Boundary Integral Equations for Viscous Flows

Transcription

Boundary Integral Equations for Viscous Flows non-Newtonian Behavior and Solid Inclusions
by
Juan P. Hernández-Ortiz
A dissertation submitted in partial fulfillment of
the requirements for the degree of
Doctor of Philosophy
(Mechanical Engineering)
at the
UNIVERSITY OF WISCONSIN-MADISON
2004
© Copyright by Juan P. Hernandez-Ortiz 2004
All Right Reserved
Boundary Integral Equations for Viscous Flows - non-Newtonian
Behavior and Solid Inclusions
Juan P. Hernández-Ortiz
Under the supervision of Professor Tim A. Osswald
At the University of Wisconsin-Madison
Direct boundary integral formulations are developed for the solution of viscous
fluid flow problems, specifically for non-Newtonian fluids and fluids containing
solids. The partial differential equations are transformed into integral equations
by Green’s identities. Here, the velocity field is represented as a combination of
hydrodynamic potentials of single- and double-layer, whose densities are the
velocity and traction fields. For non-Newtonian fluid flow problems the nonlinear terms of the original equations appear as kernels of domain integrals. The
domain grid superposition (DGS) technique is developed in order to
approximate these integrals. The technique superimposes a fixed grid with the
domain under consideration. Cells, located in the intersection between the grid
and the domain, are used to directly calculate the domain integral by cell
integration. A pressure driven flow and a Couette flow are used in order to check
the DGS technique. For viscous fluids containing solids, a direct boundary
integral formulation is used to simulate solid dynamics in viscous flows. Several
problems are solved using the direct integral formulation to compare with
approximate and/or analytical solutions.
i
To my grandfathers,
Luis Guillermo and Tito Octavio,
and my family,
papá, mamá, Angelita and Olguita
ii
“…Try to keep over your life a good piece of heaven, boy –he added, turning to me-. You
have a good soul, unusual, and with the nature of an artist, so do not deny what it
needs….”
(“…Procura guardar por encima de tu vida un buen espacio de cielo, joven –añadía,
volviéndose hacia mí-. Tienes un alma muy buena, poco usual, y una naturaleza de
artista, así que no consientas que le falte lo que necesita…”)
Marcel Proust,
In Search of Lost Time: Swann’s Way
(En Busca del Tiempo Perdido. Por el Camino de Swann)
iii
Acknowledgments
The time in Madison during these last years has been an important and special
time for me, and the people which in one way or another have influenced this
Ph.D. process, in great part contributed to make these years so special.
To begin with, I have to thank the four persons who are so important in my
life: my father, my mother and my two sisters, Angelita and Olguita. I always
thought that every person requires something that motivates him or her to
improve and progress, day after day; they have been and always will be my most
important motivation. For everything and all that they represent, I want to say
thanks.
I would like to thank Patricia Arango, Pato, who will always be an important
person in my life, and who has always been there from the distance, sharing
happiness, sadness, and giving me strength; I will always remember that with
gratitute.
I would also like to acknowledge the extended families Hernández-Mesa and
Ortiz-Uribe; grandfathers and grandmothers, uncles and aunts, and cousins,
every one plays an important part in my life. I like to thank, Alfonso and Alejo
iv
which have been like brothers to me. Luchy, Jerry and the kids who, from the
first day in Madison, have been a support and family here in United States.
Finally, in my family, I want to include my advisor Tim, and his family. Tim
has not only been an academic advisor, but he has also been a spiritual and
moral advisor. More than a student-professor relationship, we formed a deep
friendship.
From the distance, in Colombia and Europe, there is a group of friends that I
want to acknowledge, for our friendship during all these years. We have shared
ideas, laughs and tears. Especially, I have to mention Alejo Rivera, Oscar Tirado,
Pipe Toro, Jairo Montes, Cipriano Lopez, Daniel Builes. I feel fortunate to rely on
friends like them.
Madison is a great city; again, the friends that I made here are in great part
responsible for these times. Aaron Hade and Sam Woodford; good and huge
friends; Juan Sanz and family, Andrés Osorio, Humberto Rivera and family,
Alejo Roldan, Sylvana Garcia and Camilo Pardo, we created a close knit group
that shared our common roots from the distance.
There is a group of special Pofessors who have participated directly in my
academic process, not only because more than the 90% of courses were taught by
them, but they collaborated in the project with ideas and knowledge. I am deeply
thankful with them. In Mechanical Engineering to my advisor Tim Osswald and
v
Professors Jeff Giacomin and Christopher Rutland; in Chemical Engineering to
Professors Juan de Pablo, Byron Bird and Michael Graham.
In the Energy and Thermodynamics Institute of the Universidad Pontificia
Bolivariana (UPB-Medellín), I want to thank my first Professors and friends Farid
Chejne and Whady Florez, who taught me to love research and academia. And
off course, how to forget that great group of friends and colleagues: Alejo Rivera,
Juan C. Ordóñez, Daniel Builes and Jorge Hinestroza. Jorge, he will always be in
our memory, as guide and example, I know that I am never going to forget him
and my goals will always have something of him. Also, I want to thank Farid
Chejne, Alejo Rivera, Juan C. Ordóñez and Professor Adrian Bejan (Duke
University) for all these years of discussions, which are guilty of making
Thermodynamic my passion.
Finally, in Germany where I have spent a couple of summers doing research
and learning, I like to thank DaimlerChrysler AG in Ulm for the financial
support there, as well as for the first year of the project. I would also like to thank
Dr. Daniel Weiss for sharing ideas and knowledge; I consider him a great
scientist and friend.
vi
Agradecimientos
El tiempo que he pasado en Madison durante los últimos años ha sido muy
importante y especial, y pienso que lo que hace que estos años hayan sido tan
especiales es la gente, que de una manera u otra, han influido en este proceso de
estudios.
Para empezar tengo que agradecer a cuatro personas muy importantes en mi
vida: mi mamá, mi papá y mis dos hermanas, Angelita y Olguita. Siempre he
pensado que cada individuo requiere de algo que lo motive a mejorar y
progresar día tras día. Ellos han sido y siempre serán mi más importante
motivación. A todo y por todo lo que ellos representan solo me queda decirles,
Gracias.
A Patricia Arango, Pato, que siempre será una persona muy importante en mi
vida, siempre ha estado conmigo a pesar de la distancia; compartiendo alegrías,
tristezas, dándome fuerzas. Nunca lo olvidaré y siempre te lo agradeceré.
A las familias: Hernández-Mesa y Ortiz-Uribe. Abuelos y abuelas, tíos y tías,
primos y primas. Cada uno es un pedazo importante en mí. En especial a
Alfonso y Alejo que han sido como hermanos. A Luchy, Jerry y las niñas quienes,
vii
desde mi primer día en Madison, han sido un apoyo importante. Ellos han sido
mi familia en Estados Unidos.
Finalmente, en este grupo familiar de agradecimientos me voy a permitir
incluir a mi advisor Tim Osswald y a su familia. Tim no sólo ha sido un gran
guía académico, sino a su vez guía espiritual y moral. Más que una relación de
estudiante-profesor, se formó una profunda amistad.
En Colombia y Europa, hay un grupo de amigos que quiero agradecer por
compartir durante todos estos años mi amistad. Hemos compartido vivencias,
ideas, risas y tristezas. En especial hay que mencionar a Alejo Rivera, Oscar
Tirado, Pipe Toro, Jairo Montes, Cipriano López, Daniel Builes. Me siento
afortunado de haber encontrado y de contar con amigos como ustedes.
En Madison: Aaron Hade y Sam Woodford, buenos y grandes amigos; Juan
Sanz y familia, Andrés Osorio, Humberto Rivera y familia, Alejo Roldan, Sylvana
Garcia y Camilo Pardo, con quienes formé un grupo especial muy unido.
Hay un grupo de Profesores quienes han contribuido directamente en mi
crecimiento académico, no sólo porque más del 90% de los cursos que tomé
fueron dictados por ellos, sino porque participaron en el proyecto con ideas y
conocimientos. A ellos les agradezco profundamente. En Ing. Mecánica a mi
advisor Tim Osswald y a los Profesores Jeff Giacomin y Christopher Rutland; en
Ing. Química a los Profesores Juan de Pablo, Byron Bird y Michael Graham.
viii
En el Instituto de Energía y Termodinámica de la Universidad Pontificia
Bolivariana (UPB-Medellín) quiero agradecer a mis primeros profesores y
amigos Farid Chejne y Whady Florez, quienes me enseñaron a querer la
investigación y la academia. Cómo no mencionar a ese grupo de amigos y
colegas: Alejo Rivera, Juan C. Ordóñez, Daniel Builes y Jorge Hinestroza. Jorge
quien siempre estará en mi memoria como guía y ejemplo, nunca lo voy a olvidar
y mis metas y goles siempre tendrán algo para él. También, quiero agradecer a
Farid Chejne, Alejo Rivera, Juan C. Ordóñez y al Profesor Adrian Bejan (Duke
University) por todos estos años de discusiones y enseñanzas, los culpables de
hacer de la Termodinámica mi pasión.
Finalmente, en Alemania donde he pasado algunos veranos, investigando y
aprendiendo, a DaimlerChrysler AG por la financiación de los primeros años del
proyecto y por las especiales pasantías en Ulm, y al Dr. Daniel Weiss por las
ideas, conocimientos y vivencias compartidas. Lo considero un gran científico y
amigo.
ix
Contents
List of Tables
xii
List of Figures
xiii
Nomenclature
xviii
Abstract
xxv
Chapter 1
Introduction
1.1
Historical background / 3
1.1.1
1.1.2
1.2
Nonlinear flows / 3
Fluid particle motion / 6
State of the art / 13
1.2.1
1.2.2
1.3
1.4
Boundary integral equations for nonlinear viscous flows / 13
Boundary integral equations for viscous flows containing
particles / 18
Objective and motivation / 21
Overview / 21
Chapter 2
Rheological and balance equations
2.1
Flow Phenomena / 25
2.1.1
2.1.2
2.1.3
2.2
Shear flow / 29
Shearfree flow /33
Generalized Newtonian fluid/ 36
2.3.1
2.3.2
2.4
Non-Newtonian viscosity / 26
Normal stresses / 27
Elastic (memory) effects / 27
Rheological flow patterns and steady-state material functions / 28
2.2.1
2.2.2
2.3
1
The Carreau-Yasuda model / 40
The Power Law model / 40
General forms of conservation equations / 41
24
x
2.4.1
2.4.2
2.4.3
Conservation of mass / 44
Conservation of momentum / 46
Conservation of internal energy / 53
Chapter 3
Integral equation theory
3.1
3.2
3.3
3.4
3.5
56
Classification of integral equations / 57
Potentials of scalar density / 59
Direct boundary integral formulation of Poisson’s equation / 63
Direct boundary integral formulation for the momentum
equations and Hydrodynamic potentials / 69
Other direct boundary integral formulations / 74
3.5.1
3.5.2
Interface flows with surface tension / 74
Penalty-function formulation for the Navier-Stokes equations
and elastostatics / 76
Chapter 4
Boundary element method
79
4.1 Isoparametric boundary elements / 80
4.2 Evaluation of the coefficient matrix c / 87
4.3 Numerical treatment of the weakly singular integrals /88
4.4 Approximation of the domain integrals / 89
4.4.1
4.4.2
4.4.3
4.4.4
4.5
Domain grid superposition technique / 98
DGS-BEM technique for ∇ 2 u = b / 101
DGS-BEM technique for ∇ 2 u = b(x, u ) / 104
DGS-BEM in three dimensional problems / 108
Iteration scheme for non-Newtonian flow problems / 115
Chapter 5
Non-Newtonian fluid flow problems
5.1
5.2
Poiseuille flow of a power law fluid / 119
Couette flow of a power law fluid/ 130
Chapter 6
Viscous flows containing particles
6.1
6.2
119
Stokes’ law: drag on a sphere / 140
Wall effects on the motion of a single particle / 145
6.2.1
6.2.2
Sphere moving parallel to a plane wall / 145
Sphere moving perpendicular to a plane wall / 148
139
xi
6.2.3
6.2.4
6.3
Sphere moving axially in a cylindrical tube / 151
Motion of a suspended rigid fiber / 158
Particle-particle interactions / 173
6.3.1
6.3.2
Two falling rigid spheres / 173
The viscosity of particulate systems / 180
Chapter 7
Conclusions and further research
189
Appendix A
Mathematical definitions
192
A.1
A.2
A.3
A.4
A.5
A.6
Lebesgue and Hilbert spaces / 192
Lyapunov surfaces / 195
Hölder continuity /196
Harmonic functions /197
Dirichlet and Neumann boundary conditions /198
Fredholm’s theorems for integral equations / 199
Appendix B
Potential theory
B.1
B.2
B.3
Potential of a field / 203
Single-layer potential continuity on the surface / 205
Double-layer potential continuity on the surface / 207
Appendix C
Green’s functions and identities
C.1
C.2
C.3
C.4
C.5
203
210
Green’s functions for scalar operators / 211
Green’s functions for matrix operators / 214
Singular solutions for the Stokes equations / 218
Green’s identities for scalar fields / 219
Green’s identities for the momentum equations / 220
References
223
xii
List of Tables
Table 2.1:
Experimental quantities in simple shear flow
31
Table 2.2:
Special shearfree flows
34
Table 2.3:
Experimental quantities in shearfree flows
34
Table 4.1:
BEM results for the Laplace equation
101
Table 4.2:
BEM results for ∇ 2 u = −2
102
Table 4.3:
BEM results for ∇ 2 u = −u
105
Table 4.4:
BEM results for ∇ 2 u = − ∂u ∂x1
106
Table 4.5:
BEM results for ∇ 2 u = −u (∂u ∂x1 )
107
Table C.1:
Green’s functions for commonly used operators [265]
213
xiii
List of Figures
Figure 2.1:
Viscous response of non-Newtonian fluids
26
Figure 2.2:
Steady simple shear flow with constant shear rate
30
Figure 2.3:
Typical behavior of the non-Newtonian viscosity and
32
the first normal stress coefficient in a polymeric liquid
Figure 2.4:
Streamlines for elongational flow
33
Figure 2.5:
Typical behavior of elongational viscosity and non-
36
Newtonian viscosity in a polymeric liquid
Figure 2.6:
Schematic Deborah vs. deformation diagram
37
Figure 2.7:
Control volume enclosing part of an interface between
42
phase A and B
Figure 3.1:
Representation of the domain, boundary and normal
66
vector
Figure 3.2:
Internal angle at a boundary point for a “non-smooth”
67
surface in 2D
Figure 4.1:
Isoparametric 8-noded quadratic element
81
Figure 4.2:
Sub-domain
95
typical
mesh
using
discontinuous
elements
Figure 4.3:
Typical cell-BEM geometry discretization
97
Figure 4.4:
Schematic of the domain grid superposition technique
99
Figure 4.5:
Comparison between the exact and the DGS technique
103
for ∇ 2 u = −2
Figure 4.6:
DGS-BEM maximum error as a function of the internal
cells number
104
xiv
Figure 4.7:
BEM results for ∇ 2 u = −u
105
Figure 4.8:
BEM results for ∇ 2 u = − ∂u ∂x1
106
Figure 4.9:
BEM results for ∇ 2 u = −u (∂u ∂x1 )
108
Figure 4.10:
Typical mesh and internal node distribution for a 3D
109
problem
Figure 4.11:
Domain grid for: (a) (10,10,10) and (b) (10,5,3),
110
configurations
Figure 4.12:
Schematic of the angle integration for the attrition of
111
nodes
Figure 4.13:
Superimposed domain grid and final mesh structure
113
for: (a) (10,10,10) and (b) (10,5,3), configurations
Figure 4.14:
Isoparametric 20-noded prism for the cell integration
113
Figure 4.15:
Schematic of the solution methodology
116
Figure 4.16:
Schematic of the iterative scheme
117
Figure 5.1:
Poiseuille flow in a circular tube
120
Figure 5.2:
A coarse and finer surface mesh for the Poiseuille pipe
122
flow
Figure 5.3:
Poiseuille flow of a Newtonian fluid with a 7-5 mesh
123
Figure 5.4:
Influence of the NGP in the Newtonian solution with
124
the 7-5 mesh
Figure 5.5:
DGS meshes for Poiseuille flow: (15,15,5) and (20,20,8)
125
Figure 5.6:
Poiseuille flow of a non-Newtonian fluid with a
126
(15,15,5) DGS mesh
Figure 5.7:
Numerical performance for the (15,15,5) DGS mesh for
127
the Poiseuille flow
Figure 5.8:
Poiseuille flow of a non-Newtonian fluid with a
(20,20,8) DGS mesh
128
xv
Figure 5.9:
Numerical performance for the (20,20,8) DGS mesh for
129
a Poiseuille flow
Figure 5.10:
Schematic of the Couette flow problem
130
Figure 5.11:
Surface mesh for the Couette flow and internal points
131
Figure 5.12:
DGS meshes for Couette flow: (15,15,5) and (20,20,8)
133
Figure 5.13:
Newtonian solution for the Couette flow
134
Figure 5.14:
Couette flow of a non-Newtonian fluid with a (15,15,5)
135
DGS mesh
Figure 5.15:
Numerical performance for the (15,15,5) DGS mesh in
136
the Couette flow
Figure 5.16:
Couette flow for a non-Newtonian fluid with a
137
(20,20,8) DGS mesh
Figure 5.17:
Numerical performance for the (20,20,8) DGS mesh in
138
the Couette flow
Figure 6.1:
Schematic of the domain and mesh
141
Figure 6.2:
Comparison between the drag on a sphere from
143
Stokes’ law, analytical and BEM, and experimental
data
Figure 6.3:
Normalized z-velocity as a function of the distance
144
from the sphere
Figure 6.4:
Sphere settling in the presence of a plane wall
146
Figure 6.5:
Drag force for the case of a sphere moving parallel to a
147
plane wall
Figure 6.6:
Torque for the sphere moving parallel to the wall
148
Figure 6.7:
Drag force for the case of a sphere moving
150
perpendicularly to a plane wall
Figure 6.8:
Spherical particle in a circular cylinder
151
xvi
Figure 6.9:
Dimensionless force for a rigid sphere moving axially
153
in a cylindrical tube
Figure 6.10:
Error between the BEM solution with 96-290 element
154
mesh and Haberman’s approximate solution
Figure 6.11:
Correction factor from BEM for different meshes
155
Figure 6.12:
Dimensionless force for BEM and Happel and
157
Brenner’s approximate solutions as a function of the
eccentricity factor
Figure 6.13:
Dimensionless torque force for BEM and Happel and
157
Brenner’s approximate solutions as a function of the
eccentricity factor
Figure 6.14:
Prolate spheroid in shear flow
159
Figure 6.15:
Fiber representation for the BEM simulation
161
Figure 6.16:
BEM and Jeffery orientation angles for θ 0 = π 2
162
Figure 6.17:
BEM predicted fiber path for θ 0 = π 2
163
Figure 6.18:
Hydrodynamic force and torque on the fiber during
164
the simulation
Figure 6.19:
166
Figure 6.20:
167
Figure 6.21:
168
Figure 6.22:
169
Figure 6.23:
170
Figure 6.24:
171
Figure 6.25:
BEM and Jeffery orientation angle for θ 0 = π 2 and
172
two aspect ratios: (a) a r = 51 and (b) a r = 101
Figure 6.26:
Two spheres falling: (a) along their line-of-centers; (b)
perpendicular to their line-of-centers
175
xvii
Figure 6.27:
Normalized drag force for two equal-sized spheres
176
falling along their line-of-centers
Figure 6.28:
Normalized drag force for two spheres moving
177
perpendicular to their line-of-centers
Figure 6.29:
Dimensionless force as a function of the distance
179
between centers for two spheres falling parallel to
their line-of-centers
Figure 6.30:
Dimensionless force as a function of the distance
179
between centers for two spheres falling perpendicular
to their line-of-centers
Figure 6.31:
Direction of rotation and BEM torque for two spheres
180
settling beside each other
Figure 6.32:
Theoretical suspension viscosity as a function of the
182
volume concentration of spheres
Figure 6.33:
Spheres suspended in simple shear flow: 1x1x1 (Length
183
units)3 box and 40 spheres of radius of 0.05 length units
Figure 6.34:
Calculated relative viscosity for the 1x1x1 (Length
185
units)3 box with spheres with radius: (a) 0.05 length
units and (b) 0.07 length units
Figure 6.35:
Spheres suspended in simple shear flow: 0.8x0.8x0.8
186
(Length units)3 box and 40 spheres of radius of 0.05
length unit
Figure 6.36:
Calculated relative viscosity for the 0.8x0.8x0.8 (Length
187
units)3 box with spheres with radius: (a) 0.05 length
units and (b) 0.07 length units
Figure 6.37:
Calculated BEM relative viscosity
188
Figure B.1:
Inclusion of internal point source on the domain
206
xviii
Nomenclature
Scalars:
a
Dimensionless parameter in the Carreau-Yasuda model
--
Semi-major axis of a ellipse
ar
Aspect ratio for the fibers, a r =
b
Shearfree flow parameter
--
Known scalar non-homogeneous function
BS
Rate of formation per unit area
BV
Rate of formation per unit volume
c
Free coefficient in the integral representation
--
Semi-minor axis of a ellipse
CD
Drag coefficient
Cp
Specific heat at constant pressure
cof()
Cofactor of a matrix element
D
Diameter
--
Volume potential
De
Deborah number
div()
Number of divisions for the fixed grid generation
Dout
Jet or extrudate diameter
det sub()
Determinant of a sub-matrix
f
Known radial or global functions in the DRM
F13
View factor for surfaces 1 and 3
L+D
D
xix
G
Shear modulus
H
Latent heat per unit mass
H(
)
Bessel function of the third kind, also called Hankel
functions
HS
Rate of energy input per unit surface area
HV
Rate of energy input per unit volume
hT
Convection heat transfer coefficient
h
Distance between parallel plates
i
Complex variable
I
First tensor invariant
II
Second tensor invariant
III
Third tensor invariant
k
Mean curvature
KJ
Jeffery’s orbit constant
Kn
Modified Bessel functions
L
Fiber length
L2
Lebesgue space
m
Consistency index
n
Power law index
p
Pressure field
p̂ lm
Auxiliary pressure for the non-homogeneous velocity
−1
field in the DRM
q
Secondary pressure field
--
Normal derivative of a scalar potential u
r
Euclidean distance between two points
R
Radius
R1 and R2
Principal radii of curvature
xx
Re
Reynolds number
S
Domain boundary
T
Temperature
Tb
Bulk temperature of a fluid
TJ
Ellipsoid orbit period
t
Time
tp
Characteristic process time
u
Any scalar function
U
Internal energy
--
Newtonian potential
u0
Constant velocity or characteristic value for the velocity
V
Single-layer potential
W2l
Hilbert space
W
Double-layer potential
Work or force function
α
Thermal diffusivity
β
Internal angle at a boundary point
χ
Correction to Stokes’ law for the sphere motion or
dimensionless force
δ
Dirac delta function
δ ij
⎧1,
⎪
Kronecker delta, δ ij = ⎨
⎪0,
⎩
i= j
i≠ j
ε&
Elongation rate
φ
Angle between the x3 axis and the projection of
ellipsoid in the x1 − x3 plane
φ*
Green’s function or fundamental solution
γ
Variable for the Telles’ transformation
xxi
γ&
Shear rate
η
Non-Newtonian viscosity
η0
Zero-shear-rate viscosity
η∞
Infinite-shear-rate viscosity
η1 and η 2
Viscosity functions in Shearfree flows
η
Elongational viscosity
η0
Zero-elongation-rate elongational viscosity
κ
Dilatational viscosity
λ*
First Lame constant
λd
Viscosity ratio for interface flows
λp
Penalty parameter for the Navier-Stokes equations
λT
Thermal conductivity coefficient
λ
Relaxation time
−−
Parameter of the integral equation
−−
Eigenvalue of the integral equation
µ*
Second Lame constant
µ
Newtonian viscosity coefficient
σ
Surface tension coefficient
σ SB
Stefan-Boltzmann constant
θ
Angle between the major axis and the vorticity axis for
ellipsoid motion
ρ
Density
ς
Dimensionless torque
v
Poison’s ratio
ω
Relaxation parameter
Ω
Domain
Ω
Domain closure
xxii
ξ10 , ξ 20 , ξ 30
Variables that define the curvilinear coordinates in the
shape functions of an isoparametric element
ξ1 , ξ 2 , ξ 3
Set of curvilinear coordinates for the isoparametric
element
Ψ1
First normal stress coefficient
Ψ1,0
Zero-shear-rate first normal stress coefficient
Ψ2
Second normal stress coefficient
Vectors:
b
Known vector of a linear system
D
Hydrodynamic volume potential
f
Diffusive flux
F
Total flux
--
Force on a particle
FBEM
Force on a particle calculated with BEM
Force
Net force
g
Body forces vector
--
Functions for the reduced Jacobian
n
Outward normal vector
q
Diffusive flux of energy or conduction
qk
Pressure fundamental solution
t
Tangential unit vector
--
Surface tractions
T
Torque on a particle
TBEM
Torque on a particle calculated BEM
u
Velocity field
u∞
Ambient or surroundings velocity field
xxiii
û lm
Auxiliary non-homogeneous velocity field (DRM)
V
Hydrodynamic single-layer potential
v
Secondary solenoidal vector field
--
Constant value vector of displacements
w
Weight factor for the corresponding Gaussian points
W
Hydrodynamic double-layer potential
x
Cartesian coordinates of a point: x = ( x1 , x 2 , x3 ) = ( x, y, z )
--
Unknown vector in a linear system
xm
Coordinates of the collocation point in the DRM
αm
Unknown coefficients in the DRM
θl
Integrated angles for domain grid point l
ξ , ξ0
Cartesian coordinates of a point in the boundary of a
domain
∇
Gradient operator
Tensors:
A
Coefficient matrix of a linear system
C
Coefficient tensor for the integral representation
g
Kernel for the hydrodynamic single-layer potential in the
BEM equations
G
BEM matrix containing the velocity fundamental solution
h
Kernel for the hydrodynamic double-layer potential in the
BEM equations
Ĥ , H
BEM matrix containing the traction fundamental solution
J
Reduced Jacobian
J
Jacobian
K ij
Traction fundamental solution
xxiv
N
Matrix of isoparametric shape functions
u ij
Stokeslet or fundamental singular solution of the Stokes
system of equations
ri j
Rotlet fundamental solution
δ
Identity tensor
ε ijk
Permutation pseudo-tensor, ε ijk
γ&
Rate-of-strain tensor
π
Stress tensor
π*
Auxiliary stress tensor
τ
Viscous stress tensor
τ (e )
Extra stress tensor
⎧ 0, i = j , j = k , or, i = k ,
⎪
⎪
= ⎨ 1, ijk = 123,231, or 312,
⎪
⎪− 1, ijk = 132,213, or 321.
⎩
xxv
Abstract
Direct boundary integral formulations are developed for the solution of viscous
fluid flow problems, specifically for non-Newtonian fluids and fluids containing
solids. The partial differential equations are transformed into integral equations
by Green’s identities. Here, the velocity field is represented as a combination of
hydrodynamic potentials of single- and double-layer, whose densities are the
velocity and traction fields. For non-Newtonian fluid flow problems the nonlinear terms of the original equations appear as kernels of domain integrals. The
domain grid superposition (DGS) technique is developed in order to
approximate these integrals. The technique superimposes a fixed grid with the
domain under consideration. Cells, located in the intersection between the grid
and the domain, are used to directly calculate the domain integral by cell
integration. A pressure driven flow and a Couette flow are used in order to check
the DGS technique. For viscous fluids containing solids, a direct boundary
integral formulation is used to simulate solid dynamics in viscous flows. Several
problems are solved using the direct integral formulation to compare with
approximate and/or analytical solutions.
1
Chapter 1
Introduction
During processing, a substance is constantly changing its phase, state,
temperature, pressure, viscosity and, in general, all known properties. It
experiences all kinds of potential and flux differences, i.e. temperature and heat,
velocity and momentum, concentration and mass flux. To analyze processing
systems, over the last decades much work has been done in the field of transport
phenomena [28, 75], kinetic theory [27, 58, 76, 140, 177], statistical mechanics [27,
138, 139, 175, 176] and rheology [26, 27, 55, 56, 184], which gives the physical and
analytical tools needed to study all the changes that a substance undergoes
during processing. However, most systems do not have a simple model nor an
analytical solution. Normally, a model that represents a material during
processing is represented by an algebraic equation, a set of nonlinear partial
differential equations and/or an integral equation, which do not have analytical
solution. The principal purpose of numerical analysis is to provide methods for
2
obtaining useful solutions to those mathematical problems. Such methods will
give an approximate but satisfactory solution to the problem, which provides its
interpretation in terms of numbers. In the past decades and thanks to the
evolution of high-speed digital computers, numerical simulation is rapidly
evolving together with the sciences of fluid mechanics [219, 303, 316], heat
transfer [219, 303], transport phenomena, polymer rheology and processing [2, 7,
311, 116], among others. As is well known, finite differences (FDM), finite
volumes (FVM), finite elements (FEM) and boundary elements (BEM) methods
are the most widely used techniques to approximate engineering problems.
Boundary integral techniques and the boundary element method have an
advantage compared with the other techniques: the integral representation that
is generated when applying the method is an equivalent formulation to the
partial differential equations that govern the problem. Thus, once the integral
representation is achieved, approximations are only needed to find the values of
these integrals1. In addition, when applied to linear problems, the integral
representation is a boundary-only integral formula. Limitations given by the
volume discretization of the other techniques, when dealing with complex
geometries, which may have free surfaces, moving boundaries and/or solid
inclusions, are not present in linear BEM.
