp ;
64
−1e−10∗p∗pp)
70 + on (1 , u=−sin ( ( ( L−x) /(2∗ bc ) ) ∗ pi ) ,v=0)
+ on (2 , u=−1,v=0)
72 + on (3 , u=−sin (( x/(2∗ bc ) ) ∗ pi ) ,v=0)
+ on(3+ne+1,v=0)
74 + on ( l e f t , right , u=0,v=0) ;
76 // Plot Mesh and Velocity Field
plot (Th, wait=debug ,cmm= ”Mesh” , ps=” Mesh Geo A sin1sin L ”+L+” r 1 ”+r1+”
e l e 1 0 0 ”+” . eps ” ) ;
78 plot ( [ u , v ] , wait=debug ,cmm= ” Velocity Field ” , value=true , ps=”
VF Geo A sin1sin L ”+L+” r 1 ”+r1+” e l e 1 0 0 ”+” . eps ” ) ;
80 // Calculations of the surface f o r c e s
82 // Force on w2
f o r ( i =4; i<=3+ne ; i++)
84 {
f x 2 i= − int1d (Th, i ) (mu∗((2∗ dx(u)−p/mu) ∗(N. x)+(dy(u)+dx(v) ) ∗(N. y) ) ) ;
86 f y 2 i= − int1d (Th, i ) (mu∗(( dy(u)+dx(v) ) ∗(N. x) +(2∗dy(v)−p/mu) ∗(N. y) ) ) ;
fx2=fx2+f x 2 i ;
88 fy2=fy2+f y 2 i ;
}
90
// Force on w4
92 f o r ( i=3+ne+1+1;i<=3+ne+1+ne ; i++)
57

{
94 f x 4 i= − int1d (Th, i ) (mu∗((2∗ dx(u)−p/mu) ∗N. x+(dy(u)+dx(v) ) ∗N. y) ) ;
f y 4 i= − int1d (Th, i ) (mu∗(( dy(u)+dx(v) ) ∗N. x+(2∗dy(v)−p/mu) ∗N. y) ) ;
96 fx4=fx4+f x 4 i ;
fy4=fy4+f y 4 i ;
98 }
100 // Resultant f o r c e
f x h a l f= fx4−fx2 ;
102 f y h a l f= fy4−fy2 ;
104 fx=2∗ f x h a l f ;
106
108 cout<<” fx= ”<<fx<<” ; ”<<endl ;
cout<<” fy= ”<<fy<<” ; ”<<endl ;
110
112 ofstream f p l o t ( ” Geometry A sin 1 sin loop . dat” , append ) ;
114 fplot <<fx<<” t ”<<r1<<” t ”<<r2<<” t ”<<L<<endl ;
116 }
}
118 }
3 Codes/Geometry A sin 1 sin loop.edp
58

APPENDIX B
GEOMETRY B - TRAPEZOID WITH WINGS
On the following section, the code implemented to compute the problem regarding geom-
etry B and its multiple conﬁgurations and the data obtained are presented.
B.1 Code
// Geometry B − Linear Velocity Field
2
f o r ( r e a l L=0.1;L<2.1;L=L+0.1)
4 {
f o r ( r e a l d=0.5;d<=1;d=d+0.5)
6 {
f o r ( r e a l D=0.05;D<=0.5;D=D+0.05)
8 {
f o r ( int ns=0; ns <5; ns++)
10 {
f o r ( int np=0; np<5; np++)
12 {
// Create geometry
14 r e a l r1 = 0.1∗( np+1) ; // Modular Geometry : v a r i e s at each loop cycle .
r e a l r2 = 0.1∗( ns+1) ;
16 // r e a l r1 = 0 . 4 ; // When the values of r1 , r2 , d and D were f i x e d .
// r e a l r2 = r1 ;
18 // r e a l d = 0 . 2 5 ;
// r e a l D = 0 . 2 5 ;
20 r e a l h = 0 . 5 ;
bool debug = f a l s e ; // To control the online p l o t s
22
// All the borders have 1/100 dx elements (L/ nelements =1/100)
24 int ne = 50;
int nd = 100∗D;
26 int n = 100∗(L+2∗D) ;
int nm = 100∗L ;
28 int nn = 100∗d ;
int i ;
30 int [ int ] nne ( ne ) , nnd(nd) ;
r e a l mu = 1; // Viscosity
32 r e a l fx2 , fy2 , fx6 , fy6 , fx , fy , fx2i , fy2i , fx6i , fy6i , fx3 , fy3 , fx7 ,
fy7 , fx3i , fy3i , fx7i , fy7i , fxhalf , f y h a l f ;
int [ int ] r i g h t ( ne ) , l e f t ( ne ) , righth (nd) , l e f t h (nd) ;
34
// Create Geometry
36 border w1( t=L+D,−D) { x=t ; y=h+d ; l a b e l =1;}; // Top
border w2( t=h+d , h) { x=−D; y=t ; l a b e l =2;}; // L e f t V e r t i c a l
59

