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Modeling the flow of PCL fluid due to the movement of lung cilia

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Modeling the flow of PCL fluid due to the movement of lung cilia
Creator:
Chamsri, Kannanut
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Denver, CO
Publisher:
University of Colorado Denver
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English

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Degree:
Doctorate ( Doctor of philosophy)
Degree Grantor:
University of Colorado Denver
Degree Divisions:
Department of Mathematical and Statistical Sciences, CU Denver
Degree Disciplines:
Applied mathematics
Committee Chair:
Mandel, Jan
Committee Members:
Bennethum, Lynn S.
Laugou, Julien
Lee, Long
Bowler, Russell P.

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University of Colorado Denver
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Auraria Library
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Copyright Kannanut Chamsri. Permission granted to University of Colorado Denver to digitize and display this item for non-profit research and educational purposes. Any reuse of this item in excess of fair use or other copyright exemptions requires permission of the copyright holder.

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MODELING THE FLOW OF PCL FLUID DUE TO THE MOVEMENT OF LUNG
CILIA
by
Kannanut Chamsri
BS., Mathematics, Chulalongkorn University, Thailand, 2002 M.S., Applied Mathematics, Chulalongkorn University, Thailand, 2005
A thesis submitted to the Faculty of the Graduate School of the University of Colorado Denver in partial fulfillment of the requirements for the degree of Doctor of Philosophy Applied Mathematics
2012


©2012 by Kannanut Chamsri All rights reverved


This thesis for the Doctor of Philosophy degree by Kannanut Chamsri has been approved for the Applied Mathematics by
Jan Mandel, Chair Lynn S. Bennethum, Advisor Julien Laugou Long Lee Russell P. Bowler
Date
m


Chamsri, Kannanut (Ph.D., Applied Mathematics)
Modeling the Flow of PCL Fluid due to the Movement of Lung Cilia Thesis directed by Associate Professor Lynn S. Bennethum
ABSTRACT
Cilia in the human lungs are moving hairs that aid in the movement of mucus. The layer that contains the cilia is called the periciliary layer, PCL, and the liquid in that layer the PCL fluid. We consider the PCL fluid as an incompressible viscous fluid, and we consider the cilia as a periodic array of cylinders that rotate about their base with height varying as a function of the angle. We seek a three-dimensional mathematical model of the PCL fluid slowly flowing due to the movement of the cilia. We use homogenization to determine a system of equations that are then solved numerically to calculate the permeability. If r is the radius of the cilia and d is the distance between two adjacent cilia, we determine the permeability as a function of r/d and the angle 9 that the cylinders make with the bottom surface. Numerical results are obtained using the mixed finite element method of Taylor-Hood type. The numerical results for the permeability are validated by comparing the results with numerical results of Rodrigo P.A. Rocha and Manuel E. Cruz with good agreement, and when the flow aligns and is perpendicular to the array of cylinders, the results are validated with experimental data. For an initial model of the fluid flow in the PCL, we consider the overall height as a constant, the portion of the PCL with cilia as a porous medium, and the PCL fluid above the cilia as undergoing Stokes flow. To calculate the fluid velocity in the PCL layer, Stokes-Brinkman equations are applied with a fixed boundary height of the PCL layer. The numerical result is compared with an analytical solution when the top boundary fluid is moving at a constant velocity in order to validate the numerical solution. The existence and uniqueness of the numerical model is also presented.
IV


The form and content of this abstract are approved. I recommend its publication.
Approved: Lynn S. Bennethum
v


DEDICATION
This thesis is dedicated to my beloved parents, Mr. Wipas and Mrs. Chanapa Chamsri, who have never failed to give me moral support, cared for all my needs as I developed my education, and taught me that even the largest task can be accomplished if it is done one step at a time.
vi


ACKNOWLEDGMENT
I would like to thank everyone who have helped and inspired me during my doctoral study.
This thesis would not have been possible without the generous support of my advisor, Prof. Lynn S. Bennethum, for her rich patience in providing endless idea, discussions, motivation, enthusiasm and encouragement with her positive energy. Her guidance delights me in the time of research and writing this thesis.
In addition to my advisor, the support of Prof. Jan Mandel allowed me to access the forceful computing resources of the mathematical department, which provided me with computer supplies and immense knowledge. I am also indebted to Prof. Long Lee, a wonderful faculty of the University of Wyoming, for his outstanding efforts in ensureing that groundworks were kept at my fingertips. I would like to thank the rest of my thesis committee -Prof. Julien Langou and Prof. Russell P. Bowler -for their perceptive comments, and questions.
Another thank you to Dr. Bedrich Sousedik who showed great interest and support in my research and was an excellent consultant in making coding a lot easier. I appreciate his patience despite my numerous and unending questions, and his helpfulness throughout my academic years.
My special thanks also go to Prof. Howard L. Schreyer for introducing us to the Defense Advanced Research Projects Agency (DARPA) group which provides an exceptional opportunity to cooperate works with the professional team and for guiding us to work on the diverse exciting project. I would also like to thank Eric Sullivan for his helps and for the stimulating discussions we were working on together before deadlines.
I greatly appreciate and wish to thank the parents of Lynn Bateman, who donated the funds that supported my study during the Fall semester of 2011, giving me the opportunity to devote myself to conducting and strengthening my research independently and proficiently.
I would like to take this opportunity to thank Dr. Christopher Harder for providing guidance in regards to the stability of finite element schemes and his assistances.
Not forgetting to thanks also to the faculties, graduate students at the University of Colorado Denver, and friends for sharing the experiences and distracting from science.
Last but not the least, I truly thank my families and my parents for always being there for me with their support throughout my life.
Vll


TABLE OF CONTENTS
LIST OF FIGURES......................................................... x
LIST OF TABLES....................................................... xiii
CHAPTER
1. INTRODUCTION ........................................................ 1
2. HOMOGENIZATION ...................................................... 8
3. PERMEABILITY........................................................ 17
3.1 Relationship between Drag Coefficient and Permeability........ 18
3.2 Discretization of the Model Problem........................... 20
3.3 Validation of Numerical Results .............................. 24
3.4 Numerical Permeability Functions.............................. 30
3.5 Comparison with Kozeny-Carman Equation........................ 33
4. MODELING THE PCL ................................................... 44
4.1 General Form of the Momentum Equations using HMT.............. 46
4.2 The Continuity Equation....................................... 50
4.3 Darcy’s Law................................................... 52
4.4 Derivation of the Brinkman equation........................... 53
4.5 Derivation of Stokes Equation ................................ 56
4.6 Well-posedness Stokes-Brinkman................................ 57
5. NUMERICAL RESULTS FOR THE FIXED HEIGHT MODEL, TWO-DIMENSIONAL
MODEL............................................................... 71
5.1 Boundary Conditions........................................... 72
5.2 Model Discretization.......................................... 74
5.3 Validation of the Code and Numerical Results ................. 79
6. FREE BOUNDARY TWO-DIMENSIONAL MODEL................................. 86
6.1 Model Problem and Boundary Conditions......................... 87
6.2 Numerical Implementation...................................... 88
viii


7. CONCLUSION........................................... 92
APPENDICES
A. NOMENCLATURE......................................... 95
B. BASIS FUNCTIONS...................................... 102
REFERENCES.............................................. 106
IX


LIST OF FIGURES
Figure
1.1 Cartoon picture of a patchwork of goblet and ciliated cells at the epithelium . . 2
1.2 The PCL and mucous layers where Q = Hc x Lc is our computational domain
with subdomains fli = Lc x Hc\ and Q2 = Lc x Hc2............................. 4
1.3 Geometry: an ideal cell of cylinders when the angle between cylinders and horizontal plane is 90 degrees; the top view of the left figure ; the angle 9 between
the cylinder and the horizontal plane........................................ 4
2.1 A reference heterogeneities Y are periodic of period tY and their size is order of e. 9
2.2 Upscaling procedure where e > 0 is a spatial parameter....................... 9
3.1 Permeability being close to zero of fluid flow through a cell array of cylinders
when the boundary is almost not Lipschitz.................................... 26
3.2 Permeability of fluid flow through a cell array of cylinders with increasing r/d,
9 = 90°. Then r/d = 1.2077 is touching....................................... 27
3.3 Diagonal values of the permeability tensor as a function of angle 9 for r/d fixed
at 1/3.......................................................................... 27
3.4 Off-diagonal values of the permeability tensor k\2 and k2% as a function of angle
9 for r/d fixed at 1/3.......................................................... 28
3.5 Off-diagonal values of the permeability tensor k13 as a function of angle 9 for r/d
fixed at 1/3.................................................................... 28
3.6 Permeability of fluid flow through a periodic array of cylinders for 9 = 0. When the flow aligns parallel to the cylinders, figures show the experimental scalar permeabilities of Sullivan for hairs and goat wool, respectively, comparing with
our numerical 2 * fc33 and 2 * (fcn + k22) results for 9 = 7t/2.............. 31
3.7 Permeability of fluid flow through a cell array of cylinders. Figures are comparing
the experimental data of Brown and Chen with our numerical fc33 and kn + k22
when the flow is perpendicular to the array of cylinders..................... 32
x


3.8 The top left graph is the numerically generated permeability component k\\ while
the top right one is the fourth-order polynomial approximation of the top left graph; the pointwise absolute error are shown at the bottom................ 34
3.9 The diagonal components k22 and k33 of the permeability tensor.................. 35
3.10 The off-diagonal components k\2, A43 and k23 of the permeability tensor........ 36
3.11 Geometry: the left figure shows the cell array of cylinders when the angle between
the cylinders and horizontal plane is 75 degrees; the right one shows each cell consists of 2 ellipsoidal cylinders........................................ 40
3.12 Figure shows the radii of an ellipse in both major and minor axises and 9 is the
angle between the horizontal plane and the side of the cylinder while hg is the height of the periodic cell which is perpendicular to the horizontal plane. 41
3.13 The green line represents the the spherical part of the permeability tensor of the
numerical result while the blue and red ones are from the functions (3.62) and (3.63), respectively, where x-axis is the porosity depending on each angle. 43
4.1 The cartoon picture is showing the fixed boundary while the cilia is moving forward and backward making the angle 9 with the horizontal plane in the PCL. 44
5.1 The left figures shows the PCL when t = t0 where the cilia is perpendicular to
the horizontal plan while the right one displays the PCL when t = t\ where the cilia make an angle 9 to the horizontal plan, where 9 is less than 90 degrees. . . 71
5.2 A two dimensional Cartesian coordinate system with axis lines X\ and x2 and the
cartoon picture of the cilia in the PCL with boundary conditions.............. 74
5.3 Velocity profiles in the x2 direction or y-axis of the exact solution and our numerical result where uo = 1; y = 1; tl = 0.64457; Hc= 1 and Hc2 = 0.7071. ... 81
5.4 Velocity profiles of the numerical and exact solutions using our permeability
results with the corresponding angle d; uO = 1; y =1; Hc =1................... 82
5.5 Convergence of the velocity profiles of the numerical results to the exact solutions
when the angle 9 is 45°....................................................... 83
xi


5.6 Velocity profile in the x2 direction of the cilia making angles 9 = 26,30, ....90° with the horizontal plane when the shear stress is zero at the free-fluid/porous-medium interface..................................................................... 85
6.1 The cartoon picture is showing the free boundary having a unknown curve s while
the cilia is moving forward and backward making the angle 9 with the horizontal plane in the PCL........................................................... 86
6.2 A two dimensional Cartesian coordinate system with axis lines X\ and x2 and
the cartoon picture of the moving cilia creating a free boundary curve s with boundary conditions........................................................ 88
B.l Nodal numbering for the 27-variable-number-of-nodes element where the origin
is at the center of the brick................................................... 102
B.2 Nodal numbering for the 3-variable-number-of-nodes element ..................... 103
B.3 Nodal numbering for the 4-variable-number-of-nodes tetrahedron element. . . . 105 B.4 Nodal numbering for the 10-variable-number-of-nodes quadratic tetrahedron element................................................................................ 105
xii


LIST OF TABLES
Table
3.1 Permeability through the cell array of cylinders at 90 degree of the different
heights and periodic REVs providing the same permeability................... 26
3.2 Permeability through the simple cubic array of spheres with solid volume fraction
ts = 0.216 and radius = 0.5 unit; ks denotes the permeabilities in this research; 5k is the relative error of ks with respect to k^M which is the permeability calculating Rocha and Cruz (2009)......................................................... 29
3.3 Permeability through the simple cubic array of spheres with varying volume fraction of solid ts with radius 0.5 unit; ks and k^M denote the permeabilities in this research and by Rocha and Cruz (2009), respectively; 5k is the relative error of
ks with respect to kuM........................................................ 30
3.4 The fourth-order polynomial functions: a\XA + a,2X3y + a^x3 + a^y2 + a^x2y + ae,x2+a7xy3+a%xy2+a9xy + aiQX + auy4:+ a\2y3+ a\zy2+aixy + aiz approximating
hi, k22 and fc33.............................................................. 37
3.5 The fourth-order polynomial functions: a\X4 + a,2X3y + a^x3 + a^y2 + a^,x2y + a&x2+a7xy3+a%xy2+ao>xy + aiox + any4:+ a\2y3+ ai?,y2+auy + a\5 approximating
h2, hz and k2z................................................................ 38
3.6 L2-norm errors of the components of the permeability k........................ 39
4.1 Characteristic parameters..................................................... 54
5.1 L2-nc>rm error of the velocity for the Stokes-Brinkman equations where 9 is the
angle between the array of cylinders and the horizontal plane............... 81
5.2 L2-norm errors of the numerical and exact solutions of Stokes-Brinkman equa-
tions when the numbers of elements are increasing where ^ dof is the number of degrees of freedom and Ax is the uniform length of each element.......... 84
xiii


1. INTRODUCTION
Cilia are hairlike organelles that line the surfaces of the ciliated cells and beat in rhythmic waves. They provide locomotion to liquids along the epithelium, the lining of the cavity of structures throughout the body. Cilia are essentially omnipresent in the human body and play a considerable role in a number of processes. Of particular interest to us are the movement of cilia in the lungs which aids in removing fluids, pathogens, and foreign particles out of the airway and out of the lungs.
Figure 1.1 shows a portion of mucosa in bronchuses and bronchioles in the respiratory system and in the lungs. The patchwork consists of goblet cells containing mucus granules, cilia and a mucus blanket. The function of the goblet cells is to secrete mucin, heavily glycosylated proteins, to form the mucus that traps the foreign particles and form a blanket at the tip of the cilia [101]. The cilia exist in a low-viscosity fluid and together this layer is called the periciliary layer (PCL), [95, 63]. The mucus, which is a highly viscous fluid, resides above the PCL [69]. To prevent mucus accumulation in the lungs, the cilia beat in an almost two-dimensional plane. The effective bending of cilia induces the movement of the PCL fluid which in turn, propels the mucus. They do this by beating backward and forward to generate metachronal waves that give the mucus a net flow in one direction. In the forward stroke, cilia are fully extended and penetrate the mucous layer. During the backward stroke, they bend close to the epithelium, the base, and rotate back to the starting point of the forward stoke. Gueron and Levit-Gurevich [45] stated that for a single cilia beating in water, the mechanical work load done during the backward stoke is five times less than the the amount of work done during the forward stoke. The process of producing mucus to entrap foreign particles and clearing them is called mucociliary clearance or muco-ciliary transport, [65, 70, 94, 101]. Further understanding of mucociliary clearance mechanisms of the respiratory tract can be found in [93].
Damage to the mucociliary clearance mechanism can occur with the absence of sufficient mucus or excess mucus [87]. For example, when the glands that produce mucus do not
1


Periciliary Layer (PCL)
ciliated cell
goblet cell
Figure 1.1: Cartoon picture of a patchwork of goblet and ciliated cells at the epithelium
function properly, thick mucus accumulates in the lungs, leading to breathing difficulties that cause airway diseases such as cystic fibrosis, asthma, emphysema, and chronic bronchitis. If the ciliated epithelium is damaged, coughing is the only mechanism for clearance [102],
There have been several models proposed for the PCL. Several papers by John Blake analytically model the movements and the velocities of a single cilium on microorganisms, [16, 17, 18, 19, 20]. In particular, Blakes 1973 and 1977 ([17] and [20]) provide a model for mucus flows in the respiratory portion and the maximum velocities of cilia in lungs, respectively. He used the Stokes equation to model the fluid and to model the cilia. He inserted a distribution of force singularities located along the center line of the cilia and calculated the velocity of a single cilium. For an array of cylinders, he assumed the average fluid velocity is equal to the spatially averaged cilia velocity in the PCL. Other analytic studies of mucous flow caused by ciliary motion include Barton and Raynor, [5], who analytically studied the mucous flow due to the movement of a single cilium and compared the results with experimental data, and Liron [64] who considered the fluid transport by cilia using Green’s function for a surface distribution of point forces of the Stokes equation (Stokeslet) between two parallel plates. His analytical method was the same as Blake but he assumed the total velocity in a domain consists of a plane Poiseuille flow and the Stokeslet distribution on all cilia to have non-zero
2


flux where the plane Poiseuille flow was considered as disposal later on. However, the integral
equations for the force distribution is not sufficient to determine the forces uniquely.
Experimentally, Rabinovitch’s [83] applied a light-transmission method to study the ciliary movement but not the velocity of the PCL fluid while Frommer and Steele [42] constructed rows of hair cell cilia bundles found in the mammalian cochlea where the pressure was required to produce a given volume of flow rate through the obstacles. Note that in their experiments the cilia was motionless or not self-propelled.
In this study, we develop a model using an upscaling technique so that we do not have to consider the motion of each individual cilium but rather what all cilia do collectively, i.e. the cilia can be viewed as a porous medium with the self-propelled solid phase, cilia, composed of a periodic array of cylinders. Figure 1.2 illustrates a two-dimensional domain of interest, the PCL and mucous layers.The goal of this research is to model the PCL as a thin porous medium with a solid phase composed of a moving periodic array of cylinders and a liquid phase composed of the low viscosity PCL fluid. The left drawing in Figure 1.2 shows the cilia when they are perpendicular to the horizontal plane, and the right illustrates the cilia at an angle 9. The length and height of our computational cell domain are Lc and Hc, respectively. We define Hc2 as the height of the cilia, which is a function of time, and Hci = Hc — Hc2. We then define
fli = Lc x Hci (1-1)
as the domain containing the low viscosity fluid without cilia and
Q2 = Lc x Hc2 (1-2)
as the domain containing cilia with low viscosity fluid. Since fl2 contains both the fluid and solid, this domain is a porous medium. Because Hci and Hc2 change in time, Hi and fl2 are functions of time as well. Note that Hi is cannot be treated as a porous medium.
In Q2 we model the cilia as a periodic array of cylinders that rotate about their base
where the height of Q2 is a function of time. Figure 1.3 shows an example of a cell array
3


Epithelium
Figure 1.2: The PCL and mucous layers where Q = Hc x Lc is our computational domain with subdomains = Lc x Hc\ and CI2 = Lcx HC2-
Figure 1.3: Geometry: an ideal cell of cylinders when the angle between cylinders and horizontal plane is 90 degrees; the top view of the left figure ; the angle 9 between the cylinder and the horizontal plane.
of cylinders and illustrates the geometry of our model. The left cartoon displays an ideal cell of cylinders when the angle between cylinders and horizontal plane is 90 degrees. The top view of the left figure and the angle 9 between the cylinder and the horizontal plane are demonstrated in the middle and right cartoons respectively. In the middle picture, r is the radius of the cylinder and cl is the distance between cylinders. The relationship between cl and r is cl = lc — 2r where lc is the length of the square domain. If lc = 0.5, the cylinders touch when the distance cl\ in the Figure 1.3 is zero or the radius r is 0.175. To find a model of the PCL, we need to calculate the permeability, a measure of the ease with which a liquid can move through a porous material.
For an array of parallel cylinders, various experimental [105, 107, 108, 14, 26, 29, 32, 98] and analytical [48, 49, 59, 96] approaches have been performed to calculate the permeability
4


or drag for flow both perpendicular and parallel to an array of cylinders, as well as numerical studies, [2, 91]. Alcocer and Singh [2] investigated the movement of viscoelastic liquids passing through the periodic arrays of cylinders in a two-dimensional domain using a finite element method. Sangani and Acrivos [91] determined solutions, such as the drag on a cylinder, for the slow flow past a square and a hexagonal array of cylinders in a two-dimensional domain. In these, authors found the permeability when the cylinders are perpendicular or aligned with the horizontal plan. Most numerical works calculated the permeability for a two-dimensional domain. In this work, although the cilia rotate three-dimensionally, they beat in an almost two-dimensional plane. Therefore, the permeability is calculated as a function of only one angle, 9, and the cylinder density in three-dimensions.
In order to obtain the permeability, homogenization is applied to the periodic cell array of cylinders to obtain a system of equations which can be solved to numerically approximate the permeability tensor. A three-dimensional mixed finite element method using Taylor-Hood elements is employed to solve the system of equations. Moreover, we provide polynomial approximations of each entry of the permeability tensor as a function of a ratio r/d and the angle 9.
For a model of the fluid flow in the PCL, we consider a coupled free-fluid/porous-medium system of equations. Typically, Stokes or Navier-Stokes equations are used to determine the flow in domain while Darcy’s Law or the Brinkman equation is employed in Q2, [33, 73, 99]. Morandotti [73] employed the Brinkman equation to model the fluid phase of a porous medium for modeling the motion of a deformating body in a viscous fluid. Chen, Gunzburger and Wang [33] compared both the Stokes-Darcy and Stokes-Brinkman equations with the same boundary conditions. They concluded that Stokes-Darcy equations with the Beavers-Joseph condition are more precise than others. Because of our slow flow problem, Stokes equation is employed in domain Q i. For the domain since there is a transition region at the free-fluid/porous-medium interface where the porous medium and free-fluid regions are adjacent, we employ the Brinkman equation. This is because the introduction
5


of an effective viscosity parameter in the additional term of Darcy’s Law in the Brinkman equation can better handle the force from stress at the interface so that solutions from the Stokes equation can be matched with those of Darcy’ law at the interface [66]. Although the Brinkman equation is more complicated equation than Darcy’s Law, appropriated boundary conditions can be applied. Moreover, it is more convenient for coding. More information about boundary conditions of the Stokes-Brinkman and Stokes-Darcy equations can be found in [3, 30, 31, 47, 56, 57, 62, 67, 68, 75, 99],
Although in practice the PCL/mucus interface is a free boundary (with the unknown height of the PCL), as a first approximation, we propose numerically modelling the PCL fluid with a fixed boundary height. The well-posedness of the Navier-Stokes equation and the Navier-Stokes/Brinkman for constant coefficients can be found in [100] and [54], respectively. The existence and uniqueness of the Stokes-Brinkman equation for the numerical problem for a tensor coefficient are shown in this work.
In Chapter 2, we discuss the permeability tensor and how it relates to a drag coefficient. We apply the homogenization method to the periodic cell array of cylinders to obtain the system of equations which is used to determine the permeability tensor. These equations are discretized in Chapter 3 and solved using a Mixed Finite Element method and where the numerical results are verified. Because the Kozeny-Carman equation is among the more popular equations to determine the permeability, we compare 1/3 the trace of the numerical permeability tensor with the Kozeny-Carman equation in Chapter 3.
In Chapter 4, we model the PCL using the Stokes-Brinkman equation. We also show the existence and uniqueness of the discretized Stokes-Brinkman equations. The numerical results of the equations are presented in Chapter 5. In the case of a constant velocity on the top of the two-dimensional domain, we proceed to compare the numerical result with the exact solution [58]. In Chapter 6, we begin to investigate modeling the PCL with a free boundary. Finally, we summarize our results in Chapter 7.
It should be noted that to the author’s knowledge this is the first time the porous media
6


equations are being used to model a fluid flowing due to the movement of the solid phase. Classical porous-media flow problems involve a static solid phase with a liquid-phase pressure gradient inducing fluid flow.
7


2. HOMOGENIZATION
In this Chapter we develop the equations that are solved to determine the permeability for the PCL where there are cilia. We choose homogenization because it is a method used to upscale governing equations e.g. Stokes equation, from the microscopic to the macroscopic scale, and has been widely used since the 1970s [13]. The strength of this method is that for a given microscopic geometry, the coefficients in the macroscopic equations can be found explicitly. In particular, by using homogenization to upscale the Stokes equation with periodicity requirement we obtain a system of equations that can be used to determine the permeability. In this Chapter, we summarize the homogenization method as following. For more details, see e.g. [13, 34, 51], and [90].
Let Q be the periodic cell domain which consists of a fluid phase, Qp, a solid phase, Qs and a piecewise continuous liquid-solid interface T = rs CTj? where and TF are the boundaries of the solid and liquid phases respectively (see Figure 1.3). We assume slow fluid flow, a fixed solid and a viscous incompressible fluid, so that the Stokes equations are applicable:
V-v = 0 in Qp (2.1)
Vp +pAv+F=0 in Qp (2.2)
where we also assume a no-slip boundary condition on Tg; p is the dynamic viscosity; v is the velocity; p is the pressure and F is a source term. Next, we assume the diameter of the cylinders, a = 2r, see Figure 1.3, is small compared to a macroscopic scale length L which can be the length of a bronchiole in the lungs; i.e. if e = a/L, then e 0 is a spatial scale parameter. When e tends to zero, we
8