In FEM, for example, the final integral formulation comes from the scalar product of a function
and the residual of the approximation, in other words, there is an approximation involved before
the integral equation and an additional approximation is needed to solve these integrals.
1
3
The BEM has been limited to linear problems because the fundamental
solution or Green’s function is required to obtain a boundary integral formula
equivalent to the original partial differential equation of the problem. The nonhomogeneous terms accounting for nonlinear effects and body forces are
included in the formulation by means of domain integrals, making the method
lose its boundary-only character. Techniques have been developed to
approximate these domain integrals directly (cell integration [69, 216], Monte
Carlo integration [268]) or indirectly by approximation of domain integrals to the
boundary and then solving these new boundary-only integrals (dual reciprocity
[99, 100, 132, 202, 216], particular integral technique [3]).
1.1
Historical background
1.1.1
Nonlinear flows
In 1750, Leonhard Euler [89, 180, 277] derived a system of equations that
describes inviscid flows, giving birth to classical hydrodynamics. However, this
had little practical importance, since the results of classical hydrodynamics were
in glaring contradiction to everyday experience. Jean d’Alembert in 1752 [263,
276, 317] published his paradox, showing that a body immersed in an inviscid
fluid would have zero drag force. Navier (1823) [204], Cauchy (1828), Poisson
(1829), St. Venant (1843) and Stokes (1845) [292, 293] were the first to add terms
of frictional resistance to Euler’s inviscid equations. The first four wrote these
4
terms as a function of an unknown molecular function, whereas Stokes was the
first to use the coefficient of viscosity, µ. The equations containing this frictional
resistance term grouped in a molecular driven stress tensor are called the Cauchy
momentum equations, while the ones containing a constant viscosity coefficient
are the Navier-Stokes equations.
The momentum equations are a system of partial differential equations that
basically describe fluid flow; though fundamental and rigorous, they are
nonlinear, non-unique, complex and difficult to solve. They do not have a
general solution, and so far only a few particular solutions have been found,
most of these for unidirectional or nearly unidirectional flows (see for instance
Deen [75], Landau [180], White [322] and Batchelor [12]). These exact solutions
are important because basic phenomena described by the mathematical model
can be analyzed; also, they can be used as standard solutions to compare with
the approximate numerical solutions. However, in almost every practical
situation, it is necessary to use numerical methods in order to obtain a solution of
the momentum equations.
Several numerical techniques have been developed and used during the years
in order to achieve good approximate solutions of these equations. The most
common techniques are the finite differences method (FDM) [199, 303], the finite
volume method (FVM) [219, 316], the finite elements method (FEM) [17, 336] and
the boundary elements method (BEM) [38, 40, 246]. Many difficulties arise
during a numerical solution of these equations, i.e. non-linearities, non-
5
isothermal conditions, free surfaces, moving boundaries, solid inclusions and
complex geometries.
The non-linearity effects typically come from two different sources: the inertia
or convective terms, and the strain rate dependence of the viscosity, which
occurs in non-Newtonian fluids. This non-linearity has two important
consequences: first, the use of an extraordinarily high number of elements and
nodes in the discretization, depending on the problem’s parameters and
geometry. And second, more complex algebraic equations obtained from the
numerical technique, which require sophisticated algorithms to achieve
acceptable degrees of accuracy. Therefore, computational cost in terms of
computer time and memory is a restriction to the type and complexity of the
problems that can effectively be solved by common numerical techniques.
In addition, the degree of the non-linearity increases the numerical effort to
achieve satisfactory solutions. For example, when dealing with the inertia or
convective terms at constant viscosity, say the Navier-Stokes equations, higher
values of the Reynolds number require more computational effort, due to the fact
that these problems need a good distribution of internal nodes and high density
meshes. The convergence rate of the iterative methods decreases when using
such high-density meshes. Similarly, when dealing with the momentum
equations for non-Newtonian fluids, the parameters of the constitutive
rheological models have a great effect on the accuracy of the solution and the
converge rate.
6
1.1.2
Fluid particle motion
Microhydrodynamics is a part of physics where the central problem is to
determine the motion of a particle or particles in a bounded or unbounded flow.
Important theoretical treatments and reviews can be found in Happel and
Brenner [127] and Kim and Karrila [160].
The earliest study on resistance of a solid body moving relative to a fluid, in
which viscosity was considered, was published by Stokes in 1851 [294]. He
linearized the general equations for motion of a viscous incompressible fluid and
obtained a time-dependent form of the creeping motion equations. He applied
these linearized equations for the motion of a spherical pendulum and found an
equation for the force on the sphere when the frequency of the oscillation
approached to zero; this relationship is known as Stokes’ law. Lorentz [187], in
1896, following the method developed by Stokes [294], determined the motion of
a sphere in the presence of a plane wall. His technique uses reflection of the
original motion produced by the body from the surface of the wall and back
again to the body. Ladenburg [173] applied the same technique to determine the
effect of a cylindrical tube on the axial motion of a sphere, while Smoluchowski
[282-284] determined the effects of hydrodynamic interaction between two
spheres moving in a viscous fluid and studied the sedimentation of spheres
using the same Lorentz reflection technique. All these earliest contributions to
low Reynolds number hydrodynamics were summarized by Oseen in 1927 [209],
including Faxen’s (Ossen’s coworker) contributions.
7
The disturbance caused by particles suspended in a uniform shearing flow
was first analyzed by Einstein [81-83]. He developed a theory for the resistance to
shear of a suspension of small particles immersed in a continuous fluid, as a
model for large molecules in solution2. Theoretically, he showed that the increase
in viscosity of the suspending liquid is related to the volumetric concentration of
solid particles by a simple proportionality constant. In the last decades, interest
in suspension mechanics has experienced a marked increase. The basic problem
is to predict the macroscopic transport properties of a suspension, i.e. thermal
conductivity, viscosity, sedimentation rate, etc., from the micro-structural
mechanics. These flows are governed by at least three length scales: the size of
the suspended particles, the average spacing between the particles, and the
characteristic dimension of the container in which the flow occurs. A
comprehensive review of theoretical and experimental work in this area can be
found in Batchelor [15, 16], Brenner [45-47], Jeffrey and Acrivos [154] and Russel
[270]. The same basic variables that characterize the suspensions’ viscosity also
characterize sedimentation rates of solids. Several authors directed their
attention to this specific problem. Batchelor [13, 15, 16] was the first to analyze
the sedimentation of dilute suspensions of spherical particles.
Einstein’s thesis was concerned with a new method for determining the size of molecules of
chemical substances.
2
8
The behavior of flowing fiber suspensions was analyzed by Jeffery [152, 153]
and Forgacs and Mason [107, 108]. Jeffery modeled a fiber as a rigid ellipsoid; he
determined that a fiber in simple shear flow rotates in a periodic orbit while the
center of mass translates with the bulk flow. Meanwhile, Forgacs and Mason
examined fiber motion in a Couette device and observed that a fiber can undergo
a variety of complex rotational motions depending on the flexibility of the fiber.
In suspensions of many fibers, interactions between fibers perturb the fiber
motions. Such interactions may arise from hydrodynamic or colloidal forces,
excluded volume, or friction [269, 281]. Dinh and Armstrong [77], Toll and
Månson [308] and Rahnama et al. [259] studied the motion of rigid fiber
suspensions with random fiber orientation, friction between fibers and
hydrodynamic interactions. Papanastasiou and Alexandrou [213] studied the
isothermal extrusion of non-dilute short fiber suspensions in order to analyze the
coupling effects between the flow field and the fiber orientation. They used the
Dinh and Armstrong constitutive equation to model the rheology of the
suspension, and Jeffery’s equation for particles of infinite aspect ratio to describe
the fiber orientation. Later, Tucker [105] used a scaling analysis to determine the
effect of the interaction between the flow field and fiber orientation in slender
two-dimensional gaps. Then, based on orientation distribution functions, Folgar
and Tucker [105] derived a model for the orientation behavior of fibers in
concentrated suspensions. Advani and Tucker [1, 264] modified the FolgarTucker model with orientation tensors, which provided a more efficient and
9
compact way to describe the fiber orientation. Hernandez et al. [134, 135] did an
analysis of motion and loads of fibers during flow with pseudo-analytical
equations using the fundamental derivations for forces on fibers in suspensions
developed by Burgers [48].
Numerical simulations to describe the rheological and transport properties of
suspensions of solid particles have been increased in the past fifteen years. A
popular simulation algorithm was developed by Brady and coworkers [8, 35],
built on the idea of composite expansions, creating the method of Stokesian
dynamics for particulate Stokes flow. Here, hydrodynamic interactions between
remote particles are computed in terms of multi-pole expansions implemented
by Faxen’s laws, and lubrication forces developing between neighboring
particles are accounted for by local solutions developed under the support of
lubrication approximations. Generalizations of the Stokesian dynamics method
for non-spherical shapes can be found in the work of Claeys and Brady [62-64],
for Brownian suspensions in Bossis and Brady [31-33] and Banchio and Brady [8].
For non-spherical shapes, it has been found that analytical and computational
complexities have discouraged dynamic simulations. The lattice Boltzmann
formulation has been used by Ladd for dynamic simulation of sedimenting
spheres [171, 172] and to find transport coefficients of random dispersions of
hard spheres [169, 170]. Simulations of fibers in suspension have also been
studied by several authors; as mentioned above, Claeys and Brady [62-64]
10
expanded the Stokesian dynamics method for non-spherical particles, employing
a particle-level simulation which accounts for hydrodynamic interactions in
suspensions of rigid prolate spheroids. Yamane et al. [329] simulated the
dynamics of rigid cylinders, including a lubrication approximation to
hydrodynamic interactions. The interactions between flexible fibers and fluids
have also been simulated. Stockie and Green [291] used the so-called immersed
boundary method, which was originally developed by Peskin [222, 223]. This
method replaces the fluid-material interface with appropriate contributions to a
force density term in the Navier-Stokes equations. The internal boundaries can
be eliminated and a finite difference scheme is used to solve the fluid equations.
This method has also been applied to particles in suspension [106, 299]. Flexible
fibers treated as chains of rigid bodies have been studied initially by Yamamoto
and Matsuoka [325-328], where the fiber is modeled as a chain of oscillating rigid
spheres connected through springs, with additional potentials to mimic
resistance to blending and twisting. A particle-level simulation method for the
structural evolution of flexible fibers suspensions in shear flow, which is a
similar model used by Yamamoto and Matsuoka, was developed by Ross and
Klingenberg [269, 281]. They used a chain of rigid prolate spheroids connected
through ball and socket joints.
Direct simulation of particle motions has become very important in recent
years because of its many applications in real life situations and problems, such
as particle orientation in painting, fiber motion during reinforced polymer melts,
11
fluidized bed in combustion and gasification, sintering in the production of aerogels and glassy materials, etc. The meshing and mesh stability are additional
problems to be solved similarly to interfacial flows simulations. An interface is a
thin region where the pressure, density and viscosity are discontinuous, making
it difficult to simulate, since its topology changes at all times during the
simulation and needs to be followed, often making the problem ill-conditioned.
An example of this situation is the extremely large curvature during the last
stage of a drop coalescence process.
The numerical techniques for interfacial flows are the same that are used for
single phase flow, i.e. FDM, FEM, BEM. However, techniques to deal with the
presence of the interfaces, i.e. location and modeling of the fluxes on them, have
to be considered. There are two different ways to describe interface location:
fixed grid methods and moving grid methods. Apart from these two groups are
the molecular dynamics simulation (Dell’Aversana et al. 1996 [76]) and the
meshless or grid-free methods (Olson and Rothman 1977 [206]).
In fixed grid methods, the grid used to solve the bulk equations is entirely
fixed (see reviews of Bazhlekov, 2003 [20]; and Scardovelli and Zaleski 1999
[275]). Depending on how the interface is modeled, these methods can be
divided in two groups: surface-marker or front-tracking methods (Anderson
1999 [4]; Unverdi and Tryggvason 1992 [312]), where the interface is followed by
marker points; and volume-marker or volume-of-fluid methods, where a volume
fraction describes the portion of fluid in the cells of the mesh (Li et al. 2000 [181];
12
Hirt and Nichols 1981 [141]). The surface-marker methods are more accurate
with respect to the interface position while the volume-marker methods handle
interface transitions more efficiently. However, these methods cannot deal with
interface-to-interface problems at distances smaller than the element size. As a
consequence, the fixed-grid methods present a major disadvantage in the
accuracy of the solution near interfaces. And problems like drop impact, drop
coalescence or close solid-interface interactions, could be mesh dependent; unless
the mesh size, near the interfaces, is of a very small order of magnitude.
In the moving-grid methods, the mesh follows the deformation of the
interface. Some approaches were developed by Ryskin and Lean 1984 [271];
Duraiswami and Prosperetti 1992 [79]; and Bazhlekov 2003 [20]. There are,
however, two major disadvantages: mesh distortion, and thus the necessity of
mesh refinement; and numerical instability due to interfacial tension, especially
in the case of small Capillary numbers (see for instance Zinchenko et al. 1997
[338-341]; Loewenberg and Hinch 1997 [185, 186]).
Finite differences methods (Nichols and Mullins 1965 [203]) and finite
element methods (Hu et al. [143-145]; Jagota and Dawson [148-150]) have been
used to simulate some fluid particle flow problems. Again, the requirement of a
fixed domain mesh makes the solution cumbersome, and in some situations
impossible. Normally, these solutions are restricted to slow deformations or pure
viscous flows (Stokes flow). Solid projection problems, such as overlapping, are
solved using combined time schemes (explicit-implicit). The boundary integral
13
method and the BEM offer the great advantage in that for linear problems, it
reduces the spatial dimension of the problem by one, expressing the governing
equations by boundary-only equations. It has been used in several works to
simulate the dynamics of drops, like potential flows (Cheng 2000 [60]; Weiss and
Yarin 1999, 1995 [323, 330]) and homogeneous Stokes flows (Power 1994 [244246]; Pozrikidis 1992, 1990 [248, 249]; Primo et al. 2000 [256, 257]; Rallison and
Acrivos 1978, 1984 [261, 262]). The moving mesh character of the BEM overcomes
some of the difficulties in particle flow simulation, making it suitable for multiphase systems.
1.2
State of the art
1.2.1
Boundary integral equations for nonlinear viscous flows
The most common and important integral models that represent the momentum
equations were developed based on the Stokeslet technique by Ladyzhenskaya in
1963 [174]. In this representation, there are two boundary integrals that are in
terms of the velocity and traction, combined with the fundamental solutions or
Green’s function for Stokes flow. Domain integrals can appear in the
representation containing all nonlinear terms as pseudo-body forces; this was
first applied by Bush and Tanner in 1983 [52]. As was mentioned before, the nonlinearity of the viscous momentum equations comes from the strain rate
dependence of the viscosity. In order to use boundary integral techniques for
14
nonlinear flows, the domain integral containing these non-linearities has to be
solved. In fact, it is this integral that has received most of the attention of
research in the last decades. Authors have proposed several techniques, such as
analytical integration, Fourier expansions, the Galerkin vector technique, cell
integration, multiple reciprocity method, and the dual reciprocity method,
among others (see the review of Partridge et al. 1992 [216]).
The behavior of non-Newtonian fluids is strongly dependent upon the
viscosity variations within the domain. Most non-Newtonian fluids present a
shear thinning phenomena [2, 26, 28]. Polymers are some of the most important
shear-thinning fluids. The polymer viscosity can be calculated trough several
inelastic mathematical models such as the power law model [26], the Carreau
model [55, 56], the Cross model [211] and the hyperbolic tangent model [2]. The
application of the boundary integral method to a non-Newtonian problem
requires the fundamental solution or Green’s function for all the nonlinear terms
in the stress tensor. However, such a fundamental solution is not known
analytically and it is impossible to find one for a general viscosity model.
Therefore, it is necessary to lump all the nonlinear terms in a domain integral as
a pseudo-body force term.
In 1983, Bush and Tanner [52] were the first to use the boundary integral
methods to solve non-Newtonian flow problems. Later Bush, Milthorpe and
Tanner [49] combined finite element with boundary integral equations for the
extrusion of a Maxwell fluid. Bush and Phan-Thien applied the BEM for a non-
15
Newtonian Bird-Carreau-Yasuda fluid [50] and for an Oldroyd-B fluid [51].
Osswald (1986) [210] applied the boundary element method for the simulation of
compression molding, where the lubrication approximation (Hele-Shaw model)
is valid and all the nonlinear terms can be grouped in a constant domain integral,
which was solved by the particular solution method in combination with cell
integration. Barone and Osswald [10, 11] and Osswald and Tucker [212]
continued the BEM application for compression molding of fiber reinforced
polymer compounds and sheet molding compounds in non-planar parts.
Nardini
and
Brebbia
(1982)
[202]
introduced
the
dual
reciprocity
approximation (DRM), where the domain integral containing the nonlinear terms
are transformed into an equivalent series of boundary integrals. This method
requires a series of particular solutions for the equations. Several authors started
to use DRM for the nonlinear momentum equations, such as Power and
Partridge (1993, 1994) [242, 243], Power and Wrobel (1996) [246] Gomez and
Power (1997) [115]. Mätzig (1991) [189] applied the DR-BEM to non-Newtonian
problems for the solution of the 2D transient energy equation in polymer
processing applications. At the same time, Gramann [117] and Gramann and
Osswald [118] started to use the BEM for polymer mixing simulations. The use of
the DR-BEM for flow and heat transfer simulations in polymer processing was
applied for the first time in 1993 by Davis [69].
The basic idea behind the DRM is to approximate the non-homogeneous
terms as a series of known functions and in this way obtain a series of particular
16
solutions to the original equation. Thus, the correct choice of the function and
interpolation scheme used for the nonlinear terms will directly affect the results.
Two different types of functions have been used over the years: radial and
global. The first [112, 113] are functions of the Euclidean distance between points,
while the global functions [59, 60] depend directly on the global coordinates of
the points. The augmented spline, which consists of the radial basis function plus
a series of additional global functions, was introduced by Goldberg and Chen
[113]. The augmented spline was used by numerous authors [99-103, 414, 217]
and it was shown to be a very useful technique for problems involving
derivatives of the variables, such as convection terms or strain rate dependence
of viscosity. As pointed out by Partridge [215], there are criteria for selecting the
type of approximation functions, depending on the type of nonlinear term in
consideration.
The BEM, in combination with the dual reciprocity, produces full nonsymmetric matrices that are cumbersome to invert and solve. This problem can
be eliminated by decomposing the domain into smaller sub-domains. Taigbenu
(1995) [300, 301] introduced the Green Element Method (GEM) where the
domain is decomposed into smaller sub-domains (cells) surrounded by a fixed
number of boundary elements and each having different material properties.
With the GEM there are still some domain integrals that can be solved by cell
integration. Phan-Thien (1995) [224] proposed a boundary element solution for
the non-homogenous momentum equations based on the particular solution
17
approach and radial basis function interpolation for the nonlinear terms. Davis
(1995) [69] used cell integration for the pseudo-body term domain integral to
model polymers and to optimize mixing equipment. Davis and Osswald (1995)
[73] and Davis et al. (1996) [70-72] applied dual reciprocity for non-Newtonian
2D flows using the power law model for the viscosity. They found that for a
power law index below 0.8 the solutions were not satisfactory. Hernandez (1999)
[132] applied the dual reciprocity for 3D non-Newtonian flows using the power
law model and radial basis function for the approximation, and restricting the
method to high power law indexes. Mixing in single screw extrusion of a power
law fluid, using Monte Carlo techniques to solve the domain integral, was
analyzed by Rios (1995) [268]. He was able to perform simulations for low power
law indexes, however, the standard deviation of the Monte Carlo points
solutions increased as the power law index was decreased. Domain
decomposition techniques have been applied, together with a velocity-vorticity
formulation by Skerget and Samec [279, 280], to model non-Newtonian fluids
under non-isothermal conditions. Florez (2000) [99] applied the multi-domain
dual reciprocity for 2D non-Newtonian flows using the power law model. The
results obtained by Florez show that the combination of domain decomposition
and DRM is an effective way to face nonlinear problems with boundary integral
equations. Florez et al. (2002) [103] used the multi-domain dual reciprocity for
the coupled momentum and energy equations. The method gave accurate and
satisfactory results. All the studies have concluded that a domain partition is
18
required in order to use the BEM for nonlinear problems when the non-linearities
strongly affect the physical problem.
1.2.2
Boundary integral equations for viscous flows containing particles
Low Reynolds number flows with boundary integral representation have been
used to describe rheological and transport properties of suspensions of solid
spherical particles, as well as for numerical solution of different problems,
including particle-particle interaction, the motion of a particle near a fluid
interface or a rigid wall, the motion of particles in a container, etc.
As mentioned before, one of the most popular algorithms, developed by
Brady and coworkers [35, 36], is built on the idea of composite expansions, called
the method of Stokesian dynamics for particulate Stokes flow; hydrodynamic
interactions between remote particles are computed in terms of multi-pole
expansions implemented by Faxen’s laws, and lubrication forces developing
between neighboring particles are accounted for by local solutions developed
under auspices of lubrication flow. However, suspended particles appear in a
variety of shapes, making the generalization of the Stokesian dynamics for nonspherical shapes a cumbersome analytical and computational task.
An alternative approach relies on integral representation of the first and
second kind originating from the standard or generalized boundary integral
representation [160, 240, 252, 253]. In 1975, Youngren and Acrivos [333, 334] used
the integral formulae, developed by Ladyzheskaya (1963) [174] for the exterior
19
Stokes flow around a solid particle of arbitrary shape, to obtain a first kind
Fredholm integral equation for the unknown surface traction. This formulation
has been extensively used in the literature for the numerical solution of several
particle-flow problems. However, Fredholm integral equations of the first kind
give rise to unstable numerical schemes based upon discretization, and these
instabilities are manifested in the ill-conditioning of the matrix approximation of
the kernel [246]. Power and Miranda (1987) [240] obtained a second kind integral
equation for a general three-dimensional Stokes flow around a single particle. As
is known, solving an equation of the second kind is a well-posed problem. The
second kind representation can only represent flow fields corresponding to
surfaces which are force and torque free. However, the representation can be
completed by adding terms that express arbitrary total forces and torques in
suitable linear combinations, i.e. a Stokeslet and a Rotlet in the interior of the
particle. The expansion of Power and Miranda’s method to multiple particles in
an unbounded domain flow was given by Power (1987) [233] and Karrila,
Fuentes and Kim (1989) [156], and to particles in a bounded flow by Power and
Miranda (1989) [241] and Kim and Karrila (1989) [160]. This method was called
complete double layer boundary integral equation method, which was used in
several applications such as two-dimensional Stokes flow (Power 1993 [234]; Li
and Pozrikidis 2000 [182, 183]), motion of particles near a plane wall (Power and
Febres de Power 1993 [236]), flat particles in Stokes flow (Maul and Kim, 1994
[194, 195]), micro-polar fluids (Power and Ramkissoon 1994 [244, 245]), two-
20
dimensional sintering (Primo 1998 [256, 257]) and two-dimensional suspensions
of particles (Pozrikidis 2001 [254]). Recently, the method was extended to Stokes
flow with mixed boundary conditions (Power and Gomez 2001 [237]).
Dynamic simulations for multi-particle flows have been discouraged by two
main difficulties: non-uniqueness of the solution due to presence of
eigenfunctions defined over the surfaces of the individual particles, and large
computational cost required for compiling and solving the linear system of
equations arising form the boundary element implementation [249]. Integral
equations of the second kind for the density of a double-layer representation are
generally amenable to iterative solutions based on successive substitutions,
however extra cost required for evaluating and integrating higher-dimensional
kernels is a practical disadvantage. That is why dynamic simulations of large
systems based on the double-layer formulation have not been carried out. An
approach of using boundary integral methods for particulate flows was
proposed by Hernandez et al. (2002) [134, 135] where a direct formulation is used
for the deformation of viscous flows containing fibers. The surface tractions on
the fibers were integrated to compute forces in order to predict fiber damage
mechanisms in fiber reinforced polymer melts during flow. This direct
formulation generates Fredholm integral equations, for which uniqueness of the
solution is guaranteed, and it avoids the difficulties of the single-layer
formulation [246, 249].
21
1.3
Objective and motivation
This thesis is devoted to the development of a direct boundary integral
numerical technique for investigation of viscous fluid flow problems that involve
non Newtonian fluids and solid inclusions.
Typically, BEM is not used efficiently to solve nonlinear problems in fluid
mechanics. This research is intended to develop numerical strategies to
approximate the domain integrals that arise from the nonlinear terms, in order to
increase the performance of this method. Practicality is a requirement, in the
sense that moving and complex boundaries and solid inclusions must be easily
included into the analysis.
The numerical simulation of non-Newtonian fluids can be used in industries
that involve polymers and plastics. It can provide valuable information about
mixing processes, screw extrusion, molding, etc. The solid inclusions simulation
can be used as well in polymer processing applications, such as processing of
fiber reinforced polymer melts; and to predict the effect of solid size and
orientation on the rheological properties of suspensions.
1.4
Overview
Most of the physical problems can be represented mathematically by means of
partial differential equations, which can be developed from general integral
balances. Furthermore, there are additional relationships needed to complete the
22
set of equations, which associate the molecular driven tensors with the main
variables of the problem. Chapter 2 presents the rheological and balance
equations needed to analyze materials under processing conditions. The most
common features of the flow phenomena of viscous materials are described and
inelastic constitutive equations are defined. Leibniz’s integral formula is used to
find conservation equations for points in the domain and in interfaces. In
Chapter 3 a general theory of integral equations is introduced and it is explained
how the Green’s identities can be used to transform a partial differential equation
into an equivalent integral equation, which can be interpreted as a combination
between single-layer, double-layer and volume potentials. For a flow field, the
integral representation is obtained in terms of a fundamental solution known as
Stokeslet, the velocities at the boundary and the surface tractions. Direct
boundary integral formulations are presented for the Poisson’s equation,
nonlinear viscous flows, fluid flow in presence of interfaces with surface tension,
and for elastostatics problems. The boundary element method is presented in
Chapter 4. Isoparametric elements are described with the corresponding
transformation of coordinates and the methodology to calculate weakly integral
equations. The domain grid superposition technique is introduced to
approximate the domain integral containing the nonlinear terms. This method is
meant to improve the efficiency and accuracy of the BEM for nonlinear viscous
flow problems. In Chapter 5 the domain grid superposition BEM is applied to the
momentum equations for inelastic non-Newtonian fluids. Results are presented
23
for the Hagen-Poiseuille flow in a circular tube and the Couette flow. The
approximate solutions are compared with the corresponding analytical solution.
Chapter 6 is devoted to application of the direct BEM to the simulation viscous
fluid flow problems with solid inclusions. The simulated results are compared
with analytical and/or approximate solutions. Finally, some concluding remarks
and further research ideas are presented in Chapter 7.
24
Chapter 2
Rheological and balance
equations
This chapter discusses the two main parts of any transport model, the
constitutive and balance equations. The conservation equations for nonisothermal viscous flows require additional relationships that relate the viscous
stresses to the main problem potentials (velocity, pressure and/or temperature),
in the same way that the diffusive flux of heat (conduction) is related to the
gradient of temperature. For non-Newtonian fluids the equations that relate the
viscous stresses to the velocity (constitutive equations) are not as simple as
Fourier’s law for thermal conduction. This is due to the complexity of the
molecular configurations in high molecular weight substances, such as polymers
and plastics.
The first part of the chapter defines some experimental recollection of
phenomena that make non-Newtonian fluids (i.e. polymers) different than
25
materials with low molecular weight. The rheological characterization is
described, beginning with common flow patterns and their stress tensor
descriptions. Steady shear and shearfree flow material functions are defined with
corresponding behavior at low and high shear and elongation rates. Finally,
constitutive equations for the non-Newtonian viscosity are described.
After discussing some constitutive equations, the non-isothermal viscous
flow conservation equations are introduced. These will be developed from an
integral equation for a scalar potential and Leibniz integral general formula,
arriving to a general form of conservation equation, from which the conservation
equations for points within the domain and interfaces will be deduced. These
general forms will be used in order to arrive to conservation formulas for
momentum and energy, in both the domain and interfaces. Finally, the common
conservation equations for viscous fluid flow are presented.
2.1
Flow phenomena
There are three important phenomena seen in complex liquids (i.e. polymers)
that make them different from simple fluids (i.e. water): a non-Newtonian
viscosity, normal stresses in shear flow, and elastic effects.
26
2.1.1
Non-Newtonian viscosity
The most important characteristic of complex liquids is that they have a shearrate dependent viscosity or non-Newtonian1 viscosity (see Figure 2.1). Some
liquids present a decrease in viscosity when the shear rate exceeds a certain
value. These materials are referred to as shear thinning or pseudoplastic. The
viscosity of this type of material can decrease by a factor of as much as 103 or 104.
Almost all polymer solutions and melts that exhibit shear-rate dependent
viscosity are shear thinning.
Viscoplastic
Stress
Newtonian
Pseudoplastic
Dilatant
Shear-Rate
Figure 2.1:
Viscous response of non-Newtonian fluids.
Some fluids behave the opposite to shear thinning materials: their viscosity
increases with the shear rate. This is called shear thickening or dilatant behavior
[19, 267], which is exhibited by fairly concentrated suspensions of very small
The viscosity does not obey Newton’s law, where the viscosity depends on temperature and
pressure but it remains constant trough any deformation.
1
27
particles. A final different behavior is shown by some fluids that will not flow
unless acted on by some critical shear stress, called the yield stress. These fluids
are called viscoplastic. Certain paints, greases and pastes are examples of
viscoplastic fluids [26].