38 border w3( t =0 ,1; i )
{
40 x=((−D)+r e a l ( i ) ∗D/nd)+(D∗ t /nd) ;
y=h ;
42 l a b e l=2+i +1;
}; // Left Horizontal
44 border w4( t =0 ,1; i )
{
46 x=( r e a l ( i ) ∗ r1 /ne )+(r1 ∗ t /ne ) ;
y = (h−( r e a l ( i ) ∗h/ne ) )−(h∗ t /ne ) ;
48 l a b e l=i+2+nd+1;
}; // Left Slope
50 border w5( t=r1 , ( L+r2 ) ) { x=t ; y=0; l a b e l=2+nd+ne +1;}; // Base
border w6( t =0 ,1; i )
52 {
x=(L+r2 )−( r e a l ( i ) ∗ r2 /ne )−(r2 ∗ t /ne ) ;
54 y=( r e a l ( i ) ∗h/ne )+(h∗ t /ne ) ;
l a b e l=i+2+nd+ne+1+1;
56 }; // Right Slope
border w7( t =0 ,1; i )
58 {
x=((L)+r e a l ( i ) ∗D/nd)+(D∗ t /nd) ;
60 y=h ;
l a b e l=i+2+nd+ne+1+ne+1;
62 }; // Right Horizontal
border w8( t=h , h+d) { x=L+D; y=t ; l a b e l=2+nd+ne+1+ne+nd+1;}; //
Right Vertical
64
// Built Mesh
66 f o r ( i =0; i<ne ; i++)
{
68 nne [ i ] = 1;
l e f t [ i ] = i+2+nd+1;
70 r i g h t [ i ] = i+2+nd+ne+1+1;
}
72 f o r ( i =0; i<nd ; i++)
{
74 nnd [ i ] = 1;
l e f t h [ i ] = i +2+1;
76 righth [ i ] = i+2+nd+ne+1+ne+1;
}
78
mesh Th = buildmesh (w1(n)+w2(nn)+w3(nnd)+w4( nne )+w5(nm)+w6( nne )+w7(nnd)
+w8(nn) ) ;
80
60

Ph p , pp ;
86
88 int U = 1;
solve stokes ( [ u , v , p ] , [ uu , vv , pp ] ) =
90 int2d (Th) (dx(u) ∗dx(uu)+dy(u) ∗dy(uu) + dx(v) ∗dx( vv )+ dy(v) ∗dy( vv )
+ dx(p) ∗uu + dy(p) ∗vv + pp∗(dx(u)+dy(v) )
92 −1e−10∗p∗pp)
+ on (1 , u=−(y−h) ∗(U/1) ,v=0) //Top // U i s divided by the maximum d
value considered ( to be changed i f simulation inputs are modified )
94 + on (2 , u=−U∗(( y−h) /1) ,v=0) // L e f t V e r t i c a l
+ on ( le f t h , u=0,v=0) // Left Horizontal
96 + on ( l e f t , right , u=0,v=0) // Left Slope Right Slope
+ on(2+nd+ne+1,v=0) // Base
98 + on ( righth , u=0,v=0) // Right Horizontal
+ on(2+nd+ne+1+ne+nd+1,u=−U∗(( y−h) /1) ,v=0) ; // Right Vertical
100
// Plot Mesh and Velocity Field
102 plot (Th, wait=debug ,cmm= ”Mesh” , ps=”Mesh Geo B Linear ”+L+” d ”+d+” D ”+
D+” r 1 ”+r1+” r 2 ”+r2+” el e 1 00 ”+” . eps ” ) ;
plot ( [ u , v ] , wait=debug ,cmm= ” Velocity Field ” , value=true , ps=”
VF Geo B Linear ”+L+” d ”+d+” D ”+D+” r 1 ”+r1+” r 2 ”+r2+” el e 1 00 ”+” . eps ” )
;
104
// Calculations of the surface f o r c e s
106
// Force on w3
108 f o r ( i =3; i<=nd+2; i++)
{
110 f x 3 i = − int1d (Th, i ) (mu∗((2∗ dx(u)−p/mu) ∗(N. x)+(dy(u)+dx(v) ) ∗(N. y) ) ) ;
f y 3 i = − int1d (Th, i ) (mu∗(( dy(u)+dx(v) ) ∗(N. x) +(2∗dy(v)−p/mu) ∗(N. y) ) ) ;
112 fx3=fx3+f x 3 i ;
fy3=fy3+f y 3 i ;
114 }
116 // Force on w4
f o r ( i=2+nd+1; i<=ne+2+nd ; i++)
118 {
f x 2 i = − int1d (Th, i ) (mu∗((2∗ dx(u)−p/mu) ∗(N. x)+(dy(u)+dx(v) ) ∗(N. y) ) ) ;
120 f y 2 i = − int1d (Th, i ) (mu∗(( dy(u)+dx(v) ) ∗(N. x) +(2∗dy(v)−p/mu) ∗(N. y) ) ) ;
fx2=fx2+f x 2 i ;
122 fy2=fy2+f y 2 i ;
}
124
// Force on w6
126 f o r ( i=2+nd+ne+1+1;i<=2+nd+ne+1+ne ; i++)
{
61