Figure 2.1: A reference heterogeneities Y are periodic of period tY and their size is order of e.
e = 0.3 £ = 0.02
£ = 0.2
£ —> 0
y = x/£
Figure 2.2: Upscaling procedure where e > 0 is a spatial parameter.
9


have the macroscopic scale.
We consider an asymptotic expansion of v and p, in the form,
v = e“(v°(x, y) + ev1(x,y) + ...) and p = e/3(p°(x,y) + ep^x,y) + ...) (2.3)
where v* and p* are Q-periodic in y, and a and (3 are nonzero parameters that yield a physically meaningful solution. The choice of a and (3 yields different macroscopic models, and in this case we choose a and [3 to yield a nonzero macroscopic first-order pressure and a second-order velocity which can lead to obtaining Darcy’s Law. It is known to be a reasonable equation for modeling slow flow through a porous medium. Recall that, for three-dimensional
x = (x!,X2,x3) and y = (yi, y2, yz) = ^(aq,x2,x3), (2.4)
then we apply the chain rule to (2.4), the first and second derivatives with respect to Xj are
d d d dyj did dxj dxj dijj dxj dxj e dijj
and
d?_
dxj
c2 dy2
d ( d 1 d
dyj e dy.
d2 1 d2 \
| dyjdxj edy2)
d2 \ d2 1
dyjdxj J dx2'
dx3
(2.6)
10


Therefore, the vector Laplacian of the velocity held is
Av
' d2V4 d2Vo d2vq
dx2 ’ dx2 ’ dx2
' 1 d2vi
1 ( d2vi e \dxjdyj 1 / d2v2 + _ e \dxjdyj 1 / <92fs
<92tq
dyjdxj
(fvi dx2 '
d2v2
dyjdxj d2vz '
1 d2v2 t2 dy2 1 d2v3
t2 dy2 ' e \dxjdyj ' dyjdxj d2v1 d2v2 d2v3 "dyf'^yf'Jyf, d2Vl d2Vl
1
d2v2
d2v2 dx2 ’ d2va\
dx2 J
d2v2 d2v.
d2v,3
dxjdyj dyjdxj’ dxjdyj d2vl d2v2 d2v3 dx2 ’ dx2 ’ dx2 j
1 A 1 A
—Ayv+-Axyv + Axv
5j/j j ’ j dyj dyj dx3 j
where
and
AyV
A xyV
'<92Ui <92u2 <92u3\
<92m <92u2
Axv
'<92Ui <92u2 <92u2'
^ dx2 ’ dx2 ’ dx2 /
<92u2 <92u3
<92u3
/ <92ui
\dxjdyj ' dyjdxj’ dxjdyj ' dyjdxj’ dxjdyj ' dyjdxj
where a repeated index j within a single term indicates summation, i.e.
“[ dx2 '
Substituting (2.5) into (2.1) yields
d2v i dx2
„ dvj dvj 1 „ 1
0 = V • v = -r1- = -7T2- + = vx • vH—Vy ■ v,
dxj dxj e
where
_ dvj , ^
vx ■ v =-J- and V„ • v —
0X4
Similarly, substituting (2.5) and (2.6) into the Stokes equation (2.2) yields
+ y ^Asv+^A,sv + Axv^j + P= 0
vhjj ~
(2.7)
(2.8)
(2.9)
11


After some trials and errors we found that in order to find a non-zero and physically mean-
ingful solution, we let a = 2 and f3 = 0 so that
v = e2(v°(x, y) + ev1(x, y) + ...) (2.10)
p = e°(p°(x,y) + ep1(x, y) + ...) (2.11)
where, again, the function v% and p% are Q-periodic in the microscale y. Substituting (2.10) and (2.11) into (2.8) and (2.9), we have
V, . (eV(x,y) + £V(x,y) + ...) + iv, • (tV(x,y) + £V(x,y) + ...) = 0 (2.12)
and
- Vx(p°(x,y) + ep1(x,y) + ...)-^Vy(p°(x,y) + ep^y) + ...)
+ ^^Aye2(v°(x,y) + ev1(x,y) + ...) + ^Axye2(v°(x, y) + ev^x, y) + ...)
+ juAa.e2(v°(x,y) + ev1(x,y) + ...) + P = 0. (2.13)
Collecting the same orders of e (0(e) from (2.12), and 0(e_1) and 0(e°) from (2.13)), we
have the differential equations
< O V O (2.14)
Vyp°(x, y) = 0 (2.15)
-Vxp°(x,y) - Vyp\x,y) + pAyv°(x,y) + P = 0. (2.16)
Note that, from equation (2.15),
\dyi oy2 oysJ
Then p° depends only on x, i.e.
(ST o o (2.17)
For the no-slip boundary condition, we have v = 0, i.e.,
e2(v°(x,y) + ev1(x,y) + ...) = 0. 12


Hence
/° = 0 on r.
(2.18)
Since the domain is assumed to be periodic, we introduce a Hilbert space of H-periodic functions:
H(Q) = {to : to = (cui,ca2,ca3) G (H1(QF))3 : to is Q — periodic, to = 0 on Ts, Vy • to = 0}
with scalar product:
f dwj dujj
(w,a;)ff(fi) = /_ tt-tt-dy
(2.19)
(2.20)
lnF dyk dyk
where again the repeat index j and k indicate summation. Note that this is a scalar product because w and to are zero on the boundary T. Equation (2.16) can be rewritten in the indicial notation as
dp1
d2vf
- I -i~dy + y I ~pridy - I F~dy + / f*dy = °-
JnF oyi JnF ay* JnF dxi JnF
(2.21)
To obtain a weak form of the equation, we multiply (2.21) by a test function ay G H(Q) and then integrate the equation, we have
f dp1 f d2v° f dp0 f
- Ui—dy + y Ui-^-fdy- Ui—dy + f*Uidy = 0 (2.22)
jqf oyi Jo,]? JO-f 0Xi jQf
where, in this Chapter, the repeat index i refers to the number of equations while the the other repeat indices indicate summation and F = (/j8, /|, /f). Integrating by parts the first two terms, using the fact that ay = 0 on T, and p° and // are functions of x only, we have
r iduj* p —dy - n / ——dy - ------/ u%dy = 0
lnF dy.
lnF dy-j dy3
dxi
dui
Employing the divergence-free property of the test function, —— = V„ • to = 0, the first
dy*
integration is zero, and we have the simplified expression
y

'nF dy-j dy3
dXj j J fir
13


where we sum on j = 1,2,3 for each i = 1,2,3. Using the definition (2.20) of the inner product yields
p.(v°,oj)h = (fs — • I tody \/u G H(Q) (2.23)
J dp
Consequently, the problem given by equations (2.14), (2.16), and (2.18) is equivalent to the variational problem: Find v° G H(Q) satisfying equation (2.23).
To show that there exists a unique solution v° G H(Q) of (2.23) using Lax-Milgram Theorem, we define a(w,u) = , u)which is bilinear. Note that
a(u,u) = y(u,u)H(n) = y\\uj\\2H{n) Vca G H(Q) (2.24)
is also coercive and
o w,w
w
,u)\h(q) ^ ^llwllir(n)|w||ir(n) Vw,ca G H(Q)
(2.25)
can be shown to be continuous by applying the Cauchy-Schwarz inequality to the last inequality. Define the linear functional
and note that
\F(cut
F(tUi
r dxi.
f' ~ 5g) la w,dy
(2.26)
' dp
Uidy

dp0
dxi

(2.27)
so it is continuous. By applying the Lax-Milgram Theorem, we know there exists a unique v° satisfying (2.23).
The solution v° of (2.23) can be used to derive Darcy’s Law. Along the way to formulate Darcy’s Law we obtain a system of equations that can be employed to calculate the permeability. Applying the linearity property, we have
v° = 1 (f-5d) u.
p. V Bx<)
where u* is the only solution of the problem: Find u* G H(Q) such that
(2.28)
(u*,w)h(q) = / Uidy
J dp
(2.29)
14


for all to G H(Q). Thus, u* is the weak solution of the following strong formulation
Vy • U* = 0 in Qp (2.30)
Vy Qi + Ayid + = 0 in Qp (2.31)
u* = 0 on T, (2.32)
where Vy and Ay represent the gradient and laplacian operators with respect to the microscopic scale y, u* and ql are fl-periodic, and e* is the unit vector in the direction of the yi axis, i = 1,2,3. The solution u* of the system of equations (2.30)-(2.32) can be used to compute the permeability. Note that v^v^u* are defined on Qp. To obtain the Darcy’s velocity, It is natural to extend them to Q with zero values on f!g. Define
w\Lv°dti'
rl = uxt f v'dy, and u* = f u%dy. |D| JnF |D| JnF
(2.33)
Integrating (2.28) and then dividing by the volume of the domain Q, we have
which in the indicial notation, is

1
dxi
dp0
(2.34)
4—[f’-A n-
(2.35)
Equation (2.35) is Darcy’s Law and u* is the permeability tenser which depends on the geometry of the periodic domain Q. In general, we write equation (2.35) as
q =-(Vp-P
p
where
k
v
ir
m Llidy
(2.36)
(2.37)
is the permeability.
For example, with p = constant and the source term P = —pgVz where z is the z-axis in the Cartesian coordinate, [7] page 134, (2.36) becomes
pgk (p '
q =------V------h z
h \P9 ,
-K • Vh.
(2.38)
15


In (2.38), we observe that h = p/pg + z which is identical to the piezometric head from the Darcy’s experiment and K = pg\t/p is the hydraulic conductivity ([K] = L/T). Note that the variational form of (2.30)-(2.32) is: Find u* G H(i2) such that
/ Vyu* : Vyu) = / tOidy
/ Q p J Clp
(2.39)
for all to G H(£2) and then the permeability can be determined and validated from the formula (2.39) by replacing ca* with -u* and dividing by the volume of the domain £2. An consequence of the above statement is that tensor k is symmetric and positive definite matrix, meaning that the diagonal entries are all positive [90]. This also proves that the fluid moves in the direction of increasing head gradient. From equation (2.37), we observe that k depends on the actual size of the microscopic scale. Thus, the non-dimensional form of the system (2.30-2.32) is important and can be derived as following. For simplicity, we drop the superscript i in (2.30- 2.32) and note that both u and e are vectors for each i with scalar q. Let y* = y/a, iT = u/a2, q* = q/a and the domain £2 is mapped to Q* of unit size (a cube), a = diameter of the cilia. Hence, the dimensionless form is
Vy* ' U* = 0
- Vy Q* + Ay*u* + e* = 0 u* = 0
on Q*F on Q*F on r*
Similarly, the dimensionless permeability
| £2* | Jn]
u *dy*
and then the conductivity becomes
pga2 k*
p
K
because k = a2k*.
(2.40)
(2.41)
(2.42)
(2.43)
(2.44)
16


3. PERMEABILITY
In this Chapter, we use a mix finite element method to calculate the permeability of the system of equations obtained in Chapter 2.
In the past, H. Hasimoto [49] found theoretically periodic solutions of the Stoke’s equation using the Fourier series and the drag force acting on a periodic array of spheres and circular cylinders in a two-dimensional domain. The series do not diverge because the mean pressure gradient is applied. The values of c, page 323, in the solutions which depend on the sphere volume fraction and are written as a sum of the sphere volume fraction have been calculated numerically by many authors. Once the drag force is determined, the permeability coefficients are known as well. The drag coefficient is used to quantify the resistance of an object in the fluid region, while the permeability is the measure of the ability of a porous media that permits the fluid to pass through it. Zick and Homsy [111] also began with the Stoke’s equation and solved, using a different method from Hasimoto, for the drag force and then permeability of the periodic array of spheres. Although these authors had found the Stoke’s solutions theoretically, numerical constants are still needed to determine the drag coefficient.
In this Chapter, we numerically calculate the solution of the system of equations (2.40)-(2.42). A mixed finite element method is applied to the system of equations and the variational formulation is formulated in Section 3.2 while the validation of the code is presented in Section 3.3. For the sphere case we compare our results with that of Rodrigo and Manuel
[85] whose numerical solutions are checked with Zick and Homsy. Comparing the numerical results with experimental data leads to finding the relationship between the drag coefficient and permeability which is shown in Section 3.1. The numerical results and the polynomial approximations of the permeability tensors are demonstrated in Section 3.4. Finally, the famous Kozeny-Carman equation is also applied to find one of the closed forms of the numerical permeability, l/3trk, in Section 3.5.
17


3.1 Relationship between Drag Coefficient and Permeability
This section is written for the conversion of drag coefficient to permeability which is used in Section 3.3 comparing our numerical result with the experimental data from Sullivan, [98], who measured the drag coefficient of cylindrical fibers empirically. When dealing with porous media, we use the concept stated that the pressure gradient is the drag force. Before deriving the relationship, we present the drag coefficient of a sphere and cylinder. The drag force concept is commonly used to determine the force on a solid object due to the flow of fluid around it. By definition, the drag force Fd exerted on each object by fluid is given by [111]
Ft= i fp(x)^'(x)dx (3.1)
where
Uj = -pSij + + Vjti) (3.2)
is the stress tensor; p is the pressure; is the identity; Vij or VjP are the velocity gradients; rij is the outward unit normal vector and Tg is the surface of the solid region. Zick and Homsy, [111] showed that the drag force on a single sphere in a periodic array of spheres is
Ftd = 67t /irpCd(vs)i (3.3)
where
Vs
(3.4)
is the superificial velocity; u* is the velocity in the fluid region; |f!| is a volume of the cell; rp
is the radius of the sphere and C are used in the same way as those in Chapter 2.
For a periodic array of cylinders, where the cylinders are perpendicular to the flow, Lamb
[61] modified Oseens technique and linearized equations [78] to an array of cylinders. He
18


obtained the total drag force per unit length of a cylinder
Fd = 8irpC
(3.5)
where C = U/[2.0(2.0 — ln(Re))]; Re = aUp/p is the Reynolds number; U is the upstream velocity of fluid without the effect of cylinders; a is the diameter of the cylinder and p is the density of the fluid. When the flow aligns parallel to the cylinders, Iberall, [53] claimed that the drag coefficient of a cylinder in a cell is not the same as that of an isolated cylinder. Iberall [53] approximated the drag force on a cylinder enclosed by the other cylinders:
Fd = 4:7rpva (3.6)
where va = vs/tl is the average velocity of the fluid in a cell and vs is the superficial velocity of the fluid in a cell and tl is the porosity. To be able to fold a relationship between the drag coefficient and permeability, we follow a concept in the drag theory which states that the total drag force on the cylinders in a cell is the pressure drop across the cell, [32], i.e. the pressure gradient in Darcy’s Law can be replaced by the drag force. Since the drag coefficient of a cylinder in a cell array and that of an isolated cylinder are different, we employ the pressure drop equation of Chen [32], He included all of the drag forces on the cylinder in a cell as a pressure drop across the filter. His equation is:
Ap _ 2tsp 2u,a
7 ri ^dVa 9 ’
l 7tCv ai
(3.7)
where Ap is the pressure drop across a cell array of parallel, vertical, cylinders; l is the thickness of the cell; es is the solid volume fraction; aa and as are arithmetic average fiber and surface average fiber diameters of the cylinders in the cell respectively; Cd is the drag coefficient to be determined and Cv is a unit conversion factor, for example, Cv = 980 (g. mass cm.)/(g force sec.2) etc. As the fibers become more dense, the drag coefficient is modified via the volume fractions. For a periodic array of cylinder, our problem, (3.7) becomes
Ap
T
2 tsp
ira
Cdv
(3.8)
19


where Cv = 1 and aa = a2 = 1/a and a is the diameter of the cylinder. Using va = vs/tl as above, we have
A p
T
2 tsp
ira
ca-4v*
(3.9)
Applying Darcy’s Law to (3.9) and, for simplicity, assuming all variables are scalars, the permeability and drag force coefficient are related by
fma 1 tl 7r(l — ts)a2 1 27t(1 — ts)r2 1 ,Q .
= 2Vp~CdVa = 2 tsRe ~Cd = MAT ~Cd ^ ' >
where Re = pvaa/y and r is the radius of the cylinder. Hence,
A.fie = Liz o 1
(3.11)
2 es k*
where k* = k/r2 is the dimensionless permeability. Equation (3.11) will be used to calculate
the dimensionless permeability from the drag coefficient.
3.2 Discretization of the Model Problem
In this section, we numerically approximate the solution to equations (2.40)-(2.42) using a mixed finite element method. This method is employed to obtain an approximate solution to the system of equations using a variational formulation [84], Let
M^) = {? <= M^): f0 Qllv = M
(3.12)
where q is a scalar periodic function and L2(Q) = W2(Q) is the Sobolev space. In three-dimensions, the system of equations (2.40)-(2.41) can be rewritten as
d2u\
d2u2
d2u\
i
du\
dyi
d2u\
d2u2
d2u\
, du2 . 9u\ n dy2 dy3 (3.13)
d2u\ dql i dyi dyi Cl (3.14)
d2u2 dql i dyi dy2 62 (3.15)
d2ul3 dql i dyi dyz 63 (3.16)
20


where the superscript * is dropped for convenience and e* = 1 if j = i and e* = 0 if j yt i, i = 1,2,3. To preserve the symmetry of the stiffness global matrix, we multiply equation (3.13) by (-1). After the multiplication, (3.13)—(3.16) can be rewritten in the indicial notation as
du[
dyi
dq1
dVj
d2u'j
~dyf
Ji-3
1.2,3,
(3.17)
(3.18)
where again the double index l means the summation and Sij is the Kronecker delta. Note that the equation (3.18) can be rewritten as
d ( . du\\ ,
-^-\+6tj = 0, Z = 1,2,3. (3.19)
Applying the mixed finite element method, [84], we multiply the equation (3.17) and (3.19) by test functions (Q\ w') G L2(Q) x H(Q), respectively,
c 3ifi
-iQwr°- (3-20)
/Ai(u|(^--§9+*«) = 0' , = 1-2-3 <3-21>
where i indicates the number of system of equations. Employing the integration by parts
[21] to (3.21),
r ( 3iA\ r 3vf- ( 3iA\ r
- Jr - at) m + fr, ~a£ (^' -%) = -/„ ^ ' = !■2-3- <3-22)
Employing the periodic boundary condition, (3.22) becomes
r 3itA ( 3vi- \ r
/n% (=1'2'3' (3-23)
or
-Q
dw'j dw'j du'j
n dyj dyi dyi
wlA
jVij,
l = 1,2,3.
(3.24)
In the weak form, the problem becomes: Find (u*, ql) G H(Q) x Lq(Q) such that
- J^ (vy • u*J Qldy = 0 for all Ql G L20(Q),
(3.25)
21


(3.26)
f (y—qc Vy • w* + VyU* : VyW*j dy = f e* • w*ch/, for all w* G H(Q),
where (3.25) and (3.26) come from (3.20) and (3.24), respectively. At this point, we drop the superscript i for convenience. Note that the weight functions belong to finite-dimensional subspaces of the space H(Q) x Lq(Q). Let Q = Ug=ine, where Qe is a typical element [84], In each element, Uj and q are approximated by:
M / N
uj = 55 V,m(y)'^j = ( Ip I ... tpMJ
m= 1 ' '
< x\
U3
N ( \
q 55 ^n(y)qn (... n=1 ^ '
7 ,M
\Uj j
^TU
â– 3i
(3.27)
( \
Qi
W
*TQ,
(3.28)
where j = 1,2,3; M and N are the numbers of nodes of the quadratic and linear functions, respectively, and t[jm and (f)n are the interpolation functions.
Substituting both (3.27) and (3.28) into (3.20) and (3.24):
[ *if*>
'ne dyi
I ^$Tdy
lne dyj
U; = 0
Q 4
d^> d^T
-dy
u,
'PSijdy,
(3.29)
(3.30)
/He dyi dyi
we have j equations for each i with the summation under the repeat l. Another expression of (3.29) and (3.30) is
DiQ + (En + E22 + E33) iii = Fji (3.31)
D2Q + (En + E22 + E33)u2 = Fi2 (3.32)
D3Q + (En + E22 + E33)u3 = Fi3 (3.33)
—DT1u1 - Dt2u2 - Dt3u3 = 0, (3.34)
where superscript (-)T denotes a transpose of the vector or matrix and
<9’FT
D,
/ ?^-®Tdy,
lne dyj
D
T
1
ne dyj
-dy
(3.35)
22


r d^fd^fT
ne dyi dyi
(3.36)
E u
dy i Fjj
V&ijdy,
where Fjj = /Qe tUd?/ if * = j and Fjj = 0 if i 7^ j. Rewriting (3.31)—(3.34) into an explicit matrix form
E 0 0 -D^ ( \ ui ( f \ -t1 i 1
0 E 0 -d2 U2 f*2
0 0 E -d3 U3 f*3
Q 1 1 d CO 0 y
where E = En + E22 + E33 and and 4 are defined as in (3.27) and (3.28). popular form of the matrix (3.37) in articles [1, 4] is
(3.37)
Another
where
Q l < M AN
\ DT 0 J ■ ('! / \° /
(3.38)
( \ Ui W ( f ^ ^ i 1 ^E 0 0^
U = u2 D = d2 F = F*2 and A = 0 E 0 (3.39)
kFi3/ v° 0 Ey
It is well known that for Dirichlet conditions on u* the solution q is defined only up to an additive constant, which is usually fixed by imposing fQ q = 0, [[22], p.157 and [24], p.16]. Furthermore, in order to retain the symmetry of the matrix, we add one more column and row as following,
( \ / \
E -D 0 u F
DT 0 1 Q = 0
0 10 ve )
(3.40)
where e is very small after solving the matrix and the second order tensor is obtained from
the formula (2.43). Moreover, the permeability can be verified by replacing ca* in (2.39) by
23


id and using the symmetric property of the permeability. Then
k“ ~ M L is used as a consistency check with our numerical results kij from (2.43).
In order to ensure the Ladyzhenskaya-Babuska-Brezzi (LBB) consistency condition, we use the Taylor-Hood isoparametric 10-node tetrahedra, i.e. M = 10 and N = 4 for quadratic and linear functions, respectively. The discretization are generated by the open software Net-gen [92], A reference for three-dimensional finite element programming is Kwon and Bang [60] and other references on programming and implementing periodic boundary conditions can be found in [80] and [97].
3.3 Validation of Numerical Results
In case of a simple cubic array of spheres, Zick and Homsy [111] and H. Hasimoto [49] began with the slow flow as described by Stoke’s equation, and found the periodic solutions analytically for the drag coefficient acting on a periodic array of small spheres. Thereafter, Rodrigo and Manuel [85] convert the drag coefficient to be permeability and compare their numerical results with the analytical ones. Since the permeability is calculated in this work, we compare our results with Rodrigo and Manuel [85]. In case of parallel cylinders, three-difference experimental data from previous publications [26, 98, 32] are chosen to validate our numerical results. Note that, the permeabilities from both experimental data and analytical results for each porosity were expressed as a scalar which was supposed to be given as a matrix because of the anisotropic medium. Therefore, to compare our solutions with the experimental data, we average the components of the numerical permeability tensor to compete with those facts in which the graphs are shown below.
To validate the code, the periodicity assumption is verified by calculating the permeability tensors using three different periodic cell arrays of cylinders. The smallest cell consists of 5 cylinders, see the figure in Table 3.1, with the radius r = a/2 = 0.1, d = 0.3 and the height
h = 0.5. Then the dimension of the cell is 0.5 x 0.5 x 0.5. The other two cells contain 13 and
24


25 cylinders with the same radius r and the distance d as the smallest one. The dimension of the 13-cylinder cell is 1 x 1 x 1 and that of the 25-cylinder cell is 1.5 x 1.5 x 1.5. As is theoretically expected, the permeability tensors of the cells shown in Table 3.1 are the same for all of the different periodic REV’s. Next we allowed the radii of the cylinders to increase almost to a point of touching so that Tp is almost not Lipschitz (but the domain is still a Lipschitz function). Figure 3.1 is a figure of flp for this configuration and we see that the permeability:
^0.0006 0.0000 0.0000^
l.Oe — 04 *
0.0000 0.0004 0.0000 0.0000 0.0000 0.1077
(3.42)
tends to zero for this geometry. The Figure presents a periodic cell with 5 cylinders while the radii of the cylinders at the center and corners are 0.1 and 0.249, respectively. Furthermore, if the ratio r/d (see Figure 1.3) is increased then the values of the permeability in every direction, ku,i = 1,2,3 decrease as expected, see Figure 3.2. Note that when the angle between the cylinders and horizontal plane is 90 degree, kn = k22- as we can see in3 Figure 3.2. Moreover, for each ratio r/d, the values of k33 are twice that of kn or k22- That is the resistance is doubled when the cylinders are orientated orthogonal to the flow direction. This observation was previously stated in the theoretical part of the paper by Jackson and James, [55]. This helps to avoid the numerical difficulty when the permeabilities of both geometries need to be provided. The effect of the angle of the cylinders makes with the base is provided in Figures 3.3, 3.4 and 3.5. In Figure 3.3, we show the diagonal entries of the permeability tensor for the case of distinct angles which can be used as coefficients in governing equations while Figures 3.4 and 3.5 show off-diagonal components of the tensor. Because k is symmetric, only k\2,kn and k23 are presented in Figures 3.4 and 3.5.
To further validate the calculation, we compare our results with those of Rocha and
Cruz, 2009 [85], in which the geometry is a simple cubic array of spheres and the solutions
are obtained numerically. The geometry of the periodic cubic cell Q consists of a single
25