2.1.2
Normal stresses
When a simple shear flow is applied to complex liquid two extra forces appear
that are not present in a Newtonian fluid: a force that tries to separate the
moving plate and a force that tries to decrease the width of the polymeric
sample. A well known experiment to see normal stress effects is the rod-climbing
experiment. Here, rotating rods are inserted into two beakers, one containing a
Newtonian fluid and the other a non-Newtonian solution. For the Newtonian
fluid, the liquid near the rotating rod is pushed outward by the centrifugal force
(inertia effects), resulting in a dip in the liquid surface near the center of the
beaker. For the non-Newtonian fluid, on the other hand, the solution moves
toward the center of the beaker and climbs up the rod. This phenomenon, was
first described by Garner and Nissan and by Russel [2, 26], it is called the rodclimbing or Weissenberg effect.
2.1.3
Elastic (memory) effects
Normal stresses causes flow conditions to create memory in non-Newtonian
liquids. For example, consider a fluid that exits from a die gap of diameter D into
air, forming a jet of diameter Dout. For Newtonian fluids Dout will be about 13%
28
larger than D in the limit of small Reynolds number and about 13% smaller in the
limit of large Reynolds number [304]. A non-Newtonian fluid will have a Dout
about 300% of D. Extrudate dimensions of two, three, or even four times the die
gap dimensions are encountered with polymers. This phenomenon is referred to
as extrudate swell, or die swell [2, 26]. Once the complex fluid is outside the die,
the melt can no longer support the extra tension generated in the restriction and
the fluid will contract axially and expand across the free surface, in a sense,
recovering its shape before the restriction. Another experiment involves the
siphoning of Newtonian and non-Newtonian fluids, each in a separate container.
If the tubes are suddenly lifted out of the fluids, a slurping sound is heard from
the siphon that was in the Newtonian fluid as the liquid immediately empties
out of the tube, stopping the siphoning. On the other hand, the non-Newtonian
fluid continues to flow up and through the siphon [26].
Like the above experiments, there are several that relate the fact that some
non-Newtonian fluids have memory, or better, present a combination of viscous
and elastic (viscoelastic) effects [26, 184].
2.2
Rheological flow patterns and steady-state material functions
Incompressible Newtonian liquids at isothermal conditions can be characterized
by just two material constants: the density ρ and the viscosity µ. Once these
quantities are measured, the governing equations for the velocity and stress are
29
fixed for any flow system. There are many steady- and unsteady-state
experiments from which µ can de determined [313].
On the other hand, the experimental description of incompressible nonNewtonian fluids is more complicated. The density, of course, can be easily
measured. However, depending on the type of experiment that is performed on
the liquid, a host of material functions that depend on shear rate, frequency, and
time will be obtained. These material functions serve to classify fluids, and are
used to determine constants in specific non-Newtonian constitutive equations.
There are two common standard types of flow patterns used in rheology to
characterize non-Newtonian liquids: shear and shearfree flows. Material
functions are obtained from these flow patterns, depending on the specific
condition of the flow (steady, unsteady, etc.); it is then not surprising that the
material information from each type of flow is totally different [26]. Regularly,
the two flow patterns are designed to be homogeneous [26, 27, 184], in which the
velocity gradients are independent of position.
2.2.1
Shear flow
A simple shear flow is given by the velocity field,
u x = γ& yx y
uy = 0
uz = 0
(2.1)
30
in which the velocity gradient γ& yx can be a function of time. The absolute value of
the velocity gradient is called the shear rate γ& 2. A simple shear flow can be
generated between parallel plates as shown in Figure 2.2. Tube flow, axial
annular flow, tangential annular flow, flow between parallel planes, and flow
between rotating disks are common examples of shearing flows.
u0
γ = u0/h
y
h
x
Figure 2.2:
Steady simple shear flow with constant shear rate.
The total stress tensor3 for a simple shear flow has the following general form,
⎛ p + τ xx
⎜
π = pδ + τ = ⎜ τ yx
⎜ 0
⎝
τ yx
p + τ yy
0
⎞
⎟
0 ⎟
p + τ zz ⎟⎠
0
(2.2)
When the stress is measured for incompressible fluids it is impossible to separate
the pressure and the normal stresses, so normal stresses differences are used.
Thus, in simple shear flows there are only three independent, experimentally
accessible quantities summarized in Table 2.1.
2
3
The shear rate is related to the second invariant of the deformation tensor, as is discussed later.
The total stress tensor is composed of an isotropic pressure part and a viscous part.
31
Table 2.1:
Experimental quantities in simple shear flow.
Shear stress
τ yx
First normal stress difference
τ xx − τ yy
τ yy − τ zz
Second normal stress difference
For steady-state shear flow the stresses are function only of the shear rate γ& .
The viscosity η , called non-Newtonian viscosity or shear-rate dependent
viscosity, is defined analogously to the viscosity for Newtonian fluids,
τ yx = −η (γ& )γ& yx
(2.3)
In the same way, the normal stress coefficients Ψ1 and Ψ2 are defined as follows,
τ xx − τ yy = −Ψ1 (γ& )γ& yx2
(2.4)
τ yy − τ zz = − Ψ2 (γ& )γ& yx2
(2.5)
These normal stress coefficients Ψ1 and Ψ2 are called the first and second
normal stress coefficients. The non-Newtonian viscosity and the two normal
stress coefficients are known as the viscometric functions.
The non-Newtonian viscosity is the best known viscometric function. Figure
2.3 illustrates the most important features of the non-Newtonian viscosity for a
polymer melt or a polymeric liquid. At low shear rates, the shear stress ( τ yx ) is
proportional to γ& , and the viscosity approaches a constant value η 0 , called the
zero-shear-rate viscosity. At higher shear rates the viscosity of most polymeric
liquids (shear thinning materials) decreases. When plotted as logη versus log γ& ,
32
the viscosity vs. shear rate curve exhibits a linear region. Experimentally, the
slope of the linear region, or power law region, is found to be between 0.2 and 0.9
for polymeric liquids. Finally, at very high shear rates the viscosity may become
independent of the shear rate again and approaches η ∞ , the infinite-shear-rate
viscosity.
The first normal stress coefficient is also shown in Figure 2.3. Experimentally,
it is seen that Ψ1 is positive and that it has a large power law region in which, for
some polymeric liquids, Ψ1 decreases by as much as a factor of 106 [2, 26]. Like
the non-Newtonian viscosity, at low shear rates the first normal stress coefficient
is proportional to γ& 2 , so that Ψ1 tends to a constant Ψ1,0, the zero-shear-rate first
normal stress coefficient.
Ψ1,0
η0
log Ψ1
log η
log γ&
Figure 2.3:
Typical behavior of the non-Newtonian viscosity and the first normal
stress coefficient in a polymeric liquid.
33
Experimental information for the second normal stress coefficient is more
complicated to obtain. The most important facts about Ψ2 is that it is negative
and that its magnitude is much smaller than Ψ1, usually about 10% of Ψ1 [26].
However, its presence can significantly affect a flow, such as spiraling flows in
non-circular tubes and forces in wire coating processes.
2.2.2
Shearfree flow
A simple shearfree flow is given by the velocity field,
1
u x = − ε& (1 + b )x
2
1
u y = − ε& (1 − b ) y
2
u x = +ε&z
(2.6)
where b ∈ [0,1] and ε& is the elongation rate, which can depend on time. Table 2.2
presents several special shearfree flows for particular choices of the parameter b.
An elongational flow is schematically represented in Figure 2.4.
x
z
Figure 2.4:
Streamlines for elongational flow.
34
Table 2.2:
Special shearfree flows.
b = 0; ε& > 0
b = 0; ε& < 0
b =1
Elongational flow
Biaxial stretching flow
Planar elongational flow
The most general form of the total stress tensor is,
⎛ p + τ xx
⎜
π = pδ + τ = ⎜ 0
⎜ 0
⎝
0
p + τ yy
0
⎞
⎟
0 ⎟
p + τ zz ⎟⎠
0
(2.7)
For incompressible fluids, there are only two normal stress differences of
experimental interest (see Table 2.3). In the particular case of elongational and
biaxial stretching flows, for which
b = 0 , the x- and y-directions are
indistinguishable so that τ xx − τ yy = 0 and there is only one normal stress
difference to be determined.
Table 2.3:
Experimental quantities in shearfree flow.
First normal stress difference
Second normal stress difference
τ zz − τ xx
τ yy − τ xx
Since the flow is isotropic, for steady shearfree flows the stress and the
material functions depend only on the elongation rate, ε& , and the parameter b,
which defines the type of flow. Similar to shear flows, two viscosity functions η1
and η 2 describing the two normal stress differences are introduced,
τ zz − τ xx = −η1 (ε&, b )ε&
(2.8)
35
τ yy − τ xx = −η 2 (ε&, b )ε&
(2.9)
If b = 0, η 2 = 0 and η1 is equal to the elongational viscosity η ,
τ zz − τ xx = −η (ε& )ε&
(2.10)
For ε& > 0 , η describes elongational flow, and when ε& < 0 it describes biaxial
stretching. The elongational viscosity is sometimes called the Trouton or
extensional viscosity [26]. Figure 2.5 shows a schematic behavior of the
elongational viscosity as a function of the elongation rate. At low elongation
rates the elongational viscosity approaches a constant value known as the zeroelongation-rate elongational viscosity, η 0 . This value is three times the zeroshear-rate viscosity4. As the elongation rate is increased the elongational
viscosity also increases, and then it decreases at still higher elongation rates.
For each flow pattern discussed above there is a collection of experiments that
are usually performed. A long list of material functions has been defined for
these two types of flow fields, corresponding to a large variety of timedependent shear and elongation rates that can be produced experimentally. For a
general review of unsteady-state material functions see Bird, Armstrong and
Hassager [26].
4
For a Newtonian fluids η = 3µ [26].
36
3η 0
log η
η0
log η
log γ&, ε&
Figure 2.5:
2.3
Typical behavior of elongational viscosity and non-Newtonian viscosity
in a polymeric liquid.
Generalized Newtonian fluid
In the previous sections the flow phenomena and the material functions for nonNewtonian liquids were shown. As yet, there is no relation between the stresses
and the flow field parameters (velocity and pressure). Before describing the
constitutive equations, a question arises: when can a non-Newtonian fluid be
considered Newtonian, non-Newtonian or viscoelastic? A useful parameter used
to estimate the elastic and inelastic effects during flow is the Deborah number,
De , which is defined as [7, 211],
De =
λ
tp
(2.11)
37
where λ is the relaxation time of the liquid and tp is the characteristic process
time. A Deborah number of zero represents a viscous fluid and an infinite
Deborah number represents an elastic solid. Figure 2.6 illustrates a De vs.
deformation diagram, and delimits four zones: viscous fluid, linear viscoelastic
fluid, non-linear viscoelastic fluid and elastic solid. Thus, according to the
process and material, the correct constitutive equation must be selected.
The first region in Figure 2.6 is the viscous fluid zone. It was mentioned
earlier that the viscosity of polymeric liquids is non-Newtonian. Most of them
present a shear thinning behavior in which viscosity can change by a factor of 10,
100 or even 1000, making evident that such changes cannot be ignored in flow
Elasticity
Non-linear
viscoelasticity
Viscous fluid
Deformation
calculations or designs.
Linear
viscoelasticity
Deborah Number
Figure 2.6:
Schematic Deborah vs. deformation diagram.
Newton’s law of viscosity for an incompressible Newtonian fluid and any
flow field u = u(x, t ) is [180, 322],
38
τ = − µγ&
(2.12)
where µ is constant for a given temperature, pressure and composition and γ& is
the rate-of-strain tensor defined as,
t
γ& = ∇u + (∇u ) =
∂u i ∂u j
+
∂x j ∂xi
(2.13)
Rheologists created an empiricism which includes the changes of viscosity due to
the flow field in Newton’s law of viscosity [26, 184], i.e.,
τ = −ηγ&
(2.14)
which is the generalized Newtonian incompressible fluid. If the non-Newtonian
viscosity is to depend on the flow field (deformation tensor), then it must depend
on the symmetric part of the deformation tensor, the rate-of-strain tensor γ& [12,
180, 322]. Even more, the non-Newtonian viscosity must be invariant to any
coordinate change, so it must be a function on those particular combinations of
components of the tensor that are not dependent on the coordinate system. The
common invariants are5,
I = γ&ii = tr γ&
(2.15)
II = γ&ij γ& ji = tr γ& 2
(2.16)
These invariants are related to the eigenvalues of the tensor, details can be found in Strang [298]
or Arfken and Weber [6].
5
39
III = γ&ij γ& jk γ& ki = tr γ& 3
(2.17)
Incompressible fluids under shear (defined in Eq. (2.3)) will have the rate-ofstrain tensor and invariants defined by [26-28],
⎛0 1 0⎞
⎜
⎟
γ& = ⎜ 1 0 0 ⎟γ& yx ,
⎜0 0 0⎟
⎝
⎠
⎛1 0 0⎞
⎜
⎟
γ& 2 = ⎜ 0 1 0 ⎟γ& yx2 ,
⎜0 0 0⎟
⎝
⎠
⎛ 0 1 0⎞
⎜
⎟ 3
γ& 3 = ⎜ 1 0 0 ⎟γ& yx
⎜ 0 0 0⎟
⎝
⎠
(2.18)
I = tr γ& = 2(∇ ⋅ u ) = 0
II = tr γ& 2 = 2γ& yx
(2.19)
III = tr γ& 3 = 0
In other words, the first invariant is zero due to the incompressibility condition
and the third invariant is zero due to the shear flow configuration. The common
techniques to measure viscosity of non-Newtonian fluids use shear flow
arrangements, which make the viscometric functions to depend on the second
invariant only. Therefore, the generalized Newtonian fluid should be used for
shearing or nearly shearing flows, since omits the third invariant. Finally, by
convention it is preferred to use γ& , the magnitude of the rate-of-strain tensor,
γ& =
1
γ&ij γ& ji =
2
1
II
2
(2.20)
which for shear flows is called the shear rate. Thus, the final constitutive
equations for non-Newtonian viscosity are of the form η = η (γ& ) .
Several empirical non-Newtonian viscosity functions for the generalized
Newtonian fluid model can be found in literature (see for example Bird,
40
Armstrong and Hassager [26], Baird and Collias [7]), including some
recommendations and limitations. The most important models are the CarreauYasuda and Power-Law models explained below.
2.3.1
The Carreau-Yasuda model
The Carreau-Yasuda model is often used in numerical calculations, because it fits
the full flow curve. The model is defined as [26, 55, 56, 331],
[
η −η∞
a
= 1 + (λγ& )
η0 −η∞
](
n −1) a
(2.21)
where η 0 is the zero-shear-rate viscosity, η ∞ is the infinite-shear-rate viscosity, λ
is a time constant, n is the power law exponent (it describes the slope of the
viscosity in the power law region), and a is a dimensionless parameter that
describes the transition region between the zero-shear-rate region and the power
law region.
2.3.2
The Power Law model
In many industrial applications the power law region of the non-Newtonian
viscosity is the one of more importance. This region can be described by a simple
power law expression,
η = mγ& n −1
(2.22)
where the two parameters; m and n, are the consistency and power law indexes,
respectively. When n = 1 and m = µ the Newtonian fluid is recovered. If n < 1 ,
41
the fluid is shear thinning (or pseudoplastic), and if n > 1 , the fluid is shear
thickening (or dilatant). There are a wide variety of problems which have been
solved analytically using the power law model for non-Newtonian viscosity,
making it the most known and used model [2, 7, 26]. However the model
presents two major disadvantages: first, it cannot describe the viscosity for low
or high shear rates (η → ∞ as γ& → 0 and η → 0 as γ& → ∞ ), and second, since it is
an analytical model, a characteristic time and viscosity cannot be constructed
from m and n.
2.4
General forms of conservation equations
A closed region in space in which the rate of accumulation of some quantity is
equal to the net rate at which that quantity enters by crossing the boundaries or
is formed by internal sources is called a control volume [110, 277, 322]. This is the
most popular concept when formulating the general conservation equation for a
specific quantity. A general form of a control volume is shown in Figure 2.7,
which encloses a section of a moving interface between two phases, A and B (a
phase is a region where a specific quantity b is a continuous function of position
[75]). The interface divides the control volume into two regions, with volumes ΩA
and ΩB, and external surfaces SA and SB. These surfaces have unit outward
normals nA and nB, respectively, and a velocity uS(x,t). The interfacial surface
42
(inside the control volume), SI, has a unit normal nI, which points from phase A
toward phase B, and a velocity uI(x,t).
nA
nS
SA(t)
ΩA(t)
uI
Phase A
ΩB(t)
SI(t)
Phase B
nI
SB(t)
nB
Figure 2.7:
Control volume enclosing part of an interface between phase A and B.
Using the concept of control volume and Leibniz formula [12, 75, 322], a
general Leibniz formula for the conservation of any quantity b(x,t) is obtained,
∂b
∫ ∂t dΩ + ∫ (b
Ω
SI
A
− bB )u I ⋅ n I dS = − ∫ F ⋅ ndS + ∫ BV dΩ + ∫ BS dS
S
Ω
(2.23)
S
where F(x,t) is the flux of the quantity b(x,t), BV(x,t) and BS(x,t) denote the rate of
formation per unit volume and area, respectively. Equation (2.23) is an integral
representation for the conservation of the quantity b(x,t) in the control volume
illustrated in Figure 2.7. This integral representation is very useful for finding
conservation equations in points within the volume and the interface. However,
it is not appropriate as a general equation for an integral method.
43
There are two final comments about this equation. First, the external surface
velocity is incorporated in the first integral in the equation, i.e.,
∂b
d
bdΩ = ∫ dΩ + ∫ bu S ⋅ ndS
∫
∂t
dt Ω
Ω
S
(2.24)
and second, there are two conditions for the second integral to appear: the
interface must be moving and the quantity b(x,t) must be discontinuous there.
By taking the limit Ω → 0 of the Leibniz general formula, a conservation
equation valid at a given point in a continuum is obtained. First, the surface
integral accounting for the flux must be transformed into a volume integral using
the divergence theorem. The equation equivalent to Eq. (2.23) is then
⎡ ∂b
∫ ⎢⎣ ∂t + ∇ ⋅ F − B
V
Ω
⎤
⎥ dΩ = 0
⎦
(2.25)
Because the magnitude of the volume is arbitrary, the quantity inside the
brackets must vanish to satisfy equation (2.25). Thus, a relation that holds at
every point within a given phase is obtained,
∂b
= −∇ ⋅ F + BV
∂t
(2.26)
Using the methodology and arguments analogous to those used to derive Eq.
(2.25), a pointwise interfacial balance valid at any instant in time is obtained,
[(F − bu I )B − (F − bu I ) A ]⋅ n I = BS
(2.27)
44
The interfacial accumulation and transport within the plane of interface have
been neglected. These effects may arise in problems involving adsorption at solid
surfaces, or surfactants at fluid-fluid interfaces, and have to be included using
additional accumulation and flux terms [80].
The total flux F is commonly expressed as the sum of the convective and
diffusive contributions, i.e.,
F ≡ bu + f
(2.28)
where bu is the convective part and f the diffusive. The pointwise conservation
equations can be written as,
∂b
+ ∇ ⋅ (bu ) = −∇ ⋅ f + BV
∂t
(2.29)
[(f + b(u − u I ))B − (f + b(u − u I )) A ]⋅ n I = BS
(2.30)
and,
2.4.1
Conservation of mass
In this case, the conserved quantity is the total mass density, ρ. For the density
there is no net mass flow relative to the mass-average velocity, and thus there is
no diffusive flux for the total mass. Additionally, there are no sources or sinks.
Therefore, Eq. (2.29) becomes,
∂ρ
+ ∇ ⋅ (ρu ) = 0
∂t
or in Einstein’s index notation,
(2.31)
45
∂ρ ∂ρu i
+
=0
∂t
∂xi
(2.32)
This equation of local mass conservation is called the continuity equation. The
density of any pure fluid is related to the thermodynamic pressure and
temperature by an equation of state ( ρ = ρ ( p, T ) ). If the density of the fluid is
constant for a specific process, the continuity equation will reduce to,
∂u i
=0
∂xi
with
an
important
implication,
the
(2.33)
decoupling
of
density
from
the
thermodynamic pressure [12, 180].
There is an additional way to write the conservation equation for a point
within a domain of a quantity per unit mass, denoted as B ≡ b ρ . Changing
variables from b to B, the left side of equation (2.29) becomes,
⎛ ∂ρ ∂ρu i
∂b ∂bu i
+
= B⎜⎜
+
∂t
∂xi
∂xi
⎝ ∂t
⎛ ∂B
⎞
∂B ⎞
⎟
⎟⎟ + ρ ⎜⎜
+ ui
∂xi ⎟⎠
⎝ ∂t
⎠
(2.34)
According to continuity, the first parenthesis term is zero and Eq. (2.29) can be
written as,
ρ
DB
= −∇ ⋅ f + BV
Dt
(2.35)
where the new differential operator
D
∂
∂
∂
= + u ⋅ ∇ = + ui
∂t
Dt ∂t
xi
(2.36)
46
is called the material derivate or substantial derivate [28]. If density is constant
then the conservation equation for any quantity and continuity are,
Db
= −∇ ⋅ f + BV
Dt
(2.37)
∇ ⋅u = 0
(2.38)
respectively.
2.4.2
Conservation of momentum
Equations (2.29) and (2.30) were derived for any conserved quantity, without
specifying if it is a scalar or a vector quantity, in which case, its corresponding
flux is a second order tensor. There are different physical interpretations when
the general conservation equations are applied to momentum. First, the
conservation principle is derived from Newton’s second law of motion applied
to a material volume6 [28, 75, 180, 322], i.e.,
d
ρudΩ = Force
dt Ω∫
(2.39)
where Force is the net force acting on the material volume and the product ρu is
the concentration of linear momentum. Second, Leibniz’s rule for differentiating
the volume integral of the vector ρu combined with the divergence theorem and
continuity gives,
Control volume in which the exterior surface, bounding the volume, is assumed to deform with
the flow, in other words, the surface velocity is always equal to the local fluid velocity.
6
47
d
Du
ρudΩ = ∫ ρ
dΩ
∫
dt Ω
Dt
Ω
(2.40)
which is the Reynolds’ transport theorem applied to the concentration of linear
momentum. Finally, there should be an additional term in Eq. (2.39), which is a
generalized Newton’s second law for any type of control volume: a surface
integral containing the relative motion of momentum due to the difference in
velocity between the fluid and the external surface, i.e.,
∫ ρu[(u − u ) ⋅ n]dS
S
(2.41)
S
However, (ρu )(u ⋅ n ) = n ⋅ (ρuu ) and Leibniz’s rule with the divergence theorem
will give exactly the same result of Eq. (2.40). Thus, Newton’s second law for any
control volume can be written as,
Du
∫ ρ Dt dΩ = Force
(2.42)
Ω
Knowing the physical meaning of the conservation of momentum, the general
conservation can be applied to concentration of linear momentum. Written in
flux terms,
ρ
∂u
Du i
∂
(ui u j ) = − ∂π ij + ρg i
=ρ i +ρ
∂t
Dt
∂x j
∂x j
(2.43)
where the forces acting on the fluid are of two general types: forces that act on a
mass of fluid which are called body forces (i.e. gravity), and forces that act on
surfaces which are expressed in term of stresses. According to the physical
48
meaning and the description of the forces, the first term after the first equal sign
is the rate of increment (accumulation) of momentum, followed by a term
representing the addition of momentum due to a bulk motion (inertia). The
remaining terms after the second equal sign are the addition of momentum due
to molecular motion and/or molecular interactions (Brownian motions [75, 83])
and the body forces term.
It is known that a fluid at rest presents stresses associated with pressure,
which always act in a normal direction to a surface, they are isotropic7 and
positive when exert a compressive force. Stresses that only come into play when
there are velocity gradients within the fluid are called viscous or deviatoric
stresses. In general, they are neither perpendicular to the surface nor parallel to
it, but rather at some angle to the surface. Thus, the total stresses are divided into
two categories, i.e.,
π = pδ + τ
(2.44)
Multiplying p by the identity tensor (δ) ensures that pressure is a normal stress
and is isotropic. The minus sign, obtained when Eq. (2.44) is replaced into Eq.
(2.43), is needed to make positive pressures compressive8. With this
decomposition of the total stress the conservation of linear momentum in flux
terms can be re-written as,
Pressure acts equally in all directions.
In this work Professor R.B. Bird’s sign notation is used, in order to be consistent to all the
molecular constitutive equations [28].
7
8
49
ρ
Du i
∂p ∂τ ij
=−
−
+ ρg i
Dt
∂xi ∂x j
(2.45)
which are known as the Cauchy momentum equations.
For a Newtonian fluid, Eq. (2.44) reduces to [12, 75, 180],
[
]
⎛2
⎞
+
π = pδ − µ ∇u + (∇u ) − ⎜ µ − κ ⎟(∇ ⋅ u )δ
⎝3
⎠
(2.46)
This equation is a generalization of Eq (2.12) for arbitrary flows, which involves
an additional transport property κ, the dilatational viscosity. For ideal,
monoatomic gases, this is zero, while for incompressible liquids the term
containing it vanishes because of the free divergence condition ( ∇ ⋅ u = 0 ). For
incompressible Newtonian fluids the Cauchy momentum equations are reduced
to,
⎛ ∂u i
ρ ⎜⎜
⎝ ∂t
+uj
∂u i
∂x j
2
⎞
⎟ = − ∂p + µ ∂ u i + ρg i
⎟
∂xi
∂x j ∂x j
⎠
(2.47)
or in vector notation,
ρ
Du
= −∇p + µ∇ 2 u + ρg
Dt
(2.48)
which together with the continuity equation for incompressible fluids are known
as the Navier-Stokes system of equations [12, 180].
At the moment the pressure in Eq. (2.45) is the thermodynamic pressure,
again a variable defined locally by an equation of state of the form p = p (ρ , T ) . As
50
mentioned before, if the fluid is considered incompressible (Eq. (2.38)) the
pressure in Eq. (2.45) is no longer the thermodynamic pressure (the new equation
of state is ρ=constant). Thus, in incompressible flow, p is treated simply as a
mechanical variable which must help satisfy continuity and conservation of
momentum [12, 75, 180, 322].
According to Eq. (2.45) the problem under consideration can have two
distinctive forms: an inertia dominant problem,
ρ
Du i
∂p
=−
+ ρg i
∂xi
Dt
(2.49)
∂p ∂τ ij
−
+ ρg i
∂xi ∂x j
(2.50)
and a viscous dominant problem,
0=−
The pressure must be kept in order to satisfy continuity and the equation of state.
This also means that pressure can have two different types of scaling arguments
depending on the regime being analyzed. Additionally, the dropped inertia term
in Eq. (2.47) includes the time derivative, which indicates that in flows ruled by
viscous forces time appears only as a parameter9. According to the nature of
polymeric liquids the regime described in Eq. (2.50) is more appropriate to
describe them. If the viscosity remains constant for a polymeric liquid, Newton’s
From the Brownian motion idea, this type of flow responds very fast to any perturbation on the
boundary.
9
51
law of viscosity can be used and the momentum equation reduces to Stokes’
equations, i.e.,
−
∂ 2ui
∂p
+µ
+ ρg i = 0
∂xi
∂x j ∂x j
(2.51)
while for a generalized Newtonian fluid will be,
−
∂
∂p
(η (γ& )γ&ij ) + ρg i = 0
+
∂xi ∂x j
(2.52)
At interfaces there are three commonly used conditions, two of them related
with the velocity and a stress balance (using Eq. (2.30)). The first condition, from
empirical observation, is that the velocity components tangent to a fluid-solid or
fluid-fluid interface are continuous. This matching condition on the interface
between materials 1 and 2 is written as,
u t |1 = u t | 2
(2.53)
where u t = t ⋅ u and t is the tangential unit vector. This condition is called the noslip condition. The second condition is that the component of velocity normal to
an interface ( u n = n ⋅ u ) is governed by conservation of mass. Thus, for an
impermeable, inert10, stationary solid, u n = 0 in the fluid contacting the solid
surface. This condition is called the no-penetration condition.
10
Here, “inert” means without phase change [75]
52
Finally, Eq. (2.30) must be used for the conservation of momentum at the
interface; this condition is called a stress balance. For the normal and tangential
components of the total stress, the following equations are obtained [75, 249],
p1 − p 2 + τ nn | 2 −τ nn |1 +2kσ = 0
(2.54)
τ nt | 2 −τ nt |1 + t ⋅ ∇ Sσ = 0
(2.55)
where π nn = n ⋅ n ⋅ π , π nt = t ⋅ n ⋅ π , k(x,t) denotes the mean curvature, σ the surface
tension and ∇ S the gradient operator over the interface.
For a static fluid with negligible surface tension, Eq. (2.54) reduces simply to
an equality of pressures. In many fluid dynamic problems where the normal
stresses at the interface are small this is a very good approximation for static
fluids. Additionally, if the surface tension is considered, under zero normal
stresses, Laplace’s equation is obtained from Eq. (2.54), i.e.,
⎛ 1
1 ⎞
⎟⎟
p 2 − p1 = 2kσ = 2σ ⎜⎜ +
⎝ R1 R2 ⎠
(2.56)
where R1 and R2 are the principal radii of curvature of the surface. Finally, from
these balances it is seen that the curvature term affects only the normal stress
balance, whereas gradients in surface tension affect only the shear stress
condition. That extra term in the normal stress balance is called capillary
pressure.
53
2.4.3
Conservation of internal energy
Using Eq. (2.29) for the conservation of the internal energy per unit volume
( b = ρU ), the conservation of energy for a point within the domain can be
expressed in term of the fluxes as,
∂
(ρU ) + ∇ ⋅ (ρUu ) = −∇ ⋅ q − p(∇ ⋅ u ) − (τ : ∇u ) + H V
∂t
(2.57)
where the diffusive flux is now the conduction q, the pressure term represents
the reversible transformation between mechanical and thermal energy (zero for
incompressible fluids), the dyadic product between the viscous stress and the
deformation tensor is the heat generated by viscous dissipation11 and HV is the
rate of energy input from external power sources, per unit volume.