128 f x 6 i = − int1d (Th, i ) (mu∗((2∗ dx(u)−p/mu) ∗N. x+(dy(u)+dx(v) ) ∗N. y) ) ;
f y 6 i = − int1d (Th, i ) (mu∗(( dy(u)+dx(v) ) ∗N. x+(2∗dy(v)−p/mu) ∗N. y) ) ;
130 fx6=fx6+f x 6 i ;
fy6=fy6+f y 6 i ;
132 }
134 // Force on w7
f o r ( i=2+nd+ne+1+ne+1; i<=2+nd+ne+1+ne+nd ; i++)
136 {
f x 7 i = − int1d (Th, i ) (mu∗((2∗ dx(u)−p/mu) ∗(N. x)+(dy(u)+dx(v) ) ∗(N. y) ) ) ;
138 f y 7 i = − int1d (Th, i ) (mu∗(( dy(u)+dx(v) ) ∗(N. x) +(2∗dy(v)−p/mu) ∗(N. y) ) ) ;
fx7=fx7+f x 7 i ;
140 fy7=fy7+f y 7 i ;
}
142
// Resultant f o r c e
144 f x h a l f= ( fx6+fx7 )−(fx2+fx3 ) ;
f y h a l f= ( fy6+fy7 )−(fy2+fy3 ) ;
146 fx=2∗ f x h a l f ;
148
150 cout<<” fx= ”<<fx<<” ; ”<<endl ;
cout<<” fy= ”<<fy<<” ; ”<<endl ;
152
154 ofstream f p l o t ( ”Geometry B Linear . dat” , append ) ;
156 fplot <<fx<<” t ”<<d<<” t ”<<L<<” t ”<<D<<endl ;
158 // Save Mesh
savemesh (Th, ”Geometry B Linear Mesh .Th” ) ;
160
}
162 }
}
164 }
}
3 Codes/Geometry B Linear.edp
62

REFERENCES
[1] S. Guido and G. Tomaiuolo, Microconfined flow behavior of red blood cells in vitro,
C. R. Phys. 10, 751763 (2009).
[2] B. Kaoui, G. Biros, and C. Misbah, Why do red blood cells have asymmetric shapes
even in a symmetric flow? Phys. Rev. Lett. 103, 188101 (2009).
[3] J. L. McWhirter, H. Noguchi, and G. Gompper, Flow-induced clustering and
alignment of vesicles and red blood cells in microcapillaries, Proc. Natl. Acad. Sci.
U.S.A. 106, 60396043 (2009).
[4] J. L. McWhirter, H. Noguchi, and G. Gompper, Deformation and clustering of red
blood cells in microcapillary flows, Soft Matter 7, 1096710977 (2011).
[5] G. Tomaiuolo, L. Lanotte, G. Ghigliotti, C. Misbah, and S. Guido, Red blood cell
clustering in Poiseuille microcapillary flow, Phys. Fluids 24, 051903 (2012).
[6] G. Ghigliotti, H. Selmi, L. El Asmi and C. Misbah, ”Why and how does collective
red blood cells motion occur in the blood microcirculation?” Phys. Fluids 24,
101901 (2012)
[7] Giovanni Ghigliotti. Dynamics and rheology of a suspension of vesicles and red
blood cells. Data Analysis, Statistics and Probability [physics.data-an]. Université
Joseph-Fourier - Grenoble I, 2010. English. (tel-00554161)
63

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