Table 3.1: Permeability through the cell array of cylinders at 90 degree of the different heights and periodic REVs providing the same permeability.
k
0.0017 0.0000 0.0000
0.0000 0.0017 0.0000
0.0000 0.0000 0.0038
k denotes the permeability.
Figure 3.1: Permeability being close to zero of fluid flow through a cell array of cylinders when the boundary is almost not Lipschitz
26


Figure 3.2: Permeability of fluid flow through a cell array of cylinders with increasing r/cl, 9 = 90°. Then r/cl = 1.2077 is touching.
angle
Figure 3.3: Diagonal values of the permeability tensor as a function of angle 9 for r/cl fixed at 1/3.
27


10
8
6 4
'1 2
03 0
i o 0 Q_
-2 -4 -6 -8
20 30 40 50 60 70 80 90
angle
Figure 3.4: Off-diagonal values of the permeability tensor kV2 and k2i as a function of angle 9 for r/d fixed at 1/3.
- â–  k12 : i
- - - k23 y..........;......... ......... ......... ......../_
/ / /
/ / /....
IX i i / ' / / *
■»- ■ 7 l ft M ■ V * / ' ' { i' > . ' • t V / \ * /
Y \ / t
» ' i • it
angle
Figure 3.5: Off-diagonal values of the permeability tensor kri as a function of angle 9 for r/d fixed at 1/3.
28


Table 3.2: Permeability through the simple cubic array of spheres with solid volume fraction es = 0.216 and radius = 0.5 unit; Ay denotes the permeabilities in this research; Sk is the relative error of ks with respect to Ay,<>/ which is the permeability calculating Rocha and Cruz (2009).
#dof CPU time (sec) ks Sk{%)
11,329 144.31 0.03519 1.793
81,470 16,804.78 0.03480 0.665
615,812 326,804.18 0.03463 0.173
sphere of unitary diameter at the center of Q, and has a cell length of {7r/(6e's) }1//3, where es is the volume fraction of the solid sphere. Because the sphere is isotropic, the permeability tensor is a scalar multiply of the identity, AT Table 3.2 shows the permeability for solid volume fraction ts = 0.216. The variables ks and A/; •./ = 0.03457 denote the permeabilities obtained in this research and by Rocha and Cruz (2009), respectively; Sk is the relative error of ks with respect to A // •./: #dof is the number of degrees of freedom. Note that as the number of degrees of freedom, which in this case is the number of nodes used to calculate the velocities and pressure increases, the error decreases. With approximately 80,000 degrees of freedom, the relative error is less than 0.7% with CPU time about 4.6 hours. Although the error can be decreased to 0.173% more time is required -on the order 3.8 days. Different values of solid volume fraction are compared and the results are presented in Table 3.3. For ts = 0.125, the relative error is smallest, 0.096%, but this case requires the larger number of degrees of freedom, 619,656. Although we didn’t increase the number of degrees of freedom for the larger solid volume fractions, we see from Table 3.2 that the largest relative error is only 1%. The compared results are in a good agreement for a simple cubic array of spheres.
Next, we compare our results with two different sets of experimental data for an array of cylinders. One set of the data are from Sullivan [98] in which the flow is parallel to an array of cylinders and another set of data are from Brown [26] and Chen [32] in which the flow is perpendicular to the array of cylinders. Sullivan [98] presents experimental results
29


Table 3.3: Permeability through the simple cubic array of spheres with varying volume fraction of solid ts with radius 0.5 unit; ks and Icrm denote the permeabilities in this research and by Rocha and Cruz (2009), respectively; 5k is the relative error of ks with respect to kuM â– 
es ks kRM 5k{%) #dof
0.125 0.1037 0.1036 0.096 619,656
0.216 0.03463 0.03457 0.173 615,812
0.343 0.01064 0.01052 1.140 81,660
0.450 0.004419 0.004398 0.477 78,740
for flow parallel to cylindrical fibers such as glass wool, blond and Chinese hair, and goat wool. The data from Chen [32] are given in terms of the drag coefficient. For this reference we use (3.11) to fold the permeabilities. Moreover, when the array of cylinders aligns parallel with the flow, the permeability is twice that of the perpendicular orientation, i.e. 2k(6 = 7t/2) = k(6 = 7r), [55]. Then, we only calculate our numerical results when the array of cylinders is perpendicular to the flow direction to avoid the mesh generation problems and compare the results with the experimental data. Figure 3.6 showing the plots of our numerical results 2fc33 and 2(fcn + k22) and experimental data from Sullivan [98] displays the harmonization. Similarly, when 9 = ir/2, the data from Brown [26] and Chen [32] are compared with our numerical results in Figure 3.7. The data compare well with fc33 and k\\ + k22â–  Note that the data from Chen oscillate because he collected data from several publications.
3.4 Numerical Permeability Functions
In this section, we provide polynomial approximation of each entry of the permeability tensor as a function of the distance between the cylinders, the radius of the cylinder and the angle the cylinders make with the base. Figure 3.8 illustrates the idea of how the permeability varies with the geometry. The top left graph presents the numerical permeability in the
30


Sullivan (glass wool, blond and Chinese hair)
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8
solid volume fraction
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4
solid volume fraction
Figure 3.6: Permeability of fluid flow through a periodic array of cylinders for 9 = 0. When the flow aligns parallel to the cylinders, figures show the experimental scalar permeabilities of Sullivan for hairs and goat wool, respectively, comparing with our numerical 2 * fc33 and 2 * (fcn + k-22) results for 9 = tt/2.
31


Brown(Glass Wool)
0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2
solid volume fraction
Figure 3.7: Permeability of fluid flow through a cell array of cylinders. Figures are comparing the experimental data of Brown and Chen with our numerical and kn + A^2 when the flow is perpendicular to the array of cylinders.
32


direction x or ku as a function of the dimensionless quotient between the radius of a cylinder, r, and the distance between them, d (x-axis); and the angle 9 (y-axis) between cylinders and the horizontal plane defined in Figure 1.3. The variable r takes on values between 0.025 and 0.125 and the angle 9 takes on values between 26 to 90 degrees.
Next, we approximate each permeability component by a fourth-order polynomial of the form a\XA + a2x3y + a3x3 + a4x2y2 + a5x2y + a6x2 + a7xy3 + cigxy2 + a$xy + awx + any4 + a12y3 + a-132/2 + a14y + 0,15, where the coefficients a*, i = 1, 2,..., 15 are provided in Tables 3.4 and 3.5. For the right top of Figure 3.8, we estimate the numerical permeability tensor using the fourth-order polynomials. The absolute errors and the two-norm are expressed on the bottom one. Similar graphs can be obtained for the other components. Figures 3.9 and 3.10 show the numerically generated graphs of the diagonal components k22 and kgg and the off-diagonal components k\2l A 3 and k2g, respectively. The L2-norm for the difference between the numerical results and polynomial approximations for each component is presented in Table 3.6.
3.5 Comparison with Kozeny-Carman Equation
The Kozeny-Carman is a famous equation which can indicate an empirical relationship between the porosity and the isotropic, scalar, permeability [28],
k
(3.43)
from the relationship between the superficial velocity and the pressure gradient
(el)3g Ap Vs = hyS2~’
(3.44)
where k\ is a constant depending on the geometry; S is an area of particular surface per unit volume of packed space, [S'] = [1/L]. Because this expression has been widely used [15, 27, 35], we compare our numerical results with this relationship, even though the isotropy assumption for an array of parallel cylinders is clearly not valid. For the parallel cylinders,
33


11
polynomial approximation of
The pointwise error and the two-norm = 2.543e-04
x 10"
o
LU
0.5
Figure 3.8: The top left graph is the numerically generated permeability component kn while the top right one is the fourth-order polynomial approximation of the top left graph; the pointwise absolute error are shown at the bottom.
34


permeability
Figure 3.9: The diagonal components k.22 and of the permeability tensor
35


permeability permeability
Figure 3.10: The off-diagonal components ki2, k^ and k23 of the permeability tensor.
36


Table 3.4: The fourth-order polynomial functions: a\XA + a2x3y + a3x3 + a4x2j/2 + a5x2y + a^x2 + ajxy3, + a%xy2 + a$xy + aiox + any4 + ai2j/3 + a43j/2 + a44j/ + a45 approximating fcn, ^22 and fc33.
Coefficients k\\ k22 CO CO
a4 1.0198e + 000 984.1116e -003 1.3062e + 000
«2 —255.7726e -006 —2.5282e -003 —8.1104e -003
a3 — 1.3507e + 000 — 1.0963e + 000 — 1.2944e + 000
a4 —18.5001e -006 —12.9402e -006 —24.0304e -006
a§ 2.4220e -003 4.4395e -003 11.9560e -003
CLq 671.5718e -003 425.4400e -003 365.6415e -003
aj —151.3900e -009 3.2985e -009 279.1924e -009
ag 44.3630e -006 12.5010e -006 —29.5497e -006
ag —3.8002e -003 —2.8409e -003 —3.4598e -003
a4o —112.1466e -003 —50.8733e -003 —34.9274e -003
an 258.6547e -012 —2.4185e -012 176.7506e - 012
&12 —10.3378e -009 —5.9015e -009 — 170.4837e -009
—8.0786e -006 —2.1735e -006 22.2997e -006
ai4 905.0796e -006 566.7804e -006 —274.0899e -006
ai5 804.5324e -006 —2.7328e -003 8.2408e -003
where a*, i = 1,2,15 are the coefficients; x = r/d; y = 9 which is defined in Figure 1.3; r G [0.025,0.125] and 9 G [arctan(0.5), arctan(inf)], about 26 to 90 degree.
37


Table 3.5: The fourth-order polynomial functions: a\XA + a2x3y + a3x3 + a4x2j/2 + a5x2y + a^x2 + ajxy3, + a%xy2 + a$xy + cl\qx + any4 + a42t/3 + a43j/2 + a44j/ + a45 approximating k\2, fc43 and k2g.
Coefficients kn kn CO
a4 —499.8635e - 006 297.6621e -003 428.9162e — 006
a2 —9.2789e - 006 1.2256e -003 — 1.4217e — 006
a3 933.9637e — 006 —459.2711e -003 —371.3372e — 006
a4 80.3350e — 009 —36.4347e -006 —5.9070e — 009
— 1.1085e — 006 2.9797e -003 1.9071e — 006
^6 —305.3777e - 006 167.6087e -003 76.7183e — 006
aj —393.8843e - 012 — 10.4710e -009 —78.2267e - 012
ag 18.3621e — 009 31.5771e -006 17.8581e — 009
ag —251.6399e — 009 —3.1336e -003 — 1.5587e — 006
a4o 39.8263e - 006 11.6868e -003 20.6851e — 006
an 754.8963e - 015 580.3406e - 012 16.5870e — 015
&12 —33.8628e — 012 — 140.4834e -009 21.7867e — 012
^13 —2.2598e — 009 5.7977e -006 —4.9221e — 009
ai4 147.4440e - 009 251.2534e -006 349.0144e — 009
&15 —3.5492e — 006 —4.1563e -003 —6.8813e — 006
where a*, i = 1,2,15 are the coefficients; x = r/d; y = 9 which is defined in Figure 1.3; r G [0.025,0.125] and 9 G [arctan(0.5), arctan(inf)], about 26 to 90 degree.
38


Table 3.6: L2-norm errors of the components of the permeability k
components L2-norm error
k\\ 2.54e - 04
&22 1.32e - 04
CO CO 2.48e - 04
k\2 2.25e - 06
k\s 2.lie - 04
CO l.lOe — 06
there is a one-to-one relationship between the angle and porosity that can be determined analytically.
Most applications are using the porosity to implement to their problems as a primary variable to find the dependent value of the permeability. Kozeny-Carman equation indicating the relationship between velocity and gradient of pressure provides a specific form of the permeability. Geoscience communities sometime use this equation to calculate the permeability for their porous media problems. Therefore, in this section, a formula of the permeability functions associated with the Kozeny-Carman equation are expressed and compared with our numerical solutions.
To derive the formula of the permeability using Kozeny-Carman equation, it is necessary to know the volume of the cylinders changed depending on the angle that the parallel array of cylinders makes with the horizontal plane in the periodic cell. Figure 3.11, on the left, shows the cell array of cylinders with each cell consisting of 2 ellipsoidal cylinders: one in the middle and another one from the sides after bonding them together. The volume is used to determine the total volume of the cell which is applied to attain the variable S in the
39


Figure 3.11: Geometry: the left figure shows the cell array of cylinders when the angle between the cylinders and horizontal plane is 75 degrees; the right one shows each cell consists of 2 ellipsoidal cylinders.
formula (3.43):
k = ^ = ck{e'f
hS2 " A2 ’
where Ck = g/k\ is a constant and
surface area of solid staying in fluid phase total volume of the cell
Because the surface area of an ellipsoidal cylinder is
Sa
Vt'
(perimeter of shape A)L + 2(area of shape A),
(3.45)
(3.46)
(3.47)
where the length of the cylinder L and the cross-sectional area A are defined in Figure 3.12 and the fluid is past only on the side of the ellipsoidal cylinders.
Sa = 2L (perimeter of shape A), (3.48)
where only two ellipsoidal cylinders are in the cell. For the perimeter of an ellipse, there are many formulas can be used. In this work, we employ only two different ones:
3(r + ri) - \J(3r + ri)(r + 3ri) ,
7T
(3.49)
40


Figure 3.12: Figure shows the radii of an ellipse in both major and minor axises and 9 is the angle between the horizontal plane and the side of the cylinder while he is the height of the periodic cell which is perpendicular to the horizontal plane.
and
(3.50)
where r and rq are the radii of the ellipse A, defined in Figure 3.12 and r also the radius of the cylinder. The equation (3.50) is simpler but rq should not more than three times longer than r. However, equation (3.49) presented by Indian mathematician Ramanujan provides better approximation. Substituting (3.49) into (3.48), we have
5
A
2ttL
3(r + rq) - y/(3r + rq)(r + 3rq)
(3.51)
while the second formula gives
Sa = 4:irL\
IT Ti
From Figure 3.12, it’s obvious that the radius of the ellipse
he
r i =
sind
and L
sind
Therefore the volume of the 2 cylinders in the cell is
2tt
Vs = 2-Krnhe = —^r2he
sind
(3.52)
(3.53)
(3.54)
41


where he is the height of the cell which is a function of the angle 9, defined in Figure 3.12. Hence, the total volume of the periodic cell
Id



2tt
-r he
(3.55)
VsjVt es 1 — tl sin0(l — tl)
where tl and ts are fluid and solid volume fractions, respectively. Substituting (3.51), (3.53), and (3.55) into (3.46), we have
5
{2nhe/sin0) 3(r + rq) — J(3r + ri)(r + 3ri
(27rr2he/sin9(l — tl
1 -e
rsind
3 ( r + - \l ( 3r
r + 3-
sin9 J V V ' sin9 J V ' sind 3 (sind + 1) — yj(3sind + 1) (sind + 3)
(3.56)
(3.57)
(3.58)
Substituting (3.58) into (3.45), we have
k = Ck--------------------------------
(V)3r2sin 2t
(1 — t1)2 3 (sind + 1) — y (3sind + 1) (sind + 3) Similar process is applied to (3.52). Then
2 •
(3.59)
5
(4irfte/sin9)^(r2 + (^5)2)/2 v^(l - e‘) /. M , , (27r/sin= rsin# VSm °+ 1
(3.60)
Substituting (3.60) into (3.45), we have
kl = Ck i
(tl)3r2sin2t
(3.61)
2(sin2d + 1)(1 — t1)2
where Cki is a constant. Next, we apply the curve fitting method to fit the best curve to the numerical data. Least-squares regression is one technique to accomplish this objective. In this work, the curve fitting method from Matlab is used to determine the constant in the linear relationship between the numerical results and the formula 3.59 or 3.61. With small adjustment in the denominator, we have the permeability
-tr(k) = k = 1.6147-
(d)3r2sin2!
(1 — tl + 0.6)2 3 (sind + 1) — J(Ssind + 1) (sind + 3)
+ 10“4 (3.62)
42


2.5
x 10 3
porosity
Figure 3.13: The green line represents the the spherical part of the permeability tensor of the numerical result while the blue and red ones are from the functions (3.62) and (3.63), respectively, where x-axis is the porosity depending on each angle.
and
-tr(k) = kl
(e’fg (e’f
= constantv
2.0128 (el)3r2 sin2t
hS2
S2 2(1 — tl + 0.7)2(sin26l + 1)
10
-4
(3.63)
which are shown in Figure 3.13 when the radius r of the cylinders is 0.1. As we can see there is excellent agreement between the Kozeny-Carman expression and 1/3 trace of the permeability.
43


Forward stroke
Backward stroke
Mucus
Mucus
Fluid velocity
PCL
Figure 4.1: The cartoon picture is showing the fixed boundary while the cilia is moving forward and backward making the angle 9 with the horizontal plane in the PCL.
4. MODELING THE PCL
The purpose of this Chapter is to develop the governing equations that will be used to model the PCL, the layer containing both cilia and a low viscosity fluid. We assume the PCL height is fixed and consider a two-dimensional domain. Figure 4.1 illustrates the fixed height PCL and mucous layer. The red arrows indicate the cilia beating backward and forward generating metachronal waves that propel the mucus out of the lung while the green arrows demonstate the fluid flow in the PCL. For all angles except the ones close to 90°, there is a layer of PCL fluid between the tip of the cilia and the mucous layer.
Note that the densities of the PCL fluid and the mucus are different. If the difference in densities of the contiguous fluids is large, the free boundary, an interface whose location is a priori unknown, occurs at the PCL/mucus interface and modeling this will be discussed in Chapter 6. At present we assume that the PCL height is fixed.
In this model, we decompose the PCL into two domains, Hi and Q-2• The domain Hi is defined to be the region that has no cilia while n2 consists of cilia, see Figure 1.2. Because Qi is not a porous medium, there is no permeability tensor in this region. In Q-2, the permeability has a finite value. Typically, the Stokes or Navier-Stokes equations are used in Qi and Darcy’s Law with the Beavers-Joseph condition in ff2, [3, 47, 62], In this study, we
44


use the Stokes-Brinkman equations which are employed by several authors, [57, 67, 68, 75].
Next, we briefly clarify the difference between Darcy’s Law and the Brinkman equation and why the Brinkman equation is employed for this problem. Darcy’s law is typically employed where viscosity and inertial effects are negligible, and where the fluid flowing through the porous media is considered slow. Darcy’s Law [37] is
k
vd =----[Vp-pg], (4.1)
p
where vy is Darcy’s velocity; p is the fluid density and g is the gravity. Brinkman [25] claimed that in some cases the viscous shearing stresses acting on the fluid in Darcy’s law are not negligible so that an additional term should be included. To rigorously determine the form of Darcy’s Law incorporating viscous sheering stress, we use Hybrid Mixture Theory (HMT), [10, 12, 36, 103], and nondimensionalization to obtain form of the Brinkman equation, see Section 4.4:
pk”1 • (eV - eV) + Vp - y V • (2e'd') = pg, (4.2)
c
which is the equation used in domain For the divergent-free continuity equation, (4.2) is consistent with equation (61) in [106] except that (4.2) includes the term, elvs, on the left-hand side whereas in [106] it is assumed the velocity of the solid phase is zero. The term (p/e*)V • (2eldl) comes from the liquid phase stress tensor and with this term the generalized Darcy’s Law is called the Brinkman equation. This extra term helps to match the tangential stress acting on the fluid at the free-fluid/porous-medium interface.
To see the connection with the Stokes equation, if we let k-1 go to zero in domain kb, (4.2) can be rewritten as
Vp-4v-(2ehl*)=pg. (4.3)
c
If tl is a constant in space, we have
V • (2£'d') = epr + gy = epv + gy = A(eV) + e‘v(v • A (4.4)
45


Since we assume that the fluid is incompressible
V • v* = 0 (4.5)
in domain Hi. Therefore (4.3) becomes
Vp-pg= ^A(eV), (4.6)
c
which is the Stokes equation. The typical derivation of the Stokes equation is shown in Section 4.5 by beginning with the Navier-Stokes equation with the porosity tl = 1.
Coupling the Brinkman equation (4.2) with the continuity equation gives [104]
+ (eV-vs)) = 0, (4.7)
and we now have the system of equations in the porous medium domain while (4.5) and (4.6) are the equations in . Equation (4.7) is derived in Section 4.2, where il is the material time derivative of the porosity with respect to the solid phase, il = del/dt + vs • Veh In Section 4.1, we derive the general form of the macroscale momentum equation by averaging. The continuity equation (4.7) is derived in Section 4.2. Darcy’s Law and the Brinkman equation which are specific forms of the momentum equation, are derived in Sections 4.3 and 4.4, respectively. The Stokes equation is developed in Section 4.5. Finally, the well-posedness of the Stokes-Brinkman system of equations is shown in Section 4.6 for a second-order permeability tensor. Previously this had been shown only for a scalar coefficient.
4.1 General Form of the Momentum Equations using HMT
Hybrid Mixture Theory (HMT) uses averaging theorem to derive macroscopic held equations and then exploits the entropy inequality to derive constitutive equations. To transfer a microscale variable to a macroscale variable is defined in term of the intrinsic phase average. That is the average of the microscale variable weighted by the volume of the phase. In this section, we briefly describe the derivation of the macroscale conservation of mass and momentum balances for each phase using HMT [9, 12, 74, 103]. We start with the conservation
46


of mass at the microscale
—+ V-pv = 0. (4.8)
Define the indicator function
I 1 if r G 8Va
1a(r,t) = l (4.9)
|^0 if r G 8Vp, f3 7^ a,
and 5V as the representative elementary volume (REV) [6], and 8Va denotes the portion of 5V in the a—phase and 5Vp is the portion of 5V in the f3—phase. Multiplying (4.8) by 7a, integrating (4.8) with respect to 5V and dividing by the volume |£V|, we have
1 [dp If
W\ L ~sfadv + PT L v'{pvhadv = °- (4-10)
The averaging theorem [12, 38] tells us how to interchange the partial derivatives and the integral,
1
\W\
1
W\
L = m
f Vxf'yadv(£) = V,
Jsv
1
|(bV| Jsv 1 f
fjadv(C)
fladv{i)
-J2t7T7\ L /w^-n«da(C) (4.11)
8Va \dV\ J5A<*(3
fra \SV\ JsA‘
£ 1^1 fsA

(4.12)
aj3
\8V\ Jsv
where / is the quantities in the held equations, and 8Aap, wap and na are the portion of the a/3 interface within 8V, the microscopic velocity of interface a/3, and the outward unit normal vector to SVa, respectively. Applying the theorem to each term in (4.10) gives
1
f dP i cl
Jsv ot ot
d
\SV\
1
1
\SV\
Therefore (4.10) becomes d_ \ 1
m
\8V\ Jsv 1
Pladv
pvjadv
53 | tt ri [ P~wafi ■ nada, 1^1 JsA°f>
1
[ V • (pv)-fadv = V- -ry-— [ pvjadv + V —— [ pv ■ nada. Jsv [\8V\Jsv \ \8V\JsAap
(4.13)
(4.14)
|dV| Jsv 1
1^1
Pladv
V-
|dV| Jsv
pV'fadv
53 iTDT f dw«4 ■ n»da - 53 77777 / dv • n»da
\SV\ J5AU*\5V\ J5A
(4.15)
47


Define
and J>“ = 1,
\8V\
Pa = 7777-7 / Pladv
a
\5Va\ Jsv ‘
--------- f
Pa \sva\ Jsv 1 r
pV'fadv
e« = 7i«yL„,Ww“s_v)l'n“‘i0'
(4.16)
(4.17)
(4.18)
(4.19)
so that the macroscale equation of the conservation of mass (4.15) can be rewritten as
d(eap'
dt
+ V • (e“p“v“) = e"p"e
a^a
P-
(4.20)

Next, we apply the same process to the conservation of momentum at the microscale
<9(pv
dt
+ V-(pvv) - V • t - pg = 0,
(4.21)
where t is the Cauchy stress tensor. Multiplying (4.21) by 7a, integrating (4.21) with respect to SV and dividing by the volume |£D|, we have
1
f d (pv) If If
L -7T~hdv - To L v â–  + To L v â– (pvvhadv
|£D| Jsv dt
m
m
m
/ pgjadv = 0.
Jsv
(4.22)
Applying the averaging theorem to each term in (4.22), we have
1 f <9(pv) d
\SV\ Jsv dt 7“ V=dt
w\Lv'HJv=S7'
JSV V • (pv\)-fadv = V-
1
|£D| Jsv
1 r
\6V\
Therefore (4.22) becomes 1
\5V\ Jsv
1 r
|£D| Jsv
pvjadv - V —— [ pvwa/3 • nada, (4.23)
J 1^1 JsA^
tjadv + V [ tnada, (4.24)
pw'jadv + JO 77777 [„ pwnada. (4-25)
d_
dt
\SV\ Jsv
pv~fadv
- V-
\SV\ Jsv
17a du