The diffusive heat flux in a solid or a pure fluid is evaluated using Fourier’s
Law,
q = −λT ∇T
(2.58)
where λT is the thermal conductivity [22, 146]. The accumulated internal energy
is due mainly to temperature changes, U = C p T [22, 23]. Therefore, for an
incompressible fluid Eq. (2.57) becomes,
ρ
D
(C pT ) = ∇ ⋅ (λT ∇T ) − (τ : ∇u ) + H V
Dt
(2.59)
The term dissipation comes from the fact that this term represents work lost due to irreversible
transformations [322].
11
54
additionally, for a generalized Newtonian fluid it reduces to,
ρ
D
(C pT ) = ∇ ⋅ (λT ∇T ) + 1 η (γ& )(γ& : γ& ) + H V
Dt
2
(2.60)
At interfaces the energy balance from Eq. (2.30) will be,
q n | 2 −q n |1 = H S
(2.61)
where HS is the rate of energy input from an external power source, per unit
surface area. For most situations HS=0.
For situations where there is any fluid flow parallel to the interface, qn in each
phase can be evaluated using Fourier’s law, assuming that only heat transfer
normal to the interface is entirely by conduction. In a flowing fluid there is a
combination of effects between the conduction and the bulk motion [22, 146],
which is expressed in terms of a heat transfer coefficient, hT . Taking phase 2 to be
a flowing fluid, the interfacial flux in the fluid can be written as,
q n | 2 ≡ hT (T2 − Tb )
(2.62)
where T2 is the temperature in the fluid 2 at the interface and Tb is the bulk
temperature in fluid 2. Eq. (2.62) is often called Newton’s law of cooling [22, 23].
Combining this condition with Fourier’s law for phase 1, the convection
boundary condition is obtained,
− λT ,1 (n ⋅ ∇T ) = hT (T2 − Tb )
(2.63)
55
For solids separated by gases there is an additional mechanism of heat transfer,
radiation. Assuming that solids 1 and 3 are separated by a gas 2, the radiant heat
flux of energy at a surface 1 due to surface 3 is expressed as,
(
q nrad |1 = σ SB F13 T14 − T34
)
(2.64)
where σSB is the Stefan-Boltzmann constant and F13 is the view factor for these
surfaces [22, 146, 226]. If there is a phase change the energy balance at the
interface must account for the latent heat, i.e.,
q n |1 − q n | 2 = ∆Hρ1 (u n |1 −u In ) = ∆Hρ 2 (u n | 2 −u In )
(2.65)
where ∆H = H 2 − H 1 is the latent heat per unit mass [23]. Finally, two materials
are assumed to have equal temperatures at any point of contact,
T1 = T2
which is the condition of thermal equilibrium at an interface.
(2.66)
56
Chapter 3
Integral equation theory
This chapter is intended to give a general introduction to the theory of integral
equations1. The chapter is divided into two parts: scalar fields and momentum
equations. For each part definitions are given and integral equations equivalent
to the differential conservation equations discussed in Chapter 2 are developed.
Boundary integral formulations for the Poisson’s and momentum equations are
developed as combination of single-layer, double-layer and volume potential
with different densities. At the end of the chapter, direct boundary integral
formulations for flows with interfaces with isotropic surface tension and for the
Navier-Stokes equations using the penalty-function formulation are developed.
The similitude between the direct boundary integral formulation for the penaltyfunction Navier-Stokes equations and elastostatics is exposed.
For a more complete development of the theory of integral equations see Mikhlin [196], Porter
and Stirling [232], or Golberg and Chen [113].
1
57
3.1 Classification of integral equations
The theory of integral equations allows boundary value problems for partial
differential equations to be converted into integral equations, giving significant
advantages; in particular, they can be used as a basis for numerical solution of
complex problems. Integral equations go back to the work of Green at the
beginning of 18th century and much of the modern theory on integral equations
was developed by Poincaré [228], Fredholm [109] and Hilbert [136], which were
dedicated to prove the existence of solutions to these equations (see Appendix
A). Because of the wide variety of integral equations there are some classification
systems that enable to refer to a given equation in a concise manner. For integral
equations of a single variable the classical nomenclature system is based on
properties of the integrals, the location of the unknown in the equation and the
properties of the kernel [113].
Linear equations are first characterized by their kind and type. The kind
refers to the location of the unknown function and the nature of the functions
that multiply the unknown, while the type refers to the nature of the integrals
(definite or indefinite). The general form of a linear integral equation is given by,
α (x 0 )φ (x 0 ) − λ ∫ K (x 0 , x )φ (x )dS (x ) = g (x 0 )
(3.1)
S
where α (x ) , g (x ) and K (x ) are known functions in x ∈ Ω corresponding to a
closed surface S, and λ is a constant. The function K (x 0 , x ) is the kernel of the
58
integral equation, the function g (x ) is the free or non-homogeneous term, and λ
is the parameter of the equation. If the kernel is a continuous function in the
region x ∈ [a, b] , the integral equation (3.1) is a linear transformation of any
continuous function φ (x ) into another continuous function g (x ) . In particular, if
Eq. (3.1) is as follows,
λ ∫ K (x 0 , x )φ (x )dS (x ) = g (x 0 )
(3.2)
S
for x 0 ∈ S (x ) , the equation is of the first kind, here the unknown appears only
under the integral sign. On the other hand, if the unknown appears both under
and outside the integral sign, i.e.,
α (x 0 )φ (x 0 ) − λ ∫ K (x 0 , x )φ (x )dS (x ) = g (x 0 )
(3.3)
S
for x 0 ∈ S (x ) , the equation is of the second kind. In addition, α (x ) can have no
zeros on S (x ) . If α (x ) has a least one zero on S (x ) , then Eq. (3.3) is referred as an
equation of the third kind.
If the kernel K (x o ,x ) is continuous on S (x ) , then it is regular, otherwise it is
singular. For continuous kernels the integral in Eq. (3.1) can be taken in the
Riemann sense if φ (x ) is continuous as well [113]. If the kernel is discontinuous
and K (x o , x ) belongs to a Lebesgue integrable class of functions (see Appendix
A), i.e. the integral,
59
∫ ∫ K (x , x ) dS (x )dS (x)
2
0
0
(3.4)
S S
has a finite value, then the kernel is weakly singular (merely bounded). Integral
equations with continuous or weakly singular kernels are said to be of Fredholm
type. For example, a kernel of the form, K (x 0 , x ) = H (x 0 , x ) r a , where H (x o ,x ) is
a bounded function and a ∈ [0,1], in a n-dimensional space ( n ≥ 2 ), and r is the
distance between the points x0 and x, is weakly singular. If the kernel is not
continuous or weakly singular, it is possible to define the integral in some special
way that the kernel is transformed and the new integral is interpreted as a
Cauchy Principal Value providing that φ (x ) is Hölder continuous (see Appendix
A) [113, 249, 201].
3.2 Potentials of scalar density
The potential φ * (x ) generated by a source point over a homogeneous and
isotropic medium, mathematically defined as a function at least twice
differentiable with respect to the coordinates, which satisfies Laplace’s equation
at all points except the point of application of the source x o , i.e.,
∇ 2φ * (x 0 , x ) = 0 for x ≠ x 0
(3.5)
is called the fundamental solution of the equation. It represents the Green’s
function, and it is also known as the free-space Green’s function. In a threedimensional domain it is,
60
φ * (x ) =
1 1
4π r
(3.6)
The linear character of the Laplace equation allows the calculation of the
potential at a point x induced by several point sources xi of intensity Qi, as the
superposition of the potentials of the individual source,
φ (x 1 ,...x n ; x ) =
1
4π
n
Qi
∑r
i
(3.7)
i
where ri is the distance between the source point xi and the field point x. This
potential is a continuous function, together with its derivatives, everywhere
except at the source points. A more general definition of a potential is described
in Appendix B.
Considering a distribution of simple sources of volume density ρ (x ) , the
potential associated with this distribution, is defined by generalizing Eq. (3.7),
D(x 0 , ρ ) =
1
4π
∫ r (x , x ) ρ (x)dΩ
1
Ω
(3.8)
0
which is a continuous function of x, differentiable to all orders, at all points of
free space2, while, for points x located inside the domain Ω, the integrand of the
volume potential contains a singularity. However, if the density ρ (x ) is bounded
throughout Ω, the potential D exists at all points x ∈ Ω and is everywhere
continuous and differentiable throughout the space [158]. In other words, the
2
Points located outside the domain Ω.
61
derivatives of the first order of D may be obtained by differentiating under the
integral in Eq. (3.13). However the same is not valid for the second order
derivative. In fact, the continuity of the density does not suffice the existence of
these derivatives. Therefore, it is necessary to impose that the density satisfies a
Hölder condition (Appendix A) [151, 158, 246].
Harmonic functions, like Eq. (3.8), can also be generated by a distribution of
potentials along the surface of the domain, S. A potential, associated with a
continuous distribution of simple sources extending over a surface S and density
σ (x ) , of the form,
V (x 0 , σ ) =
1
4π
∫ r (x , x ) σ (x )dS
1
S
(3.9)
0
is called a single-layer potential, which is a solution of Laplace’s equation as well
[246]. A very important feature of this surface potential is that it is continuous as
the point x crosses the surface S (Appendix B), i.e.,
V (ξ, σ ) =
1
4π
∫ r (ξ, x ) σ (x )dS
1
(3.10)
S
for ξ ∈ S . A second type of surface potential can be obtained as the limit of two
single layers of opposite signs, it is called a double-layer potential and it is
defined as,
W (x 0 ,ψ ) = ∫ K (x 0 , x )ψ (x )dS
S
(3.11)
62
where the function ψ (x ) is the surface density or moment of the double layer,
and the kernel is of the form,
K (x 0 , x ) =
1 ∂ ⎛1⎞
1 1 ∂r
1 (x o − x ) j
=
n j (x )
⎜ ⎟=−
4π ∂nx ⎝ r ⎠
4π r 2 ∂nx 4π
r3
(3.12)
The double-layer potential at a surface point ξ ∈ S , in a smooth surface, coming
from the interior domain is (Appendix B),
1
W (ξ,ψ )( i ) = − ψ (ξ ) + ∫ K (ξ, x )ψ (x )dS
2
S
(3.13)
However, the potential for a surface point ξ ∈ S , in a smooth surface, coming
from the exterior domain is,
1
W (ξ,ψ )( e ) = ψ (ξ ) + ∫ K (ξ, x )ψ (x )dS
2
S
(3.14)
The sign change is due to the direction of the normal. Equations (3.13) and (3.14)
show that the double-layer potential W (x,ψ ) has a discontinuity of ψ (ξ ) as the
point x crosses the surface S. Despite the discontinuity of the double-layer
potential, when the point crosses the surface, its normal derivative is continuous,
which is the Lyapunov-Tauber theorem for the continuity of the normal
derivative of a double-layer potential [151, 246].
63
3.3 Direct boundary integral formulation of Poisson’s equation
An effective method of formulating the boundary-value problems of potential
theory is to represent the harmonic function by a single-layer or a double-layer
potential generated by continuous source distributions, of initially unknown
density, over the boundary S, and forcing these potentials to satisfy the
prescribed boundary conditions of the problem. This procedure leads to the
formulation of integral equations which define the source densities concerned.
This method is usually called indirect method, and can be in terms of a singlelayer potential (equation of the first kind) or a double-layer potential (equation of
a second kind) [113, 246, 249].
In engineering applications it is often convenient to obtain integral
representations which directly involve the field and its fluxes, rather than
equations for single- or double-layer densities. This methodology is commonly
called the direct method. For Poisson’s equation this can be done using the
Green’s identities for scalar fields (Appendix C). Poisson’s equation is an
important equation in physics and engineering, it is widely used in transport
phenomena and it is defined as,
∇ 2 u (x, t ) =
∂ 2 u (x, t )
= b(u, x, t )
∂x j ∂x j
(3.15)
For example, for a constant thermal conductivity, λT , the energy conservation
equation (2.60) can be written in this form for the temperature, i.e.,
64
∇ 2T = b(T , x, t )
(3.16)
where the non-homogeneous term b(T , x, t ) is defined by,
b(T , x, t ) =
ρ D
(C pT ) − 1 η (γ& )(γ& : γ& ) + H V
λT Dt
2λ
λT
(3.17)
where ρ is the density, Cp the specific heat, η the non Newtonian viscosity and
HV the heat generation per unit volume. If the non-homogeneous term b(T , x, t )
only includes the material derivative, then equation (3.16) can be written as,
α∇ 2 T =
∂T
+ u ⋅ ∇T
∂t
(3.18)
where α is a diffusion coefficient (in this particular case the thermal diffusivity
α = λT ρ C p ), equation (3.18) is called the convection-diffusion equation. In
addition, for a zero velocity field it is reduced,
α∇ 2 T =
∂T
∂t
(3.19)
which is commonly referred as the heat equation. Finally, for steady-state it
reduces to Laplace’s equation, i.e.,
∇ 2T = 0
(3.20)
To find a direct integral equation of Poisson’s equation (3.15) the second
Green’s identity is written for two regular functions, differentiable at least to the
65
second order. A function u (x ) which is a solution of Laplace’s equation and the
fundamental solution or Green’s function φ * (x ) , i.e,
∇ 2φ * = −δ (x − x 0 )
(3.21)
where δ (x − ξ ) is the Dirac delta function with its peak at the point x 0 [78]
(Appendix C). The second Green’s identity is then reduced to,
⎛
∂φ * (x 0 , x ) ⎞
∂u (x )
⎟⎟dS
u (x 0 ) = ∫ ⎜⎜ φ * (x 0 , x )
− u (x )
n
n
∂
∂
x
x
⎠
S⎝
(3.22)
for every point x 0 ∈ Ω . This equation represents a harmonic function u (x ) as the
superposition of a single-layer potential of density ∂u ∂n and a double-layer
potential of density u , i.e.,
u (x 0 ) = V (x 0 , ∂u ∂n ) − W (x 0 , u )
(3.23)
When the point approaches to a point in the surface ξ ∈ S , it is known that the
single-layer potential is continuous, while the double-layer potential presents a
jump, or discontinuity (Appendix B), i.e. [246, 249],
1
u (ξ ) = V (ξ, ∂u ∂n ) − W (ξ, u )
2
(3.24)
1
∂u (x )
∂φ * (ξ, x )
u (ξ ) = ∫ φ * (ξ, x )
dS − ∫ u (x )
dS
2
∂nx
∂n x
S
S
(3.25)
or in integral form,
66
These equations are an equivalent integral formulation for Laplace’s equation in
a domain Ω with a Lyapunov type closed surface S (Figure 3.1), which can have
Neumann (i.e. constant temperature), Dirichlet (i.e. constant heat flux), Robin
(i.e. convection heat condition), or any type of boundary conditions on S (see
Appendix A).
n
Ω
u(x,t)
S
Figure 3.1:
Representation of the domain, boundary and normal vector.
One important issue regarding the direct integral equation formulation is
that, for Neumann boundary conditions, the boundary integral equation is a
Fredholm integral equation of the second kind while for Dirichlet problems, it
becomes a Fredholm integral equation of the first kind3.
3
For integral equations of the first and second kind Fredholm theorems are valid.
67
n
Ω
β
n
S
Figure 3.2:
Internal angle at a boundary point for a “non-smooth” surface in 2D.
Integral equations (3.24) and (3.25) can be generalized for “non-smooth”
surfaces in the form4 [246, 249],
c(ξ )u (ξ ) = V (ξ, ∂u ∂n ) − W (ξ, u )
(3.26)
where the free coefficient c(ξ ) is given, in a two-dimensional problem, by (Figure
3.2),
c(ξ ) =
β
2π
(3.27)
which means that c(ξ ) ∈ [0,1] . It is seen in Figure 3.2 that the surface is not of the
Lyapunov type, it looks like a Kellogg regular surface. However, if the normal is
assumed to be continuous, in other words zooming the corner will look like a
The analysis for the continuity of the double-layer potential that is performed in the Appendix B
was done for a smooth surface. The only change is on the upper limits on the integrals over the
hemisphere.
4
68
smooth surface, then the analysis and formulations are valid in this type of
geometries.
The integral formulation of Poisson’s equation is found in the same way that
the Laplace’s equation, except that the second volume integral is kept in Green’s
second identity. For a point x 0 in the domain the integral formulation will be a
superposition of a single-layer, a double-layer and a volume potential of density
b(u , x, t ) , i.e.,
u (x 0 ) = V (x 0 , ∂u ∂n ) − W (x 0 , u ) + D(x 0 , b )
(3.28)
or in integral notation,
u (x 0 ) = ∫ φ * (x 0 , x )
S
∂φ * (x 0 , x )
∂u (x )
dS − ∫ u (x )
dS
∂nx
∂nx
S
+ ∫ φ * (x 0 , x )b(u, x, t )dΩ
(3.29)
Ω
And for a point in the surface S, the discontinuity of the double-layer potential
must be taken into account, i.e.,
c(ξ )u (ξ ) = V (ξ, ∂u ∂n ) − W (ξ, u ) + D(ξ, b(u , x, t ))
c(ξ )u (ξ ) = ∫ φ * (ξ, x )
S
∂u (x )
∂φ * (ξ, x )
dS − ∫ u (x )
dS
∂nx
∂nx
S
+ ∫ φ * (ξ, x )b(u, x, t )dΩ
(3.30)
(3.31)
Ω
These equations are equivalent to the differential form of Poisson’s equation
(3.15).
69
3.4 Direct boundary integral formulation for the momentum
equations and Hydrodynamic potentials
From the differential form of the momentum equations, the Navier-Stokes and
the low Reynolds number non-Newtonian flow equations can be written as
follows,
∂u i
=0
∂xi
(3.32)
∂ 2ui
∂p
−
+µ
= gi
∂xi
∂x j ∂x j
(3.33)
where u is the velocity field, p the pressure or the modified pressure, depending
if gravity is included in the analysis. Equation (3.32) is the continuity equation
for incompressible fluids [12, 180, 322]. For the Navier-Stokes equations, the
pseudo-body force term g in equation (3.33) is defined as,
⎛ ∂u
∂u
gi = ρ⎜ i + u j i
⎜ ∂t
∂x j
⎝
⎞
⎟
⎟
⎠
(3.34)
while for the low Reynolds number flow of a non-Newtonian fluid,
gi = −
∂τ ij(e )
∂x j
(3.35)
where τ (e) is the extra stress tensor that represents the non-Newtonian effects in
the stress tensor. For inelastic generalized Newtonian fluids this stress tensor is
defined as,
70
τ ij( e ) (γ& ) = (η (γ& ) − µ )γ&ij
(3.36)
In this case µ is an arbitrary constant, chosen as the zero shear rate viscosity. The
expression for the non-Newtonian viscosity is a constitutive equation for a
generalized Newtonian fluid, like the power law or Ostwald-de-Waele model [2,
26],
η (γ& ) = mγ& n −1
(3.37)
where m is called the consistency index and n ∈ [0,1], the power law index.
In order to obtain an integral representation for the momentum equations
(3.32) and (3.33) for the flow field (u, p ) , Green’s formulae for the momentum
equations are used together with the fundamental singular solution of Stokes’
equations (Appendix C), i.e.,
µ
∂ 2 u ik (x 0 , x ) ∂q k (x 0 , x )
−
= −δ (x − x 0 )δ ik
∂x j ∂x j
∂xi
∂u ik (x 0 , x )
=0
∂xi
(3.38)
(3.39)
to get an integral representation formulae for the velocity fields. For a point
x ∈ Ω the integral representation is given as [174],
71
(
)
u k (x 0 ) = ∫ π ij* u k (x 0 , x ), q k (x 0 , x ) u i (x )n j (x )dS
S
− ∫ u (x 0 , x )π ij (u(x ), p(x ))n j (x )dS + ∫ u (x 0 , x )g i (x )dΩ
k
i
k
i
(3.40)
Ω
S
where
u ik (x 0 , x ) = −
1 ⎛ δ ik ( x0 − x )i ( x0 − x )k
⎜
+
8πµ ⎜⎝ r
r3
⎞
⎟⎟
⎠
(3.41)
is the fundamental singular solution of the Stokes system of equations or Green’s
fundamental solution, known as the Stokeslet, located at the point x and oriented
in the k-th direction, with a corresponding pressure
q k (x 0 , x ) = −
1 ( x 0 − x )k
1 ∂ ⎛1⎞
=
⎜ ⎟
3
4π
4π ∂nk ⎝ r ⎠
r
(3.42)
and
⎛ ∂u ik ∂u kj ⎞
⎟
π u (x 0 , x ), q (x 0 , x ) = q δ ij + µ ⎜
+
⎜ ∂x j ∂xi ⎟
⎝
⎠
3 ( x0 − x )i ( x0 − x ) j ( x 0 − x )k
=−
4π
r5
*
ij
(
k
k
)
k
(3.43)
is the symmetric component of a Stokes doublet, which is a fundamental
singularity call Stresslet (Appendix C). The inner product between the Stresslet
and the normal vector gives the traction fundamental solution,
K ij (x 0 , x ) = −
3 ( x0 − x )i ( x0 − x ) j ( x 0 − x )k
nk
4π
r5
and equation (3.40) can be written as,
(3.44)
72
u i (x 0 ) = ∫ K ij (x 0 , x )u j (x )dS
S
− ∫ u i (x 0 , x )π kj (u(x ), p (x ))n k (x )dS + ∫ u i (x 0 , x )g j (x )dΩ
j
j
(3.45)
Ω
S
This equation suggests that, similarly to scalar fields, there are hydrodynamic
potentials for vector fields, which expressions are given in this integral
representation. The hydrodynamic single-layer potential of density ψ (x ) is
defined as,
Vi (x 0 , ψ ) = − ∫ u ij (x 0 , x )ψ j (x )dS
(3.46)
S
the hydrodynamic double-layer potential of density φ(x ) is,
Wi (x 0 , φ ) = ∫ K ij (x 0 , x )φ j (x )dS
(3.47)
S
and the hydrodynamic volume potential of density ρ(x ) is,
Di (x 0 , ρ ) = ∫ u ij (x 0 , x )ρ j (x )dΩ
(3.48)
Ω
Thus, the velocity field u(x ) can be written as a superposition of single-layer
potential of density t (x ) , double-layer potential of density u(x ) and volume
potential of density g(x ) as,
u i (x ) = Wi (x, u ) + Vi (x, t ) + Di (x, g )
for any x ∈ Ω , and t (x ) are the surface tractions defined as,
(3.49)
73
t i (x ) = π ij (u, p )n j (x )
(3.50)
According to the definition of the hydrodynamic potentials, they are expected to
behave similar to the single- and double-layer potentials when a point
approaches to the surface S. In fact, the hydrodynamic single-layer potential is
continuous while the hydrodynamic double-layer potential presents a jump or
discontinuity. For a general surface equation (3.49) is,
cij (x 0 )u j (x 0 ) = Wi (x 0 , u ) + Vi (x 0 , t ) + Di (x 0 , g )
(3.51)
or in integral form,
cij (x 0 )u j (x 0 ) = ∫ K ij (x 0 , x )u j (x )dS
S
− ∫ u ij (x 0 , x )π kj (u(x ), p(x ))n k (x )dS + ∫ u ij (x 0 , x )g j (x )dΩ
(3.52)
Ω
S
for x 0 ∈ S , where cij (x 0 ) is a second order tensor defined as,
⎛ ci
⎜
cij = ⎜ 0
⎜0
⎝
0
ci
0
0⎞
⎟
0⎟
ci ⎟⎠
(3.53)
where ci ∈ [0,1] .
For the Stokes system of equations, i.e. g = 0 , equation (3.51) reduces to a
simple superposition of hydrodynamic single-layer and double-layer potentials,
cij (x 0 )u j (x 0 ) = Wi (x 0 , u ) + Vi (x 0 , t )
(3.54)
74
For a second boundary-value problem, i.e. given surface tractions, the above
equation is a Fredholm integral equation of the second kind, for which
Fredholm’s theory is valid (Appendix A). In the case of a first boundary-value
problem, i.e. given surface velocity, it reduces to Fredholm integral equation of
the first kind, while for mixed boundary-value problems, becomes a mixed
integral equation. There is no general theory for the last two cases [109, 246], and
the analysis must be performed to every particular case.
3.5 Other direct boundary integral formulations
3.5.1
Interface flows with surface tension
Consider an interface, without impurities or surfactants, between an external
fluid with viscosity µ and an internal fluid with viscosity λ d µ , characterized by
an isotropic surface tension σ. The external fluid at infinity is made to flow with
velocity u i∞ (x ) that causes it to shear; consequently, the interface will deform
continuously. The integral representation formulae for the velocity fields are
found from Green’s formulae for Stokes equations given by Ladyzhenskaya
[174]. For the exterior problem it will be,
u i (x 0 ) + ∫ K ij (x 0 , x )(u j (x ))e dS =
S
u (x ) + ∫ u i (x 0 , x )(π jk (u(x )))e nk (x )dS
∞
i
j
S
(3.55)
75
for every x ∈ Ω e , where (u(x ))e and (π ij (u(x )))e are the values of the velocity field
u and of the stress π ij (u(x )) , respectively, at a point x ∈ S coming from Ωe. For
the internal fluid, the Green’s representation formulae is given by,
u i (x 0 ) − ∫ K ij (x 0 , x )(u j (x ))i dS =
S
−
1
λd
j
∫ ui (x 0 , x)(π jk (u(x )))i nk (x)dS
(3.56)
S
for every x ∈ Ω i , where (u(x ))i and (π ij (u(x )))i are the values of the velocity field
u and of the stress π ij (u(x )) , respectively, at a point x ∈ S coming from Ωi. On the
interface there are two additional conditions: the matching condition for the
velocity and the discontinuity of the stress tensor which is function of the local
curvature and surface tension. Considering that the free surface is smooth and
letting a point in the exterior domain approach to the surface, the following
equation is obtained from equation (3.55),
1
u i (x 0 ) + ∫ K ij (x 0 , x )u j (x )dS =
2
S
u (x ) + ∫ u i (x 0 , x )(π jk (u(x )))e nk (x )dS
∞
i
(3.57)
j
S
Similarly, as a point in the internal fluid approaches the surface equation (3.56) is
reduced to,
1
u i (x 0 ) − ∫ K ij (x 0 , x )u j (x )dS =
2
S
−
1
λd
∫ u (x , x)(π (u(x ))) n (x )dS
j
i
S
0
jk
i
k
(3.58)
76
Multiplying equation (3.57) by λ d and adding to equation (3.58) Rallison and
Acrivos (1978) [260-262] found the following second kind Fredholm integral
equation for the unknown surface velocity,
2(1 − λ d )
K (x , x )u (x )dS = Fi (x )
(1 + λd ) ∫S ij 0 j
(3.59)
⎤
2 ⎡ ∞
j
⎢u i (x ) + ∫ u i (x 0 , x ){2k (x )σ + (ρ e − ρ i )g}dS ⎥
(1 + λd ) ⎣
S
⎦
(3.60)
u i (x ) +
for x ∈ S , where
Fi (x ) =
and k (x ) = 0.5(k1 + k 2 ) is the mean surface curvature. They found this equation for
the specific case of a viscous drop immersed in a different viscous fluid. The
homogeneous form of equation (3.59) has only one eigen-solution when λ d = 0 ,
and if λ d = ∞ the six rigid-body motions for the drop are all eigen-solutions.
Therefore, it follows that the Fredholm integral equation (3.75) does not admit a
unique solution at these two poles of the resolvent (see Appendix A). However,
Power (1987) [246] proved, analytically, that this integral equation possesses a
unique continuous solution u(x) for any F(x) when 0 < λ d < ∞ .
3.5.2
Penalty-function formulation for the Navier-Stokes equations and
elastostatics
The Navier-Stokes system of equations or the momentum equations for an
Incompressible Newtonian fluid can be written in the following form
[white,landau,batchelor],
77
ρ
∂u i
∂u
∂p
∂
+ ρu j i = −
+µ
∂t
∂x j
∂xi
∂x j
⎛ ∂u i ∂u j
⎜
+
⎜ ∂x
∂xi
j
⎝
⎞
⎟
⎟
⎠
(3.61)
which in some references is called the velocity-pressure formulation. Sometimes,
a penalty parameter is employed to eliminate the pressure term from this
equation. That is, the pressure is approximated by,
p = −λ p
∂u i
∂xi
(3.62)
where λ p is the penalty parameter. Since p has a finite value, a large value of
the penalty parameter will make the divergence of the velocity approach zero,
enforcing in the limit λ p → ∞ the automatic satisfaction of the continuity
equation for incompressible fluids. For numerical calculations, a large but finite
value of λ p is selected, so that a slight compressibility is included in the analysis.
The penalty-function formulation for the steady-state Navier-Stokes equations is,
∂ 2u j
∂ 2 ui
∂u
(λ p + µ )
+µ
= ρu j i
∂xi ∂x j
∂x j ∂x j
∂x j
(3.63)
These equations are written in the following form,
∂ 2u j
∂ 2 ui
(λ * + µ *)
+µ*
= −bi
∂xi ∂x j
∂x j ∂x j
(3.64)
where λ * and µ * are the first and second Lame constants and b(x ) is a body
force, then, Equation (3.64) represents the Navier equations for elasticity [178, 38,
40]. The boundary integral representation formulae corresponding to the Navier
78
equations of elastostatics can be developed in a similar way as for the velocitypressure formulation of the momentum equations. It is given by [38, 40, 246],
cij (x 0 )u i (x 0 ) + ∫ p ij* (x 0 , x )u j (x )dS =
S
∫ u (x , x ) p (x )dS + ∫ u (x , x )b (x )dΩ
*
ij
0
*
ij
j
0
(3.65)
j
Ω
S
where the kernels of the integrals, u * and p * , are the Kelvin’s fundamental
solutions, which are of the form,
u ij* (x 0 , x ) =
(x0 − x )i (x0 − x ) j ⎤
⎡ (3 − 4v )
1
δ ij +
⎢
⎥
16π (1 − v )G ⎣ r
r3
⎦
(3.66)
and
⎧⎡
(x0 − x )i (x0 − x ) j ⎤ ∂r ⎫
⎪⎢(1 − 2v )δ ij + 3
⎪
⎥
1
⎪⎣
∂nx ⎪
r2
*
⎦
pij (x 0 , x ) = −
⎨
⎬
8π (1 − v )r 2 ⎪ (1 − 2v )
(x0 − x )i n j − (x0 − x ) j ni ⎪⎪
⎪⎩− r
⎭
(
)
(3.67)
The coefficients G and v are the shear modulus and Poisson’s ratio, respectively.
