Abstract
In this thesis, we want to find the degree of impact stenosis
geometric features, especially the axial symmetry, can have on the
pressure drop along the side walls of stenotic arterial vessels.
Computational fluid dynamics (CFD) simulations are applied to
solve the three-dimensional steady Navier-Stokes equations, a
time-independent system characterizing incompressible, Newtonian,
homogeneous flow and commonly used in blood flow models.
The numerical solutions are computed using Finite Element
approximation, and linearization of the system is done by Picard
iteration method. As a result, we found that pressure
drop can serve as an indicator for the geometric shape, especially
the axial symmetricity, in cases of significant arterial
stenosis. While for mild stenosis, pressure drop is
not sufficiently informative in distinguishing between axisymmetric
and nonsymmetric stenosis geometries.
Table of Contents
Acknowledgement III
1 Introduction 1
1.1
Background and Motivation . . . . . . . . . . . . . . . . . . . . .
. . . 1
1.2 Related work . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . 2
1.3 Outline of the thesis . . . . . . . . . . . . . . . . . . . . .
. . . . . . . 3
2 The 3-D
Navier-Stokes Equations 4
2.1
Navier-Stokes problem at a glance . . . . . . . . . . . . . . . . .
. . . . 4
2.1.1 Preliminary theorems . . . . . . . . . . . . . . . . . . . .
. . . . 5
2.1.2 Conservation of mass . . . . . . . . . . . . . . . . . . . .
. . . . 6
2.1.3 Conservation of momentum . . . . . . . . . . . . . . . . . .
. . 6
2.2 Steady Navier-Stokes equations . . . . . . . . . . . . . . . .
. . . . . . 8
3 Numerical
Approaches 10
3.1 The Weak
Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . .
10
3.2 Finite Element Approximation . . . . . . . . . . . . . . . . .
. . . . . . 11
3.3 Linearization: Picard Iteration . . . . . . . . . . . . . . . .
. . . . . . . 14
3.3.1 Incremental analysis . . . . . . . . . . . . . . . . . . . .
. . . . 14
3.3.2 Relaxation . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . 15
4 Modeling and
CFD Simulations 16
4.1
Stenosis geometries . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . 16
4.1.1 Geometry description . . . . . . . . . . . . . . . . . . . .
. . . . 16
4.1.2 Construction and meshing . . . . . . . . . . . . . . . . . .
. . . 19
4.2 Boundary conditions . . . . . . . . . . . . . . . . . . . . . .
. . . . . . 21
4.3 Numerical solution . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . 23
5 Results,
Discussion and Follow-ups 24
5.1 Results and Discussion . . . . . . . . . . . . . .
. . . . . . . . . . . . . 24
5.2 Comparison with experimental results .
. . . . . . . . . . . . . . . . . . 28
5.3 Future works and applications . . . . . . . . . . . . . . . . .
. . . . . . 28
Bibliography
30
About this Honors Thesis
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