V-
|(iD| JsAa
\SV\ Jsv
pW'Yadv
48


'\SV\
[ pgjadv = V [ pvwa/3 • nada + V [ tnada
Jsv \SV\ JsAap \SV\ JsAap
Ppa
~ J2 r^Ti [ pvvnada.
Ua \sv\ JsA
Defining
Da d „ „
— =----1- v“ • V
Dtdt
1
pgjadv,
Pa \sva\ Jsv‘
■ / t'yadv - —— / pvvjadv + p“v“vc JdV \dva\ Jsv
\sva
1
L ft
Pa \dVa\ JdAap equation (4.26) can be rewritten as d
/ [t + pv(wa/3 - v)] • nada - v“e
JSAae
.a za
'4>
(4.26)
(4.27)
(4.28)
(4.29)
(4.30)
dt
p“v“) + V-(e“p“v“v“) - V-(e"t") - e“p“g“ = 53 e“p“ (f “ + e^v“) , (4.31)

where
(4.32)
I] e“p“ (T“ + e^v“) = ^ ( [t + pv(wa/3 - v)] • nada.
p+cc P+a \0V\J5Aal3
To simplify the macroscopic scale of the linear momentum equation (4.31), we begin by applying the chain rule to the first two term of (4.31) and then employing the macroscale continuity equation (4.20) to replace the time derivative and advection terms, d (e"p") /dt + V-(e“p“v"), by the interactive quantity at the interface a/3, e“p“e|), as follows.
d
53 e“p“ (TO + epa) = — (e“p“v“) + V-(e“p“v“v“) - V-(e“t“) - e“p
Ppa

+ (e“p“) + V-(e“p“v“)v“ + e“p“v“ • Vv“ - V-(e“t“) - e“p°
5 (e"pc
dT
■ V-(e“p“v“'
5
v“ + (e“p“)-v“ + e“p“v“ • Vv" - V-(e“t“) - e“pc
44“
53 e“p“eOv“ + (e“p“) ^v“ + e“p“v“ ' Vv“ - V-(e“t“) - e“p
.qj.q\ a
(4.33)
Canceling 53 e“p“eOv" from both sides of (4.33) and applying the material time derivative definition (4.27), yields
e“p“
Davc
~df
- V-(e"t") - e“p“g“ = 53 e“p“T
(4.34)
«44
49


which is the general form of the macroscopic scale of the momentum equation.
4.2 The Continuity Equation
The continuity equation is the conservation of mass, and the goal of this section is to derive equation (4.7) by using simplifying assumptions. We first begin with the macroscale continuity equation
d(ea.na)
V 1 J + V • (e“p“v“) = ^ e“p“e“.
cJt /34«
Applying the chain rule to the second term of (4.35), we obtain
d(eapc
(4.35)
dt
+ v"V • e“p“ + e"p"V • v“ = XI e“p“e
a^a
4'
(4.36)
44“
Using the material time derivative definition (4.27) to (4.36), we have
Da(eapc
dt
+ e"p"V • v“ = J] tapae/j.
44“
(4.37)
Our first assumption is that there is no chemical reaction between the liquid and solid phases, so that the right-hand side of (4.37) is zero and it becomes
Da(tap°
dt
+ e"p"V • v“ = 0.
(4.38)
We next assume both the liquid and solid phases are incompressible so that Dapa/dt = 0. Then (4.38) written for both liquid and solid phases becomes
Dlel
~Dt
Dsts
~Dt~
+ tlV â–  V1 = 0,
esV • vs = 0.
(4.39)
(4.40)
Since the volume of the solid can be written in terms of the liquid volume fraction, es = l — tl, we can eliminate es from (4.40)
Dsel
~W
+ (1
- el)V â–  vs
0.
(4.41)
â– 50


Adding (4.39) and (4.41), we obtain
Dltl Dstl Dt dt
+ e*(V • v* — V • vs) + V • vs = 0.
(4.42)
From the definition of the material time derivative:
Dl d , „ Ds
— = — + v • V and — —
Dt dt Dt

(4.43)
we have
Dltl dt1 , Dsd
----= — + V • Ve' and------------
Dt dt Dt

(4.44)
Adding the two equations of (4.44), we get
Dlel Dsel Dt Dt
Dlel Dsel
(V - vA • Vel
Eliminating the term
Dt Dt
by combining (4.45) and (4.42) together gives
(4.45)
(V - vs) • Ve* + e'V • (v' - vs) + V • vs = 0,
(4.46)
or
V • eV - vs) + V • vs = 0.
(4.47)
So we have
V • vs = —V • e*(v* — vs), (4.48)
and substituting (4.48) into (4.41), we get
^-+ (1 - e')V • eV - vs) = 0, (4.49)
which is the form of the continuity equation for an incompressible solid and liquid phases that we will use.
51


4.3 Darcy’s Law
Darcy’s Law is a widely used concept used to model flow through a porous medium. In this section, we derive a generalized form of Darcy’s Law using HMT. We assume we have only two phases, solid and liquid. Thus equation (4.34) can be written as
ey
DW
~Dt
- V-(eV) -eVg‘ = t'p'T
l J„l
l
(4.50)
where T(, is the rate at which momentum is exchanged from the solid phase to the liquid phase. Note that at this point we haven’t incorporated any constitutive equations. By exploiting the entropy inequality and linearizing T(, about the variable vl — vs where l and s stand for liquid and solid phases [103, 109], we obtain
elpltls = pVt1 - e'R • (v* - vs)
(4.51)
where R is second-order tensor resulting from linearization. Now, we consider the macroscale Cauchy stress tensor on the left-hand side of (4.50). The equation for the stress tensor (4.29) involves microscale variables. Darcy’s Law is a macroscale equation so we cannot use (4.29) to obtain Darcy’s Law. By exploiting the entropy equation we obtain that the constitutive equation for the liquid stress tensor is given by
t' = -pl + 2pd', (4.52)
where p and cb are the pressure and the rate of deformation for the liquid phase. Substituting (4.51) and (4.52) into (4.50), we have

elpl—-----V-(—e'pl + el2pdl) - elplgl = pVt' - e*R • (v* - vs
Dt
(4.53)
Note that
V • (elpl) = tlV ■ (pi) + pi • (Ve*) = e'Vp + pVt1. (4.54)
52


Thus, (4.53) can be rewritten as
Canceling plVtl from both sides and neglecting the inertial and viscosity terms, we have
tlVp - t'plgl = -e'R • (v* - vs)
(4.56)
where g* is the gravity vector g = (0,0,— g). Dividing both sides by tl and letting R ptlk_1, we have Darcy’s law:
k
• (Vp - pg)
(4.57)
where the supersript l is dropped from the gravity term in order to be consistent with the notation in the previous sections.
By employing the general form of the macroscopic scale of the momentum equation
to the liquid phase, we now have the widely used model, Darcy’s Law. Next, we show that Darcy’s law can be extended to the Brinkman equation.
4.4 Derivation of the Brinkman equation
In this section, we derive the Brinkman equation by using HMT and nondimensional-ization. We begin with the version of Darcy’s Law given by (4.55):
(4.34) and linearizing the rate, T(,, at which momentum is exchanged from the solid phase
Subtracting pVt1 from both sides, dividing both sides by tl and letting ptlk 1 = R, we have
(4.59)
or by rearranging the terms,
53


Scaling Parameter Description Primary Dimensions
L characteristic length [L]
f characteristic frequency m
Vo characteristic speed [L/t]
Po reference pressure [M/{Lt2)]
go gravitational acceleration [L/t2]
Table 4.1: Characteristic parameters
where + v* • V is the material time derivative. Next, we normalize equation (4.60)
and show that some terms in the equation can be neglected with respect to our problem. To normalize (4.60), we choose scaling parameters defined in Table 4.1. Define the following dimensionless variables:
v‘ p ~ g p = —, g = —, (4.61)
Vo Vo O O
TV, f = ft, A = L2 A. (4.62)
Substituting (4.61)-(4.62) into (4.60), we get
dvl ~dt
1 ILV o
2
pfvo^r + ^ • Vv* + V0jjkr\tlvl - elvs) +
L
V • (2eldl)
ti p2 v ^ ^ > P9og-where dl = ^ ((Vv)T + (Vv)j. Multiplying (4.63) by k/pvo, we have
eV - eV
k dV fju o
pv\l. w
(4.63)
<4ii4>
p" ' tJt L Lv0 ~ v0 el L2
For our slow flow problem, we choose the reference time t to be the time it takes the cilia to go through one cycle, L to be the height of the cilia, p and p, to be the density and dynamic viscosity of water at temperature 40 °C and g0 to be the Earth’s gravity. Then
t ~ 1.029 s, L & 7 x 10 6 m, p & 992.2 kg/m3 g0 = 9.81 m/s2
(4.65)
54


p & 0.653 x 10 3 kg/(m • s), p0 = 10 1 kg/(m • s2)
(4.66)
where the units are in International System of Units, SI. The values of time t and length L are from [77] and [20], respectively. Then the characteristic speed at the tip of the cilia is
T t
v0 = — = (7 x 10-6)(tt/4)/1.029 w 5.34 x 10“6 m/s, (4.67)
where 9 is the angle between the cilia and the horizontal plane. For the permeability and porosity, we employ the maximum values from our numerical results. Therefore the scalar permeability and porosity are
k = 10“14 m2, tl = 1, (4.68)
respectively. For our convenience to determine the values of each coefficient, we rewrite (4.64) as
C\{t'vl - elvs) = -C2^ - c^1 • Vv* - C4Vp + C5g + U6V • (2tldl), (4.69)
where the coefficients after substituted by the values from (4.65), (4.66), (4.67) and (4.68) are
C\
c2
Co
C4
C5
1
kpf
p
kpv o pL
kpo
pv0L
kpgo
pv o
k
Tie
1.47 x 10“8 1.15 x 10“8 = 4.09 x 10“2 2.79 x 10“2 2.04 x 10“4.
(4.70)
(4.71)
(4.72)
(4.73)
(4.74)
(4.75)
The coefficients C2 and C3 are relatively small in comparison with the others. Neglecting these terms, (4.60) becomes
pk-1 • (eV - eV) + Vp - ^V • (2eldl) = pg.
c
(4.76)
55


which is the Brinkman equation.
4.5 Derivation of Stokes Equation
In this section, we nondimensionalize the Navier-Stokes equation to show that some terms in this equation are negligible for our problem in domain fR We begin with the Navier-Stokes equation which is the differential equation for conservation of momentum for a fluid with stress given by a constitutive equation for a Newtonian fluid. For incompressible flow, it can be written in the form
(<9v \
p ( -7^- + v • Vv 1 = - Vp + pg + pAv (4.77)
Using the same characteristic parameters as given in Table 4.1, the nondimensional variables are defined to be
v
V
v
Vo'
p
p_
Po'
g
go'
LV, t = ft, A = L2A.
(4.78)
(4.79)
Substituting (4.78)-(4.79) into (4.77), we have
/<9v\ pv2 / p0f7. . ~ . pv0~
pfyo ( -gf ) + -j- ' VvJ = --Vp + pp0g +
(4.80)
Multiplying (4.80) by L2/pv0, we obtain pfL2 (dv\ pLv0
v • Vv
P°-^X7~ , P9oL2 ~ .
-Vp 4------g + Av.
(4.81)
p \dt J p v y pv0 pv0
Note that the coefficient in front of v • Vv is the Reynolds number. Using the values from (4.65)-(4.67), we obtain the coefficients:
PfL2 p
pLv0
p
PqL pv 0
7.23 x 10“5, 5.67 x 10“5, 2.00 x 102,
(4.82)
(4.83)
(4.84) 56


—— = 1.36 x 102. (4.85)
/iv0
Neglecting the terms associated with coefficients (4.82)-(4.83), equation (4.77) becomes
Vp — pg = p Av, (4.86)
which is the Stokes equation.
4.6 Well-posedness Stokes-Brinkman
Ingram [54] proved the well-posedness of the Stokes-Brinkman system of equations for a positive constant permeability. Here we show the existence and uniqueness for a permeability tensor k. Recall the Brinkman and continuity equations are
pk"1 • (eV - eV) + Vp - -^V • (2e'd') = pg, in Q (4.87a)
t' + (l-e')V-(e'(v'-vs)) = 0 in Q (4.87b)
where d* = 1/2(Vv* + (Vv*)T) is the rate of deformation and the superscript T is the transpose. The velocity of the liquid vl and the pressure p are unknown. Since the solid velocity vs, dynamic viscosity p, porosity tl and inverse of the permeability k-1 are known, we move pk_1 • elvs to the right-hand side and equation (4.87a) becomes
pk”1 • eV + Vp - • (2e'd') = pg + pk”1 • eV, (4.88)
c
which is the Stokes equation when k-1 is zero. Define the spaces
L20(Q) = {qE L2(Q) : f qdQ = 0} (4.89)
J n
Hq(Q,) = (w G H1^) : w|an = 0} (4.90)
Hl(Q) = (w G Hl(Q) : w|9n = s} (4.91)
H~l{Q) = (77q(Q)) ,the dual of Hq(Q) (4.92)
V = (w G H1(f2) : w|an = 0 and V • w = 0} (4.93)
VJ~ = (w1 G Hq(Q) : f w-1 • w = 0 Vw EV} (4.94)
Jo,
57


V"° = {w' e H ^U) : (w',w)H-i(n)xHi(n) = 0 Vw G V} (4.95)
where V1- denotes the orthogonal of V in Hq(Q) associated with the H1^1) seminorm | - |iji(r2)5 V° is the polar set of id; (•, ■)represents the duality pairing between ih-1(il) and Hq(Q), and the trace theorem 4.5, below, ensures the existence of the function s G Hl^2{di1) used in the dehnition of seminorm. Note that for a three-dimensional domain, w G Hl({l)3 and Vw G Hl({l)3x3. However, for simplicity, we write w G Hl(Q) in any case and the meaning follows from the context. Let the vector ^ = pg + pk_1 • tl\s, where vs is a bounded continuous function. Let fi G H~l(Q) with the following norm:
where
|fl||iT 1(n) SUPw6iTg(^)>wdO
(fi,w)

llwllirpn)
|iji(n) represents the standard norm for Hl(Vl). Note that
(4.96)
V • (2eldl) = A(eV) + V(V • eV
(4.97)
Assume tl is hxed in space and define
v
eV.
(4.98)
Then equations (4.88) and (4.87b) can be rewritten as
pk”1 • v + Vp - ^Av = ^ + AV(V • v

c V-v
(1 -d
V • eV,
(4.99a)
(4.99b)
where the right-hand side of (4.99b) is known. Using (4.99b) to eliminate the last term in (4.99a) and letting / = — il/(1 — el)+'V-el\s and f = fi+p/dV/, we get the Stokes-Brinkman equations in the following form
pk 1 • v + Vp
^Av = f c (4.100a)
V • v = /. (4.100b)
58


Define the linear and bilinear functionals
a(v, w) = / ^Vv : Vw + / p(k 1 • v) • w, J n e J n (4.101)
&(v>9) = ~ • v, J n (4.102)
Ci(w) = (fl,w)H-l(n)xHl(n) - J - w> (4.103)
a 1 (4.104)
Then, the weak formulation of (4.100) can be expressed as
Problem 4.1 (Weak Stokes-Brinkman) Find v G H)(Q) and p G Lq(Q) such that
Vw G ihg(n), a(v, w) + b(w,p) = ci(w), (4.105a)
VgGh^D), b{\,q) = G2{q). (4.105b)
Before we prove the existence and uniqueness of the Stokes-Brinkman equation, we first introduce the following norms, seminorms and theorems, [23].
Definition 4.2 (Weak derivative) We say that a given function v G L}0C(VL) has a weak derivative, Drfv, provided there exists a function w G L}oc{i1) such that
f w{x){x)dx = (—1)'"' f v{x)'a\x)dx \/ G CZ°(Q) (4.106)
J Q J Q
where is the set of functions with compact support in Q; Lj0C(Q) = {v : v G
Ll(K) V compact K C interior 12}. If such a w exists, we define Dfv = w.
Definition 4.3 (Sobolev norm) Let k be a non-negative integer, and let f G Lj0C(Q). Suppose that the weak derivatives Dff exists for all |cr| ^ k. Define the Sobolev norm
/ \
\\f\\w£(n)
in the case 1 ^ p < oo, and in the case p
E IK/IIEn)
\||a|sCfc = OO
(4.107)
WfWw^(n) max ||D“/||Loo(n).
|q|
(4.108)
59


In either case, we define the Sobolev spaces via
W*((l) -.= {/ e LlM : ||/||w?(n) < 00} . (4.109)
Definition 4.4 (Seminorm) For k a non-negative integer and f E W*(Q), let
( \',p
\|a|=fc J
(4.110)
in the case 1 ^ p < 00, and in the case p = 00
l/lw*(fi) = max ||D“/||LOo(n). (4.111)
The space Wâ„¢{dQ) = Hm(dQ) can be introduced for the boundary dQ. In case of m = 1/2, the space Hm(dQ) is equipped with a norm [40]
Irp/qan) — llslll2(9n) + lsli/2,an
(4.112)
where
lsli/2,9Q = f \x-y\-{n+l)\s(x)-s(y)\2dxdy (4.113)
and n is the dimensional number.
The following theorems are important to prove the existence and uniqueness of the weak solutions. The proofs are in the citations referred below. The first theorem is referred to as the inverse trace theorem and it ensures that if s E H1^2(dQ), then there exists vs G H1(Q) such that the trace of vs on dQ is s, [40].
Theorem 4.5 (Direct and Inverse Trace Theorem for H1^)) There exist positive constants K and K' such that, for each w G Hl{Q), its trace on dQ belongs to Hlt2(dQ) and ||w||ii-i/2('9n) ^ w||#i(Q). Conversely, for each given function s G H1^2(dQ), there
exists a function vs E H1(Q) such that its trace on dQ coincides with s and
vs||i?i(n) ^ K‘
H1/2(9Q)
(4.114)
60


The next theorem states that the divergence operator is an isomorphism between Lq(Q) and V[44], and the Ladyzhenskaya-Babuska-Brezzi, LBB, condition, which is needed for the stability of the mixed finite element method, is mentioned [24],
Theorem 4.6 Let Q be connected. Then
1) the operator grad is an isomorphism of Lq(Vl) onto V°
2) the operator div is an isomorphism ofVL onto Lq(Q)
Moreover, there exists f3 > 0 such that
inf sup -t:—rj—~^ f3 > 0 (4.115)
qeL20(n) wetfi(o) II wlliv!(o) IMIl2(q)
and for any q E Lq(Q), there exists a unique v G V1- C H((Q) satisfying
||v||Hi(n) ^ P~1M\l2(q)- (4.116)
To prove the existence and uniqueness, Lax-Milgram is the main theorem used in the proof. We begin with some definitions [23].
Definition 4.7 A bilinear form on a normed linear space H is said to be bounded
(or continuous^ if'3C < oo such that
\a(v,w)\^C\\v\\H\\w\\H Vv,weH (4.117)
and coercive on E C H if3a> 0 such that
a(v,v) ^ ck|| Vu G E. (4.118)
In order to prove the existence and uniqueness of Problem (4.1), we require operator notations. Recall that (L^Q))' = Lq(Q) and = i7-1(Q) where (L^Q))' and
are the dual spaces of Lq(Q) and ih_1(l1), respectively.
Definition 4.8 Let v, w G H1(f2) and q G Lq(Q). Define linear operators A : H((Q) —>• H-^Q) or A E C(H((Q); ih_1(n)) and B E L20(Q)) by
(Rv, w)Hi(n)xH-i(n) := a(v, w) Vv, w G H((Q)
(4.119)
61


(^v,g)Hi(n)xi2(n) := b(y,q) Vv G H^(Q),Wq G L%(fl) (4.120)
Let B' G £(Lq(Q); i/_1(n)) 6e i/ie dual operator of B, i.e.
(B'q, v) = (q, Bv) := 6(v, q) E H((Q),Wq E L(,(Q). (4.121)
With these operators, Problem (4.1) is equivalently written in the form:
Problem 4.9 Find v G H)(fl),p G Lq(Q) such that
Av + B'p = f m H~1(Q) (4.122a)
£v = / in Lq(Q). (4.122b)
Next, we introduce Lax-Milgram theorem which is one of the most important theorems used to prove the existence and uniqueness of the weak solution.
Theorem 4.10 (Lax-Milgram) Given a Hilbert space (H, (•, •)) a continuous, coercive bilinear form a(-, •) and a continuous linear functional F G H', there exists a unique v G H such that
a(v,w) = F(w) VweH. (4.123)
The following inequalities are applied in the proof of continuities and coercivity, [44, 23]
Theorem 4.11 (Poincare inequality) IfQ is connected and bounded at least in one direction, then for each integer m ^ 0, there exists a constant K = K(m, Q) > 0 such that
or for space H((Q)
w\\Hâ„¢(n) ^ K\w\hâ„¢(ci) VweHâ„¢(Q),
(4.124)
IMkdn) ^ K\\Vw\\L2{q) Ww E Hq(Q). (4.125)
Theorem 4.12 (Holder’s Inequality) Fori A, p, q ^ oo such that 1 = 1/p+l/q, if f G Lp( Q) and g G Lq(Q), then fg G L1(Q) and
ll/S'lkyn) ^ ||/||lp(o)IM|l«(o)- (4.126)
62


Theorem 4.13 (Cauchy-Schwarz’s Inequality) This is simply Holder’s inequality in the spe-
cial case p = q = 2. If f,g G L2(Q), then fg G L1(Q) and
fn | f(x)g(x)\dx ^ ||/"||l2(q)II^IIl2(n)
(4.127)
for vectors v,w G L2(Q), we also have
(4.128)
We next show that the L2(f!)-norm of divergence of a function in Hl{Q) is less than or equal to a multiplication of a constant and the i71(fl)-seminorm of the function and also provide the proof. This theorem will be used to prove Theorem 4.15.
Theorem 4.14 Let Q C Rn and w G Hl{12). Then
V • w||L2(n) ^ Vn|w|Hi(n)-
(4.129)
Proof: Let ip = dwi/dxi where w = (uq, w2,wn). Since \u\ + u2 + ...,un|2 ^ n(u\ + u\ + ... + u2n), integrating and taking square root both sides, we have
Then ||V • w||i2(n) ^ \Sh\w\Hi(n). ■
The following theorems show that the linear functionals ci(w), c2{q) and bilinear functionals a(-, •), b(-, •) are continuous and a(-, •) is coercive.
Theorem 4.15 The linear functionals ci(w); c2(q) and bilinear functionals a(-, ■), b(-, •) are continuous anda(-,-) is coercive, i.e.,
(4.130)
a(w, w) ^ Cc\(w
2
(4.131)
iTpQ)
where Cc = min{p/e, pC^}; Ck is a positive number. In particular,
t
Vw eH\n), (4.132)
63


(4.133)
(4.134)
(4.135)
c2(g) ^ WfWmnMmn), Vq e L2(tt) b(v,q) ^ Vn|v|Hi(n)||g||L2(n), Vv G H\n), Vq G L2(Q) a(v,w) ^ C'allvHfl-i(n)||w||jH'i(o), Vv G H\Q), Vw g H\Q), where n is the dimensional number; Ca = iaax{/i/el, I k^\}-
Proof: The linearity of Ci(w) and c2{q) and bilinearity of a(v, w) and b(v,q) are obvious. Next we show they are continuous. Let v, w G H1(Q) and q G L2(Q), then
|ci(w)| = |(fi,w)H-i(n)xHi(n) — Jn ^y/V • w|
f
^ |(fi,w)ir-1(n)xir01(n)l +
(fi,w)

w
ll^Vn)
w
h1{ n)
+ ~l f /v • w
c J n
^ IlfilliT-yfyllwIliryn) + ^||/||L2(n)||V • w||L2(Q) ^ l|fi Hiv-1(o) IIwlliv1(n) + Vn^j\\f\\L2^\w\Hi;n;
^ (llfilln-yn) + H/llnyo))||w||iTi(n)
(4.136)
(4.137)
(4.138)
(4.139)
(4.140)
(4.141)
where we apply the definition of H~l(Vl)-norm, (4.96), and Cauchy-Schwarz’s inequality, Theorem 4.13, for the third inequality. For the fourth inequality we use the fact that ||V • wIIl2(q) ^ xA^wlirpn) where | • | is the seminorm and n is the dimensional number. For the continuity of c2(q), it is obvious that
|c2(?)| =
fq
^ ll/l|L2(Q)lkl|L2(n),
(4.142)
where Cauchy-Schwarz’s inequality has been applied. To show continuity of 6(v, q), we have
\b(v,q)\ = J^q'V-v ^ IM|l2(q)||V • v||L2(n) ^ V^|v|iri(n)||g||L2(n), (4.143)
where Cauchy-Schwarz’s inequality and the fact that || V • w||L2(q) ^ v^wI-HTh) are applied for the last two inequalities; n denotes the dimensional number. To prove the continuity of a(v, w) for a two-dimensional domain, let v = (iq,^)- We first consider
Ik”1
rlll2(n) = fQ(k nV + k12lv2)2 + fn(k2iVi + k2^v2f
64


^ /(Kivi)2 + 2 + feV)2 + (k£v2)2
Jn
< 2 iSfJfgi X (2t,l +2"2 + 2|l>lU2|)
WsVAV2/n(3tt + 3<)
= 6ig3f2lA«1l2/n(t,f + t'0
= 6 .max, l*'ijTllvlli2(i2), (4.144)
where Young’s inequality ab ^ ap/p + 69/g where a,b ^ 0, p,q > 0 and l/p + l/q = l is applied at the third inequality. The proof that a(v, w) is continuous is completed with the following:
|a(v, w
^ Vv : Vw

in c
f h
h(k
-i
v • w
—Vv : Vw
in c
p(k
-l
V • w
^ 4ll Vv||L2(n) || Vw||L2(n) + p.||k 1 • v||L2(n)||w||L2(n)
^ ^jllVvIU2(n)||Vw||L2(n) + max |^d|||v||L2(n)||w||L2(n)
iy*,jy2