79
Chapter 4
Boundary element method
Analytical solutions to the Fredholm integral equations or to the combination of
single-, double-layer and volume potentials that arise from the integral
representations are possible for a limited number of boundary geometries and
types of flow. Therefore the original integral equations must be transformed into
an equivalent algebraic form that can be solved by a suitable numerical
approach. This methodology is known as the boundary element method (BEM),
named after the fact that the boundary is divided into segments or elements in
which the integral is evaluated.
This chapter describes the BEM. Initially, the boundary discretization is
explained with the corresponding numerical quadrature for the integrals. The
numerical evaluation of weakly singular equations is described as well as the
numerical evaluation of the boundary coefficients. Then, the proposed domain
grid superposition technique is explained for the numerical evaluation of the
80
domain integrals containing the nonlinear terms. Finally, the iterative numerical
algorithm for the non linear BEM system of equations is described.
4.1 Isoparametric boundary elements
The first step of the BEM is to discretize the boundary into a series of elements
over which the velocity and traction are assumed to vary according to some
interpolation functions [38, 216, 255]. Here, the single-layer and double-layer
potential are going to be approximate, which are the boundary integrals in the
direct formulation discussed in the previous chapter.
For the sake of understanding the basis of the integral formulation, the Stokes
equations will be used, i.e.,
cij (x 0 )u j (x 0 ) = ∫ K ij (x 0 , x )u j (x )dS − ∫ u ij (x 0 , x )t j (x )dS
S
(4.1)
S
Once the boundary is divided into NE elements, which can be performed
according to the definition of a Lyapunov surface (see Appendix A), equation
(4.1) will be equivalent to,
cij u j = ∫ K ij u j dS k − ∫ uij t j dS k
Sk
(4.2)
Sk
where k = 1,..., NE .
Each element is defined by a number of points or nodes where the unknown
values of the velocity of traction are sought. The number of nodes in each
81
element defines its type and order (for a detailed description of elements and
their order see Chandrupatla and Belegundu [57] or Kardestuncer [155]). In this
work, the boundary was discretized into isoparametric 8 noded quadratic
elements (see Figure 4.1).
ξ2
4
7
8
x3
r
5
1
3
4
7
3
6
2
6
8
ξ1
x2
x1
Figure 4.1:
1
5
2
Isoparametric 8 noded quadratic element.
Isoparametric elements present many advantages compared with other types
of elements. First, the isoparametric interpolation, which is used for any variable
within the element (coordinates, velocity, temperature, etc), will give the same
weight or importance to every node in the element. Second, any distortion of the
element in the regular Cartesian coordinates will not affect the interpolation; this
ensures that a relative small number of elements can represent a complex form of
the type which is liable in real problems. With isoparametric interpolation not
only can two-dimensional elements be distorted into others in two dimensions,
but the mapping of these can be taken into three dimensions as illustrated in
82
Figure 4.1. In addition, the new curvilinear coordinates ( ξ1 , ξ 2 ) are in the range
[− 1,+1] , thus for any integration, Gaussian quadratures can be used, which are
the most efficient way to approximate an integral [137, 255].
The value of any variable at any point within the element is defined in terms
of the node’s values according to the isoparametric interpolation. For instance,
the coordinates and the velocity field for each element can be written as follows,
x = Ni xi
(4.3)
u = Niui
(4.4)
where i = 1,...,8 and N i are interpolation or shape functions given in terms of the
local coordinates. Further, the points with coordinates x will lie at approximate
points of the element boundary; as from the general definitions of the shape
functions, they have a value of unity at the point in question and zero elsewhere.
For the 8-noded quadratic element the interpolation or shape functions for the
corner nodes are defined by [85, 147, 336, 337],
Ni =
1
(1 + ξ10 )(1 + ξ 20 )(ξ10 + ξ 20 − 1)
4
(4.5)
where the new variables are defined as,
ξ10 = ξ1ξ1,i
ξ 20 = ξ 2ξ 2,i
and the shape functions for a typical mid-side node are,
(4.6)
83
(
)
1
1 − ξ12 (1 + ξ 20 )
2
1
N i = 1 + ξ10 1 − ξ 22
2
ξ1,i = 0,
Ni =
(
ξ 2,i = 0,
)(
(4.7)
)
Equation (4.3) in matrix form will be,
⎛ x1 ⎞ ⎛ N 1
⎜ ⎟ ⎜
⎜ x2 ⎟ = ⎜ 0
⎜x ⎟ ⎜ 0
⎝ 3⎠ ⎝
0
N1
0
0
N2
0
N1
0
0
0
N2
0
0
0
N2
... N 8
...
...
0
0
0
N8
0
⎛ x11 ⎞
⎜ 1⎟
⎜ x2 ⎟
⎜ x1 ⎟
⎜ 3⎟
⎜ x12 ⎟
0 ⎞⎜ 2 ⎟
⎟ x
0 ⎟⎜ 22 ⎟
⎜x ⎟
N 8 ⎟⎠⎜ 3 ⎟
⎜ ... ⎟
⎜ x8 ⎟
⎜ 18 ⎟
⎜ x2 ⎟
⎜ x8 ⎟
⎝ 3⎠
(4.8)
or more compact as,
x = Nx j
(4.9)
where N is the matrix of isoparametric shape functions and x j = xij is the vector
of nodal coordinates of the eight element nodes, where j denotes the node and i
the direction. In the same way the velocity and traction fields can be expressed
as,
u = Nu j
(4.10)
t = Nt j
(4.11)
84
For any point in the domain and boundary, the kernels in the boundary
integrals of equation (4.2) can be written in matrix form as,
⎛ K 11
⎜
K ij = h = ⎜ K 21
⎜K
⎝ 31
K 12
K 22
K 32
K 13 ⎞
⎟
K 23 ⎟
K 33 ⎟⎠
(4.12)
and
⎛ u11
⎜
u ij = g = ⎜ u 12
⎜ u1
⎝ 3
u12
u 22
u
2
3
u13 ⎞
⎟
u 23 ⎟
u 33 ⎟⎠
(4.13)
By substitution of equations (4.10) to (4.13) into the equation (4.2) the boundary
integral formula can be written as follows,
⎧⎪
⎫⎪
⎧⎪
⎫⎪
cu i = ⎨ ∫ hNdS j ⎬u j − ⎨ ∫ gNdS j ⎬t j
⎪⎩S j
⎪⎭
⎪⎩S j
⎪⎭
(4.14)
where the summation j = 1,..., NE is carried out over the NE elements, S j is the
surface of element j, and u j and t j are the nodal velocities and tractions in that
element. Integrals in equation (4.14) can then be solved numerically after the
coordinates on each element have been transformed into the local system defined
by ξ1 and ξ 2 .
Equation (4.9) defines a transformation of coordinates from a global system to
a local system of coordinates over each boundary element, f : x → (ξ1 , ξ 2 ) . To
85
evaluate the integrals in equation (4.14), the differential of length on the
boundary has to be redefined as,
dS j = J j dξ1 dξ 2
(4.15)
where J is the reduced Jacobian whose magnitude J is given by the expression
(
J = g12 + g 22 + g 32
)
12
(4.16)
where
g=
∂x j ∂x k
∂x ∂x
×
= ε ijk
∂ξ1 ∂ξ 2
∂ξ1 ∂ξ 2
(4.17)
and ε ijk is the permutation pseudo-tensor. Equation (4.14) is then reduced to,
⎧ +1 +1
⎫ j ⎧+1 +1
⎫
cu = ⎨ ∫ ∫ hN J j dξ1 dξ 2 ⎬u − ⎨ ∫ ∫ gN J j dξ1 dξ 2 ⎬t j
⎩−1 −1
⎭
⎩−1 −1
⎭
i
(4.18)
After Gaussian quadratures [100, 137] are applied to each integral, this
equation will be,
{(
cu i = hN J
)
j lk
} {(
wl wk u j − gN J
)
j lk
}
wl wk t j
(4.19)
where the internal summation (over k and l) is for every Gaussian point in ξ1 and
ξ 2 , wk and wl are the weight factors at those points and the functions (hN J )
and (gN J ) are evaluated at each integration point. Equation (4.19) must be
applied for each node (i) on the surface or inside the domain, once integrated it
can be written as follows,
86
ˆ ij u j − G ij t j
cu i = H
(4.20)
where the matrices Ĥ and G are defined from equation (4.18) as,
Ĥ ij = ∫ hNdS j
Sj
G ij = ∫ gNdS j
(4.21)
Sj
The velocity is a continuous function; therefore there is a unique value of u in
every node. Generally, this is not true for the traction vector. However, for a
Lyapunov surface, where the normal is continuous, the tractions are also
continuous. Equation (4.20) will be,
Hu = Gt
(4.22)
ˆ − c . For a problem with N boundary nodes and NI internal points
where H = H
H ∈ M (3( N + NI ),3( N + NI )) and G ∈ M (3( N + NI ),3 N ) . Consequently, there are
3( N + NI ) velocity unknowns and 3 N traction unknowns. This makes equation
(4.22) a system of 3( N + NI ) equations with 3( N + NI ) + 3 N unknowns. Each
boundary nodal point has either traction or velocity specified for each direction
as a boundary condition, thus the system in equation (4.22) can ultimately be
arranged into a solvable system of equations as,
Ax = b
(4.23)
87
where the coefficient matrix A contains columns of matrices H or G; it is fully
populated and non-symmetric. The vector x has unknown traction or velocity
and b is the vector obtained from the multiplication of the boundary conditions
with the corresponding coefficients in H or G.
4.2 Evaluation of the coefficient matrix c
If the boundary is smooth the coefficient matrix is c = (1 2 )δ , when non-smooth a
possibility in 2D is to use equation (4.35). However, in three-dimensional
domains it is cumbersome or nearly impossible to use this equation. Another
way to calculate the diagonal terms of matrix H is to use the fact that when a
uniform potential is applied over a bounded region, all the derivatives must be
zero (see Appendix A, the Dirichlet theorem). Hence, from equation (4.21) it is
obtained,
Hv = 0
(4.24)
where v is a vector of constant value. Thus, the sum of all elements of any row of
H ought to be zero, and the values of the diagonal can be easily calculated once
the off-diagonal coefficients are known, i.e.
N
H ii = − ∑ H ij
j =1
(i ≠ j )
This means that the coefficient matrix c need not be calculated explicitly.
(4.25)
88
4.3 Numerical treatment of the weakly singular integrals
The Green’s functions or fundamental solutions in the matrices H and G go to
infinity as the distance between the source and field point decreases, i.e. the
Euclidean distance r → 0 . The previous section illustrated how the singular
coefficients in the H matrix can be calculated from a constant potential over the
surface or a rigid body motion. However, the weak singularity of the Stokeslet in
the kernel of the double-layer potential (matrix G) needs a special treatment.
The weak singularity of the Stokeslet is of the order O(log r ) which can be
dealt with a self-adaptive coordinate transformation called the Telles’
transformation [305]. For example, consider the evaluation of an integral
+1
∫ f (ξ )dξ
(4.26)
−1
where the function f (ξ ) is weakly singular at ξ 0 . The singularity can be
cancelled off by forcing its Jacobian to be zero at the singular point in a new
Telles space defined as [305],
ξ = aγ 3 + bγ 2 + cγ + d
(4.27)
where the constants in this third order polynomial are given by,
a=
1
Q
3γ 2
c=
Q
b=−
3γ
Q
3γ
d =+
Q
(4.28)
89
with
Q = 1 + 3γ 2
(
γ = ξ 0ξ * + ξ *
) + (ξ ξ
13
*
0
− ξ*
)
13
+ ξ0
(4.29)
ξ * = ξ 02 − 1
The integral is then calculated in terms of γ as follows,
(
) ⎞⎟ 3(γ − γ )
⎛ (γ − γ )3 + γ γ 2 + 3
∫−1 f ⎜⎜⎝
Q
+1
⎟
⎠
Q
2
dγ
(4.30)
Now the integral can be evaluated by using the standard Gauss quadrature.
After the transformation all the standard Gauss points of the numerical
quadrature are biased towards the singularity where the Jacobian is zero.
4.4 Approximation of the domain integrals
In Chapter 3, it was mentioned that the continuity and momentum equations,
∂u i
=0
∂xi
−
∂ 2ui
∂p
+µ
= gi
∂xi
∂x j ∂x j
(4.31)
(4.32)
have an integral representation, which is a superposition of three hydrodynamic
potentials: a single-layer, a double-layer and a volume with specific densities,
i.e.,
90
cij (x 0 )u j (x 0 ) = ∫ K ij (x 0 , x )u j (x )dS
S
− ∫ u i (x 0 , x )π kj (u(x ), p(x ))n k (x )dS + ∫ u i (x 0 , x )g j (x )dΩ
j
S
j
(4.33)
Ω
for x 0 ∈ S and where the pseudo-body force vector, g , includes all non-linear
terms and body forces of the problem under consideration.
Previously, it was explained how to deal with the hydrodynamic potentials of
single- and double-layer, i.e. the surface integrals in equation (4.33). Several
methods have been developed to approximate the domain integral in this
equation. As a matter of fact in the international conferences on boundary
elements, organized every year since 1978 [216], numerous papers on different
and novel techniques to approximate the domain integral have been presented in
order to make the BEM applicable to complex non-linear and time dependent
problems. Many of these papers were pointing out the difficulties of extending
the BEM to such applications. The main drawback in most of the techniques was
the need to discretize the domain into a series of internal cells to deal with the
terms not taken to the boundary by application of the fundamental solution, such
as non-linear terms.
Some of these methods approximate the domain integrals to the boundary in
order to eliminate the need for internal cells, i.e. boundary-only formulations.
The dual reciprocity method (DRM) introduced by Nardini and Brebbia [202] is
one of the most popular techniques. The method is closely related to the method
of particular integrals technique (PIT), introduced by Ahmad and Banerjee [3],
91
which also transforms domain integral to boundary integrals. In the PIT method
a particular solution satisfying the non-homogeneous PDE is first found and then
the remainder of the solution, satisfying the homogeneous PDE, is obtained by
solving the corresponding integral equations. The boundary conditions for the
homogeneous PDE must be adjusted to ensure that the total solution satisfies the
boundary conditions of the original problem [3, 9, 230, 231]. The DRM also uses
the concept of particular solutions, but instead of obtaining the particular
solution and the homogeneous solution separately, it applies the divergence
theorem to the domain integral terms and converts the domain integral into
equivalent boundary integrals [216]. A brief description of the DRM starts with
the expansion of the non-linear term g using radial or global interpolation
functions, i.e.
N
(
)
g i (x ) = ∑ f x, x m α lmδ il
(4.34)
m =1
The coefficients α lm are unknown terms that can be solved by application of
equation (4.34) at each of the collocation nodes located on both the boundary,
and domain, where the non-linear terms are approximated. The functions
(
)
f x, x m depend only on geometry. There are two types, radial basis functions
and global functions [113,216,246]. As pointed out by Partridge [215] there are
criteria for selecting the type of the approximation functions. Examples of radial
basis functions the thin-plate spline (TPS),
92
(
)
f x, x m = r 2 log(r )
(4.35)
)
(4.36)
and expansions of the type,
(
f x, x m = 1 + r + r 2 + r 3 + ... + r n
(
)
where r = r x, x m is the Euclidean distance between the field point x and the
collocation point x m . Equation (4.34) when applied to the N collocation nodes
generates 3N linear equations with 3N unknowns. With the approximation given
in (4.34), the domain integral in equation (4.33) becomes,
j
∫ ui (x 0 , x )g i (x )dΩ =
Ω
∑ α ∫ u (x , x ) f (x, x )δ
N+A
j =1
m
l
m
j
i
0
il
dΩ
(4.37)
Ω
To reduce the last domain integral to a boundary integral, a new auxiliary nonhomogeneous Stokes’ field is defined for each interpolation function as follows,
∂uˆ ilm
=0
∂xi
∂ 2 uˆ ilm (x ) ∂pˆ lm (x )
−
µ
= f
∂x j ∂x j
∂xi
(4.38)
m
(x )δ il
(4.39)
where û ilm is an the auxiliary non-homogeneous velocity field with the
corresponding pressure p̂ lm . Applying Green’s identities to the flow field
(uˆ (x), pˆ (x ))
lm
i
lm
and substituting the resulting domain integral into equations
(4.33), obtains the following system of Fredholm’s integral equations, in terms of
boundary only integrals,
93
cij (x 0 )u i (x 0 ) − ∫ K ij (x 0 , x )u j (x )dS x + ∫ u ij (x 0 , x )σ ij (u(x ), p(x ))n j (x )dS x =
S
N
∑α
m =1
m
S
⎧
⎫
lm
lm
j
lm
m
m
⎨cij (x 0 )uˆ j (x 0 ) − ∫ K ij (x 0 , x )uˆ j x, x dS x + ∫ u i (x 0 , x )tˆ j x, x dS x ⎬
S
S
⎩
⎭
(
)
(
)
(4.40)
The analytical expression for the auxiliary Stokes flow field
(uˆ (x), pˆ (x ))
lm
i
lm
corresponding to the interpolation functions can be found by the approach
suggested by Power and Wrobel [99,100,216].
Two major disadvantages were encountered when applying the DRM and
PIT to non-linear flow problems. First, the lack of convergence as the non linear
terms in the problem become dominant: for the Navier-Stokes equations Cheng
et al. [59] and Power and Partridge [242, 243] reported problems when the
Reynolds number was higher than 200, for non-Newtonian fluid flow Davis [69]
and Hernandez [132] faced problems when the shear-thinning exponent was
lower than 0.8 and for thermal convection problems (non-isothermal) when the
Rayleigh number was higher than 103 [238, 239, 273, 274]. Second, in the PIT and
DRM the resulting algebraic system consists of a series of matrix multiplications
of fully populated matrices, which generates expensive computing times for
complex problems.
When dealing with the BEM solution of large problems, it is usual to use the
method of domain decomposition, in which the original domain is divided into
subregions, and in each of them the full integral representation formula is
94
applied. At the interfaces between adjacent subregions, continuity conditions are
enforced. Some authors refer to the subregion BEM formulation as the Green
element method (GEM; see Taigbenu [300] and Taigbenu and Onyejekwe [301]).
Popov and Power [230, 231] found that the DRM approximation of the volume
potential of a highly nonlinear problem can be substantially improved by using
the domain decomposition scheme. This decomposition technique solved the
problems that were previously encountered in the DRM and PIT, i.e. high
Reynolds number [99, 100, 238, 239], low shear-thinning exponents [102] and
high Rayleigh numbers [103]. Although the method keeps the boundary-only
character, it is necessary to construct internal divisions in the domain, which in
some situations is close to a finite element mesh. The corresponding matching
conditions, that are necessary to keep the system closed, i.e. continuity of the
velocity and equilibrium of tractions between adjacent sub-domains, will lead to
cumbersome over-determined systems or complicated discontinuous elements
will be necessary for the internal mesh [230, 231, 238, 239]. In simple twodimensional problems the discontinuous elements will not be an impediment,
while in full three dimensional domains the domain decomposition will be a
difficult task. Figure 4.2 illustrates domain decomposition using discontinuous
elements for a pipe flow in 3D.
95
Figure 4.2:
Sub-domain typical mesh using discontinuous elements.
As a consequence, the application of these types of methods for complex nonlinear problems is limited. Both methods intend to keep the boundary-only
character, which is perfect for small order nonlinearities, but require domain
decomposition (complicated internal meshes) when dealing with high order
96
nonlinearities. Techniques that directly approximate the domain integral have
been developed over the years: Fourier expansions, the Galerkin vector
technique, the multi reciprocity method, Monte Carlo integration and cell
integration. In the early boundary element analysis the evaluation of the domain
integrals was mostly done by cell integration. The technique is effective and
general, but causes the method to lose its boundary-only nature. It is the simplest
way of computing the domain term by subdividing the domain into a series of
internal cells, on each of which a numerical integration scheme, such as Gauss
quadratures, can be applied. Several authors applied the technique for
Newtonian, non-Newtonian and non-isothermal problems with very accurate
results and without the restrictions funded by the techniques that approximate
the domain integrals into boundary integrals [69, 162, 162, 165, 279, 280]. The
domain discretization for the Cell-BEM technique is done by dividing the
boundary into a specific type of elements, while the domain will have a different
type of mesh. The internal cells are not required to be discretized all the way to
the boundary in order to avoid the discontinuity of the kernels in the boundary
and to avoid the necessity of recording which nodes are in the boundary and
domain at the same time. As reported by several authors [38-40, 69, 70] this does
not affect the accuracy of the technique, in fact it is considered an advantage.
Figure 4.3 illustrates a typical mesh for the cell-BEM, where the boundary is
divided into isoparametric 8 noded elements and the domain into isoparametric
97
10 noded tetrahedrons. The gaps between the boundary and domain mesh have
been exaggerated for visualization.
Figure 4.3:
Typical cell-BEM geometry discretization.
The difference between boundary and domain discretization increases the
time needed for pre-processing of a specific problem. In addition, in moving
98
boundary problems there is the necessity of re-meshing the internal cells, which
implies the record of internal solutions and interpolations for transient problems.
In the cell-BEM the integral formulation is applied for both, the boundary and
internal nodes, and for every node, the internal cells are used to approximate the
domain integral (volume potential). A set of nonlinear equations are formed for
the boundary and internal unknowns, and the equations are solved by successive
iterations or by Newton’s method [69, 100, 132, 268]. In conclusion, the big
inconvenience of the cell-BEM technique is the cumbersome pre-processing, two
different meshes, and the re-meshing in moving boundary problems.
4.4.1
Domain grid superposition technique
The proposed domain grid superposition (DGS) technique avoids separating preprocessing for the boundary and domain and the re-meshing in moving
boundary problems. The DGS technique will superimpose a grid with the
domain under consideration, then the cells that are not included in the
intersection between the grid and domain will be excluded from the analysis.
Figure 4.4 illustrates the DGS technique in a simple 2D geometry. The internal
grid is used to directly approximate the domain integral by means of cell
integration. The nodes generated by the domain grid superposition can be solved
for or not, depending of the necessities of the problem. For example, in the
problem illustrated in Figure 4.4, the solution is sought for the boundary and the
internal nodes, the value of the unknowns associated with the cells do not have
99
to be solved for explicitly. On the other hand, in transient problems or in
problems where a refined domain solution is needed (for example, in polymer
mixing applications), the internal cells can be also solved for, which will simply
increases the computational effort.
Figure 4.4:
Schematic of the domain grid superposition technique.
100
Due to the fact that the kernels of the hydrodynamic potentials are a function
of the distance between points, these functions only need to be calculated once
for the grid domain, thus, saving computational time. For moving boundaries, in
every step the technique includes or excludes the internal cells that are needed
without additional re-meshing, which offers the advantage of keeping the
moving boundary character of the BEM for interfacial flow problems as exposed
in Chapter 1.
The ellipse shown in Figure 4.4 is used to check the DGS-BEM for the 2D
Poisson’s equation, i.e.,
∂ 2u
= b(x, u )
∂x j ∂x j
(4.41)
The ellipse has a semi-major axis of length 2 and semi-minor axis of length 1. It is
discretized using 50 quadratic boundary elements and it has 105 internal nodes.
Initially, the accuracy of the BEM technique is demonstrated with the solution of
Laplace equation, b(x, u ) = 0 , using the non-homogeneous boundary condition,
u (ξ ) = x1 (ξ ) + x 2 (ξ )
(4.42)
for every point ξ ∈ S . Results for the solution of the Laplace equation are
summarized in Table 4.1 for some of the internal nodes. The maximum error for
this problem and the chosen discretization was 0.00933938%, which certifies BEM
as the most accurate method for linear problems, i.e. the integral representation
101
is equivalent and only the numerical error calculating the boundary integrals
affect the solution.
Table 4.1:
4.4.2
BEM results for the Laplace equation.
Node
x1
x2
BEM
Exact
Error (%)
101
102
103
160
161
162
201
202
203
204
205
1.320000
1.309673
1.279783
-0.816908
-0.748328
-0.682783
0.000000
0.500000
-0.500000
1.500000
-1.500000
0.000000
0.082398
0.161674
-0.518426
-0.543692
-0.564847
0.000000
0.000000
0.000000
0.000000
0.000000
1.319936
1.391995
1.441373
-1.335251
-1.291941
-1.247553
0.000000
0.499981
-0.499981
1.499923
-1.499923
1.320000
1.392070
1.441458
-1.335334
-1.292020
-1.247629
0.000000
0.500000
-0.500000
1.500000
-1.500000
0.0048485
0.0053877
0.0058968
0.0062157
0.0061145
0.0060916
0.0000000
0.0038000
0.0038000
0.0051333
0.0051333
Domain grid superposition BEM technique for ∇ 2 u = b
The ellipse shown in Figure 4.3 is represented by the equation,
x12 x 22
+
=1
a2 c2
(4.43)
This equation will appear in the exact solution of the Poisson’s equation defined
by,
∂ 2u ∂ 2u
+
= −2
∂x12 ∂x 22
(4.44)1
which, for example, represents the problem of Saint-Venant torsion of a member of constant
cross-section [40, 216].
1
102
with homogeneous Dirichlet boundary conditions, u (ξ ) = 0 for ξ ∈ S . The exact
solution for the scalar potential u (x ) and its normal derivative q (x ) are,
⎞
⎛ x2 x2
u (x ) = −0.8⎜⎜ 12 + 22 − 1⎟⎟
c
⎠
⎝a
q (x ) =
(4.45)
∂u ∂x1 ∂u ∂x 2
+
= −0.2 x12 + 8 x 22
∂x1 ∂n ∂x 2 ∂n
(
)
(4.46)
The results for a boundary discretization of 50 quadratic isoparametric elements
and 2809 linear 4-noded isoparametric internal cells (50-2809 mesh) are
summarized in Table 4.2, where a comparison between the dual reciprocity, the
domain grid superposition and the exact solution is presented. For this mesh and
grid the maximum error for the potential of the internal nodes was 0.54%. It can
be seen that the DRM results are better for u (x ) in this specific mesh, however
the values for q (x ) are much better with the DGS-BEM technique.
Table 4.2: BEM results for ∇ 2 u = −2 .
Variable
x1
x2
DRM
[216]
DGS
Exact
q
2.000000
1.345674
1.237740
0.000000
0.000000
0.300000
0.900000
1.500000
0.600000
0.000000
0.739791
0.785494
-1.000000
0.000000
0.000000
0.000000
0.000000
-0.450000
-0.68000
---1.5880
0.8070
0.7890
0.6430
0.3490
0.5730
-0.763183
-1.292776
-1.352725
-1.604682
0.815919
0.797567
0.650636
0.356759
0.577191
-0.800000
-1.237832
-1.293600
-1.600000
0.800000
0.782000
0.638000
0.350000
0.566000
U
103
Figure 4.5 shows a comparison between the exact and DGS-BEM solutions of
the internal nodes distributed in a small concentric ellipse; these nodes have a
constant value of the scalar potential u. Results show great accuracy for the
proposed technique in this type of problem. Figure 4.6 shows the dependence of
the internal cells in the maximum error for the internal nodes. The error
decreases quickly as the number of cells is increased, but this behavior slows
down as the number of cells is too large. It should be pointed out that a coarse
cell discretization of 500 cells results in an error of only 5%.
Figure 4.5:
Comparison between the exact and the DGS technique for ∇ 2 u = −2 .
104
Figure 4.6:
4.4.3
DGS-BEM maximum error as a function of the internal cells number.
DGS-BEM technique for ∇ 2 u = b(x, u )
Three different problems are used to check the technique for non-homogeneous
vectors that depend in the scalar potential u (x ) . Initially, Table 4.3 and Figure 4.7
show the results for the case,
∇ 2 u = −u
(4.47)
with a non-homogeneous boundary condition u (ξ ) = sin ( x1 ) for ξ ∈ S , which is a
particular solution for equation (4.47), and constitutes a solution of the problem
because of its imposition over the boundary. The DGS-BEM solution yields a
105
maximum error in the internal nodes, with the 50-2809 mesh, of a 2.2%, however
the DRM gives better results than the DGS-BEM.
Table 4.3: BEM results for ∇ 2 u = −u .
x1
x2
DRM [216]
DGS
Exact
0.000000
0.300000
0.900000
1.500000
0.600000
0.000000
0.000000
0.000000
0.000000
-0.450000
0.0000
0.2940
0.7800
0.9940
0.5620
0.000000
0.267851
0.721524
0.950846
0.562754
0.000000
0.295520
0.783327
0.997495
0.564642
Figure 4.7:
BEM results for ∇ 2 u = −u .
Table 4.4 and Figure 4.8 present the results for the convective case,
∇ 2u = −
∂u
∂x1
(4.48)
106
A particular solution for this case is u (ξ ) = exp(− x1 ) for ξ ∈ S , which, when
imposed as a boundary condition, constitutes the exact solution of the problem.
The maximum error for the internal nodes solution, with the 50-2809 mesh, is
4.6%. In this case and for the 50-2809 mesh, the DGS and DRM results are
comparable.
Table 4.4: BEM results for ∇ 2 u = − ∂u ∂x1 .
x1
x2
DRM [216]
DGS
Exact
0.000000
0.300000
0.900000
1.500000
0.600000
0.000000
0.000000
0.000000
0.000000
-0.450000
1.0110
0.7250
0.3630
0.2140
0.5230
0.979122
0.753710
0.344598
0.213345
0.438679
1.000000
0.740818
0.406570
0.223130
0.548812
Figure 4.8:
BEM results for ∇ 2 u = − ∂u ∂x1 .
Finally, Table 4.5 and Figure 4.9 show the results for the Burger’s equation,
107
∇ 2 u = −u
∂u
∂x1
(4.49)
This is a nonlinear problem which has a particular solution u (ξ ) = 2 x1 ; if
imposed as a boundary condition, it will be the exact solution for the equation
(4.49). To avoid the singularity of this particular solution at x1 = 0 , the origin of
the Cartesian system in Figure 4.3 was displaced to the point (3,0 ) . The shown
results are for the 50-2809 mesh and the maximum error for the internal nodes
was 1.7%. Here, DRM and DGS are comparable.
Table 4.5: BEM results for ∇ 2 u = −u (∂u ∂x1 ) .
x1
x2
DRM [216]
DGS
Exact
3.000000
3.300000
3.900000
4.500000
3.600000
0.000000
0.000000
0.000000
0.000000
-0.450000
0.6750
0.6140
0.5190
0.4460
0.5630
0.674501
0.605264
0.519061
0.442013
0.548928
0.666667
0.606061
0.512821
0.444444
0.555556
As shown in the previous examples, the technique has proven to give
accurate results for various combinations of Poisson’s equation. When compared
with DRM, a known approximation technique for the domain integral, the DGSBEM, gives comparable results for linear problems and simple nonlinear
problems. Additionally, it has to be pointed out that the error for the convective
case is higher than the error for the nonlinear case, due to the fact that the
derivates are more crucial for this convective case. The linear 4-noded
isoparametric cells are not an efficient nor accurate way to calculate the
derivatives. Therefore, for complex problems or problems with high order
108
nonlinearities which involve derivatives, the order of the internal cells must be
increased (quadratic or cubic elements) or the derivatives must be calculated
with more accurate approaches, such as radial basis functions [112, 113].
Figure 4.9:
4.4.4
BEM results for ∇ 2 u = −u (∂u ∂x1 ) .
DGS-BEM in three dimensional problems
In three dimensional geometries the methodology of the DGS is exactly the same
as in two dimensional geometries. However, the criterion for the selection and
rejection of the intersected cells is more difficult. Figure 4.10 shows a circular
cylinder of constant radius that can be used, for example, for the solution of a
pressure driven flow. The figure presents the boundary mesh, which can be
109
generated with in-house meshing codes or with commercial codes (i.e.
Hypermesh), as well as the set of internal nodes where the solution is sought.
The first step for the generation of the domain grid is to find the maximum and
minimum points of the domain under interest; with these points a rectangular
prism is generated between the domain limits minus some percentage, which
will indicate the gap between the boundary and internal meshes.
Figure 4.10:
Typical mesh and internal node distribution for a 3D problem.
Three domain grid parameters, which consist in the number of divisions in
every direction of the rectangular prism (div( x1 ), div( x 2 ), div( x3 )) , will state how
the internal grid will be generated. For example, in Figure 4.11 two different
110
types of domain grids, a (10,10,10), which means that the rectangular prisms
containing the domain will be divided into 10 cells in every dimension, and a
(10,5,3) are shown.
Figure 4.11:
Domain grid for: (a) (10,10,10) and (b) (10,5,3), configurations.
The rules for the domain grid generation are the same rules as for any
internal domain discretization. For example, the number of divisions in each
direction must be selected according to the nature of the problem. Specifically,
for the domain in Figure 4.10, it is known that the most important gradients are
in the x1 − x 2 plane, this plane must be divided into more cells than the x3 -
111
direction. In addition, the problem has symmetry in the same x1 − x 2 plane, thus
the internal cells must have the same number in the x1 and x 2 directions.
After the domain grid is generated for the rectangular prism, the attrition of
nodes and cells begins. It is a geometric methodology which consists in checking
if every node in the domain grid is included or not inside the domain of interest.
Because it is a three dimensional geometry, three different planes must be
checked, i.e. x1 − x 2 − , x1 − x3 − and x 2 − x3 − planes. The idea is to integrate the
angle that is formed between the domain grid point and all the points in the
boundary. If the angle is 0 the point is excluded, and if the angle is 2π the point
is included (see Figure 4.12).
S
x2
θ2,k
x1
xj
θ1,k
n
xi
Figure 4.12:
Schematic of the angle integration for the attrition of nodes.
Three angles have to be calculated for each plane, i.e.,
(ξ − x l ) ⋅ (ξ − x l ) ⎟
θ l ,k = ⎢ ∫ cos −1 ⎜⎜ ref
⎟dS ⎥
−
−
ξ
x
ξ
x
l
l
⎝ ref
⎠ ⎦⎥
⎣⎢ξ
⎡
ζ
ref
⎛
⎞
⎤
(4.50)
xi − x j − plane
112
for every point l in the domain grid, with ξ ∈ S and ξ ref a reference point in the
surface from where the integration begins. For a point to be in the intersection
between the domain grid and the problem geometry it must satisfy,
θ l ,1 = θ l , 2 = θ l ,3 = 2π
(4.51)
Figure 4.12 shows schematically how, in the xi − x j − plane, the angle θ l ,k is
calculated for point x 1 and x 2 . After the integration is done for every point in the
boundary, the angle for point x 1 will be 2π , which means that the point is inside
and must be included in the plane. However, for point x 2 the angle will be 0,
which means that it falls outside and it must be excluded from the analysis.
Figure 4.13 illustrates the domain grid after the attrition of nodes for the two grid
configurations of Figure 4.12, (10,10,10) and (10,5,3). For geometries with solid
inclusions, internal interfaces (Couette flow), the attrition criterion is inverted for
the internal surfaces. In other words, all the angles, θ l ,k , must be 0 for the node to
be included in the intersection.
The cell integration will use isoparametric quadratic 20-noded elements (see
Figure 4.14) to directly approximate the domain integral or the hydrodynamic
volume potential in the integral formulation, i.e.,
∫ u (x , x )g (x )dΩ
j
i
Ω
0
j
(4.52)
113
Figure 4.13:
Superimposed domain grid and final mesh structure for: (a) (10,10,10)
and (b) (10,5,3), configurations.
ξ3
ξ2
ξ1
20 nodes
Figure 4.14:
Isoparametric 20-noded prism for the cell integration.
114
The shape or interpolation functions for the 20-noded isoparametric element
are similar to the shape functions for the 2D 8-noded isoparametric element [85,
147, 336, 337]. For the corner nodes they are defined as,
Ni =
1
(1 + ξ10 )(1 + ξ 20 )(1 + ξ 30 )(ξ10 + ξ 20 + ξ 30 − 2)
8
(4.53)
where the new variables are defined as,
ξ10 = ξ1ξ1,i
ξ 20 = ξ 2ξ 2,i
ξ 30 = ξ 3ξ 3,i
(4.54)
while for a typical mid-side node are,
ξ1,i = 0,
Ni =
ξ 2,i = ±1,
ξ 3,i = ±1,
1
(
1 − ξ12 )(1 + ξ 20 )(1 + ξ 30 )
4
(4.55)
When ξ 3 = ξ 30 = 1 the above expressions reduce to the shape functions for the 2D
8-noded isoparametric element (equations (4.5) to (4.7)). For every cell the
integration is carried out in the new curvilinear isoparametric variables. Similar
to the hydrodynamic potentials of single- and double-layer the volume element
must be affected by the coordinate transformation,
dΩ = dx1 dx 2 dx3 = J dξ1 dξ 2 dξ 3
(4.56)
where J is the magnitude of the Jacobian matrix, which is defined in the
coordinate transformation as,
115
⎧ ∂N i ⎫ ⎡ ∂x1
⎪ ⎢
⎪
⎪ ∂ξ1 ⎪ ⎢ ∂ξ1
⎪ ∂N i ⎪ ⎢ ∂x1
⎬=⎢
⎨
⎪ ∂ξ 2 ⎪ ⎢ ∂ξ 2
⎪ ∂N i ⎪ ⎢ ∂x1
⎪ ∂ξ ⎪ ⎢ ∂ξ
⎩ 3⎭ ⎣ 3
⎧ ∂N i ⎫
∂x3 ⎤ ⎧ ∂N i ⎫
⎪
⎪
⎪
⎥⎪
∂ξ1 ⎥ ⎪ ∂x1 ⎪
⎪ ∂x1 ⎪
∂x3 ⎥ ⎪ ∂N i ⎪
⎪ ∂N i ⎪
⎬
⎬ = J⎨
⎨
⎥
∂x 2 ⎪
∂ξ 2 ⎪ ∂x 2 ⎪
⎪
∂x3 ⎥⎥ ⎪ ∂N i ⎪
⎪ ∂N i ⎪
⎪ ∂x ⎪
⎪
⎪
∂ξ 3 ⎥⎦ ⎩ ∂x3 ⎭
⎩ 3⎭
∂x 2
∂ξ1
∂x 2
∂ξ 2
∂x 2
∂ξ 3
(4.57)
The magnitude of the Jacobian matrix can be calculated from,
⎛ ∂x ∂x
×
J = ⎜⎜
∂
ξ
⎝ 1 ∂ξ 2
⎞ ∂x
⎟⎟ ⋅
⎠ ∂ξ 3
(4.58)
with this transformation the integral in equation (4.52) can be evaluated with
regular Gaussian quadratures described previously in this chapter. In addition,
the integration is done in the internal cells previously selected. However, the
integrated cell volume is less than the real geometry volume; therefore a
correction must be made to be consistent with the problem’s parameters,
∫ u (x , x )g (x )dΩ
j
∫ u (x , x )g (x )dΩ ≈
j
i
Ω
0
j
i
0
j
Ω cells
∫ dΩ
× Ω domain
(4.59)
Ω cells
4.5 Iteration scheme for non-Newtonian flow problems
Since the integral representation for the momentum equations of a shear-rate
dependent viscosity fluid is nonlinear, an iterative solution must be used. The
algorithm is schematically shown in Figures 4.15 and 4.16. Figure 4.15 illustrates
the pre-processing and solution, while Figure 4.16 shows the iterative process.
116
START
Geometry, physical data
and boundary conditions
Domain grid superposition
BEM for boundary
and internal nodes
H, G
u*, t*
BEM for domain
grid
Hg, Gg
Newtonian solution
Iterative process (Figure 4.15)
END
Figure 4.15:
Schematic of the solution methodology.
The solution starts by reading the domain geometry, as well as the physical
and numerical data. The domain grid superposition is then applied to the
geometry. The BEM equations are calculated for the boundary and internal
nodes ( H, G ), as well as for the grid domain ( H g , G g ). The initial guess for the
solution is then found from the Newtonian problem for the boundary and
internal nodes.
117
H, G
u*, t*
Hg, Gg
Domain grid velocity
solution
ug
Temporal non-Newtonian viscosity
and extra stress for domain grid
τ (e)
g
Domain integral for domain grid
and new domain grid velocity
ug
Non-Newtonian viscosity and
extra stress for domain grid
τ (e)
g
Domain integral for boundary,
internal nodes and grid
g
Solution for boundary
and internal nodes
u, t
u*=u
t*=t
Convergence?
YES
Figure 4.16:
Schematic of the iterative scheme.
The iteration starts by finding a temporary velocity field for the domain grid
using the guess for the velocity and tractions over the boundary. This temporary
velocity field is used to find a non-Newtonian viscosity, extra stress and domain
118
integral term for the domain grid. This domain term will be used to calculate a
new velocity field for the domain grid. Thus, an updated non-Newtonian
viscosity, extra stress tensor and domain integrals for both, the boundary and
grid domain, are obtained. These domain terms are used to find an updated
velocity and traction field for the boundary and internal nodes. A converge
criterion is used to verify if the guessed velocity and traction fields are equal to
the updated ones. When the guessed and updated fields are different, relaxation
for the iteration can be used, i.e.,
u* = ωu * +(1 − ω )u
t* = ωt * +(1 − ω )t
(4.60)
where ω ∈ [0,1] is a relaxation parameter. For the scheme shown in Figure 4.16
the relaxation parameter was 1.
In addition to relaxation techniques for the iteration process, power law
indexes or shear-thinning exponents affect the speed of convergence. Lower
exponents, i.e. n < 0.5 , need more iterations and computational time. A way to
decrease the number of iterations and computational time is to use better guesses
for the velocity and traction fields. This can be done by relaxing the power law
exponent: starting with a nearly Newtonian power law exponent ( n = 0.95 ) the
solution is obtained from the Newtonian guesses fields. Then, the power law
exponent is decreased by an interval ( dn = 0.05 ) and the solution is sought from
the previous non-Newtonian solution. This is repeated until the target power law
exponent has been reached.
119
Chapter 5
Non-Newtonian fluid flow
problems
In this chapter two different non-Newtonian problems are solved using the
mapped boundary element technique: a Poiseuille flow and a Couette flow. The
analytical solutions for a power law fluid are compared with DGS-BEM. These
are used to asses the accuracy of the technique.
5.1 Poiseuille flow of a Power Law fluid
Poiseuille flow is a pressure gradient driven flow, with application primarily to
tubes and pipes (see Figure 5.1). They are named after J.L.M. Poiseuille who
experimented with low-speed flow in tubes [229].
Ignoring entrance effects in the tube, the analytical solution of this type of
flow can be obtained by integrating the momentum in the radial direction with a
120
varying viscosity given by the Power Law model [26, 28]. This solution can be
expressed as,
s
s +1
⎛τ R ⎞ R ⎡ ⎛ r ⎞ ⎤
u (r ) = ⎜ ⎟
⎢1 − ⎜ ⎟ ⎥
⎝ m ⎠ s + 1 ⎣⎢ ⎝ R ⎠ ⎦⎥
(5.1)
where s = 1 n and the shear stress at the tube wall, τ R , is given by
τR =
( p0 − p L )
2L
(5.2)
R
In the equations n is the power law index, m the consistency index, p 0 the
pressure at z=0, p L the pressure at z=L, L the tube length and R the tube radius.
As mentioned in Chapter 2, n=1 corresponds to a Newtonian fluid and for most
polymeric liquids 0.2 < n < 0.6 .
r
R
z
L
z=0
p0
Figure 5.1:
Poiseuille flow in a circular tube.
z=L
pL
121
For the simulation, the physical dimensions of the geometry have been set to
1.0 length units length and a diameter of 0.5 length units. The selected power law
fluid had m = 1000.0 force-timen/area units, with n varying from 1.0 to 0.2. The
pressure drop was ∆p = 100.0 force/area units. The accuracy for the approximation
of the surface integrals depends in the number of divisions in r and z, and the
number of gauss points (NGP). The solution of the linear and non-linear problem
will be affected by these parameters. Figure 5.2 illustrates two different surface
meshes used: one with 7 r-divisions and 5 z-divisions (320 elements with 962
nodes) and one with 10 r-divisions and 8 z-divisions (638 elements with 1916
nodes). The figure also shows the internal node distribution.
Figure 5.3 shows the Newtonian solution ( n = 1 ) for the 7-5 mesh and 10 NGP,
where the z-velocity is normalized with the theoretical maximum velocity value.
The maximum error for this solution was 0.25%. The influence of the NGP in the
solution is shown in Figure 5.4, according to the figure the optimal NGP for the
7-5 mesh is 15, while 10 NGP for a 10-5 mesh. The correct selection of the
divisions (elements and nodes) and the NGP will later affect the accuracy of the
non-linear solutions. The NGP not only influences the accuracy but also
determines the computational time for the BEM matrices calculation.
122
Figure 5.2:
A coarse and finer surface mesh for the Poiseuille pipe flow.
123
Figure 5.3:
Poiseuille flow of a Newtonian fluid with a 7-5 mesh.
For the DGS technique, i.e. non-linear problems ( n < 1 ), the number of
divisions for the internal grid and the NGP for the cell integration will also affect
the solution, being more significant the dependence on the grid divisions. Figure
5.5 shows two different internal meshes: a (15,15,5) mesh (715 cells and 3810
nodes) and a (20,20,8) mesh (2272 cells and 10997 nodes). Increasing the NGP
will result in a longer computational time, whereas the accuracy is not increased
in a notorious way. The number of internal divisions will affect directly the
calculation of the domain integral, thus affecting the accuracy of the solution.
There are two main reasons for the effect of the internal cells in the solution: first,
they are used to calculate the internal derivatives of the problem (velocity and
124
extra stress tensor), and second, they are used in the cell integration for the
approximation of the domain integral.
Figure 5.4:
Influence of the NGP in the Newtonian solution with the 7-5 mesh.
125
Figure 5.5:
DGS meshes for Poiseuille flow: (15,15,5) and (20,20,8).
126
Figure 5.6 presents the results for the Poiseuille flow of a power law fluid,
using the 7-5 mesh for the surface and the (15,15,5) DGS mesh, the numerical
performance (error and iterations) is shown in Figure 5.7. For this DGS mesh the
technique gives accurate results down to a power law index of 0.7, i.e. the error is
less than 10%. For n = 0.6 , the solution converged to a solution with poor
accuracy. For lower power law indexes the technique fails to approximate the
derivatives and the domain integral, given results that does not agrees in value
nor trend. For the (15,15,5) DGS mesh the pre-processing computational time
(BEM matrices for the surface and DGS cells) was of about 2 hours and every
iteration took approximately 1 hour.
Figure 5.6:
Poiseuille flow of a non-Newtonian fluid with a (15,15,5) DGS mesh.
127
Figure 5.7:
Numerical performance for the (15,15,5) DGS mesh in the Poiseuille flow.
128
Figure 5.8 shows results for lower power law indexes using a (20,20,8) DGS
mesh and Figure 5.9 shows the numerical performance. The pressure drop was
increased to ∆p = 5000.0 force/area units to increase the maximum velocity value.
The results present high degree of accuracy, i.e. error less than 5%. However,
these simulations were extremely expensive from a computational point of view;
they took 12 hours of pre-processing and 3 hours per iteration. The whole
iteration process takes 9 hours for n = 0.9 and 96 hours (4 days) for n = 0.2 .
Figure 5.8:
Poiseuille flow of a non-Newtonian fluid with a (20,20,8) DGS mesh.
129
Figure 5.9:
Numerical performance for the (20,20,8) DGS mesh in the Poiseuille flow.
130
5.2 Couette flow of a Power Law fluid
Consider the steady flow maintained between two concentric cylinders by steady
angular velocity of one or both cylinders. These flows are named after M.
Couette [65], who performed experiments on the flow between fixed and moving
concentric cylinders. This problem is of special interest because the high velocity
gradient that appears near the rotor, particularly for inelastic non-Newtonian
fluids.
ω
R2
R1
Figure 5.10:
Schematic of the Couette flow problem.
The velocity field for the Couette flow problem (Figure 5.10) is onedimensional in the radial direction and an analytical solution can be obtained by
integrating the radial component of the momentum equations in cylindrical
coordinates [7, 28]. For a Power Law fluid the expression for the radial velocity is
given by,
u (r ) =
(R2
ω
R1 )
2n
⎡ R22 n − r 2 n ⎤
⎢
⎥
2 n −1
− 1 ⎣⎢ r
⎦⎥
(5.3)
131
where r is the radial coordinate from the rotation center.
Figure 5.11:
Surface mesh for the Couette flow and internal points.
132
For the simulation, the physical dimensions of the geometry have been set to
2.0 length units length, an outer diameter of 1.0 length units and an inner diameter
of 0.5 length units. The selected power law fluid had m = 1000.0 force-timen/area
units, with n varying from 1.0 to 0.2. The inner cylinder angular velocity was
ω = 1.0 . Two surface meshes are shown in Figure 5.11: a 3-5 (432 elements with
1320 nodes) and a 5-8 (1120 elements with 3396 nodes). In Figure 5.12 a (15,15,5)
(440 cells with 2840 nodes) and a (20,20,8) (1440 cells with 7964 nodes) are shown.
Figure 5.13 illustrates the Newtonian solution with a 3-5 mesh (432 elements
and 1320 nodes) and 10 NGP. There are two important issues to be noticed: first,
the accuracy of the BEM for the linear case, i.e. error less than 2.0%, and the
symmetry given by the BEM solution, which can be noticed by the shape of the
error in both sides of the Couette geometry. The non-Newtonian fluid solution
for power law indexes from 0.9 to 0.6 is shown in Figure 5.14. The DGS mesh for
these results was (15,15,5), the pre-processing time was about 1 hour and each
iteration took 0.5 hours approximately. The numerical performance is shown in
Figure 5.15 for these simulations. Similar to the Poiseuille flow, the DGS mesh
was unable to obtain results for power law indexes lower than 0.6. In fact,
although the n = 0.6 solution is satisfactory (error less than 10%), the error starts
to oscillate between the internal points; this phenomena is increased for lower
power law indexes.
133
Figure 5.12:
DGS meshes for Couette flow: (15,15,5) and (20,20,8).
134
Figure 5.13:
Newtonian solution for the Couette flow.
135
Figure 5.14:
Couette flow of a non-Newtonian fluid with a (15,15,5) DGS mesh.
136
Figure 5.15:
Numerical performance for the (15,15,5) DGS mesh in the Couette flow.
In order to obtain accurate results for lower power law indexes, a DGS mesh
of (20,20,8) was used. Again, these simulations were computationally expensive,
however, the results were satisfactory as presented in Figures 5.16 and 5.17. The
pre-processing (BEM matrices for boundary and DGS grid) time for these
simulations was 10 hours and 4 hours per iteration. The iteration process took 12
hours for n = 0.9 and 164 hours (7 days) for n = 0.2 .
137
Figure 5.16:
Couette flow for a non-Newtonian fluid with a (20,20,8) DGS mesh.
138
Figure 5.17:
Numerical performance for the (20,20,8) DGS mesh in the Couette flow.
139
Chapter 6
Viscous flows containing
particles
In this chapter the interaction between particles of simple shapes, like spheres, is
studied using the direct boundary integral formulation exposed in Chapter 3.
Initially, the flow field over a sphere is solved. The surface tractions are
integrated in order to find the drag force and this value is compared with Stokes’
analytical solution. Interaction between spheres is analyzed and the force is
compared with Faxen’s laws. The effects of a rigid wall are also studied. The
effective viscosity of a fluid with suspended spheres is calculated and compared
with Einstein’s formula and the experimental model developed by Guth and
Simha [121]. The final part of the chapter is focused in particles of cylindrical
shape in shear flow. Jeffery’s orbits are predicted and compared with the
analytical solution.
140
6.1 Stokes’ law: drag on a sphere
The drag on a sphere of radius R moving steadily with velocity u 0 through an
unbounded fluid is, by Stokes’ law [294], which derives from the pure viscous
momentum equations (Stokes equations),
FStokes = 6πµRu 0
(6.1)
Although the Stokes’ law is strictly valid for Re << 1 , it agrees with experiment
up to about Re = 1 [258]. The drag coefficient in the sphere is defined as
CD =
2F
πρR 2 u 02
(6.2)
Several cases corresponding to different values of the Reynolds number were
solved using the direct boundary integral formulation. Figure 6.1 shows a
schematic diagram of the domain with a simple mesh. The solution was
performed using a bounded domain with periodic boundary conditions in a way
that the external walls do not affect the solution. There are two different, but
equivalent, ways to set up the boundary conditions: zero inlet fluid velocity with
constant sphere velocity, or constant inlet fluid velocity with zero sphere
velocity. According to the boundary integral representation defined in Chapter 3,
with all known velocities the system is reduced to a Fredholm integral equation
of the first kind for the unknown surface tractions (equation (3.70), a simple
extension of Youngren and Acrivos’ equation [333, 334]).
141
Figure 6.1:
Schematic of the domain and mesh.
The surface tractions must be integrated in order to obtain the forces and
torques on the sphere as follows,
FBEM = ∫ tdS
(6.3)
TBEM = ∫ x × tdS
(6.4)
S
and
S
Figure 6.2 is a plot of the drag coefficient versus the sphere Reynolds number,
based on the Stokes’ law, the direct boundary integral formulation and actual
experimental data [110, 221]. The physical dimensions of the box have been set to
50.0 length units length and the sphere radius to 0.2 length units. The fluid
142
viscosity is 1000.0 force-time/area units and a density of 1000.0 mass/volume units.
The box was divided into 96 quadratic elements (8 noded) with 290 nodes, while
the sphere into 216 elements with 650 nodes. The error between the predicted
drag force and Stokes’ law is defined by,
% Error =
FBEM − FStokes
FStokes
100
(6.5)
As expected, the error depends directly on the mesh but it is totally independent
of the Reynolds number. In particular case of Figure 6.2 the error was 1.0628%. In
the problem of determining the low Reynolds number incompressible viscous
flow due to the motion of a solid particle, with boundary surface S of the
Lyapunov type, and with a no-slip boundary condition on S, i.e.,
u(x 0 ) = u 0
(6.6)
for all x 0 ∈ S , the velocity field must decay at infinity in as,
( )
u(x ) = 0 ≈ r −1
(6.7)
where r is the distance from the particle to a point x. This will give an additional
way to check the BEM solution; the velocity field must decay at least as r −1 for all
Reynolds numbers. Figure 6.3 illustrates the normalized z-component of the
velocity field as a function of the normalized distance from the sphere for the
BEM solution. The dotted line is the r −1 function. As shown the velocity follows
the decay as r → ∞ .
143
Figure 6.2:
Comparison between the drag on a sphere from Stokes’ law, analytical
and BEM, and experimental data.
144
Figure 6.3:
Normalized z-velocity as a function of the distance from the sphere.
145
6.2 Wall effects on the motion of a single particle
The interaction of a particle with walls will depend on the particle shape,
orientation, and position, as well as the geometry of the containing walls. There
are some approximate solutions that can be used to compare the BEM results.
The effect of containing walls on a particle is important because it is similar to
the effect of secondary particles which will be analyzed in the next section.
6.2.1
Sphere moving parallel to a plane wall
The motion of a sphere parallel to an external plane wall (Figure 6.4) was treated
by Faxen [91-93]. Extensions of the theory to non-spherical bodies, and to shear
and parabolic flows, have been developed by generalization of Faxen’s original
technique [127]. For the case of a sphere and a single plane wall Faxen obtained
[94, 127],
FFaxen =
− δ 2 (6πµRu 0 )
1 − (9 16)(R L ) + (1 8)(R L ) − (45 256)(R L ) − (1 16 )(R L )
3
4
5
(6.8)
In 1964, O’neill [207] solved exactly the translational motion of a sphere
parallel to a wall using a general bipolar coordinate solution of the Stokes’
system of equations employed by Stimson and Jeffery [290]. Goldman et al. [114]
computed the force due to the parallel motion numerically using O’Neill’s series
solution. Figure 6.5 shows a comparison between the dimensionless force, as a
function of the distance, on the sphere by Faxen (equation (6.8)), Goldman et al.
146
solution and the BEM simulation. Faxen’s approximate solution agrees with
Goldman et al. exact solution when L R > 1 , the BEM results show good
agreement with the exact solution.
L
x2
u0
x1
Figure 6.4:
Sphere settling near a plane wall.
If the sphere is free to rotate it will do so in the same direction as if it were
rolling along the wall [127], Figure 6.6 illustrates the torque on the sphere as a
function of the distance to the wall.
147
Figure 6.5:
Drag force for the case of a sphere moving parallel to a plane wall.
148
Figure 6.6:
6.2.2
Torque for the sphere moving parallel to the wall.
Sphere moving perpendicular to a plane wall
The motion of a spherical particle toward or away form a rigid single plane
surface, which serves as the bottom of the container, has been treated analytically
by Lorentz [187], Brenner [42] and Wakiya [318-321]. Lorentz [187] solved for the
case where the sphere diameter is small compared with the distance to the plane,
Brenner [42] solved the problem without Lorentz’ restriction using the general
bipolar coordinate solution of the Stokes’ system of equations employed by
Stimson and Jeffery [290]. The force on the sphere can be written as,
F = (6πµRu 0 )χ (R L )
(6.9)
149
where χ (R L ) is a correction to Stokes’ law, which according to Brenner is given
by the expression [42, 127],
n(n + 1)
n =1 (2n − 1)(2n + 3)
∞
4
3
χ = sinh α ∑
(6.10)
⎡ 2 sinh (2n + 1)α + (2n + 1)sinh 2α
⎤
− 1⎥
⎢
2
2
2
⎣ 4 sinh (n + 1 2 )α − (2n + 1) sinh α ⎦
Here, the parameter α = cosh −1 (L R ) . Experimental confirmation of this
correction has been reported by MacKay, Suzuki and Mason [188]. Using the
method of reflections Wakiya found an approximation up to O(R L ) given by
3
[318-321],
FWakiya =
Figure
6.7
illustrates
a
6πµRu 0
(6.11)
1 − (9 8)(R L ) + (1 2)(R L )
3
comparison
between
the
correction
factor
(dimensionless force) of Brenner and Wakiya with the calculated BEM solution.
The results show that Wakiya’s approximate solution agrees very closely with
the values computed by Brenner when the dimensionless distance L R > 1 . On
the other hand the BEM simulation does not have this restriction.
150
Figure 6.7:
Drag force for the case of a sphere moving perpendicularly to a plane
wall.
151
6.2.3
Sphere moving axially in a cylindrical tube
The motion of a single spherical particle parallel to the longitudinal axis of a long
cylinder through a viscous fluid (Figure 6.8) has been the subject of many studies
[30, 97, 122-127].
z-direction
Radius: R
b
u0
R0
Figure 6.8:
Spherical particle in a circular cylinder.