(4.145)
(4.146)
(4.147)
(4.148)
(4.149)
where Ca = maxj/i/e) -\/6^ maxiy*,jy2 |/q/|} and (4.144) is employed to the third inequality. Before proving the coercivity of the bilinear form a(w, w), we consider, for two-dimensions,
(k-1 • w) • w = kiiw\ + k^wl + 2k^2W\W2 (4.150)
where w = {w\,w2) and k-1 is symmetric. Note that (k-1 • w) • w = wTk-1w as matrix
multiplication where T is the transpose. For our problem k-1 is a positive definite matrix
and its diagonal entries are positive numbers. Then k^wf + k^w^ > 0 when w > 0. We
next need to focus on 2ki2W\W2. If 2ki2W\W2 ^ 0, it’s easy to see that (k-1 • w) • w ^
kiiw\ + k^wl ^ minj/cf/, + WV) = Cfc(wi + w^) where Ck > 0. If 2ki2W\w2 < 0,
65


we have two cases:
Case 1: ki2 > 0 and uqu>2 < 0.
Thus, (k 1 • W) • W = k^wj + k22w2 — 2kl2\w\W2\-
We now apply Young’s inequality: ab < a2/2 + b2/2 where a,b > 0. Then \wiw2\ = |uq||u>2| < |^i|2/2 + \w2\2/2.
> k^wj + k22wl - 2ki2{\w\\2/2 + |w2|2/2)
= k^wj + k22wl - k^2{\wi\2 + |w2|2)
= (fcn - k^)w2 + (k22 - ki2)w2.
For our problem, k_1 is diagonally dominant.
Then, Ck = min{(k^ - ki2), (k22 - ki2)} > 0.
Therefore, (k_1 • w) • w > Ck(wf + w2).
Case 2: ki2 < 0 and uqu>2 > 0.
Since k_1 is bounded, 3 Cb such that ki2 = —Cb where Cb > 0. Thus, (k-1 • w) • w = k^wj + k22w2 — 2CbW\W2
= kiiw\ + k22w\ — 2Cb\w\W2\ = kiiw\ + k22w\ — 2Cb\w\\\w2\ We now apply Young’s inequality: |u>i||u>2| < \w\\2/2 + |u>2|2/2 Therefore, k~[iw\ + k22w2 — 2Cb\w\\\w2\
> k^wj + k22wl - 2C'6(|u>i|2/2 + |w2|2/2)
= k^wj + k22wl - C'6(|u>i|2 + |w2|2)
= (k{i - Cb)w\ + (k22 - Cb)w2 ^ min{(fcn - Cb), {k22 - Cb)}(wf + wf)
For our problem, Cb is less than the diagonal entries of k_1.
Thus, Ck = mm{(kii - Cb), (k22 - Cb)} > 0 Hence, 3 Ck > 0, (k_1 • w) • w > Ck(w\ + w2)
From all of the cases above, we have
Thus, k^wj + k22w\
2kl2\wiw2\.
3Ck>0, (k-1
• w) • w ^ Ck{wl + w2)
(4.151)
66


Hence
Jq{k 1 • w) • w ^ Ck J^(wl + w22) = C'fc||w|||2(0). (4.152)
We are now ready to show coercivity. We begin with the formula (4.101). Then
a(w, w)= / ^Vw : Vw+ / p,(k 1 • w) • w J n c J n (4.153)
^ ^l|Vw|||2(n) + pA||w|||2(n) (4.154)
^ C'c||w||^i(n), (4.155)
where Cc = min{fi/e, fiCk}- â– 
Next it is necessary to know the following formulation so that it will be used in the proof of the next theorem. Assume that fi G ih_1(H),s G Hlt2(dVl) and / G L2(fl). By the inverse trace theorem 4.5, we have
3vs G Hl{Q) such that vs|9n = s, (4.156)
and
3KS>0 such that ||vs||Hi(n) ^ Ks||s||Hi/2(9n). (4.157)
Therefore, we have the following equalities,
//=/v-v=/ v-n = / s-n = / vs • n = / V • vs, (4.158)
J n J n Ja n Jan Jan J n
where we applied the divergence theorem to the second and last equalities and n is the unit outward pointing normal of the boundary <912. The equalities show that
[ (/ — V • vs) = 0 (4.159)
J n
which implies that / — V • vs G Lq(12). From Theorem 4.6, then
3! v0 G V1 C Hq(Q) such that V • v0 = / — V • vs. (4.160)
We now can prove the existence and uniqueness of the Stokes-Brinkman equations.
67


Theorem 4.16 (Well-posedness of the Stokes-Brinkman equations) Assume that fi G f, / G L2(Sl) and s G Hll2{dVl). There exists a unique v G H)(Tl),p G L2(Q) satisfying Problem f.l, equations (4.105). Moreover,
vl|JH'1(0)
i iitf-ho)
+ Vn^\\f\\Lpn)^ + + l)
vl|JH'1(0)
(4.161)
where v = vs + v0 and
||p||l2(q) ^ + Vn^j\\f\\L2(n)^J + -^Ca\\\\\Hpn). (4.162)
where f3 is the constant in (4-115).
Proof: Let fi G f, / G L2(fl) and s G H1l2(dfl). Let v = vs + Vo- For any
w G V, let F(w) = Ci(w) — a(v, w). From Theorem 4.15, ci(-) and a(-,-) are linear and bilinear respectively. Then F(-) is linear. Moreover, the continuities of ci(-) and a(-,-) implies that F(-) is continuous. By applying the Lax-Milgram Theorem, there exists a unique v G V C H((Q) such that a(v,w) = F(w). Let v = v + v = v + vs + v0. Therefore, v|9n = v|9n + vs|9n + v0|9q = 0 + s + 0 = s because v0 G P-1 C and
v G V C Moreover, V-v = V- v + V-vs + V-vo = 0 + V-vs + / — V • vs = /
since v G V and V • v0 = / — V • vs, (4.160). Therefore, v is in H)(Q) and satisfies the continuity equation. To show that v is unique, we apply the coercivity of a(-,-). Since a(v, w) = F(w) = ci(w) — a(v, w) and v = v + v, a(v,w) = ci(w). Let Vi and v2 satisfy a(vi,w) = ci(w) and a(v2, w) = ci(w). Then a(vi — v2,w) = 0 for any w G V. Thus, a(vi - v2, vi - v2) = 0. Therefore, 0 = a(vi - v2, Vi - v2) ^ T7C||Vi - v2||^i(n) ^ 0. Since, Cc > 0, ||Vi — v2||^i(Q) = 0. Then vi = v2 in the iF(f2)-norm. Hence, there exists a unique v such that a(v, w) = Next, we show that there exists p G L2(Q) satisfying Problem 4.1 or 4.9. Let (Fi, w) = Ci(w). Since a(v, w) = V°. From Theorem (4.6) and the isomorphic property, there exists a unique p G L2(Q) such that


B'p = F\ — A\ — Av = F\ — A\ or A\ + B'p = F\. Hence, there exists a unique v G and p G Lq(Q) satisfying Problem 4.1.
In order to prove (4.161), we first consider ||v0||iji(r2)- Using / — V • vs G Lq(Q) and applying equations (4.160) and (4.116), we have
llvo||iri(n) ^ P~l\\f ~ V ' vJl2(q)
^ P~l (||/||l2(q) + ||V • vs||L2(n))
^ P~l (||/||l2(q) + V^llvsllirqn))
1 (||/I|l2(q) + V/^'(Us||s||ii-i/2(9n)) , (4.163)
where Theorem 4.14 is applied to the third inequality and Theorem 4.5 is applied to the fourth inequality. Then
||v||iri(n) — || vs + v0 ||iji(n)
mi 2(9Q) + P 1 (||/IIl2(q) + v^C'sllsll mi2 (an))
= ft 1||/||l2(q) + (1 + V^/9 1)Us||s||ii-i/2('9n). (4.164)
Next, we apply the coercivity of a(-, •) to obtain a bound for v:
CcWvWnpn) < a(v,v) = ci(v) - a(v, v)
^ (IIfillir-pn) + \Ai“||/||L2(n))||v||ij-i(n) + Ca||v||^i(Q)||v||jji(q), (4.165)
where Theorem 4.15 is applied to the second inequality. Dividing both sides by C'c||v||fl-i(Q), we have
IMIirpn) ^ IIfilliv-1(n) +
Using (4.165), we have
p
n—
e
L2( Q)
Ca
C,
\m(n)-
llvllirqn) — Ilv + v||#i(n)
^ ~C + V^—ll/llztyn)^ + ^r111^IIivx(n) + ||v||hi(q)
69


Next, we employ condition (4.115) to obtain a bound for pressure p:
MJm SUP
weiTi(Q)
w||_H-i(Q)
sup
weiiRn)
w
upnjIbllL^n)
^ P > 0.
(4.167)
Rearranging we have
WpWlPQ) < P 1
sup
weirRn)
&(w,j?)
Iwllirpn)
(4.168)
Note that
6(w,p) = ci(w) — a(v, w)
^ ^llfillir-pn) + \A^||/||L2(n)^ HwlliJ1(si) + Co.||v||^1(^-2)||w||jH-i(£-2)• (4.169)
Substituting (4.169) into (4.168), we obtain
Iblkyn) ^ P 1 SUP ||fillir-pn) + ||/||l2(q) + ^allvllirpn)
weiTRn) e
— P 1 y fi Lu-^n) + 11/\\l2(q)j 1 + CaP 1 V Hl(n), (4.170)
and we have complete the proof of the well-posedness of the Stokes-Brinkman Theorem 4.16. We now have a system of equations: equations, â– 
pkr1 • (eV - eV) + Vp - ^V • (2tldl) = pg, c in Q (4.171)
v* = s on dQ, (4.173)
which are well-posed for a fixed numerical domain Q and boundary conditions defined by the function s G Hl^2{dQ).


Epithelium
t = t1
Figure 5.1: The left figures shows the PCL when t = to where the cilia is perpendicular to the horizontal plan while the right one displays the PCL when t = t\ where the cilia make an angle 9 to the horizontal plan, where 9 is less than 90 degrees.
5. NUMERICAL RESULTS FOR THE FIXED HEIGHT MODEL, TWO-DIMENSIONAL MODEL
In this Chapter, we apply a mixed finite element method to the system of the Stokes-Brinkman equations and the continuity equation:
Hk-1 • (eV - eV) + Vp - 4'V • (2e'd') = pg, in Q1 U Q2 (5.1)
c
t'+ (1 — e')V • (eV - V5)) = 0. infUUfU (5.2)
where the Stokes equation is applied in domain Q\(t) while the Brinkman equation is employed in n2(t). Figure 5.1 shows the domains. The left one represents the domain when the cilia are perpendicular to the horizontal plane at time t = to- In this case, we have only porous medium, fl2, in the PCL so only the Brinkman equation is applied. The right figure shows the PCL when the cilia make an angle 9 to the horizontal plane at time t = t.\. In this study the angle 9 is between arctan(0.5) to arctan(inf) or about 26 to 90 degrees. Whenever the angle 9 is less than 90°, both subdomains, fli and f22, have nonzero areas and the Stokes-Brinkman equations are applied. The boundary conditions for the domains are discussed in Section 5.1. The variational formulation and the model discretization are presented in Section 5.2, while the numerical results and the validation are provided in Section 5.3.
71


5.1 Boundary Conditions
In this section, appropriately physically meaningful boundary conditions that also guarantee the existence of a unique solution are discussed. To be well-posed either velocity or the traction vector (t • n) must be prescribed everywhere on the boundary. Furthermore, for a two-dimensional domain, we need two scalar equations at each boundary for the velocity [11]. For the pressure, we consider the bilinear functional (4.102):
b(w,p) = — pV • w J n
= / w • grad p dfl — wp ■ ncff
J n Jr
(5.3)
= / w - grad p dQ (5.4)
J n
where we use Green’s first identity to get the second equality and w is in Hq(Q) so it is zero on the boundary. This implies that b(w,p) does not change if we add a constant function to p. Note that (5.1) only determines pressure up to an additive constant. This constant is usually fixed by enforcing the normalization [21]
pdQ = 0. (5.5)
J n
We divide the boundary into five pieces: the free-fluid/porous-mediurn interface, £2in£22, the free-fluid/fluid interface between the PCL and mucous layer at the top of U C2, the left and right sides of U C2 and the bottom of the PCL, to be able to solve the systems of equations numerically. On the sides, we assume periodicity for both velocities and pressure. The velocities are assumed zero at the bottom.
Let us first consider the boundary conditions at the free-fluid/porous-medium interface. Many authors [3, 47, 62] who use the Darcy equation in C2 and the Stokes equation in apply the Beavers-Joseph condition [8] which states that the velocity component parallels to the interface can slip according to
duv
dx2
a.
m / v
\fk
^slip ^

(5.6)
72


where am is a material dependent, dimensionless, parameter; k is the permeability; uv and vP are the velocities of the viscous and porous flows, respectively and uvslip is the slip velocity depending on the structure of permeable material within the boundary region, height of the
result there is not a natural way to balance the shear stresses. The Beavers-Joseph boundary condition provides a transition model from the viscous to the porous flows [43, 46, 75, 88]. If the porosity is closed to one at the interface, the viscous stress in the viscous flow is completely transferred to the fluid in the porous media.
Alternately, if one uses the Brinkman equation in fl2 and the Stokes equation in Hi, the deviatoric part of the stress tensors can be considered. For this case, both continuous and discontinuous shear stress boundary conditions have been considered [39, 68]. We first assume continuity of the normal component of stress, i.e.
normal vector pointing outward of the boundary. If the normal component of the velocity is continuous across the interface, we automatically have (5.8).
For the top boundary condition we first consider the shear stress. Instead of mucus, air is first assumed at the top of PCL. Therefore the boundary condition on the top of Hi is
domain and fluid viscosity. The difference of the tangential components of the velocities in (5.6) is the jump due to a nominal boundary layer at the interface. Recall that in Darcy’s law there is no viscous term, 2pdl = 0, which is the deviatoric part of the stress tensor. As a
t* • n|i = t* • n|2,
(5.7)
where t* = —pi + 2pdl. We also need to conserve mass across the interface (mass flux must be equal) [11]:
pv • n|i = pv • n|2
(5.8)
where the subscripts 1 and 2 refer to the domain fR and fl2, respectively and n is the unit
(5.9)
73


no shear
periodic
Stokes
Cilia moving.
Brihkmian
periodic
v = 0
Figure 5.2: A two dimensional Cartesian coordinate system with axis lines x\ and x-2 and the cartoon picture of the cilia in the PCL with boundary conditions.
where vl = (u(,t4). The equation (5.9) can be rewritten as
dv[ dvl2
dx-2 dxi
Since we assume that the vertical velocity is zero at this interface,
dv\ dvk —- = —- = 0.
dx'2 dxi
Hence, our initial model as shown in Figure 6.2 is
p.k”1 • (eV - eV) + Vp - • (2tldl) = pg,
e
il + {l-tl)V â–  (e'(v'-vs)) = 0. v1 = 0
vl and p are periodic
in fh U 122 in 121 U 122 at the bottom of 122 on the sides
vl is continuous
dv\ dvi
d^2 = d^i =
at 121 fl 121 on the top of 121.
(5.10)
(5.11)
(5.12)
(5.13)
(5.14)
(5.15)
(5.16)
(5.17)
5.2 Model Discretization
The appropriate discretization of the governing equations is the first step to obtain a
numerical solution using the mixed finite element method. We begin by deriving the weak
formulation of the Stokes-Brinkman equations (5.12)-(5.13) which are equivalent to (5.1)
74


and (5.2): Find (v,p) G H1^) x Lq(Q) such that
(5.18)
(5.19)
p k 1 • v + Vp — -^Av = pg + pk 1 • tlvs + ^-V/
e( e(
V • v = /,
with the boundary conditions (5.14)-(5.17) where / = — 5/(1 — d) + V • eVs and Q is a computational domain. Note that every term on the right hand side of (5.18) is given since the velocity of the cilia or solid phase is known, see e.g. [20]. Writing (5.18) as scalar equations in two-dimensions gives:
p
p
knVi + k12v2
k2lvl + k22v2
p 02 to cS 02 to CS
d 5x2 5x2
p d2v2 d2v2
d dx\ dx2
dp
dx\
dp
dx2
pe
knvt + k12v2
— 1 „,S
d df
d dx i
-pg + pe1 k2^v[ + k22vs
pd£_
el dx2 ’
(5.20)
(5.21)
where gravity is given by g = (0, — g). To find the weak formulation, we multiply (5.20)-(5.21) by test functions G Hq(Q), i = 1,2 and integrate over the domain Q = Qi U SI2. This yields
WidQ
(5.22)
p
k2}v i + k22v 2
p "52x2 52x2"
e* 5x2 ' 5x2
-pg + pe' + k22v2

p^/
e* 5x2
- XT--) w2dtt dx 2 /
w2dQ.
(5.23)
Integrating by parts the pressure term, the second-order term and the source term /, we have the weak formulation: Find (v,p) G Hl{Q) x L2(Q) such that
p
kn Vi + k12 v2 widQ + — /
J e‘ Jn
p
5uq 5uq
5xi dxi 5x2 5x2


pe
Jn 5xi
— / puqnidr + w / Jr e* Jr

uqdfl — ^
f^P-dn
dv i 5xi
-Ui
5xi
5x-
-n2
e- j n dxi
uqdr + w / /winidr, V««i G TTq (ST) e* Jr
(5.24)
75



k2}vi + k22v2 w2dQ + —7
c Jfl
dv2 dw2 dv2 dw2
dQ
I = I
In ox 2 J n
- / pw2n2dV + ^
Jr c Jr
dx\ dx\ dx2 dx2
r\
pg + pel \k21V{ + k^v^j w2dtt Jq f~^Td^ dv2
dx
-ni
dv2
dx2
-n2
w2dV + ^7 / fw2n2dV, Vw2e Ho(ty (5.25) c Jr
where [n\, n2) is the outward unit normal vector and T is the boundary of the domain Q. Let Th be a triangulation of domain Q and
Vh = {veH\n): v\k is quadratic,\/K ETh} Hh = {q E L%(Q) : q\K is linear,WK E Th}.
(5.26)
(5.27)
be finite-dimensional subspaces of H1(Q) and Lq(Q), respectively. In hnite element method we approximate the solutions (u*,p) EVhX Hh by letting
M
Vi X
E ^
x V,:
^Tv,.
m= 1 L
p(x) = E 1=1
(5.28)
(5.29)
where i^m and 4>i are called basis functions while 'F and $ are their vector forms; V* and P are vectors of the velocities and pressure, respectively, and the numbers M and L are determined by the interpolation function. For example, for a tetrahedral element, M = 10 for quadratic function for u* and L = 4 for linear function for p. Substituting (5.28) and (5.29) into (5.24) and (5.25) and the basis function 'F into W\ and w2, we have
p
f k^wTdnv 1 + f k^mvTdQv2
J n J n
r
In dxi dxi
dQ
<94- <9^T
In dx2 dx2
dQ Vi
- 4 f
- f y^mdrp + 4
r d^T
/ y—nidr-
/ r ox 1
f d^T A
/ ^—n2dr)Vl
/r ox2 /
p
[ k^wTdnv1+ [ k221mvTdQv2
J n J n
p
r d^ d^T
In dxi dxi
el Jn dxi
+ w / f^rtidT, (5.30)
e Jr
<94- d^T
dQ
In dx2 dx2
â– dQ V,
<9^
dx2
$TdQP
-pg + ptl (k2iv[ + k22V2))
P r .dV
If
e j n
dx2
dQ
76


- / Tn2dTP + 4
/ f—mdr
/r OTi
r 5Tt \
/ 'b——n2^r v2
/ r + ^JfVn2dT. (5.31)
To form an element matrix for the finite element method in fl2
r 5T5Tt
A = /
Jn%
K,,-
Ini dxi dxj
Fi = fit1
k^v{ + ^>2)
â– dflg Q*
- r 5Tt
I $^—dne2,
HD <7Xi
5T
5 Jn% cixi

- / TTVdHP + ^
/• 5TT / T— /rs 5xi
/• 5TT \
/ T —n2dT| Vi
/ry 5x2 /
+ -t f fv^dri
c Jri
PQ + he* feiX + k22 v2)) 'M2 - yr / /
i fL-r.S i L-1„.sAA ,Tr jne h f r hfl
- / T$Tn2dnP + 4
+ 4/
e‘ .try
d JniJ dx2 2
t5Tt f .5TT
T ——nidTg
/ry <9x
/ry 5x2
n2<*n V2
(5.32)
(5.33)
(5.34)
where fl| is the element domain such that fl2 = [J Q|. Then (5.30) and (5.31) become
Mii1 A Vi + Aifc^AVa + (h/e*)(Kn + K22)V1 - QfP = F1} (5.35)
h^/AY, + h^AVa + (h/e*)(Kn + K22)V2 - Q^P = F2, (5.36)
where the superscript T denotes the transpose. Applying the same process to the continuity equation (5.19):
dv\ dv2 il t dvf t dv2
, e—- + e dx\ 8x2 1 — e dx\ dx
(5.37)
we have the weak form
5TT
5Tt
- / $ ——dQ2Vi - / $ ——dQ2V2 = -
il tdv{ tdv2
Ini dxi
dx2
+ e5AT + eA5r (5-38)
Ini \ 1 — tl dx\ dx
Let
F, = -
el i dvf tdvs2
t ——- + e
TcAT.
*ni V 1 — 5 ' 5xi ' 8x2 1 2
(5.39)
77


Then (5.38) becomes
-QlVi - Q2V;
2 — ^3-
(5.40)
Writing the system of equations (5.35), (5.36) and (5.40) in element matrix form, we have
(
\^\
V
V!
v2
Vp/
f2
VFV
(5.41)
[ikiiA. + (/i/d)(Kn + K22) ^12 A — Qf
jik2i A l^k22 A + (/r/e^Kn + K22) —Qf1
—Qi —Q2 0 ^
We now have the matrix form of the discrete system of equations in domain Q2.
We next find the element matrix form for domain Note that the momentum equations in Q! are the same as those in except there are no velocity terms and the porosity is one. Applying the same process as that applied to obtain (5.41), we have

r <9T<9Tt Ini dxi dxi
\T,
dQ\
r <9T<9Tt
Ini dx2 dx2
dQ\ V!
<9T

Ini dxi

- / w mdre, p + —j
Jr? c
r <9Tt
/ y—nidri
Iri ox i
r <9Tt \
(5.42)

r <9T<9Tt
Ini dxi dxi
dn\
<9T <9TT
In? 0x2 dx2
dn\ v2
<9T
$Tdf^P
In? 0x2
- I T$Tn2dnP + ^
I T^e

r <9Tt / T—mdT?

<9TT
/rf 5x2
n2 (5.43)
Let 12^ be the element domain such that Hi = (J 12)4 Writing (5.42), (5.43) and (5.38) in
e
the matrix form, we have
(pyd)(Kn + K22) 0 -Q
0 (p,/d)( Kn + K22)—Q'
V
—Qi
where
B!

- / w mdrrp +
Jr? c
and
B,

- / W n2driP + ^ Jr? c
—q2
r <9Tt
/
/r? OTi
/ T—mdH
/rf OTi
0
Ad
V2
vp/

b2
VFV
(5.44)
/r? ctr2
n2 (5.45)
/ t—mh v2
/r? (5.46)
78


Note that the velocities and pressure in Hi and Q2 are not the same but we still use the same notation for simplicity. Since the normal and shear stresses are continuous across the interface fU fl fl2 where can share the nodes between the pure-fluid and porous medium domains, the velocity and pressure terms in the surface integrals Bi and Fi are equal in magnitude but have opposite signs which cancel each other in the final discretized equation, [43, 89, 99]. Similarly, the velocity and pressure terms in the surface integrals B2 and F2 can be canceled in the final discretized equation. Therefore, by using finite element method, the the stress-continuity condition is automatically satisfied and no special procedure is needed to impose such interfacial condition.
5.3 Validation of the Code and Numerical Results
Before we provide the Stokes-Brinkman numerical results for our problem, we validate our results by comparing with an exact solution for which the boundary conditions are simple enough to determine the analytic solution. The boundary condition imposed is that at x2 = Hci + Hc2 = Hc, see Figure 5.1, the liquid is dragged with a constant velocity u0 by an impermeable plate. Using the Brinkman equation to model fluid flow in Q2, the general form for the velocity is [58]
u(x2)
Cc + Cfc(:r2 — Hc2) if x2 > Hc2 Cetm-H„)fVi if.r 2 (5.47)
where Cc, Cand C are constants and k is the scalar permeability. Applying the boundary condition
u(Hc) = u0 (5.48)
and the continuities of the horizontal component of the velocity and shear stress across the free-fluid/porous-medium interface, i.e.
u
h:
u
H
+
c2
(5.49)
79


, du
^ tWX2=h*
du
dx2
(5.50)
V 1x2=H+>
where + and — refer to the free fluid and porous medium respectively and \j! is the effective viscosity which is a parameter matching the shear stress at the free-fluid/porous-medium interface [68], we obtain expressions for the coefficients Cc, Ck and C:
u0
Cr
l + (/i'/li)((Hc-Hc2)/Vk)
Cc
H \Jk C=Cc.
Eliminating Cc,Ck and C from (5.47) using (5.51)-(5.53), we have
if x2 > Hc2
(5.51)
(5.52)
(5.53)
u{x2) =
U0
-t i d / Hc Hc 2,
V Vk '
n' (x2 - Hc2
Vk ,
(x2 - H,
c2j
(5.54)
\Jk
e v a, if x2 < Hc2.
Figure 5.3 shows the velocity prohles of the exact solution and our numerical result when the velocity u0 = 1, the porosity tl = 0.64457, the effective viscosity \j! = n/tl, the scalar permeability k = 9.5660e — 04 and the height Hc = 1 and Hc2 = 0.7071. The number of elements and degrees of freedom are 12,800 and 58,403, respectively, and the L2-norm error is 0.4385.
We next employ our numerical permeabilities that are a function of the angles 9 from about 26 to 90 degrees to compare the velocity prohles to the exact solutions where the variable Hc2 is calculated from Hcsm9 and k is the component k33 of the permeability tensor k, see Figure 3.3. The results are given in Figure 5.4 for 800 elements and 3803 degrees of freedom for both pressure and velocities while the L2-norm errors are shown in Table 5.1 for 11 different angles. For 9 = 45°, Figure 5.5 shows the convergence of the velocity prohles of our numerical results to the exact solution when the number of elements is increasing from 200 to 12800. The L2-norm errors of the graphs are demonstrated in Table 5.2.
Next, we calculate the huid velocity prohle when the cylinders move as a pendulum. We
employ an angular velocity wa = d9/dt and assume that the maximum velocity is at the
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1
- - - numerical result 1 - â–  - â–  exact solution
GO
C§ 0.5 I
0.4
0.3!
0.2
0.1
0*------1------1-----1------1------1-----1------1------1------1------
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
velocity
Figure 5.3: Velocity profiles in the x-2 direction or y-axis of the exact solution and our numerical result where uq = 1; p. = 1; tl = 0.64457; Hc = 1 and HC2 = 0.7071.
Table 5.1: La-norm error of the velocity for the Stokes-Brinkman equations where 9 is the angle between the array of cylinders and the horizontal plane.
angle 9° L2-norm errors
26.5 0.0466
30.9 1.0092
34.9 0.7471
38.6 0.7548
41.9 1.0390
45.0 1.5918
51.3 0.8077
56.3 0.9084
63.4 0.6294
75.9 0.9683
90.0 1.2073
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y-axis it*angle = 30.9638 y-axis ** angle = 34.992
velocity velocity velocity

angle = 38.6598 c§ angle = 41.9872 c§ angle = 45
;L ;L
velocity velocity velocity
f
angle = 51.3402 if) X angle = 56.3099 if) X angle = 63.4349
i i
velocity velocity velocity
f f
angle = 75.9638 X angle = 90
i
velocity velocity
Figure 5.4: Velocity profiles of the numerical and exact solutions using our permeability results with the corresponding angle 9; uO = 1; fi =1; Hc =1.
82


axis
Figure 5.5: Convergence of the velocity profiles of the numerical results to the exact solutions when the angle 9 is 45°
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Table 5.2: L2-norm errors of the numerical and exact solutions of Stokes-Brinkman equations when the numbers of elements are increasing where ^ dof is the number of degrees of freedom and Ax is the uniform length of each element.
#ele # dof Ax L2-norm errors
200 1003 0.1 1.8434
800 3803 0.05 1.5918
3200 14803 0.025 1.1824
12800 58403 0.0125 0.4385
angle 9 = 90°. Our initial numerical result, Figure 5.6, is calculated when the shear stress at the free-fluid/porous-medium interface is assumed to be zero. This means no shear force on the free fluid and only the tip of cilia moves the fluid. The lines from the left to right are the increasing angle 9 between the array of cylinders and the horizontal plane. As can be seen from Figure 5.6 the velocity in the porous medium (n2) increases as 9 increases. The velocity of the Stokes fluid at fl Q2 is the same as the velocity in the porous medium due to the continuity of normal and shear stresses at the interface.
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degree approx = [26 30 34 38 41 45 51 56 63 75 90]
Figure 5.6: Velocity profile in the x-2 direction of the cilia making angles 9 = 26, 30, ....90° with the horizontal plane when the shear stress is zero at the free-fluid/porous-medium interface.
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Forward stroke
Backward stroke
Mucus
curve s
PCL
Fluid velocity
Solid velocity
Mucus
PCL
Figure 6.1: The cartoon picture is showing the free boundary having a unknown curve s while the cilia is moving forward and backward making the angle 9 with the horizontal plane in the PCL.
6. FREE BOUNDARY TWO-DIMENSIONAL MODEL
The purpose of this Chapter is to propose a model for the free boundary problem in the two-dimensional domain Q = U 02- Figure 6.1 illustrates the domain of interest and the free boundary between the PCL and mucous layers, whose height is a priori unknown. The curve s, which is a function of x and t, is an unknown and changes due to the movement of the cilia.
There exit many approaches for simulating free boundary problems including the immersed boundary method [72, 81], the volume of fluid (VOF) method [50, 76] and the level set method [79]. For example, Hirt and Nichols [50] uses the volume of fluid method to treat complicated free boundary configurations, which is proposed originally by Nichols et al. [76]. The idea of the immersed boundary method is provided in [72] and [81] where it is used to model systems of elastic structures (or membranes) deforming and interacting with fluid flows. In this approach, the fluid is represented in an Eulerian coordinate frame and the structures in a Lagrangian coordinate frame. A disadvantage of this method is that imposing of the boundary conditions is not straightforward compared to the others. For the level set method, the curves at the interface are tracked on a fixed Cartesian grid (Eulerian approach). The method is a great tool for modeling time-varying objects such as a drop of oil floating in water [79]. One of advantages of level set method is two to three dimensions
can be coded quickly which is time comsuming for the VOF method. However, one of the
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considerable advantages of the VOF method is that they conserve mass well while the level set method does not. For steady-state problems, authors in [82, 110] employed new finite element methods to solve free boundary problems. Moreover, Saavedra and Scott [86] presented theoretical error analysis of a finite element method of a free boundary problem for viscous liquid. For the steady-state Laplace equation, Zhang and Babuska [110] solved the free boundary problem in a fixed domain exactly, as well as an exponential rate of convergence. They proved that the sequence of solutions of free boundary problems converges to the solution of the given free boundary problem. In addition, Peterson et al. [82] provides a new iterative method for two-dimensional free-surface problems with arbitrary initial geometries. They used an elastic deformation of the mesh to preserve the continuity. Next, we provide our model and outline the method from [82] to get an idea to be able to apply to our free boundary problem.
6.1 Model Problem and Boundary Conditions
We summarize the model and boundary conditions in this section. We assume that the the free surface at the free-fluid/porous-medium interface is represented by a curve s(s,t) = (x(s,t),y(s,t)) parametrized by arc length s and depended on time t. At the free boundary, curve s, the kinematic boundary condition at which a material point on the boundary remains on the boundary can be expressed as [82]
n-v = n'lW C6-1)
where n is the outward free-surface normal vector and v = dvb The absence of a material derivative in the steady-state momentum equation (5.1) allows the problem to be solved using quasi-steady-state methods. The free-surface location can be updated explicitly using the kinematic boundary condition. Before we mention the numerical method, the primary system of equations of our computational domains Qi and is
Hk 1 • (tlvl — tlvs) + Vp — ^-V • (2tldl) = pg, in U Q2, t > 0 (6.2)
c
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Full Text

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MODELINGTHEFLOWOFPCLFLUIDDUETOTHEMOVEMENTOFLUNG CILIA by KannanutChamsri BS.,Mathematics,ChulalongkornUniversity,Thailand,2002 M.S.,AppliedMathematics,ChulalongkornUniversity,Thailand,2005 Athesissubmittedtothe FacultyoftheGraduateSchoolofthe UniversityofColoradoDenverinpartialfulllment oftherequirementsforthedegreeof DoctorofPhilosophy AppliedMathematics 2012

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c 2012byKannanutChamsri Allrightsreverved ii

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ThisthesisfortheDoctorofPhilosophydegreeby KannanutChamsri hasbeenapprovedforthe AppliedMathematics by JanMandel,Chair LynnS.Bennethum,Advisor JulienLaugou LongLee RussellP.Bowler Date iii

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Chamsri,KannanutPh.D.,AppliedMathematics ModelingtheFlowofPCLFluidduetotheMovementofLungCilia ThesisdirectedbyAssociateProfessorLynnS.Bennethum ABSTRACT Ciliainthehumanlungsaremovinghairsthataidinthemovementofmucus.Thelayer thatcontainstheciliaiscalledthepericiliarylayer,PCL,andtheliquidinthatlayerthe PCLuid.WeconsiderthePCLuidasanincompressibleviscousuid,andweconsider theciliaasaperiodicarrayofcylindersthatrotateabouttheirbasewithheightvarying asafunctionoftheangle.Weseekathree-dimensionalmathematicalmodelofthePCL uidslowlyowingduetothemovementofthecilia.Weusehomogenizationtodetermine asystemofequationsthatarethensolvednumericallytocalculatethepermeability.If r istheradiusoftheciliaand d isthedistancebetweentwoadjacentcilia,wedeterminethe permeabilityasafunctionof r=d andtheangle thatthecylindersmakewiththebottom surface.NumericalresultsareobtainedusingthemixedniteelementmethodofTaylorHoodtype.Thenumericalresultsforthepermeabilityarevalidatedbycomparingtheresults withnumericalresultsofRodrigoP.A.RochaandManuelE.Cruzwithgoodagreement,and whentheowalignsandisperpendiculartothearrayofcylinders,theresultsarevalidated withexperimentaldata.ForaninitialmodeloftheuidowinthePCL,weconsiderthe overallheightasaconstant,theportionofthePCLwithciliaasaporousmedium,and thePCLuidabovetheciliaasundergoingStokesow.Tocalculatetheuidvelocity inthePCLlayer,Stokes-Brinkmanequationsareappliedwithaxedboundaryheightof thePCLlayer.Thenumericalresultiscomparedwithananalyticalsolutionwhenthetop boundaryuidismovingataconstantvelocityinordertovalidatethenumericalsolution. Theexistenceanduniquenessofthenumericalmodelisalsopresented. iv

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Theformandcontentofthisabstractareapproved.Irecommenditspublication. Approved:LynnS.Bennethum v

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DEDICATION Thisthesisisdedicatedtomybelovedparents,Mr.WipasandMrs.ChanapaChamsri, whohaveneverfailedtogivememoralsupport,caredforallmyneedsasIdevelopedmy education,andtaughtmethateventhelargesttaskcanbeaccomplishedifitisdoneone stepatatime. vi

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ACKNOWLEDGMENT Iwouldliketothankeveryonewhohavehelpedandinspiredmeduringmydoctoral study. Thisthesiswouldnothavebeenpossiblewithoutthegeneroussupportofmyadvisor, Prof.LynnS.Bennethum,forherrichpatienceinprovidingendlessidea,discussions, motivation,enthusiasmandencouragementwithherpositiveenergy.Herguidancedelights meinthetimeofresearchandwritingthisthesis. Inadditiontomyadvisor,thesupportofProf.JanMandelallowedmetoaccessthe forcefulcomputingresourcesofthemathematicaldepartment,whichprovidedmewithcomputersuppliesandimmenseknowledge.IamalsoindebtedtoProf.LongLee,awonderful facultyoftheUniversityofWyoming,forhisoutstandingeortsinensureingthatgroundworkswerekeptatmyngertips.Iwouldliketothanktherestofmythesiscommittee-Prof. JulienLangouandProf.RussellP.Bowler-fortheirperceptivecomments,andquestions. AnotherthankyoutoDr.BedrichSousedikwhoshowedgreatinterestandsupportin myresearchandwasanexcellentconsultantinmakingcodingaloteasier.Iappreciatehis patiencedespitemynumerousandunendingquestions,andhishelpfulnessthroughoutmy academicyears. MyspecialthanksalsogotoProf.HowardL.SchreyerforintroducingustotheDefenseAdvancedResearchProjectsAgencyDARPAgroupwhichprovidesanexceptional opportunitytocooperateworkswiththeprofessionalteamandforguidingustoworkon thediverseexcitingproject.IwouldalsoliketothankEricSullivanforhishelpsandforthe stimulatingdiscussionswewereworkingontogetherbeforedeadlines. IgreatlyappreciateandwishtothanktheparentsofLynnBateman,whodonatedthe fundsthatsupportedmystudyduringtheFallsemesterof2011,givingmetheopportunityto devotemyselftoconductingandstrengtheningmyresearchindependentlyandprociently. IwouldliketotakethisopportunitytothankDr.ChristopherHarderforproviding guidanceinregardstothestabilityofniteelementschemesandhisassistances. Notforgettingtothanksalsotothefaculties,graduatestudentsattheUniversityof ColoradoDenver,andfriendsforsharingtheexperiencesanddistractingfromscience. Lastbutnottheleast,Itrulythankmyfamiliesandmyparentsforalwaysbeingthere formewiththeirsupportthroughoutmylife. vii

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TABLEOFCONTENTS LISTOFFIGURES....................................x LISTOFTABLES.....................................xiii CHAPTER 1.INTRODUCTION...................................1 2.HOMOGENIZATION.................................8 3.PERMEABILITY....................................17 3.1RelationshipbetweenDragCoecientandPermeability...........18 3.2DiscretizationoftheModelProblem......................20 3.3ValidationofNumericalResults........................24 3.4NumericalPermeabilityFunctions.......................30 3.5ComparisonwithKozeny-CarmanEquation..................33 4.MODELINGTHEPCL................................44 4.1GeneralFormoftheMomentumEquationsusingHMT...........46 4.2TheContinuityEquation............................50 4.3Darcy'sLaw...................................52 4.4DerivationoftheBrinkmanequation......................53 4.5DerivationofStokesEquation.........................56 4.6Well-posednessStokes-Brinkman........................57 5.NUMERICALRESULTSFORTHEFIXEDHEIGHTMODEL,TWO-DIMENSIONAL MODEL.........................................71 5.1BoundaryConditions..............................72 5.2ModelDiscretization...............................74 5.3ValidationoftheCodeandNumericalResults................79 6.FREEBOUNDARYTWO-DIMENSIONALMODEL................86 6.1ModelProblemandBoundaryConditions...................87 6.2NumericalImplementation...........................88 viii

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7.CONCLUSION.....................................92 APPENDICES A.NOMENCLATURE...................................95 B.BASISFUNCTIONS..................................102 REFERENCES.......................................106 ix

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LISTOFFIGURES Figure 1.1Cartoonpictureofapatchworkofgobletandciliatedcellsattheepithelium..2 1.2ThePCLandmucouslayerswhere= H c L c isourcomputationaldomain withsubdomains 1 = L c H c 1 and 2 = L c H c 2 ................4 1.3Geometry:anidealcellofcylinderswhentheanglebetweencylindersandhorizontalplaneis90degrees;thetopviewoftheleftgure;theangle between thecylinderandthehorizontalplane.........................4 2.1AreferenceheterogeneitiesYareperiodicofperiod Y andtheirsizeisorderof .9 2.2Upscalingprocedurewhere > 0isaspatialparameter..............9 3.1Permeabilitybeingclosetozeroofuidowthroughacellarrayofcylinders whentheboundaryisalmostnotLipschitz.....................26 3.2Permeabilityofuidowthroughacellarrayofcylinderswithincreasing r=d , =90 .Then r=d =1 : 2077istouching.......................27 3.3Diagonalvaluesofthepermeabilitytensorasafunctionofangle for r=d xed at1 = 3..........................................27 3.4O-diagonalvaluesofthepermeabilitytensor k 12 and k 23 asafunctionofangle for r=d xedat1 = 3.................................28 3.5O-diagonalvaluesofthepermeabilitytensor k 13 asafunctionofangle for r=d xedat1 = 3.......................................28 3.6Permeabilityofuidowthroughaperiodicarrayofcylindersfor =0.When theowalignsparalleltothecylinders,guresshowtheexperimentalscalar permeabilitiesofSullivanforhairsandgoatwool,respectively,comparingwith ournumerical2 k 33 and2 k 11 + k 22 resultsfor = = 2............31 3.7Permeabilityofuidowthroughacellarrayofcylinders.Figuresarecomparing theexperimentaldataofBrownandChenwithournumerical k 33 and k 11 + k 22 whentheowisperpendiculartothearrayofcylinders..............32 x

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3.8Thetopleftgraphisthenumericallygeneratedpermeabilitycomponent k 11 while thetoprightoneisthefourth-orderpolynomialapproximationofthetopleft graph;thepointwiseabsoluteerrorareshownatthebottom............34 3.9Thediagonalcomponents k 22 and k 33 ofthepermeabilitytensor.........35 3.10Theo-diagonalcomponents k 12 ;k 13 and k 23 ofthepermeabilitytensor.....36 3.11Geometry:theleftgureshowsthecellarrayofcylinderswhentheanglebetween thecylindersandhorizontalplaneis75degrees;therightoneshowseachcell consistsof2ellipsoidalcylinders...........................40 3.12Figureshowstheradiiofanellipseinbothmajorandminoraxisesand isthe anglebetweenthehorizontalplaneandthesideofthecylinderwhile h isthe heightoftheperiodiccellwhichisperpendiculartothehorizontalplane.....41 3.13Thegreenlinerepresentsthethesphericalpartofthepermeabilitytensorofthe numericalresultwhiletheblueandredonesarefromthefunctions.62and .63,respectively,wherex-axisistheporositydependingoneachangle.....43 4.1Thecartoonpictureisshowingthexedboundarywhiletheciliaismoving forwardandbackwardmakingtheangle withthehorizontalplaneinthePCL.44 5.1TheleftguresshowsthePCLwhen t = t 0 wheretheciliaisperpendicularto thehorizontalplanwhiletherightonedisplaysthePCLwhen t = t 1 wherethe ciliamakeanangle tothehorizontalplan,where islessthan90degrees...71 5.2AtwodimensionalCartesiancoordinatesystemwithaxislines x 1 and x 2 andthe cartoonpictureoftheciliainthePCLwithboundaryconditions.........74 5.3Velocityprolesinthe x 2 directionor y -axisoftheexactsolutionandournumericalresultwhere u 0 =1; =1; l =0 : 64457; H c =1and H c 2 =0 : 7071....81 5.4Velocityprolesofthenumericalandexactsolutionsusingourpermeability resultswiththecorrespondingangle ;u0=1; =1; H c =1...........82 5.5Convergenceofthevelocityprolesofthenumericalresultstotheexactsolutions whentheangle is45 ................................83 xi

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5.6Velocityproleinthe x 2 directionoftheciliamakingangles =26 ; 30 ;:::: 90 withthehorizontalplanewhentheshearstressiszeroatthefree-uid/porousmediuminterface....................................85 6.1Thecartoonpictureisshowingthefreeboundaryhavingaunknowncurve s while theciliaismovingforwardandbackwardmakingtheangle withthehorizontal planeinthePCL...................................86 6.2AtwodimensionalCartesiancoordinatesystemwithaxislines x 1 and x 2 and thecartoonpictureofthemovingciliacreatingafreeboundarycurve s with boundaryconditions..................................88 B.1Nodalnumberingforthe27-variable-number-of-nodeselementwheretheorigin isatthecenterofthebrick.............................102 B.2Nodalnumberingforthe3-variable-number-of-nodeselement..........103 B.3Nodalnumberingforthe4-variable-number-of-nodestetrahedronelement....105 B.4Nodalnumberingforthe10-variable-number-of-nodesquadratictetrahedronelement.........................................105 xii

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LISTOFTABLES Table 3.1Permeabilitythroughthecellarrayofcylindersat90degreeofthedierent heightsandperiodicREVsprovidingthesamepermeability............26 3.2Permeabilitythroughthesimplecubicarrayofsphereswithsolidvolumefraction s =0 : 216andradius=0.5unit; k s denotesthepermeabilitiesinthisresearch; k istherelativeerrorof k s withrespectto k RM whichisthepermeabilitycalculating RochaandCruz................................29 3.3Permeabilitythroughthesimplecubicarrayofsphereswithvaryingvolumefractionofsolid s withradius0.5unit; k s and k RM denotethepermeabilitiesinthis researchandbyRochaandCruz,respectively; k istherelativeerrorof k s withrespectto k RM ................................30 3.4Thefourth-orderpolynomialfunctions: a 1 x 4 + a 2 x 3 y + a 3 x 3 + a 4 x 2 y 2 + a 5 x 2 y + a 6 x 2 + a 7 xy 3 + a 8 xy 2 + a 9 xy + a 10 x + a 11 y 4 + a 12 y 3 + a 13 y 2 + a 14 y + a 15 approximating k 11 ;k 22 and k 33 .....................................37 3.5Thefourth-orderpolynomialfunctions: a 1 x 4 + a 2 x 3 y + a 3 x 3 + a 4 x 2 y 2 + a 5 x 2 y + a 6 x 2 + a 7 xy 3 + a 8 xy 2 + a 9 xy + a 10 x + a 11 y 4 + a 12 y 3 + a 13 y 2 + a 14 y + a 15 approximating k 12 ;k 13 and k 23 .....................................38 3.6L2-normerrorsofthecomponentsofthepermeability k .............39 4.1Characteristicparameters..............................54 5.1 L 2 -normerrorofthevelocityfortheStokes-Brinkmanequationswhere isthe anglebetweenthearrayofcylindersandthehorizontalplane...........81 5.2 L 2 -normerrorsofthenumericalandexactsolutionsofStokes-Brinkmanequationswhenthenumbersofelementsareincreasingwhere#dofisthenumberof degreesoffreedomand x istheuniformlengthofeachelement.........84 xiii

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1.INTRODUCTION Ciliaarehairlikeorganellesthatlinethesurfacesoftheciliatedcellsandbeatinrhythmic waves.Theyprovidelocomotiontoliquidsalongtheepithelium,theliningofthecavity ofstructuresthroughoutthebody.Ciliaareessentiallyomnipresentinthehumanbody andplayaconsiderableroleinanumberofprocesses.Ofparticularinteresttousarethe movementofciliainthelungswhichaidsinremovinguids,pathogens,andforeignparticles outoftheairwayandoutofthelungs. Figure1.1showsaportionofmucosainbronchusesandbronchiolesintherespiratory systemandinthelungs.Thepatchworkconsistsofgobletcellscontainingmucusgranules, ciliaandamucusblanket.Thefunctionofthegobletcellsistosecretemucin,heavily glycosylatedproteins,toformthemucusthattrapstheforeignparticlesandformablanket atthetipofthecilia[101].Theciliaexistinalow-viscosityuidandtogetherthislayer iscalledthepericiliarylayerPCL,[95,63].Themucus,whichisahighlyviscousuid, residesabovethePCL[69].Topreventmucusaccumulationinthelungs,theciliabeatin analmosttwo-dimensionalplane.Theeectivebendingofciliainducesthemovementof thePCLuidwhichinturn,propelsthemucus.Theydothisbybeatingbackwardand forwardtogeneratemetachronalwavesthatgivethemucusanetowinonedirection.In theforwardstroke,ciliaarefullyextendedandpenetratethemucouslayer.Duringthe backwardstroke,theybendclosetotheepithelium,thebase,androtatebacktothestarting pointoftheforwardstoke.GueronandLevit-Gurevich[45]statedthatforasinglecilia beatinginwater,themechanicalworkloaddoneduringthebackwardstokeisvetimesless thanthetheamountofworkdoneduringtheforwardstoke.Theprocessofproducingmucus toentrapforeignparticlesandclearingthemiscalledmucociliaryclearanceormuco-ciliary transport,[65,70,94,101].Furtherunderstandingofmucociliaryclearancemechanismsof therespiratorytractcanbefoundin[93]. Damagetothemucociliaryclearancemechanismcanoccurwiththeabsenceofsucient mucusorexcessmucus[87].Forexample,whentheglandsthatproducemucusdonot 1

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Figure1.1: Cartoonpictureofapatchworkofgobletandciliatedcellsattheepithelium functionproperly,thickmucusaccumulatesinthelungs,leadingtobreathingdicultiesthat causeairwaydiseasessuchascysticbrosis,asthma,emphysema,andchronicbronchitis.If theciliatedepitheliumisdamaged,coughingistheonlymechanismforclearance[102]. TherehavebeenseveralmodelsproposedforthePCL.SeveralpapersbyJohnBlake analyticallymodelthemovementsandthevelocitiesofasingleciliumonmicroorganisms,[16, 17,18,19,20].Inparticular,Blakes1973and1977[17]and[20]provideamodelformucus owsintherespiratoryportionandthemaximumvelocitiesofciliainlungs,respectively.He usedtheStokesequationtomodeltheuidandtomodelthecilia.Heinsertedadistribution offorcesingularitieslocatedalongthecenterlineoftheciliaandcalculatedthevelocityofa singlecilium.Foranarrayofcylinders,heassumedtheaverageuidvelocityisequaltothe spatiallyaveragedciliavelocityinthePCL.Otheranalyticstudiesofmucousowcausedby ciliarymotionincludeBartonandRaynor,[5],whoanalyticallystudiedthemucousowdue tothemovementofasingleciliumandcomparedtheresultswithexperimentaldata,and Liron[64]whoconsideredtheuidtransportbyciliausingGreen'sfunctionforasurface distributionofpointforcesoftheStokesequationStokesletbetweentwoparallelplates. HisanalyticalmethodwasthesameasBlakebutheassumedthetotalvelocityinadomain consistsofaplanePoiseuilleowandtheStokesletdistributiononallciliatohavenon-zero 2