When the sphere is at the cylinder axis, b = 0 , Bohlin using an extension to
the method of reflections developed by Faxen [91-96], carried an approximation
for the correction factor, χ (R R0 ) , of Stokes’ law as follows [30],
χ Bohlin
⎡1 − 2.10443(R R0 ) + 2.08877(R R0 )3 − 0.94813(R R0 )5
=⎢
6
8
10
⎢⎣1.372(R R0 ) + 3.87(R R0 ) − 4.19(R R0 ) + ...
−⎤
⎥
⎥⎦
−1
(6.12)
Haberman and Sayre considered the problem and they employed general
solutions of the Stokes’s system of equations in terms of the stream function and
152
they found an infinite set of linear algebraic equations for evaluating the
coefficients appearing in the expansion of the stream function [122-125, 127].
Keeping only two equations of the infinite set, the correction factor by Haberman
is [122-125],
χ Haberman =
1 − 0.75857(R R0 )
5
1 − 2.1050(R R0 ) + 2.0865(R R0 ) + 0.72603(R R0 )
3
6
(6.13)
According to Fidleris and Whitmore [98] experimental data equations (6.12)
and (6.13) give good numerical values up to
(R
R0 ) = 0.6 . For the BEM
simulation the physical dimensions of the cylinder have been set to 100.0 length
units length and the radius to 1.0 length units. The fluid viscosity is 1000.0 forcetime/area units and the sphere velocity is 1.0 length/time units. Figure 6.9 gives a
comparison of the values obtained by Haberman’s approximate and exact
solutions and Bohlin approximate solution with the values obtained with the
boundary integral formulation when the cylinder was divided into 96 quadratic
elements (8 noded) with 290 nodes and the sphere into 216 elements with 650
nodes. According to Figure 6.9 the boundary integral solution, with this mesh,
gives accurate results for radius ratios below 0.5. However, the BEM solution,
with this mesh, is comparable to Haberman’s approximate solution (equation
(6.13)).
153
Figure 6.9:
Dimensionless force for a rigid sphere moving axially in a cylindrical
tube.
Figure 6.10 shows the error of the BEM solution when compared with
Hamerban’s approximate solution as a function of the radius ratio. As expected
in this type of problems the number of elements is an important issue for a
precise solution. Different meshes were tested for the most extreme case,
R R0 = 0.8 , the results are shown in Figure 6.11, Haberman’s approximate and
exact solutions are shown as a comparison.
154
Figure 6.10:
Error between the BEM solution with 96-290 element mesh and
Haberman’s approximate solution.
According to Figure 6.11 the value of the correction factor will depend in the
domain discretization or mesh. For poor meshes the value although different to
the exact solution will be close to Haberman’s approximate value. If the number
of elements is increased the BEM solution approaches to the exact solution. The
practical difference between the solutions, for the different meshes, is the
computational time; being a couple of minutes for the 54-54 mesh while a couple
of hours for the case of 600-384. In practical terms the selection of the mesh, for
problems with big difference between Haberman’s solutions, will be determined
155
by the balance between the time needed for the calculation, the desired accuracy
and the ratio between the solid inclusion and domain characteristic size.
Figure 6.11:
Correction factor from BEM for different meshes.
For the case when the eccentricity is different from zero, b ≠ 0 , Happel and
Brenner used the method of reflections and found the following expressions for
the force and torque shown in dimensionless form [127],
χ=
F
= 1+
6πµRu 0
⎛ b
f ⎜⎜
⎝ R0
⎞⎛ R
⎟⎟⎜⎜
⎠⎝ R0
⎛ b ⎞⎛ R
T
= g ⎜⎜ ⎟⎟⎜⎜
ς=
2
8πµR u 0
⎝ R0 ⎠⎝ R0
⎞
⎟⎟ + ...
⎠
(6.14)
⎞
⎟⎟ + ...
⎠
(6.15)
2
156
where the functions f (b R0 ) and g (b R0 ) can be evaluated as a power series of
the dimensionless eccentricity, (b R0 ) . For small values of the eccentricity, they
can be obtained by numerical integration [127],
⎛ b
f ⎜⎜
⎝ R0
⎛ b
⎞
⎟⎟ = 2.10444 − 0.6977⎜⎜
⎝ R0
⎠
⎛ b
g ⎜⎜
⎝ R0
⎛ b
⎞
⎟⎟ = 1.296⎜⎜
⎝ R0
⎠
2
⎛ b ⎞
⎞
⎟⎟ + O⎜⎜ ⎟⎟
⎝ R0 ⎠
⎠
⎛ b
⎞
⎟⎟ + O⎜⎜
⎝ R0
⎠
⎞
⎟⎟
⎠
4
(6.16)
3
(6.17)
A cylinder divided into 208 eight noded quadratic elements with a sphere
discretizied in 294 elements (208-294 mesh) was used in order to compare the
force and torque obtained by the integral equation to Happel and Brenner’s
approximate solution. A ratio between the sphere and cylinder radius was
R R0 = 0.6 . Figures 6.12 and 6.13 show the comparison between the BEM and the
approximate solutions given in equations (6.14) and (6.15). As expected the
solutions are comparable when the eccentricity factor is small, b R0 → 0 .
According to the BEM solution both, the force and torque, have an asymptotic
behavior when the sphere is close to the walls.
157
Figure 6.12:
Dimensionless force for BEM and Happel and Brenner’s approximate
solutions as a function of the eccentricity factor.
Figure 6.13:
Dimensionless torque force for BEM and Happel and Brenner’s
approximate solutions as a function of the eccentricity factor.
158
6.2.4
Motion of a suspended rigid fiber
The motion of ellipsoids in uniform, viscous shear flow in a Newtonian fluid was
analyzed by Jeffery [152-153] in 1922. For a prolate spheroid of aspect ratio a r
(defined as the ratio between the mayor axis and the minor axis) in simple shear
flow, u ∞ = (x3γ&,0,0) , the angular motion of the spheroid is described by [152-153],
tan θ =
K J ar
a cos 2 φ + sin 2 φ
2
r
(6.18)
and
⎛
t
tan φ = a r tan ⎜⎜ 2π
⎝ TJ
⎞
⎟⎟
⎠
(6.19)
where θ is the angle between the fiber’s major axis and the vorticity axis, i.e. x 2
axis, φ is the angle between the x3 axis and the x1 − x3 projection of the fiber axis
(see Figure 6.14), TJ is the orbit period,
TJ =
2π
γ&
⎛
1 ⎞
⎜⎜ a r + ⎟⎟
ar ⎠
⎝
(6.20)
and K J is the orbit constant, determined by the initial orientation by,
K J = tan θ 0
sin 2 φ 0
cos φ 0 +
ar
2
(6.21)
These equations predict that the spheroid will repeatedly rotate through the
same orbit, the particle will not migrate across the streamline, and that the orbit
period is independent of the initial orientation.
159
x3
φ
x2
u∞
θ
x1
Figure 6.14:
Prolate spheroid in shear flow.
The direct boundary integral formulation was implemented for the motion of
a single rigid cylindrical fiber in simple shear. To avoid discontinuities in the
normal vector (according to the Lyapunov surface definition) semi-spheres of the
same cylinder radius were used to cap cylinder, as schematically shown in
Figure 6.15. The aspect ratio for the fiber is redefined as,
ar =
L+D
D
(6.22)
The simulation of the fiber motion differs from the previous sphere
simulations in the sense that the fiber is suspended in the liquid, which means
that due to small time scales given by the pure viscous nature of the flow, the
hydrodynamic force and torque on the particle are identically or approximately
zero [127, 249]. Numerically, this means that the velocity and traction fields on
160
the particle are unknown, which differs from the previous simulations where the
velocity field was fixed and the integral equations were reduced to equations of
the first kind for the traction unknowns. Although expensive in the
computational point direct integral formulations are an effective way to find the
velocity and traction fields for suspended particles with a simple iterative
procedure, where the tractions are assumed initially and then corrected until the
hydrodynamic force and torque are zero. In addition, non-uniqueness of
solutions does not have to be considered contrary to indirect integral
formulations. As pointed out by different authors the major inconvenient of
direct particle simulation is the computational time once the number of particles
is considerable, however the rapid evolution of digital computers will soon
mitigate this problem [35, 36, 254].
For the BEM simulations the fiber length was set to 2.0 length units, the
diameter to 0.2 length units and the shear rate to 2.0 reciprocal time units. These
data implies an aspect ratio a r = 11 and an orbit period TJ = 34.843 time units.
Figure 6.16 shows the evolution of θ and φ in time for a fiber initially
perpendicular to the vorticity axis, i.e. θ 0 = π 2 . This simulation requires a large
number of elements on the fiber surface (500 elements with 1200 nodes) and a
small time step (0.01 time units); it is computationally expensive (10 minutes per
time step) but it agrees with Jeffery’s prediction. The path of the fiber during the
simulation is illustrated in Figure 6.17. While Figure 6.18 shows the
161
hydrodynamic force and torque on the fiber as a function of time, as it is shown
they are practically zero through the simulation, which confirms that the fiber is
suspended in the flow.
Figure 6.15:
Fiber representation for the BEM simulation.
162
Figure 6.16:
BEM and Jeffery orientation angles for θ 0 = π 2 .
163
Figure 6.17:
BEM predicted fiber path for θ 0 = π 2 .
164
Figure 6.18:
Hydrodynamic force and torque on the fiber during the simulation.
165
Figures 6.19 to 6.24 show the evolution of the orientation angles as a function
of time and the fiber path for θ 0 = π 3 (60o), θ 0 = π 6 (30o) and θ 0 = π 36 (5o),
respectevely. In order to follow the fiber through the rotation, the time step for
these simulations was 0.001 time units, a bigger time step can deviate the fiber
from its orbit. In fact, this issue is what causes the simulations to be so expensive.
However, the direct simulation of the fiber motion is significant for multiple fiber
suspensions and for fiber reinforced polymer melts. The performance BEM
technique for two different aspect ratios is shown in Figure 6.25, where fibers
where initially placed under shear with an orientation θ 0 = π 2 (90o). As the orbit
period increases the time step can be increased, however for a bigger the aspect
ratio the number of surface elements must also increased, thus incrementing the
computational time.
166
Figure 6.19:
167
Figure 6.20:
168
Figure 6.21:
169
Figure 6.22:
170
Figure 6.23:
171
Figure 6.24:
172
Figure 6.25:
BEM and Jeffery orientation angle for θ 0 = π 2 and two aspect ratios: (a)
a r = 51 and (b) a r = 101 .
173
6.3 Particle-particle interactions
The hydrodynamic interaction between particles of simple shapes, such as
spheres, spheroids, etc., has been the subject of many studies. Most of them have
been based on the Stokes equation, giving a first-order approximation for the
interaction of particles that are close to one another and move with small relative
velocity in a fluid at rest. In addition, it is assumed that the particles are
sufficiently distant from boundary walls for the surrounding fluid to be regarded
as unbounded. The magnitude of the interaction among the particles is governed
by their shapes and sizes, the distance between them, their orientations with
respect each other, the individual orientation relative to the direction of a specific
field and their velocities relative to the fluid at infinity [127, 246].
6.3.1
Two falling rigid spheres
In 1911, Smoluchowski used the method of reflections to solve the interaction
between a systems of spheres suspended in a viscous fluid. The method amounts
to seek a systematic scheme of successive approximations, whereby the
boundary-value problem can be solved to any degree of approximation by
considering the boundary conditions associated to each particle at a time [282284]. In order to satisfy the boundary conditions for each body a general solution
is necessary making the technique cumbersome for bodies of arbitrary shape
[246]. As a consequence, the method has been applied only for the case of two
spheres in a specific position relative to each other: say two spheres moving
174
along the line of centers (Figure 6.26a) and two spheres moving perpendicular to
the line of centers (Figure 6.26b). These problems have been the subject of many
studies; Happel and Brenner give a good literature review on this topic [127].
For two equal-sized spheres moving along the line of centers (Figure 6.26a)
the drag exerted on the leading sphere for (R L ) < 1 is given by [246, 282-284],
F
6πµR
(
)
= u1 1 + 9(R L ) + 93(R L ) + 1197(R L ) + 19821(R L ) −
2
(
4
6
8
) (
u 2 3(R L ) + 19(R L ) + 387(R L ) + 5331(R L ) + 76115(R L ) + O (R L )
3
5
7
9
10
)
(6.23)
where R is the radius of the sphere, u1 is the magnitude of the velocity of the
leading sphere, u 2 is the magnitude of the velocity of the trailing sphere. The
drag on the trailing sphere is obtained by exchanging the velocities in equation
(6.23).
Figure 6.27 illustrates a comparison between the normalized force for the leading
and trailing spheres computed using equation (6.23) and the direct BEM. The
force is normalized with the Stokes drag force of the leading sphere (sub-index
1). The velocity of the leading sphere was chosen to be two times the value of the
trailing sphere. The conditions for the simulation were setup in a way that the
limitations given in the equation (6.23) are satisfied, i.e. (R L ) < 1 . The results
show satisfactory agreement between the BEM simulation and the approximate
175
solution even for the most extreme case (R L ) = 1 . In addition, the torque on both
spheres were calculated and equal to zero, which was expected [127].
R1
u1
L
R1
L
R2
R2
u2
(a)
Figure 6.26:
u2
u1
(b)
Two spheres falling: (a) along their line-of-centers; (b) perpendicular to
their line-of-centers.
In the case of two equal-sized spheres moving perpendicular to the line of
centers (Figure 6.26b), the drag exerted on the right sphere for (R L ) < 1 is given
by [246, 282-284],
9
465
⎛
2
(R L )4 ⎞⎟ −
= u1 ⎜1 + (R L ) +
6πµR
256
⎠
⎝ 16
59
15813
⎛3
3
(R L )5 ⎞⎟ + O (R L )6
u 2 ⎜ (R L ) + (R L ) +
64
7168
⎠
⎝4
F
(
)
(6.24)
176
where u1 is the velocity of the right sphere and u 2 the velocity of the left sphere;
the drag on the left is found by interchanging the velocities in the equation,
similar to the first case.
Figure 6.27:
Normalized drag force for two equal-sized spheres falling along their
line-of-centers.
Figure 6.28 shows the comparison between the approximate solution
(equation (6.24)) and the boundary element solution for two equal-sized spheres.
In this case the velocity of the right sphere is two times the velocity of the left
sphere. The conditions where fixed in a way that (R L ) < 1 , in order to assure the
177
used of equation (6.24). The drag force is normalized with the Stokes drag force
of the right sphere.
Figure 6.28:
Normalized drag force for two spheres moving perpendicular to their
line-of-centers.
Experimental data for both cases are available from the work of Eveson, et. al.
[90] and Bart [18]. The spheres were of equal size and moving with same
velocity. It is necessary to consider that experimentally, spheres are dropped in a
vessel, usually cylindrical, and not in an infinite medium. This is especially
important for the effect of the wall in the motion of the spheres [127]. Figures 6.29
and 6.30 shows the comparison between experimental data from Bart [18], taken
at Reynolds numbers less than 0.05, and the BEM simulation as a function of the
178
dimensionless distance between the centers. Experimental values are more
accurate when the spheres are close to each other or when they are touching
[127]. This is why a good value to compare is the correction factor to Stokes force
when the spheres touch. In the case of to spheres falling parallel to their line of
centers (Figure 6.26) Bart’s experimental value is χ = F 6πµRu1 = 0.647 [18, 127]
while the BEM value is χ = 0.657 , for an error of 1.37%. For two spheres falling
perpendicular to their line of centers (Figure 6.30) the experimental value is
χ = 0.707 [18, 127] and the BEM calculated value is χ = 0.735 , for an error of
3.3%. When the spheres are falling perpendicular to their line of centers they will
also experience a torque, which in the case of two equal-sized spheres moving
with the same velocity should be equal in magnitude with different sign. Figure
6.31 shows the torque experienced by each sphere calculated with the BEM, as
illustrated once the spheres are being at a more distance the torque decreases
because the interaction between the spheres decreases.
179
Figure 6.29:
Dimensionless force as a function of the distance between centers for two
spheres falling parallel to their line-of-centers.
Figure 6.30:
Dimensionless force as a function of the distance between centers for two
spheres falling perpendicular to their line-of-centers.
180
Figure 6.31:
6.3.2
Direction of rotation and BEM torque for two spheres settling beside each
other.
The viscosity of particulate systems
The basic problem of suspension mechanics is to predict the macroscopic
transport properties of a suspension, i.e. thermal conductivity, viscosity,
sedimentation rate, etc., from the micro-structural mechanics. These flows are
governed by at least three length scales: the size of the suspended particles, the
average spacing between the particles, and the characteristic dimension of the
container in which the flow occurs. A number of excellent reviews of the general
subject of suspension rheology are available [84, 127]. Of special interest is the
hydrodynamic treatment of the problem by Frisch and Simha [84] and Hermans
181
[131]. Numerous models have been proposed to estimate the suspension
viscosity, most of them are a power series of the form,
µ
= 1 + a1φ + a 2φ 2 + a3φ 3 + ...
µ0
(6.25)
where φ is the volume concentration of the suspended solids. For dilute systems
of spheres of equal size, where interaction effects are neglected, Einstein [81, 82]
arrives at the following formula,
µ
= 1 + 2.5φ
µ0
(6.26)
Einstein’s formula holds for any type of linear viscometers, and can be derived
by different methods [48, 127, 154]. For dilute systems considering the first-order
effect of the spheres interacting with one another Guth and Simha [121] gave,
µ
= 1 + 2.5φ + 14.1φ 2
µ0
(6.27)
Simha [278] reduced the last term in equation (6.27) to 12.6φ 2 when the volume
occupied by the spheres is considered. Vand [314, 315] considered the collision
effect, neglecting the Brownian motion effect and obtained,
µ
= 1 + 2.5φ + 7.349φ 2
µ0
(6.28)
In a treatment of higher-order concentration effects, Kynch [166, 167] found an
equation similar to that given by Guth and Simha, and Vand,
182
µ
= 1 + 2.5φ + 7.5φ 2
µ0
(6.29)
Figure 6.32 illustrates the relative suspension viscosity as a function of the
volumetric concentration of spheres for the different theoretical approaches.
Figure 6.32:
Theoretical suspension viscosity
concentration of spheres.
as
a
function
of
the
volume
The direct boundary integral formulation was used to simulate suspended
spheres in simple shear flow. The viscosity was then calculated by integration of
the surface tractions on the moving wall. Figure 6.33 shows a typical mesh for the
domain and spheres for these simulations, in this mesh the box has dimensions
of 1x1x1 (Length units)3 and 40 spheres of radius of 0.05 length units. Initially, the
spheres are positioned randomly in the box, periodic boundary conditions are
183
used in the x- and y-direction and non-slip on the z-direction. The spheres moves
according to the flow field and the viscosity is calculated for several time steps,
for each configuration an average suspension viscosity if then obtained.
Figure 6.33:
Spheres suspended in simple shear flow: 1x1x1 (Length units)3 box and 40
spheres of radius of 0.05 length units.
The box was divided into 216 elements with 650 and each sphere into 96
elements with 290 nodes. The computational time depends, as for any particulate
simulation, on the number of spheres. Two different sphere radii were used in
the simulations: 0.05 length units and 0.07 length units. In the same way, the box
dimensions were set to 1x1x1 (Length units)3 and 0.8x0.8x0.8 (Length units)3. Each
case was simulated with 10, 20, 30 and 40 spheres. Figure 6.34 shows the
184
suspension viscosity calculated with the BEM for the 1x1x1 (Length units)3 box,
Einstein’s formula and Guth and Simha power expansion are shown as a
comparison.
Following the power expansion model, the BEM relative viscosity for spheres
of radius of 0.05 length units is approximately,
µ
≈ 1 + 2.2006φ + 32.946φ 2
µ0
(6.30)
while for spheres of radius of 0.07 length units by,
µ
≈ 1 + 3.1292φ + 3.5883φ 2
µ0
(6.31)
These formulas are plotted in Figure 6.34 as dotted lines.
To increase even further the volume concentration of spheres the box
dimensions were decreased to 0.8x0.8x0.8 (Length units)3. Figure 6.35 shows the
geometry and mesh for 40 suspended spheres. The results for the BEM relative
viscosity is shown in Figure 6.36 for two different radii: 0.05 length units and 0.07
length units. Einstein’s formula and Guth and Simha’s power expansion are also
shown as a comparison.
185
Figure 6.34:
Calculated relative viscosity for the 1x1x1 (Length units)3 box with spheres
with radius: (a) 0.05 length units and (b) 0.07 length units.
186
Figure 6.35:
Spheres suspended in simple shear flow: 0.8x0.8x0.8 (Length units)3 box
and 40 spheres of radius of 0.05 length units.
For this box geometry, the power expansion model for the BEM relative
viscosity for spheres of radius of 0.05 length units is approximately,
µ
≈ 1 + 2.6166φ + 8.2184φ 2
µ0
(6.32)
while for spheres of radius of 0.07 length units by,
µ
≈ 1 + 2.636φ + 10.589φ 2
µ0
(6.33)
187
Figure 6.36:
Calculated relative viscosity for the 0.8x0.8x0.8 (Length units)3 box with
spheres with radius: (a) 0.05 length units and (b) 0.07 length units.
188
The numerical correlations given by the direct BEM simulations are similar to
the expressions mentioned at the beginning of the section. In fact, the first
coefficient in the power expansion is close to the one predicted by Einstein [81].
The second coefficient in the power expansion is between the value suggested by
Guth and Simha [121] and Vand [314], except for the simulation with the lowest
sphere volume concentration, i.e. the 1x1x1 (Length units)3 box with spheres with
radius of 0.05 length units. In Figure 6.37 the calculated BEM relative viscosity is
collapsed for all cases. The BEM power expansion for the viscosity is,
µ
≈ 1 + 2.5463φ + 11.193φ 2
µ0
Figure 6.37:
Calculated BEM relative viscosity.
(6.34)
189
Chapter 7
Conclusions and further research
Within this thesis, direct boundary integral equations and boundary element
methods (BEM) have been developed for applications in viscous fluids,
emphasizing the solution of non-Newtonian fluid flow problems and viscous
fluids containing solid inclusions.
The domain grid superposition technique, proposed here, basically selects
cells within the domain and uses them to directly approximate the domain
integrals by means of cell integration. This technique allows the BEM to be
satisfactorily used for non linear problems. Even though this thesis uses the
power law model as a constitutive equation, the DGS-BEM technique can be
used for any non linear problem in fluid mechanics, such as viscoelastic
constitutive equations or the Navier-Stokes equations. The technique was
applied to solve Poiseuille flow of a power law fluid and the non-Newtonian
Couette flow problem. Both were used due to the availability of their analytical
190
solution and the latter is additionally interesting because of its mixed boundary
conditions, namely, periodicity at the end of the cylinders and velocities at the
cylinders’ surfaces.
Direct boundary integral formulations were applied on several situations of
fluids containing solid inclusions. Wall effects, particle-particle interactions and
drag coefficients were analyzed and satisfactorily compared with analytical or
approximate solutions. The dynamics of a single fiber in shear flow were also
simulated and its orbit was compared with the expressions given by Jeffery in
1922. Finally, the direct formulation was used to predict the suspension viscosity
of a fluid containing equally sized spheres.
In this study, methodologies for non linear flows and viscous fluids
containing solids were developed and successfully applied in specific problems.
However, there are many areas for future research. Some of these can be
summarized as follows:
•
Non-isothermal problems, reactive systems, free surfaces, elastic solids
and in general, problems that require coupled solutions of balance
equations.
•
Future research is necessary to approximate the domain integrals. Even
though the technique developed in this study was satisfactory, it is
computationally expensive. Methods that approximate this domain
191
integral to the surface need additional attention to overcome this obstacle.
Specially, the dual reciprocity technique and the basis functions used in it
need the most amount of attention.
•
More robust algorithms for the solutions of non linear systems of
equations can accelerate techniques, as the ones developed in this thesis,
such as the DGS technique. The different iterative solvers, with their
preconditioners, should be thoroughly investigated to avoid using direct
iterative methods.
•
Parallel computing for the pre-processing of the BEM and the non linear
solution can also be applied to speed up the solution process. The parallel
Newton-Krylov method can be used for this purpose.
192
Appendix A
Mathematical definitions
A.1
Lebesgue and Hilbert spaces
Throughout the entire thesis, functions of a point x = ( x1 , x 2 , x3 ) of threedimensional Euclidean space (E3) are going to be considered; which may also
depend on time t. Let Ω denote a domain of the space E3 and S its boundary. The
closure, denoted by Ω , is Ω = Ω + S . All the functions are real and locally
summable in the sense of Lebesgue, while all derivatives are interpreted in the
generalized sense [285, 286].
The Hilbert space ( W2l (Ω )
l = 0,1,2,... ) consists of all functions u (x ) which are
measurable on Ω, have derivatives D k u with respect to x of all orders k ≤ l , and
are such that both the function and the all these derivatives are square-integrable
193
over Ω (by means of a inner product and a norm). For l = 0 , the space W2l (Ω ) is a
Lebesgue space, usually denoted by L2 (Ω ) .
The scalar (inner) product ( f , g ) of two functions f (x ) and g ( x ) in a L2 space
is defined by,
b
( f , g ) = ∫ f (x )g (x )dx
(A.1)
a
which have a few simple properties that arise from its definition:
(a) The inner product of functions that are the sums of several terms is
performed according to the rule of multiplication of polynomials, i.e.,
( f , g1 + g 2 ) = ( f , g1 ) + ( f , g 2 )
(A.2)
(b) Permutation of the factors in a inner product has the effect of replacing it
by its complex conjugate,
( f , g ) = (g , f )
(A.3)
(c) A constant multiplying the first function may be taken outside the inner
product,
(αf , g ) = α ( f , g )
(A.4)
(d) A constant multiplying the second function may be taken outside the
inner product after first taking its complex conjugate,
( f , αg ) = α ( f , g )
(A.5)
194
(e) The inner product of a function with itself is a non-negative quantity,
b
(f, f)= ∫ f
2
(x ) dx ≥ 0
(A.6)
a
it vanishes if, and only if, f ( x ) = 0 .
In addition, two functions f (x ) and g ( x ) are called orthogonal in the interval
(a, b ) if,
b
( f , g ) = ∫ f (x )g (x )dx = 0
(A.7)
a
If the functions are real, then this condition of orthogonality is simplified to,
b
( f , g ) = ∫ f (x )g (x )dx = 0
(A.8)
a
According to these conditions the inner product between two functions in L2 can
be interpreted as the degree of similarity (alignment) between two functions.
From property (e) a quantity
(f , f )
can be defined (sense of length,
measurability), it is called the norm of f and is normally symbolized as f . It
satisfies all norm properties, i.e.,
αf = α f
f ≥0
f + g ≤ f + g (triangle inequality)
(A.9)
195
Therefore, a Lebesgue space (L2) are functions in a region (a, b ) with an inner
product, defined in equation (A.1); measurable by a norm,
, and that are
square-integrable,
b
f = ∫ f ( x ) f ( x )dx < ∞
(A.10)
a
A.2
Lyapunov surfaces
A Lyapunov surface is a smooth surface possessing a tangent plane and a
normal, but not necessary a curvature, at each point. Which implies the existence
of local coordinates at any point of the surface, z-axis along the normal, x- and yaxes in the tangent plane, such that a portion of the surface has the equation
z = z ( x, y )
where
the
partial
derivatives
∂z ∂z
, ,
∂x ∂y
but
not
necessarily
∂2z ∂2z ∂2z
,
,
, exist and are continuous. As a consequence the surface can be
∂x 2 ∂y 2 ∂x∂y
decomposed into a finite number of overlapping pieces and its normal varies
continuously (i.e. the ellipsoid).
If n 1 , n 2 are the unit normal vector at any points x 1 , x 2 of a Lyapunov surface,
it is required that [151, 174]
cos −1 (n 1 ⋅ n 2 ) ≤ D x 1 − x 2
α
(A.11)
196
where D > 0 and 0 < α ≥ 1 . This condition characterizes the Hölder continuity
(see section A.3) and it holds for the surface defined by z = z ( x, y ) if
∂z ∂z
,
are
∂x ∂y
Hölder continuous over the surface [151].
Lyapunov surfaces are less general than those considered by Kellogg [158],
which could have corners or edges provided that they are not too sharp. For
instance, a cube is a Kellogg regular surface although a cone is not. Kellogg
regularity guarantees the fundamental existence-uniqueness theorems of
harmonic function theory [151, 158, 227]. However, the restriction to Lyapunov
surfaces is necessary for formulations of boundary-value problems via potential
theory.
A.3
Hölder continuity
A function f (x ) satisfies a Hölder condition in the interval a ≤ x ≤ b , if for any
two distinct points x1 , x 2 ∈ [a, b ] [151, 158],
f ( x 2 ) − f ( x1 ) < D x 2 − x1
α
(A.12)
where D > 0 and 0 < α ≥ 1 . If α = 1 the Hölder condition becomes the Lipschitz
condition, symbolized f ( x ) ∈ L[a, b].
If f ( x ) is differentiable in the interval a ≤ x ≤ b then f (x ) satisfies the
Lipschitz condition, but not conversely. For example, the absolute value of x
197
⎡ 1 1⎤
( f ( x ) = x ) is Hölder continuous in ⎢− , ⎥ but it is not differentiable at x = 0 .
⎣ 2 2⎦
This indicates that differentiability as a stronger condition than Hölder
continuity.
A.4
Harmonic functions
A function φ (x ) is said to be harmonic within a three-dimensional domain Ω,
bounded by a closed surface S, if it satisfies the following conditions:
(a)
φ (x ) is continuous in Ω + S ,
(b)
φ (x ) is differentiable to at least the second order in Ω,
(c)
φ (x ) satisfies the Laplace’s equation in Ω, i.e. ∇ 2φ (x ) = 0 .