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uxwheretheplanePoiseuilleowwasconsideredasdisposallateron.However,theintegral equationsfortheforcedistributionisnotsucienttodeterminetheforcesuniquely. Experimentally,Rabinovitch's[83]appliedalight-transmissionmethodtostudythe ciliarymovementbutnotthevelocityofthePCLuidwhileFrommerandSteele[42]constructedrowsofhaircellciliabundlesfoundinthemammaliancochleawherethepressure wasrequiredtoproduceagivenvolumeofowratethroughtheobstacles.Notethatin theirexperimentstheciliawasmotionlessornotself-propelled. Inthisstudy,wedevelopamodelusinganupscalingtechniquesothatwedonothave toconsiderthemotionofeachindividualciliumbutratherwhatallciliadocollectively, i.e.theciliacanbeviewedasaporousmediumwiththeself-propelledsolidphase,cilia, composedofaperiodicarrayofcylinders.Figure1.2illustratesatwo-dimensionaldomain ofinterest,thePCLandmucouslayers.ThegoalofthisresearchistomodelthePCLasa thinporousmediumwithasolidphasecomposedofamovingperiodicarrayofcylinders andaliquidphasecomposedofthelowviscosityPCLuid.TheleftdrawinginFigure1.2 showstheciliawhentheyareperpendiculartothehorizontalplane,andtherightillustrates theciliaatanangle .Thelengthandheightofourcomputationalcelldomainare L c and H c ,respectively.Wedene H c 2 astheheightofthecilia,whichisafunctionoftime,and H c 1 = H c )]TJ/F20 11.9552 Tf 11.955 0 Td [(H c 2 .Wethendene 1 = L c H c 1 .1 asthedomaincontainingthelowviscosityuidwithoutciliaand 2 = L c H c 2 .2 asthedomaincontainingciliawithlowviscosityuid.Since 2 containsboththeuidand solid,thisdomainisaporousmedium.Because H c 1 and H c 2 changeintime, 1 and 2 are functionsoftimeaswell.Notethat 1 iscannotbetreatedasaporousmedium. In 2 wemodeltheciliaasaperiodicarrayofcylindersthatrotateabouttheirbase wheretheheightof 2 isafunctionoftime.Figure1.3showsanexampleofacellarray 3

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Figure1.2: ThePCLandmucouslayerswhere= H c L c isourcomputationaldomain withsubdomains 1 = L c H c 1 and 2 = L c H c 2 . Figure1.3: Geometry:anidealcellofcylinderswhentheanglebetweencylindersand horizontalplaneis90degrees;thetopviewoftheleftgure;theangle betweenthe cylinderandthehorizontalplane. ofcylindersandillustratesthegeometryofourmodel.Theleftcartoondisplaysanideal cellofcylinderswhentheanglebetweencylindersandhorizontalplaneis90degrees.The topviewoftheleftgureandtheangle betweenthecylinderandthehorizontalplaneare demonstratedinthemiddleandrightcartoonsrespectively.Inthemiddlepicture, r isthe radiusofthecylinderand d isthedistancebetweencylinders.Therelationshipbetween d and r is d = l c )]TJ/F16 11.9552 Tf 12.273 0 Td [(2 r where l c isthelengthofthesquaredomain.If l c =0 : 5,thecylinders touchwhenthedistance d 1 intheFigure1.3iszeroortheradius r is0 : 175.Tondamodel ofthePCL,weneedtocalculatethepermeability,ameasureoftheeasewithwhichaliquid canmovethroughaporousmaterial. Foranarrayofparallelcylinders,variousexperimental[105,107,108,14,26,29,32,98] andanalytical[48,49,59,96]approacheshavebeenperformedtocalculatethepermeability 4

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ordragforowbothperpendicularandparalleltoanarrayofcylinders,aswellasnumericalstudies,[2,91].AlcocerandSingh[2]investigatedthemovementofviscoelasticliquids passingthroughtheperiodicarraysofcylindersinatwo-dimensionaldomainusinganite elementmethod.SanganiandAcrivos[91]determinedsolutions,suchasthedragonacylinder,fortheslowowpastasquareandahexagonalarrayofcylindersinatwo-dimensional domain.Inthese,authorsfoundthepermeabilitywhenthecylindersareperpendicularor alignedwiththehorizontalplan.Mostnumericalworkscalculatedthepermeabilityfora two-dimensionaldomain.Inthiswork,althoughtheciliarotatethree-dimensionally,they beatinanalmosttwo-dimensionalplane.Therefore,thepermeabilityiscalculatedasa functionofonlyoneangle, ,andthecylinderdensityinthree-dimensions. Inordertoobtainthepermeability,homogenizationisappliedtotheperiodiccellarrayof cylinderstoobtainasystemofequationswhichcanbesolvedtonumericallyapproximatethe permeabilitytensor.Athree-dimensionalmixedniteelementmethodusingTaylor-Hood elementsisemployedtosolvethesystemofequations.Moreover,weprovidepolynomial approximationsofeachentryofthepermeabilitytensorasafunctionofaratio r=d andthe angle . ForamodeloftheuidowinthePCL,weconsideracoupledfree-uid/porous-medium systemofequations.Typically,StokesorNavier-Stokesequationsareusedtodetermine theowindomain 1 whileDarcy'sLawortheBrinkmanequationisemployedin 2 , [33,73,99].Morandotti[73]employedtheBrinkmanequationtomodeltheuidphaseof aporousmediumformodelingthemotionofadeformatingbodyinaviscousuid.Chen, GunzburgerandWang[33]comparedboththeStokes-DarcyandStokes-Brinkmanequations withthesameboundaryconditions.TheyconcludedthatStokes-Darcyequationswiththe Beavers-Josephconditionaremoreprecisethanothers.Becauseofourslowowproblem, Stokesequationisemployedindomain 1 .Forthedomain 2 ,sincethereisatransition regionatthefree-uid/porous-mediuminterfacewheretheporousmediumandfree-uid regionsareadjacent,weemploytheBrinkmanequation.Thisisbecausetheintroduction 5

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ofaneectiveviscosityparameterintheadditionaltermofDarcy'sLawintheBrinkman equationcanbetterhandletheforcefromstressattheinterfacesothatsolutionsfromthe StokesequationcanbematchedwiththoseofDarcy'lawattheinterface[66].Althoughthe BrinkmanequationismorecomplicatedequationthanDarcy'sLaw,appropriatedboundary conditionscanbeapplied.Moreover,itismoreconvenientforcoding.Moreinformation aboutboundaryconditionsoftheStokes-BrinkmanandStokes-Darcyequationscanbefound in[3,30,31,47,56,57,62,67,68,75,99]. AlthoughinpracticethePCL/mucusinterfaceisafreeboundarywiththeunknown heightofthePCL,asarstapproximation,weproposenumericallymodellingthePCL uidwithaxedboundaryheight.Thewell-posednessoftheNavier-Stokesequationandthe Navier-Stokes/Brinkmanforconstantcoecientscanbefoundin[100]and[54],respectively. TheexistenceanduniquenessoftheStokes-Brinkmanequationforthenumericalproblem foratensorcoecientareshowninthiswork. InChapter2,wediscussthepermeabilitytensorandhowitrelatestoadragcoecient. Weapplythehomogenizationmethodtotheperiodiccellarrayofcylinderstoobtainthe systemofequationswhichisusedtodeterminethepermeabilitytensor.Theseequations arediscretizedinChapter3andsolvedusingaMixedFiniteElementmethodandwhere thenumericalresultsareveried.BecausetheKozeny-Carmanequationisamongthemore popularequationstodeterminethepermeability,wecompare1/3thetraceofthenumerical permeabilitytensorwiththeKozeny-CarmanequationinChapter3. InChapter4,wemodelthePCLusingtheStokes-Brinkmanequation.Wealsoshow theexistenceanduniquenessofthediscretizedStokes-Brinkmanequations.Thenumerical resultsoftheequationsarepresentedinChapter5.Inthecaseofaconstantvelocityon thetopofthetwo-dimensionaldomain,weproceedtocomparethenumericalresultwith theexactsolution[58].InChapter6,webegintoinvestigatemodelingthePCLwithafree boundary.Finally,wesummarizeourresultsinChapter7. Itshouldbenotedthattotheauthor'sknowledgethisisthersttimetheporousmedia 6

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equationsarebeingusedtomodelauidowingduetothemovementofthesolidphase. Classicalporous-mediaowproblemsinvolveastaticsolidphasewithaliquid-phasepressure gradientinducinguidow. 7

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2.HOMOGENIZATION InthisChapterwedeveloptheequationsthataresolvedtodeterminethepermeability forthePCLwheretherearecilia.Wechoosehomogenizationbecauseitisamethodusedto upscalegoverningequationse.g.Stokesequation,fromthemicroscopictothemacroscopic scale,andhasbeenwidelyusedsincethe1970s[13].Thestrengthofthismethodisthat foragivenmicroscopicgeometry,thecoecientsinthemacroscopicequationscanbefound explicitly.Inparticular,byusinghomogenizationtoupscaletheStokesequationwithperiodicityrequirementweobtainasystemofequationsthatcanbeusedtodeterminethe permeability.InthisChapter,wesummarizethehomogenizationmethodasfollowing.For moredetails,seee.g.[13,34,51],and[90]. Letbetheperiodiccelldomainwhichconsistsofauidphase, F ,asolidphase, S andapiecewisecontinuousliquid-solidinterface)-310(=)]TJ/F21 7.9701 Tf 282.114 -1.793 Td [(S [ )]TJ/F21 7.9701 Tf 7.314 -1.793 Td [(F where)]TJ/F21 7.9701 Tf 41.359 -1.793 Td [(S and)]TJ/F21 7.9701 Tf 30.304 -1.793 Td [(F arethe boundariesofthesolidandliquidphasesrespectivelyseeFigure1.3.Weassumeslow uidow,axedsolidandaviscousincompressibleuid,sothattheStokesequationsare applicable: r v =0in F .1 r p + 4 v + f s = 0 in F .2 wherewealsoassumeano-slipboundaryconditionon)]TJ/F21 7.9701 Tf 292.869 -1.793 Td [(S ; isthedynamicviscosity; v is thevelocity; p isthepressureand f s isasourceterm.Next,weassumethediameterof thecylinders, a =2 r ,seeFigure1.3,issmallcomparedtoamacroscopicscalelength L whichcanbethelengthofabronchioleinthelungs;i.e.if = a=L ,then 1.Let x beamacroscopicvariableanddenethemicroscopicvariableorthestretchedcoordinate y = x = .Thedynamicviscositycoecient isassumedxedandindependentof .Figure 2.1showsareferencecell Y whichisperiodicandtheperiodis Y .Figure2.2demonstrates theupscalingprocedurewhere > 0isaspatialscaleparameter.When tendstozero,we 8

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Figure2.1: AreferenceheterogeneitiesYareperiodicofperiod Y andtheirsizeisorder of . Figure2.2: Upscalingprocedurewhere > 0isaspatialparameter. 9

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havethemacroscopicscale. Weconsideranasymptoticexpansionof v and p ,intheform, v = v 0 x ; y + v 1 x ; y + ::: and p = p 0 x ; y + p 1 x ; y + ::: .3 where v i and p i are-periodicin y ,and and arenonzeroparametersthatyielda physicallymeaningfulsolution.Thechoiceof and yieldsdierentmacroscopicmodels, andinthiscasewechoose and toyieldanonzeromacroscopicrst-orderpressure andasecond-ordervelocitywhichcanleadtoobtainingDarcy'sLaw.Itisknowntobe areasonableequationformodelingslowowthroughaporousmedium.Recallthat,for three-dimensional x = x 1 ;x 2 ;x 3 and y = y 1 ;y 2 ;y 3 = 1 x 1 ;x 2 ;x 3 ; .4 thenweapplythechainruleto.4,therstandsecondderivativeswithrespectto x j are d dx j = @ @x j + @ @y j @y j @x j = @ @x j + 1 @ @y j .5 and d 2 dx 2 j = @ @x j @ @x j + 1 @ @y j ! + @ @y j @ @x j + 1 @ @y j ! @y j @x j = @ 2 @x 2 j + 1 @ 2 @x j @y j + 1 @ 2 @y j @x j + 1 @ 2 @y 2 j ! = 1 2 @ 2 @y 2 j + 1 @ 2 @x j @y j + @ 2 @y j @x j ! + @ 2 @x 2 j : .6 10

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Therefore,thevectorLaplacianofthevelocityeldis v = d 2 v 1 dx 2 j ; d 2 v 2 dx 2 j ; d 2 v 3 dx 2 j ! = 1 2 @ 2 v 1 @y 2 j + 1 @ 2 v 1 @x j @y j + @ 2 v 1 @y j @x j ! + @ 2 v 1 @x 2 j ; 1 2 @ 2 v 2 @y 2 j + 1 @ 2 v 2 @x j @y j + @ 2 v 2 @y j @x j ! + @ 2 v 2 @x 2 j ; 1 2 @ 2 v 3 @y 2 j + 1 @ 2 v 3 @x j @y j + @ 2 v 3 @y j @x j ! + @ 2 v 3 @x 2 j ! ; = 1 2 " @ 2 v 1 @y 2 j ; @ 2 v 2 @y 2 j ; @ 2 v 3 @y 2 j # + 1 " @ 2 v 1 @x j @y j + @ 2 v 1 @y j @x j ; @ 2 v 2 @x j @y j + @ 2 v 2 @y j @x j ; @ 2 v 3 @x j @y j + @ 2 v 3 @y j @x j # + " @ 2 v 1 @x 2 j ; @ 2 v 2 @x 2 j ; @ 2 v 3 @x 2 j # = 1 2 y v + 1 xy v + x v where y v = @ 2 v 1 @y 2 j ; @ 2 v 2 @y 2 j ; @ 2 v 3 @y 2 j ! ; x v = @ 2 v 1 @x 2 j ; @ 2 v 2 @x 2 j ; @ 2 v 2 @x 2 j ! and xy v = @ 2 v 1 @x j @y j + @ 2 v 1 @y j @x j ; @ 2 v 2 @x j @y j + @ 2 v 2 @y j @x j ; @ 2 v 3 @x j @y j + @ 2 v 3 @y j @x j ! wherearepeatedindex j withinasingletermindicatessummation,i.e. d 2 v 1 dx 2 j = v 1 ;jj = 3 X j =1 d 2 v 1 dx 2 j . Substituting.5into.1yields 0= r v = dv j dx j = @v j @x j + 1 @v j @y j = r x v + 1 r y v ; .7 where r x v = @v j @x j and r y v = @v j @y j : .8 Similarly,substituting.5and.6intotheStokesequation.2yields r x p )]TJ/F16 11.9552 Tf 10.494 8.088 Td [(1 r y p + 1 2 y v + 1 xy v + x v + f s = 0 : .9 11

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Aftersometrialsanderrorswefoundthatinordertondanon-zeroandphysicallymeaningfulsolution,welet =2and =0sothat v = 2 v 0 x ; y + v 1 x ; y + ::: .10 p = 0 p 0 x ; y + p 1 x ; y + ::: .11 where,again,thefunction v i and p i are-periodicinthemicroscale y .Substituting.10 and.11into.8and.9,wehave r x 2 v 0 x ; y + 3 v 1 x ; y + ::: + 1 r y 2 v 0 x ; y + 3 v 1 x ; y + ::: =0.12 and )-222(r x p 0 x ; y + p 1 x ; y + ::: )]TJ/F16 11.9552 Tf 10.494 8.088 Td [(1 r y p 0 x ; y + p 1 x ; y + ::: + 1 2 y 2 v 0 x ; y + v 1 x ; y + ::: + xy 2 v 0 x ; y + v 1 x ; y + ::: + x 2 v 0 x ; y + v 1 x ; y + ::: + f s = 0 : .13 Collectingthesameordersof O from.12,and O )]TJ/F18 7.9701 Tf 6.587 0 Td [(1 and O 0 from.13,we havethedierentialequations r y v 0 x ; y =0.14 r y p 0 x ; y =0.15 r x p 0 x ; y )-222(r y p 1 x ; y + y v 0 x ; y + f s = 0 : .16 Notethat,fromequation.15, @p 0 @y 1 ; @p 0 @y 2 ; @p 0 @y 3 ! = ; 0 ; 0 : Then p 0 dependsonlyon x ,i.e. p 0 = p 0 x : .17 Fortheno-slipboundarycondition,wehave v = 0 ,i.e., 2 v 0 x ; y + v 1 x ; y + ::: =0 : 12

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Hence, v 0 =0on)]TJ/F20 11.9552 Tf 68.584 0 Td [(: .18 Sincethedomainisassumedtobeperiodic,weintroduceaHilbertspaceof-periodic functions: H = f ! : ! = ! 1 ;! 2 ;! 3 2 H 1 F 3 : ! is )]TJ/F16 11.9552 Tf 11.955 0 Td [(periodic, ! =0on)]TJ/F21 7.9701 Tf 45.754 -1.793 Td [(S ; r y ! =0 g .19 withscalarproduct: w ;! H = Z F @w j @y k @! j @y k dy .20 whereagaintherepeatindex j and k indicatesummation.Notethatthisisascalarproduct because w and ! arezeroontheboundary.Equation.16canberewritteninthe indicialnotationas )]TJ/F25 9.9626 Tf 11.291 14.058 Td [(Z F @p 1 @y i dy + Z F @ 2 v o i @y 2 j dy )]TJ/F25 9.9626 Tf 11.956 14.058 Td [(Z F @p 0 @x i dy + Z F f s i dy =0 : .21 Toobtainaweakformoftheequation,wemultiply.21byatestfunction ! i 2 H and thenintegratetheequation,wehave )]TJ/F25 9.9626 Tf 11.291 14.058 Td [(Z F ! i @p 1 @y i dy + Z F ! i @ 2 v o i @y 2 j dy )]TJ/F25 9.9626 Tf 11.955 14.058 Td [(Z F ! i @p 0 @x i dy + Z F f s i ! i dy =0.22 where,inthisChapter,therepeatindex i referstothenumberofequationswhilethethe otherrepeatindicesindicatesummationand f s = f s 1 ;f s 2 ;f s 3 .Integratingbypartstherst twoterms,usingthefactthat ! i =0on)]TJ/F20 11.9552 Tf 45.754 0 Td [(; and p o and f s i arefunctionsof x only,wehave Z F p 1 @! i @y i dy )]TJ/F20 11.9552 Tf 11.955 0 Td [( Z F @v o i @y j @! i @y j dy )]TJ/F25 9.9626 Tf 11.955 17.534 Td [( @p 0 @x i )]TJ/F20 11.9552 Tf 11.955 0 Td [(f s i ! Z F ! i dy =0 Employingthedivergence-freepropertyofthetestfunction, @! i @y i = r y ! =0 ; therst integrationiszero,andwehavethesimpliedexpression Z F @v o i @y j @! i @y j dy = f s i )]TJ/F20 11.9552 Tf 13.151 8.088 Td [(@p 0 @x i ! Z F ! i dy; 13

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wherewesumon j =1 ; 2 ; 3foreach i =1 ; 2 ; 3.Usingthedenition.20oftheinner productyields v 0 ;! H = f s )-222(r x p 0 Z F !dy 8 ! 2 H .23 Consequently,theproblemgivenbyequations.14,.16,and.18isequivalenttothe variationalproblem:Find v 0 2 H satisfyingequation.23. Toshowthatthereexistsauniquesolution v 0 2 H of.23usingLax-Milgram Theorem,wedene a w ;! = w ;! H whichisbilinear.Notethat a !;! = !;! H = k ! k 2 H 8 ! 2 H .24 isalsocoerciveand j a w ;! j = j jj w ;! j H 6 k w k H j ! k H 8 w ;! 2 H 2.25 canbeshowntobecontinuousbyapplyingtheCauchy-Schwarzinequalitytothelastinequality.Denethelinearfunctional F ! i = f s i )]TJ/F20 11.9552 Tf 13.151 8.087 Td [(@p 0 @x i ! Z F ! i dy .26 andnotethat j F ! i j = f s i )]TJ/F20 11.9552 Tf 13.151 8.088 Td [(@p 0 @x i ! Z F ! i dy 6 k f s i k + k @p 0 @x i k ! k ! i k H 1 ! ; .27 soitiscontinuous.ByapplyingtheLax-MilgramTheorem,weknowthereexistsaunique v 0 satisfying.23. Thesolution v 0 of.23canbeusedtoderiveDarcy'sLaw.AlongthewaytoformulateDarcy'sLawweobtainasystemofequationsthatcanbeemployedtocalculatethe permeability.Applyingthelinearityproperty,wehave v 0 = 1 f s i )]TJ/F20 11.9552 Tf 13.151 8.088 Td [(@p 0 @x i ! u i .28 where u i istheonlysolutionoftheproblem:Find u i 2 H suchthat u i ;! H = Z F ! i dy .29 14

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forall ! 2 H : Thus, u i istheweaksolutionofthefollowingstrongformulation r y u i =0in F .30 )-222(5 y q i + 4 y u i + e i =0in F .31 u i =0on)]TJ/F20 11.9552 Tf 148.575 0 Td [(; .32 where r y and 4 y representthegradientandlaplacianoperatorswithrespecttothemicroscopicscale y ; u i and q i are{periodic,and e i istheunitvectorinthedirectionofthe y i axis, i =1 ; 2 ; 3.Thesolution u i ofthesystemofequations.30-2.32canbeusedto computethepermeability.Notethat v 0 ; v 1 ; u i aredenedon F .ToobtaintheDarcy's velocity,Itisnaturaltoextendthemtowithzerovalueson S .Dene f v 0 = 1 j j Z F v 0 dy; f v 1 = 1 j j Z F v 1 dy; and f u i = 1 j j Z F u i dy: .33 Integrating.28andthendividingbythevolumeofthedomain,wehave f v 0 = 1 f s i )]TJ/F20 11.9552 Tf 13.151 8.088 Td [(@p 0 @x i ! f u i .34 whichintheindicialnotation,is f v 0 j = 1 f s i )]TJ/F20 11.9552 Tf 13.151 8.088 Td [(@p 0 @x i ! f u i j : .35 Equation.35isDarcy'sLawand f u i j isthepermeabilitytenserwhichdependsonthe geometryoftheperiodicdomain.Ingeneral,wewriteequation.35as q = )]TJ/F32 11.9552 Tf 10.494 8.088 Td [(k r p )]TJ/F32 11.9552 Tf 11.955 0 Td [(f s .36 where k ij = f u i = 1 j j Z F u i j dy .37 isthepermeability. Forexample,with =constantandthesourceterm f s = )]TJ/F20 11.9552 Tf 9.298 0 Td [(g r z where z isthez-axis intheCartesiancoordinate,[7]page134,.36becomes q = )]TJ/F20 11.9552 Tf 10.494 8.088 Td [(g k r p g + z ! = )]TJ/F32 11.9552 Tf 9.299 0 Td [(K r h: .38 15

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In.38,weobservethat h = p=g + z whichisidenticaltothepiezometricheadfromthe Darcy'sexperimentand K = g k = isthehydraulicconductivity[ K ]=L/T.Notethat thevariationalformof.30{.32is:Find u i 2 H suchthat Z F r y u i : r y ! = Z F ! i dy .39 forall ! 2 H andthenthepermeabilitycanbedeterminedandvalidatedfromtheformula .39byreplacing ! i with u i j anddividingbythevolumeofthedomain.Anconsequence oftheabovestatementisthattensor k issymmetricandpositivedenitematrix,meaning thatthediagonalentriesareallpositive[90].Thisalsoprovesthattheuidmovesinthe directionofincreasingheadgradient.Fromequation.37,weobservethat k dependson theactualsizeofthemicroscopicscale.Thus,thenon-dimensionalformofthesystem.302.32isimportantandcanbederivedasfollowing.Forsimplicity,wedropthesuperscript i in.30-2.32andnotethatboth u and e arevectorsforeach i withscalar q .Let y = y =a , u = u =a 2 , q = q=a andthedomainismappedto ofunitsizeacube, a =diameter ofthecilia.Hence,thedimensionlessformis r y u =0on F .40 )-222(5 y q + 4 y u + e =0on F .41 u =0on)]TJ/F23 7.9701 Tf 76.969 4.936 Td [( .42 Similarly,thedimensionlesspermeability k = 1 j j Z F u dy .43 andthentheconductivitybecomes K = ga 2 k .44 because k = a 2 k . 16