If S is a Lyapunov surface, it is possible to determine φ (x ) throughout Ω in
terms of suitable prescribed information over S. Therefore, for a given arbitrary
values of φ (x ) over S exists a unique φ (x ) in Ω which assumes these values. This
is the Dirichlet existence theorem of harmonic function theory [151]. For
example, if φ (x ) = 1 over S implies that φ (x ) = 1 in Ω. Additionally, for arbitrary
values of
∂φ (x )
over S, which satisfies the Gauss condition [174, 246],
∂ni
∫
S
∂φ (x )
dS = 0
∂n
(A.13)
198
there exists a unique φ (x ) in Ω whose normal derivative assumes these values.
This is the Neumann existence theorem of harmonic function theory [151]. As a
consequence of this theorem a
∂φ (x )
= 0 over S implies φ (x ) = c (an arbitrary
∂ni
constant) in Ω. Finally, harmonic functions may also exits in the domain exterior
to S. Dirichlet and Neumann theorems remain valid providing that the harmonic
function φ (x ) is regular in the exterior domain, i.e.,
φ (x ) = O (r −1 ) as r → ∞
A.5
(A.14)
Dirichlet and Neumann boundary conditions
The Dirichlet and Neumann boundary conditions are particular cases of a
prescribed linear relation
α (x )φ (x ) + β (x )
between φ (x ) and
∂φ (x )
= f (x )
∂n
(A.15)
∂φ (x )
at each point of S. Defining f (x ) as a continuous
∂n
function over S, the Dirichlet condition is defined by
α (x ) = 1 and β (x ) = 0
(A.16)
and the Neumann condition is defined by
α (x ) = 0 and β (x ) = 1
Finally, a Robin boundary condition is defined by
(A.17)
199
α (x ) < 0 and β (x ) = 1
where α (x ) and
(A.18)
f (x ) must be Hölder continuous functions over S. The
inequality α (x ) < 0 is commonly chosen to be consistent with the convention that
the normal derivative,
∂φ (x )
, is directed into the domain Ω under consideration.
∂n
Similar to the Dirichlet and Newmann existence theorems there is also an
existence-uniqueness theorem for Robin conditions and for difficult mixed
boundary conditions such as
α (x ) = 1 and β (x ) = 0 in S1
(A.19)
α (x ) = 0 and β (x ) = 1 in S 2
(A.20)
where S = S1 + S 2 [151, 310].
A.6
Fredholm’s theorems for integral equations
In general, an integral equation cannot be solved in closed form. Thus, as a rule it
solution involves the use of approximate methods [196, 246]. However, these
methods can only be applied when the solvability of the equation has been
established. This analysis is performed using the general theorems on integral
equations established by Fredholm. Similar to algebraic equations, the solvability
of integral equations is directly related to its eigenvalue problem, that is why
200
each of Fredholm’s theorem is related to certain well-known propositions of
linear algebra.
For integral equations of the second kind such,
b
f (ξ ) − λ ∫ K (ξ , x ) f ( x )dx = g (ξ )
(A.21)
a
there is an eigenvalue problem associated to the homogeneous equation,
b
f (ξ ) = λ ∫ K (ξ , x ) f ( x )dx
(A.22)
a
in which values of the parameter λ are found in such a way that the equation has
a non-trivial solution, i.e. f ( x ) ≠ 0 . These values are called the eigenvalues of the
kernel K (ξ , x ) , and the corresponding function f ( x ) is the eigenfunction. The
conjugate or adjoint of Eq. (A.21) is,
b
h(ξ ) − λ ∫ K ( x, ξ )h(x )dx = y (ξ )
(A.23)
a
where the kernel conjugate, K ( x, ξ ) , is obtained by permuting the arguments and
taking the complex conjugate.
Theorem 1:
In the finite portion of the complex λ plane there exits no more than a finite
number of eigenvalues of Fredholm’s integral equation (A.21).
201
Theorem 2:
To each eigenvalue there corresponds at least one eigenfunction. The number of
linear independent eigenfunctions, corresponding to a given eigenvalue, is finite.
Theorem 3:
If λ0 is an eigenvalue of the kernel K (ξ , x ) , then λ0 is an eigenvalue of the
conjugate kernel K ( x, ξ ) . The number of linearly independent eigenfunctions of
equation (A.21) and of its conjugate (A.23), is one and the same.
Theorem 4:
Let λ0 be an eigenvalue of the kernel K (ξ , x ) . In order that the inhomogeneous
equation,
b
f (ξ ) − λ0 ∫ K (ξ , x ) f (x )dx = g (x )
(A.24)
a
should have a solution, it is necessary and sufficient that its right-hand side g ( x )
be orthogonal to all the eigenfunctions of the adjoint homogeneous equation,
b
h(ξ ) − λ ∫ K ( x, ξ )h( x )dx = 0
(A.25)
a
Fredholm’s alternative:
Either the inhomogeneous equation (A.21) possesses a unique solution f ( x ) ,
whatever its right-hand side g ( x ) may be, or the corresponding homogeneous
202
equation has a non-trivial solution in which case, according to the previous
theorems, equation (A.21) will have non-unique solutions if, and only if, its righthand side is orthogonal to all the eigenfunctions of the adjoint homogeneous
equation.
In terms of algebraic equations these theorems are equivalent to:
(a)
If the determinant of a system is different from zero1, the system and
its conjugate are solvable; moreover, the solutions are unique,
whatever the free terms the system may have. The homogenous
solution has only the trivial solution.
(b)
If the determinant of the system is zero, then the homogeneous system
has only non-trivial solutions. The number of linearly independent
solutions of two homogeneous systems is one and the same: any vector
in the null space of the corresponding matrix.
(c)
If the determinant of the system is zero, the non-homogeneous system
is solvable if, and only if, the known vector b is orthogonal to all the
solutions of the homogeneous conjugate system.
1
The columns of the corresponding matrix are linearly independent.
203
Appendix B
Potential theory
B.1 Potential of a field
A particle of mass m is subject to the force of a specific field F(x ) = ( f1 , f 2 , f 3 ) ,
which will move according to Newton’s second law of motion, i.e.,
m
d 2 xi
= fi
dt 2
(B.1)
where i=1,2,3. For a conservative field [158, 178] the function
W (x o , x ) =
x
∫ f dx
i
i
(B.2)
xo
is the work function of the specific filed, which is determined by the field only.
Since the work function is independent of the axis system, the component of the
field in any direction is equal to the derivative of the work in that direction, i.e.,
204
∂W (x )
∂xi
fi =
(B.3)
Thus, a conservative field can be specified by a single function W (x ) , whereas a
general field requires three functions ( f1 , f 2 , f 3 ) [178]. Because the work
determines the field in this way, it is commonly called the force function. In other
words, any field which has a force function, with continuous derivatives, is
conservative [158]. The generalization of this concept to attractive and repulsive
force fields will be as follows: in vector analysis, a field F(x ) = ( f1 , f 2 , f 3 ) is called
the gradient of the potential1 U (x ) ,
( f1 , f 2 , f 3 ) = ∇U (x ) = ⎜⎜ ∂U (x ) , ∂U (x ) , ∂U (x ) ⎟⎟
⎞
⎛
⎝ ∂x1
∂x 2
∂x3 ⎠
(B.4)
Sometimes the potential coincides with the force function and in others to the
negative of the force function. In Newtonian fields, the potential (called
Newtonian) at x due to a unit source at x 0 is defined as,
U (x ) =
1 1
4π r (x )
(B.5)
which has by convention the following characteristics:
(a)
the potential is the force function, and the negative of the potential
energy, if the force is attractive (i.e. gravitation),
__________________________________________
1
Called potential by Green in 1828 and by Gauss in 1813 [158].
205
(b)
the potential is the negative of the force function, and identical to the
potential energy, if the force repels (i.e. electricity and magnetism).
The Newtonian potential is a continuous function, differentiable to all orders
and it satisfies Laplace’s equation,
∇ 2U (x 0 , x ) = 0
(B.6)
everywhere except at the source point x 0 , which characterizes U (x ) as a
harmonic function of x everywhere except at x = x 0 . Formally, U (x ) satisfies
Poisson’s equation,
∇ 2U (x 0 , x ) = −δ (x − x 0 )
(B.7)
where δ is the Dirac’s delta function centered upon x = x 0 [78].
Continuous distributions of simple sources over a line, a surface and a
volume generate Newtonian potentials of line, surface and volume as follows,
U (x o , λ ) = ∫
1
λ (x )dC
(
)
r
x
,
x
0
C
U (x o , σ ) = ∫ ∫
S
1
σ (x )dS
r (x 0 , x )
U (x o , ρ ) = ∫ ∫ ∫
Ω
1
ρ (x )dΩ
r (x 0 , x )
B.2 Single-layer potential continuity on the surface
The single-layer potential of density σ (x ) is defined as,
(B.8)
206
V (x 0 , σ ) =
1
4π
∫ r (x , x ) σ (x )dS
1
(B.9)
0
S
Because its kernel has a singularity as x 0 approaches the surface point ξ ∈ S , the
singular point is included in a hemisphere centered at x 0 , the integral is solved
for as the limit at its radius goes to zero (see Figure B.1), i.e.,
V (ξ, σ ) =
n
⎡
⎤
1
1
1
lim ⎢ ∫ σ (x )dS + ∫ σ (x )dS ⎥
r
4π e→0 ⎢⎣ S − S * r
⎥⎦
Se
(B.10)
Se
e
S*
S
Figure B.1:
Ω
Inclusion of internal point source on the domain.
In the limit when e → 0 , the first term in the above expression recovers the
original surface S. For the hemisphere, in the second term, dS = e 2 cos φdφdθ , thus
the second integral is,
⎡π π 2 1
⎤
1
lim ⎢ ∫ ∫ e 2σ (x ) cos φdφdθ ⎥ = 0
4π e→0 ⎢⎣ 0 −π 2 e
⎥⎦
(B.11)
Finally, the single-layer potential at a surface point ξ ∈ S is,
V (ξ, σ ) =
1
4π
∫ r (ξ, x ) σ (x )dS
1
S
(B.12)
207
which shows that the single-layer potential is continuous as the point crosses the
surface.
B.3 Double-layer potential continuity on the surface
The double-layer potential of density ψ (x ) is defined as,
W (x 0 ,ψ ) =
1
4π
∂ ⎛1⎞
⎜ ⎟ψ (x )dS
x ⎝r⎠
∫ ∂n
S
(B.13)
Similar to the analysis performed for a single-layer potential, as the point x 0
approaches the surface point ξ ∈ S , the singular point is included by a
hemisphere centered at x 0 , and then analyze the limit at its radius goes to zero
(see Figure B.1),
W (x 0 ,ψ ) =
⎡
1
∂
lim ⎢ ∫
4π e→0 ⎢⎣ S − S * ∂nx
∂
⎛1⎞
⎜ ⎟ψ (x )dS + ∫
∂nx
⎝r⎠
Se
⎤
⎛1⎞
⎜ ⎟ψ (x )dS ⎥
⎝r⎠
⎥⎦
(B.14)
However, for the case of the double-layer potential it is important to make the
distinction between a point that approaches the surface from the interior domain
Ω (i ) or from the exterior domain Ω (e ) , because the kernel of the double-layer
potential has different signs when coming from the interior or exterior, since the
normal vector is always considered to point outward the domain. In the limit
when e → 0 , the first term in the above expression recovers the original surface
208
S. In order to evaluate the second integral, a new term is added and subtracted as
follows,
∂
1
lim ∫
4π e→0 Se ∂nx
⎛1⎞
⎜ ⎟ψ (x )dS =
⎝r⎠
⎡ ∂ ⎛1⎞
∂ ⎛1⎞ ⎤
1
lim ⎢ ∫
⎜ ⎟(ψ (x ) − ψ (x 0 ))dS + ψ (x 0 ) ∫
⎜ ⎟dS ⎥
∂n x ⎝ r ⎠ ⎦⎥
4π e→0 ⎢⎣ Se ∂nx ⎝ r ⎠
Se
(B.15)
On taking the limit, the first integral on the right hand tends to zero due to the
continuity of the density. The kernel of the second integral, for a point x 0 → ξ
(
)
from Ω (i ) , is of the form − (1 4π ) 1 r 2 because at any point in the hemisphere
r = e and ∂r ∂n = 1 since r and n have the same direction. Therefore, the second
integral becomes,
⎡π π 2 1 2
⎤
1
1
ψ (x 0 ) lim ⎢ ∫ ∫ − 2 e cos φdφdθ ⎥ = − ψ (ξ )
e→0
4π
2
⎢⎣ 0 −π 2 e
⎥⎦
(B.16)
Thus, the final expression for the double-layer potential at a surface point ξ ∈ S ,
coming from the interior domain Ω (i ) is,
1
W (ξ,ψ )( i ) = − ψ (ξ ) + ∫ K (ξ, x )ψ (x )dS
2
S
(B.17)
Following the same methodology, the double layer-potential at a surface point
ξ ∈ S , coming from the exterior domain Ω (e ) is,
1
W (ξ,ψ )( e ) = ψ (ξ ) + ∫ K (ξ, x )ψ (x )dS
2
S
(B.18)
209
These results indicate that the double-layer potential
W (x 0 , x )
has a
discontinuity, or jump, of the density ψ (ξ ) as the point crosses the surface S.
210
Appendix C
Green’s functions and identities
The Green’s function or fundamental solution is defined, in the simplest physical
way, as the response due to a unit source in an infinite problem. For example, if
u ij (ξ, x ) is the Green’s function for the velocity in a viscous flow problem that
means: u ij (ξ, x ) is the velocity at point x in the direction j due to a unit point
force applied at ξ in the i –direction. In addition, u ij (ξ, x ) is a kernel between two
points, which, according to Betti’s reciprocal theory, satisfies the symmetry
property [265, 266],
uij (ξ, x ) = uij (x, ξ )
(C.1)
physically, this symmetry property gives a relation between the flow due to a
point force with pole at ξ and the flow due to another point force with pole at x .
Similar symmetry properties exist for potential flow, elastostatics and scalar
potentials [249, 287, 288].
211
C.1
Green’s functions for scalar operators
Mathematically, the fundamental solution of a problem is the solution of the
governing differential equation when the Dirac delta is acting as a forcing term
[168, 265, 266]. Due to the infinite nature of the problem no boundary conditions
are needed and providing that the Dirac delta or delta function posses a singular
nature, the Green’s function or fundamental solution is also singular. It is
defined by,
L u (ξ, x ) = −δ (ξ, x )
(C.2)
where L is a scalar differential operator and δ (ξ, x ) is the Paul Dirac delta
function [78], in which ξ is the source point and x the field point. For matrix
operators, equation (C.2) is re-written as follows,
Lij u kj (ξ, x ) = −δ (ξ, x )δ ki
(C.3)
where δ ki is the Kronecker delta function. From the Dirac delta definition, i.e.,
⎧∞ when ξ = x
⎩ 0 when ξ ≠ x
δ (ξ, x ) = ⎨
(C.4)
there are two useful properties that are used in the fundamental solution
derivation,
lim ∫ δ (ξ, x )dΩ = 1
Ω →0 Ω
(C.5)
212
∫ δ (ξ, x )F(x )dΩ = F(ξ )
(C.6)
Ω
where Ω is an arbitrary domain. Typically, it is chosen a circle or a sphere for
two- and three-dimensional domains, respectively [287, 288].
The most common technique for the derivation of fundamental solutions is to
use integral transforms, such as, Fourier, Laplace or Hankel transforms [168]. For
simple operators, such as the Laplacian, direct integration and the use of the
properties of the Dirac delta are typically used to construct the fundamental
solution. The starting point is to solve for the homogeneous equation,
L u (ξ, x ) = 0
(C.7)
using simple techniques, such as direct integration in polar coordinates,
separation of variables, variation of parameters. The constants that appear due to
the integration procedures are solved for using the property cited in equation
(C.5), i.e.,
lim ∫ Lu (ξ, x )dΩ = −1
(C.8)
Ω →0 Ω
For example, the Laplacian operator in a three-dimensional domain, can be
written as,
1 ∂ ⎛ 2 ∂⎞
⎜r
⎟u (ξ, x ) = 0
r 2 ∂r ⎝ ∂r ⎠
By direct integration, it is obtained that,
(C.9)
213
u (ξ, x ) = a +
b
r
(C.10)
dθdφ = −1
(C.11)
And equation (C.8) will be,
2π 2π
∂ ⎛
b⎞
lim ∫ ∫ ∂n ⎜⎝ a + e ⎟⎠e
e →0
0
2
0
For a sphere ∂e ∂n = 1 and dS = e 2 dθdφ , then a is an arbitrary constant and
b = 1 4π . The fundamental solution or Green’s function will be,
u (ξ, x ) = a +
1 1
4π r
(C.12)
Table C.1 presents the most common fundamental solutions which are used as
basics for many problems in computational mechanics.
Table C.1:
Green’s functions for commonly used operators [265, 266].
Equation
Laplace
∇ 2 u = −δ (ξ, x )
Helmholtz
(∇
2
)
+ λ2 u = −δ (ξ, x )
Modified Helmholtz
(∇
2
− λ2 )u = −δ (ξ, x )
Bi-harmonic
∇ 4 u = −δ (ξ, x )
1
Where
1D
u=−
x
r
2D1
1
u=−
ln r
2π
3D
1 1
u=−
4π r
u=−
1
sin (λ x )
2λ
1 (1)
H (λr )
4i
u=−
1 1
exp(− iλr )
4π r
u=−
1
1
sin (− iλ x ) u = −
K 0 (λr )
2π
2λ
u=−
1 1
exp(λr )
4π r
u=−
r
8π
u=−
u=−
1 2
r ln r
8π
H (1) and K 0 are Hankel and Bessel functions respectively.
214
C.2
Green’s functions for matrix operators
For matrix operators, more sophisticated techniques must be used in order to
find the fundamental solutions. The operator decoupling technique to
breakdown matrix operators to simple or compound scalar operators is one of
the most used methods. This method is due to Lars Hörmander [142], the
technique is simple and depends on the understanding of simple definitions of
matrix algebra. Consider the following matrix,
⎡ a11
a = a ij = ⎢⎢a 21
⎢⎣ a31
a12
a 22
a32
a13 ⎤
a 23 ⎥⎥
a33 ⎥⎦
(C.13)
The cofactor of any element is defined as follows [86-88],
cof (aij ) = (− 1)
(i + j )
det sub(aij )
(C.14)
The matrix composed of the cofactor elements is called cofactor matrix. The
Hörmander method is a technique to decompose the matrix operators to simple
scalar operators (i.e. Table C.1), for which the fundamental solutions can be
obtained easily. Once, the fundamental solutions or the scalar potentials of these
simple operators are obtained, the Hörmander technique provides a backward
procedure to construct the fundamental solution for the original matrix operator.
Consider the following general differential equation,
Lu = b
(C.15)
215
where L is the matrix-type differential operator, b is the body force vector and u
the problem variable. The fundamental solution u ij (ξ, x ) is required to be used in
a relevant boundary integral formulation. The steps of the Hörmander technique
are as follows [141, 265, 266],
(a) Compute the adjoint operator, because after the boundary integral
formulation is set up the fundamental solution of,
Ladj u k = −δδ
(C.16)
must be calculated. Here, Ladj is the adjoint of the original operator, δ is
the delta function, δ is the identity matrix and u k is the desired
fundamental solution,
[ ( )]
(b) Compute the cofactor matrix of the adjoint operator, cof Ladj , and its
[ ( )] ,
transpose, cof Ladj
t
(c) Compute the determinant of the transpose of the cofactor matrix,
[
]
det cof (Ladj ) ,
t
(d) Compute the scalar potential, Φ , which is the solution of the following
equation,
[
]
det cof (Ladj ) Φ = −δ
t
(C.17)
Here, instead of computing the fundamental solution for the original
operator L, it has been decomposed into a new scalar operator
[
]
det cof (Ladj ) , which is simple or compound, and can be dealt with easily.
t
216
(e) Finally, compute the fundamental solution using,
( )
u k = cof Ladj Φ
(C.18)
Consider the following Navier governing differential equations [171],
Lij u kj* = −δ (ξ, x )δ ki
(C.19)
where the matrix operator is defined as,
Lij = G
∂2
G ∂ ∂
G
= G∇ 2δ ij +
∂i∂ j
δ ij +
∂x j ∂x j
1 − 2v ∂xi ∂x j
1 − 2v
(C.20)
where G is the modulus and v the Poisson’s ratio. This operator is self-adjoint
and the cofactor matrix can be obtained as follows,
( )
cof Ladj
ij
G
⎡
2
⎢G∇ + 1 − 2v ∂ 2 ∂ 2
=⎢
G
⎢ −
∂ 1∂ 2
1 − 2v
⎣
G
⎤
∂ 2 ∂1 ⎥
1 − 2v
⎥
G
2
G∇ +
∂ 1∂ 1 ⎥
1 − 2v
⎦
−
(C.21)
The determinant of the transpose of the cofactor matrix can be computed as,
[ ( )]
det cof Ladj
ij
t
G
G
⎞
⎞⎛
⎛
∂ 1∂ 1 ⎟ −
= ⎜ G∇ 2 +
∂ 2 ∂ 2 ⎟⎜ G∇ 2 +
1 − 2v
1 − 2v
⎠
⎠⎝
⎝
G
G
⎞
⎞⎛
⎛
∂ 2 ∂1 ⎟
∂ 1∂ 2 ⎟⎜ −
⎜−
⎠
⎠⎝ 1 − 2v
⎝ 1 − 2v
(C.22)
which becomes,
[ ( )]
det cof Ladj
ij
t
=
2G (1 − v ) 4
∇
1 − 2v
(C.23)
217
Thus, according to Hörmander a potential Φ is needed, in way that satisfies,
2G 2 (1 − v ) 4
∇ Φ (ξ, x ) = −δ (ξ, x )
1 − 2v
(C.24)
A fundamental solution for this equation will be (see Table D.1),
Φ=
− (1 − 2v ) 1 2
r ln r + f
2G 2 (1 − v ) 8π
(C.25)
where
f = ar 2 + b ln r + c
(C.26)
in which, a,b,c are arbitrary constants [265, 266]. Thus, Φ represents a Galerkin
tensor and f is a complementary solution for the bi-harmonic operator, which can
be omitted by setting all constants to zero. The final step in the method says that
the fundamental solution can be written as,
u ij* =
∂ 2Φ
G ⎛⎜
2(1 − v )δ ij ∇ 2 Φ −
∂xi ∂x j
1 − 2v ⎜⎝
⎞
⎟
⎟
⎠
(C.27)
with equations (C.25) and (C.26) the fundamental solution will be,
(1 − v ) ⎧⎪− δ
⎡ 7 − 8v
⎤ ∂r ∂r ⎫⎪
+ (3 − 4v ) ln r ⎥ +
⎨ ij ⎢
⎬+
8πG ⎪⎩
⎣ 2
⎦ ∂xi ∂x j ⎪⎭
∂r ∂r ⎤ ⎫⎪
G ⎧⎪
b ⎡
⎥⎬
⎨2a(3 − 4v )δ ij − 2 ⎢δ ij + 2
∂xi ∂x j ⎥⎦ ⎪⎭
1 − 2v ⎪⎩
r ⎢⎣
u ij* =
If the constants in equation (C.26) are defined by,
(C.28)
218
a=
(1 − 2v )(7 − 8v )
32πG 2 (1 − v )(3 − 4v )
(C.29)
b=0
the Kelvin fundamental solution for 2D will be obtained, i.e.
u ij* =
C.3
⎧⎪
1
∂r ∂r ⎫⎪
⎨− δ ij (3 − 4v ) ln r +
⎬
8πG (1 − v ) ⎪⎩
∂xi ∂x j ⎪⎭
(C.30)
Singular solutions for the Stokes equations
The singular solutions for the Stokes equations are solutions to the following
non-homogeneous Stokes equations,
∂ 2 u ik
∂p k
−
µ
= 8πµα ik D xmδ (x − x 0 )
∂x j ∂x j ∂xi
(C.31)
and the continuity equation
∂u ik
=0
∂xi
(C.32)
where the differential operator D xm is defined as,
D x0 = 1
⎛ ∂ ⎞
⎟⎟
D 1x = β l ⎜⎜
∂
x
⎝ l⎠
⎛ ∂2
D x2 = γ j β l ⎜
⎜ ∂x ∂x
⎝ j l
(C.33)
⎞
⎟
⎟
⎠
219
where the vectors α j , β, γ are constant vectors defining the orientation of the
singularities. The solution of equations (C.32) and (C.33) with m = 0 and k = 1,2,3
is the fundamental solution u ik (x 0 , x ) , known as Stokeslet, located at point x (Eq.
(3.57)). The solution with m = 1 corresponds to a dipole or Stokes doublet and it
has two components. The symmetric component gives a fundamental singularity
called Stresslet (Eq. (3.59)). Its antisymmetric component, known as Rotlet, can be
found directly from the equations when m = 1 and D x = (1 2)ε ijl δ jl (∂ ∂x j ) . It is
defined by [160, 246],
ri j (x o , x ) =
1 ε ilk δ lj ( x0 − x )k
8πµ
r3
(C.34)
Solutions with m ≥ 2 are the higher multipoles.
C.4
Green’s identities for scalar fields
Let Ω be a region in space bounded by a closed surface S of Lyapunov type, and
F(x ) be a vector field acting on this region [119, 246]. The divergence (Gauss)
theorem establishes that the total flux of the vector field across the closed surface
must be equal to the volume integral of the divergence of the vector,
∂Fi
∫ F n dS = ∫ ∂x
i
S
i
Ω
dΩ
(C.35)
i
Substituting F(x ) = φ (x )∇ψ (x ) into Gauss theorem and using the chain rule for
the divergence of the vector, the so-called Green’s first identity is obtained,
220
∂ψ
∂ψ ∂φ
∂ 2ψ
∫S φ ∂n dS = Ω∫ ∂xi ∂xi dΩ + Ω∫ φ ∂xi ∂xi dΩ
(C.36)
where ∂ ∂n = ∇ ⋅ n . This identity is also valid when interchanging φ (x ) and ψ (x ) ,
i.e.,
∫ψ
S
∂φ
∂ψ ∂φ
∂ 2φ
dS = ∫
dΩ + ∫ψ
dΩ
∂n
∂xi ∂xi
∂xi ∂xi
Ω
Ω
(C.37)
Subtracting equation (C.37) from (C.36) gives the Green’s second identity,
⎛ ∂ 2ψ
∂ 2φ ⎞
∂φ ⎞
⎛ ∂ψ
⎜
−
−
=
φ
ψ
φ
ψ
dS
⎜
⎟
∫S ⎝ ∂n ∂n ⎠ Ω∫ ⎜⎝ ∂xi ∂xi ∂xi ∂xi ⎟⎟⎠dΩ
(C.38)
The functions φ (x ) and ψ (x ) must be differentiable at least to the orders that
appear in the integrands.
C.5
Green’s identities for the momentum equations
In order to obtain the Green’s identities for the flow field (u, p ) , a vector z is
defined as the dot product of the stress tensor π(u, p ) and a second solenoidal
vector field v. And the divergence or Gauss’ theorem is applied to the vector z,
∂
∫ ∂x (π v )dΩ = ∫ π
ij i
Ω
j
v n j dS
ij i
(C.39)
S
where the stress tensor π(u, p ) is defined for an incompressible Newtonian fluid
by,
221
⎛ ∂u
∂u j ⎞
⎟
π ij (u, p ) = − pδ ij + µ ⎜⎜ i +
⎟
x
x
∂
∂
j
i
⎝
⎠
(C.40)
The chain rule for differentiating in the volume integral of Eq. (C.39) and the
following identities [246],
∂π ij
⎛ ∂ 2ui
∂p ⎞⎟
−
vi = ⎜ µ
vi
⎜ ∂x ∂x
∂x j
∂xi ⎟⎠
j
j
⎝
π ij
∂vi µ ⎛⎜ ∂u i ∂u j
=
+
∂x j 2 ⎜⎝ ∂x j ∂xi
⎞⎛ ∂vi ∂v j
⎟⎜
+
⎟⎜ ∂x
⎠⎝ j ∂xi
(C.41)
⎞
⎟
⎟
⎠
(C.42)
gives,
µ ⎛⎜ ∂u i
∫ 2 ⎜ ∂x
Ω
⎝
+
j
∂u j ⎞⎛ ∂vi ∂v j
⎟⎜
+
∂xi ⎟⎠⎜⎝ ∂x j ∂xi
⎞
⎟dΩ +
⎟
⎠
⎛ ∂ 2ui
∂p ⎞
∫Ω ⎜⎜ µ ∂x j ∂x j − ∂xi ⎟⎟vi dΩ = ∫S π ij vi n j dS
⎝
⎠
(C.43)
which is the Green’s first identity for the flow field (u, p ) . This identity can be
also applied to a flow field (v, q ) and then subtracted from Eq. (C.43),
⎡⎛ ∂ 2 u i
⎛ ∂ 2 vi
∂q ⎞ ⎤
∂p ⎞⎟
⎜µ
−
v
−
⎢
∫Ω ⎢⎜ ∂x j ∂x j ∂xi ⎟ i ⎜⎜ µ ∂x j ∂x j − ∂xi ⎟⎟ui ⎥⎥ dΩ
⎠ ⎦
⎝
⎠
⎣⎝
[
]
(C.44)
= ∫ π ij vi n j − π ij* u i n j dS
S
which is the so called Green’s second identity, where the auxiliary stress field
π * (v, q ) is defined as,
222
⎛ ∂v
∂v j ⎞
⎟
π ij* (v, q ) = −qδ ij + µ ⎜⎜ i +
⎟
x
x
∂
∂
j
i
⎠
⎝
(C.45)
223
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Boundary Integral Equations for Viscous Flows

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