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3.PERMEABILITY InthisChapter,weuseamixniteelementmethodtocalculatethepermeabilityofthe systemofequationsobtainedinChapter2. Inthepast,H.Hasimoto[49]foundtheoreticallyperiodicsolutionsoftheStoke'sequationusingtheFourierseriesandthedragforceactingonaperiodicarrayofspheresand circularcylindersinatwo-dimensionaldomain.Theseriesdonotdivergebecausethemean pressuregradientisapplied.Thevaluesof c ,page323,inthesolutionswhichdependonthe spherevolumefractionandarewrittenasasumofthespherevolumefractionhavebeen calculatednumericallybymanyauthors.Oncethedragforceisdetermined,thepermeabilitycoecientsareknownaswell.Thedragcoecientisusedtoquantifytheresistanceof anobjectintheuidregion,whilethepermeabilityisthemeasureoftheabilityofaporous mediathatpermitstheuidtopassthroughit.ZickandHomsy[111]alsobeganwiththe Stoke'sequationandsolved,usingadierentmethodfromHasimoto,forthedragforceand thenpermeabilityoftheperiodicarrayofspheres.Althoughtheseauthorshadfoundthe Stoke'ssolutionstheoretically,numericalconstantsarestillneededtodeterminethedrag coecient. InthisChapter,wenumericallycalculatethesolutionofthesystemofequations.40.42.AmixedniteelementmethodisappliedtothesystemofequationsandthevariationalformulationisformulatedinSection3.2whilethevalidationofthecodeispresented inSection3.3.ForthespherecasewecompareourresultswiththatofRodrigoandManuel [85]whosenumericalsolutionsarecheckedwithZickandHomsy.Comparingthenumerical resultswithexperimentaldataleadstondingtherelationshipbetweenthedragcoecient andpermeabilitywhichisshowninSection3.1.Thenumericalresultsandthepolynomial approximationsofthepermeabilitytensorsaredemonstratedinSection3.4.Finally,the famousKozeny-Carmanequationisalsoappliedtondoneoftheclosedformsofthenumericalpermeability,1 = 3tr k ,inSection3.5. 17

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3.1RelationshipbetweenDragCoecientandPermeability Thissectioniswrittenfortheconversionofdragcoecienttopermeabilitywhichisused inSection3.3comparingournumericalresultwiththeexperimentaldatafromSullivan,[98], whomeasuredthedragcoecientofcylindricalbersempirically.Whendealingwithporous media,weusetheconceptstatedthatthepressuregradientisthedragforce.Beforederiving therelationship,wepresentthedragcoecientofasphereandcylinder.Thedragforce conceptiscommonlyusedtodeterminetheforceonasolidobjectduetotheowofuid aroundit.Bydenition,thedragforce F d exertedoneachobjectbyuidisgivenby[111] F d i = Z )]TJ/F22 5.9776 Tf 5.289 -1.339 Td [(S t ij x n j x d x .1 where t ij = )]TJ/F20 11.9552 Tf 9.299 0 Td [(p ij + v i;j + v j;i .2 isthestresstensor; p isthepressure; ij istheidentity; v i;j or v j;i arethevelocitygradients; n j istheoutwardunitnormalvectorand)]TJ/F21 7.9701 Tf 214.251 -1.794 Td [(S isthesurfaceofthesolidregion.Zickand Homsy,[111]showedthatthedragforceonasinglesphereinaperiodicarrayofspheresis F d i =6 r p C d v s i .3 where v s = 1 j j Z F v i x d x .4 isthesupericialvelocity; v i isthevelocityintheuidregion; j j isavolumeofthecell; r p istheradiusofthesphereand C d isthedragcoecient.Notethatthenotationsand F areusedinthesamewayasthoseinChapter2. Foraperiodicarrayofcylinders,wherethecylindersareperpendiculartotheow,Lamb [61]modiedOseenstechniqueandlinearizedequations[78]toanarrayofcylinders.He 18

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obtainedthetotaldragforceperunitlengthofacylinder F d =8 C .5 where C = U= [2 : 0 : 0 )]TJ/F20 11.9552 Tf 12.091 0 Td [(ln Re ]; Re = aU= istheReynoldsnumber; U istheupstream velocityofuidwithouttheeectofcylinders; a isthediameterofthecylinderand isthe densityoftheuid.Whentheowalignsparalleltothecylinders,Iberall,[53]claimedthat thedragcoecientofacylinderinacellisnotthesameasthatofanisolatedcylinder. Iberall[53]approximatedthedragforceonacylinderenclosedbytheothercylinders: F d =4 v a .6 where v a = v s = l istheaveragevelocityoftheuidinacelland v s isthesupercialvelocity oftheuidinacelland l istheporosity.Tobeabletondarelationshipbetweenthe dragcoecientandpermeability,wefollowaconceptinthedragtheorywhichstatesthat thetotaldragforceonthecylindersinacellisthepressuredropacrossthecell,[32],i.e. thepressuregradientinDarcy'sLawcanbereplacedbythedragforce.Sincethedrag coecientofacylinderinacellarrayandthatofanisolatedcylinderaredierent,we employthepressuredropequationofChen[32].Heincludedallofthedragforcesonthe cylinderinacellasapressuredropacrossthelter.Hisequationis: p l = 2 s C v C d v 2 a a a a 2 s ; .7 where p isthepressuredropacrossacellarrayofparallel,vertical,cylinders; l isthe thicknessofthecell; s isthesolidvolumefraction; a a and a s arearithmeticaverageber andsurfaceaverageberdiametersofthecylindersinthecellrespectively; C d isthedrag coecienttobedeterminedand C v isaunitconversionfactor,forexample, C v =980 g.masscm./gforcesec. 2 etc.Asthebersbecomemoredense,thedragcoecient ismodiedviathevolumefractions.Foraperiodicarrayofcylinder,ourproblem,.7 becomes p l = 2 s a C d v 2 a .8 19

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where C v =1and a a = a 2 s =1 =a and a isthediameterofthecylinder.Using v a = v s = l as above,wehave p l = 2 s a C d v a l v s : .9 ApplyingDarcy'sLawto.9and,forsimplicity,assumingallvariablesarescalars,the permeabilityanddragforcecoecientarerelatedby k = a 2 s 1 C d l v a = )]TJ/F20 11.9552 Tf 11.955 0 Td [( s a 2 2 s Re 1 C d = 2 )]TJ/F20 11.9552 Tf 11.956 0 Td [( s r 2 s Re 1 C d ; .10 where Re = v a a= and r istheradiusofthecylinder.Hence, C d 2 Re = )]TJ/F20 11.9552 Tf 11.955 0 Td [( s s 1 k ; .11 where k = k=r 2 isthedimensionlesspermeability.Equation.11willbeusedtocalculate thedimensionlesspermeabilityfromthedragcoecient. 3.2DiscretizationoftheModelProblem Inthissection,wenumericallyapproximatethesolutiontoequations.40-.42using amixedniteelementmethod.Thismethodisemployedtoobtainanapproximatesolution tothesystemofequationsusingavariationalformulation[84].Let L 2 0 = f q 2 L 2 : Z qdy =0 g ; .12 where q isascalarperiodicfunctionand L 2 = W 0 2 istheSobolevspace.Inthreedimensions,thesystemofequations.40-.41canberewrittenas @u i 1 @y 1 + @u i 2 @y 2 + @u i 3 @y 3 =0.13 @ 2 u i 1 @y 2 1 + @ 2 u i 1 @y 2 2 + @ 2 u i 1 @y 2 3 = @q i @y 1 )]TJ/F20 11.9552 Tf 11.955 0 Td [(e i 1 .14 @ 2 u i 2 @y 2 1 + @ 2 u i 2 @y 2 2 + @ 2 u i 2 @y 2 3 = @q i @y 2 )]TJ/F20 11.9552 Tf 11.955 0 Td [(e i 2 .15 @ 2 u i 3 @y 2 1 + @ 2 u i 3 @y 2 2 + @ 2 u i 3 @y 2 3 = @q i @y 3 )]TJ/F20 11.9552 Tf 11.955 0 Td [(e i 3 .16 20

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wherethesuperscript isdroppedforconvenienceand e i j =1if j = i and e i j =0if j 6 = i , i =1 ; 2 ; 3.Topreservethesymmetryofthestinessglobalmatrix,wemultiplyequation .13by-1.Afterthemultiplication,.13{.16canberewrittenintheindicialnotation as )]TJ/F20 11.9552 Tf 13.151 8.087 Td [(@u i l @y l =0 ; .17 )]TJ/F20 11.9552 Tf 13.695 8.088 Td [(@q i @y j + @ 2 u i j @y 2 l + ij =0 ;l =1 ; 2 ; 3 ; .18 whereagainthedoubleindex l meansthesummationand ij istheKroneckerdelta.Note thattheequation.18canberewrittenas )]TJ/F20 11.9552 Tf 14.908 8.087 Td [(@ @y l q i jl )]TJ/F20 11.9552 Tf 13.151 8.888 Td [(@u i j @y l ! + ij =0 ;l =1 ; 2 ; 3 : .19 Applyingthemixedniteelementmethod,[84],wemultiplytheequation.17and.19 bytestfunctions Q i ; w i 2 L 2 0 H ,respectively, )]TJ/F25 9.9626 Tf 11.955 14.059 Td [(Z Q i @u i l @y l =0 ; .20 Z w i j )]TJ/F20 11.9552 Tf 14.908 8.087 Td [(@ @y l q i jl )]TJ/F20 11.9552 Tf 13.151 8.888 Td [(@u i j @y l ! + ij ! =0 ;l =1 ; 2 ; 33.21 where i indicatesthenumberofsystemofequations.Employingtheintegrationbyparts [21]to.21, )]TJ/F25 9.9626 Tf 11.291 14.059 Td [(Z )]TJ/F20 11.9552 Tf 7.779 8.579 Td [(w i j q i jl )]TJ/F20 11.9552 Tf 13.151 8.889 Td [(@u i j @y l ! n l + Z @w i j @y l q i jl )]TJ/F20 11.9552 Tf 13.15 8.889 Td [(@u i j @y l ! = )]TJ/F25 9.9626 Tf 11.291 14.058 Td [(Z w i j ij ;l =1 ; 2 ; 3 : .22 Employingtheperiodicboundarycondition,.22becomes Z @w i j @y l q i jl )]TJ/F20 11.9552 Tf 13.151 8.889 Td [(@u i j @y l ! = )]TJ/F25 9.9626 Tf 11.291 14.059 Td [(Z w i j ij ;l =1 ; 2 ; 3 ; .23 or Z )]TJ/F20 11.9552 Tf 9.298 0 Td [(q i @w i j @y j + @w i j @y l @u i j @y l = Z w i j ij ;l =1 ; 2 ; 3 : .24 Intheweakform,theproblembecomes:Find u i ;q i 2 H L 2 0 suchthat )]TJ/F25 9.9626 Tf 11.291 14.059 Td [(Z r y u i Q i dy =0forall Q i 2 L 2 0 ; .25 21

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Z )]TJ/F20 11.9552 Tf 9.298 0 Td [(q i r y w i + r y u i : r y w i dy = Z e i w i dy; forall w i 2 H ; .26 where.25and.26comefrom.20and.24,respectively.Atthispoint,wedropthe superscript i forconvenience.Notethattheweightfunctionsbelongtonite-dimensional subspacesofthespace H L 2 0 .Let= [ J e =1 e ,where e isatypicalelement[84]. Ineachelement, u j and q areapproximatedby: u j = M X m =1 m y u m j = 1 ::: M 0 B B B B B B @ u 1 j : u M j 1 C C C C C C A = T u j ; .27 q = N X n =1 n y q n = 1 ::: N 0 B B B B B B @ q 1 : q N 1 C C C C C C A = T Q ; .28 where j =1 ; 2 ; 3; M and N arethenumbersofnodesofthequadraticandlinearfunctions, respectively,and m and n aretheinterpolationfunctions. Substitutingboth.27and3.28into.20and.24: )]TJ/F25 9.9626 Tf 11.955 17.534 Td [(" Z e @ T @y l dy # u l =0 ; .29 )]TJ/F25 9.9626 Tf 11.955 17.534 Td [(" Z e @ @y j T dy # Q + " Z e @ @y l @ T @y l dy # u j = Z e ij dy; .30 wehave j equationsforeach i withthesummationundertherepeat l .Anotherexpression of.29and.30is )]TJ/F32 11.9552 Tf 9.298 0 Td [(D 1 Q + E 11 + E 22 + E 33 u 1 = F i 1 .31 )]TJ/F32 11.9552 Tf 9.298 0 Td [(D 2 Q + E 11 + E 22 + E 33 u 2 = F i 2 .32 )]TJ/F32 11.9552 Tf 9.298 0 Td [(D 3 Q + E 11 + E 22 + E 33 u 3 = F i 3 .33 )]TJ/F32 11.9552 Tf 9.298 0 Td [(D T 1 u 1 )]TJ/F32 11.9552 Tf 11.955 0 Td [(D T 2 u 2 )]TJ/F32 11.9552 Tf 11.956 0 Td [(D T 3 u 3 =0 ; .34 wheresuperscript T denotesatransposeofthevectorormatrixand D j = Z e @ @y j T dy; D T j = Z e @ T @y j dy .35 22

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E ll = Z e @ @y l @ T @y l dy; F ij = Z e ij dy; .36 where F ij = R e dy if i = j and F ij =0if i 6 = j .Rewriting.31{.34intoanexplicit matrixform 0 B B B B B B B B B B @ E 00 )]TJ/F32 11.9552 Tf 9.298 0 Td [(D 1 0 E 0 )]TJ/F32 11.9552 Tf 9.298 0 Td [(D 2 00 E )]TJ/F32 11.9552 Tf 9.298 0 Td [(D 3 )]TJ/F32 11.9552 Tf 9.298 0 Td [(D T 1 )]TJ/F32 11.9552 Tf 9.298 0 Td [(D T 2 )]TJ/F32 11.9552 Tf 9.298 0 Td [(D T 3 0 1 C C C C C C C C C C A 0 B B B B B B B B B B @ u 1 u 2 u 3 Q 1 C C C C C C C C C C A = 0 B B B B B B B B B B @ F i 1 F i 2 F i 3 0 1 C C C C C C C C C C A .37 where E = E 11 + E 22 + E 33 and and aredenedasin.27and.28.Another popularformofthematrix.37inarticles[1,4]is 0 B B @ A )]TJ/F32 11.9552 Tf 9.298 0 Td [(D )]TJ/F32 11.9552 Tf 9.298 0 Td [(D T 0 1 C C A 0 B B @ U Q 1 C C A = 0 B B @ F 0 1 C C A .38 where U = 0 B B B B B B @ u 1 u 2 u 3 1 C C C C C C A D = 0 B B B B B B @ D 1 D 2 D 3 1 C C C C C C A F = 0 B B B B B B @ F i 1 F i 2 F i 3 1 C C C C C C A and A = 0 B B B B B B @ E 00 0 E 0 00 E 1 C C C C C C A : .39 ItiswellknownthatforDirichletconditionson u i thesolution q isdenedonlyuptoan additiveconstant,whichisusuallyxedbyimposing R q =0,[[22],p.157and[24],p.16]. Furthermore,inordertoretainthesymmetryofthematrix,weaddonemorecolumnand rowasfollowing, 0 B B B B B B @ E )]TJ/F32 11.9552 Tf 9.299 0 Td [(D0 )]TJ/F32 11.9552 Tf 9.299 0 Td [(D T 01 01 0 1 C C C C C C A 0 B B B B B B @ U Q 1 C C C C C C A = 0 B B B B B B @ F 0 0 1 C C C C C C A .40 where isverysmallaftersolvingthematrixandthesecondordertensorisobtainedfrom theformula.43.Moreover,thepermeabilitycanbeveriedbyreplacing ! i in2.39by 23

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u j andusingthesymmetricpropertyofthepermeability.Then k ij = 1 j j Z F u j i dy = 1 j j Z F r u j : r u i dy = j j u j ; u i H .41 isusedasaconsistencycheckwithournumericalresults k ij from.43. InordertoensuretheLadyzhenskaya-Babuska-BrezziLBBconsistencycondition,we usetheTaylor-Hoodisoparametric10-nodetetrahedra,i.e. M =10and N =4forquadratic andlinearfunctions,respectively.ThediscretizationaregeneratedbytheopensoftwareNetgen[92].Areferenceforthree-dimensionalniteelementprogrammingisKwonandBang [60]andotherreferencesonprogrammingandimplementingperiodicboundaryconditions canbefoundin[80]and[97]. 3.3ValidationofNumericalResults Incaseofasimplecubicarrayofspheres,ZickandHomsy[111]andH.Hasimoto[49] beganwiththeslowowasdescribedbyStoke'sequation,andfoundtheperiodicsolutions analyticallyforthedragcoecientactingonaperiodicarrayofsmallspheres.Thereafter, RodrigoandManuel[85]convertthedragcoecienttobepermeabilityandcomparetheir numericalresultswiththeanalyticalones.Sincethepermeabilityiscalculatedinthiswork, wecompareourresultswithRodrigoandManuel[85].Incaseofparallelcylinders,threedierenceexperimentaldatafrompreviouspublications[26,98,32]arechosentovalidateour numericalresults.Notethat,thepermeabilitiesfrombothexperimentaldataandanalytical resultsforeachporositywereexpressedasascalarwhichwassupposedtobegivenas amatrixbecauseoftheanisotropicmedium.Therefore,tocompareoursolutionswith theexperimentaldata,weaveragethecomponentsofthenumericalpermeabilitytensorto competewiththosefactsinwhichthegraphsareshownbelow. Tovalidatethecode,theperiodicityassumptionisveriedbycalculatingthepermeabilitytensorsusingthreedierentperiodiccellarraysofcylinders.Thesmallestcellconsistsof 5cylinders,seethegureinTable3.1,withtheradius r = a= 2=0 : 1, d =0 : 3andtheheight h =0 : 5.Thenthedimensionofthecellis0 : 5 0 : 5 0 : 5.Theothertwocellscontain13and 24

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25cylinderswiththesameradius r andthedistance d asthesmallestone.Thedimension ofthe13-cylindercellis1 1 1andthatofthe25-cylindercellis1 : 5 1 : 5 1 : 5.Asis theoreticallyexpected,thepermeabilitytensorsofthecellsshowninTable3.1arethesame forallofthedierentperiodicREV's.Nextweallowedtheradiiofthecylinderstoincrease almosttoapointoftouchingsothat)]TJ/F21 7.9701 Tf 204.097 -1.793 Td [(F isalmostnotLipschitzbutthedomainisstilla Lipschitzfunction.Figure3.1isagureof F forthiscongurationandweseethatthe permeability: k =1 : 0 e )]TJ/F16 11.9552 Tf 11.955 0 Td [(04 0 B B B B B B @ 0 : 00060 : 00000 : 0000 0 : 00000 : 00040 : 0000 0 : 00000 : 00000 : 1077 1 C C C C C C A .42 tendstozeroforthisgeometry.TheFigurepresentsaperiodiccellwith5cylinderswhilethe radiiofthecylindersatthecenterandcornersare0.1and0.249,respectively.Furthermore, iftheratio r=d seeFigure1.3isincreasedthenthevaluesofthepermeabilityinevery direction, k ii ;i =1 ; 2 ; 3decreaseasexpected,seeFigure3.2.Notethatwhentheangle betweenthecylindersandhorizontalplaneis90degree, k 11 = k 22 .aswecanseein3Figure 3.2.Moreover,foreachratio r=d ,thevaluesof k 33 aretwicethatof k 11 or k 22 .Thatis theresistanceisdoubledwhenthecylindersareorientatedorthogonaltotheowdirection. ThisobservationwaspreviouslystatedinthetheoreticalpartofthepaperbyJacksonand James,[55].Thishelpstoavoidthenumericaldicultywhenthepermeabilitiesofboth geometriesneedtobeprovided.Theeectoftheangleofthecylindersmakeswiththe baseisprovidedinFigures3.3,3.4and3.5.InFigure3.3,weshowthediagonalentriesof thepermeabilitytensorforthecaseofdistinctangleswhichcanbeusedascoecientsin governingequationswhileFigures3.4and3.5showo-diagonalcomponentsofthetensor. Because k issymmetric,only k 12 ;k 13 and k 23 arepresentedinFigures3.4and3.5. Tofurthervalidatethecalculation,wecompareourresultswiththoseofRochaand Cruz,2009[85],inwhichthegeometryisasimplecubicarrayofspheresandthesolutions areobtainednumerically.Thegeometryoftheperiodiccubiccellconsistsofasingle 25

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Table3.1: Permeabilitythroughthecellarrayofcylindersat90degreeofthedierent heightsandperiodicREVsprovidingthesamepermeability. k 0.00170.00000.0000 0.00000.00170.0000 0.00000.00000.0038 k denotesthepermeability. Figure3.1: Permeabilitybeingclosetozeroofuidowthroughacellarrayofcylinders whentheboundaryisalmostnotLipschitz 26

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Figure3.2: Permeabilityofuidowthroughacellarrayofcylinderswithincreasing r=d , =90 .Then r=d =1 : 2077istouching. Figure3.3: Diagonalvaluesofthepermeabilitytensorasafunctionofangle for r=d xedat1 = 3. 27

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Figure3.4: O-diagonalvaluesofthepermeabilitytensor k 12 and k 23 asafunctionof angle for r=d xedat1 = 3. Figure3.5: O-diagonalvaluesofthepermeabilitytensor k 13 asafunctionofangle for r=d xedat1 = 3. 28

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Table3.2: Permeabilitythroughthesimplecubicarrayofsphereswithsolidvolumefraction s =0 : 216andradius=0.5unit; k s denotesthepermeabilitiesinthisresearch; k isthe relativeerrorof k s withrespectto k RM whichisthepermeabilitycalculatingRochaand Cruz. #dofCPUtimesec k s k % 11,329144.310.035191.793 81,47016,804.780.034800.665 615,812326,804.180.034630.173 sphereofunitarydiameteratthecenterofandhasacelllengthof f = s g 1 = 3 ,where s isthevolumefractionofthesolidsphere.Becausethesphereisisotropic,thepermeability tensorisascalarmultiplyoftheidentity, k I .Table3.2showsthepermeabilityforsolid volumefraction s =0 : 216.Thevariables k s and k RM =0 : 03457denotethepermeabilities obtainedinthisresearchandbyRochaandCruz,respectively; k istherelative errorof k s withrespectto k RM ;#dofisthenumberofdegreesoffreedom.Notethatasthe numberofdegreesoffreedom,whichinthiscaseisthenumberofnodesusedtocalculatethe velocitiesandpressureincreases,theerrordecreases.Withapproximately80,000degreesof freedom,therelativeerrorislessthan0.7%withCPUtimeabout4.6hours.Althoughthe errorcanbedecreasedto0.173%moretimeisrequired-ontheorder3.8days.Dierent valuesofsolidvolumefractionarecomparedandtheresultsarepresentedinTable3.3.For s =0 : 125,therelativeerrorissmallest,0.096%,butthiscaserequiresthelargernumberof degreesoffreedom,619,656.Althoughwedidn'tincreasethenumberofdegreesoffreedom forthelargersolidvolumefractions,weseefromTable3.2thatthelargestrelativeerroris only1%.Thecomparedresultsareinagoodagreementforasimplecubicarrayofspheres. Next,wecompareourresultswithtwodierentsetsofexperimentaldataforanarray ofcylinders.OnesetofthedataarefromSullivan[98]inwhichtheowisparalleltoan arrayofcylindersandanothersetofdataarefromBrown[26]andChen[32]inwhichthe owisperpendiculartothearrayofcylinders.Sullivan[98]presentsexperimentalresults 29

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Table3.3: Permeabilitythroughthesimplecubicarrayofsphereswithvaryingvolume fractionofsolid s withradius0.5unit; k s and k RM denotethepermeabilitiesinthisresearch andbyRochaandCruz2009,respectively; k istherelativeerrorof k s withrespectto k RM . s k s k RM k %#dof 0.1250.10370.10360.096619,656 0.2160.034630.034570.173615,812 0.3430.010640.010521.14081,660 0.4500.0044190.0043980.47778,740 forowparalleltocylindricalberssuchasglasswool,blondandchinesehair,andgoat wool.ThedatafromChen[32]aregivenintermsofthedragcoecient.Forthisreferenceweuse.11tondthepermeabilities.Moreover,whenthearrayofcylindersaligns parallelwiththeow,thepermeabilityistwicethatoftheperpendicularorientation,i.e. 2 k = = 2= k = ,[55].Then,weonlycalculateournumericalresultswhenthearray ofcylindersisperpendiculartotheowdirectiontoavoidthemeshgenerationproblems andcomparetheresultswiththeexperimentaldata.Figure3.6showingtheplotsofour numericalresults2 k 33 and2 k 11 + k 22 andexperimentaldatafromSullivan[98]displays theharmonization.Similarly,when = = 2,thedatafromBrown[26]andChen[32]are comparedwithournumericalresultsinFigure3.7.Thedatacomparewellwith k 33 and k 11 + k 22 .NotethatthedatafromChenoscillatebecausehecollecteddatafromseveral publications. 3.4NumericalPermeabilityFunctions Inthissection,weprovidepolynomialapproximationofeachentryofthepermeability tensorasafunctionofthedistancebetweenthecylinders,theradiusofthecylinderandthe anglethecylindersmakewiththebase.Figure3.8illustratestheideaofhowthepermeability varieswiththegeometry.Thetopleftgraphpresentsthenumericalpermeabilityinthe 30

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Figure3.6: Permeabilityofuidowthroughaperiodicarrayofcylindersfor =0.When theowalignsparalleltothecylinders,guresshowtheexperimentalscalarpermeabilities ofSullivanforhairsandgoatwool,respectively,comparingwithournumerical2 k 33 and 2 k 11 + k 22 resultsfor = = 2. 31

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Figure3.7: Permeabilityofuidowthroughacellarrayofcylinders.FiguresarecomparingtheexperimentaldataofBrownandChenwithournumerical k 33 and k 11 + k 22 when theowisperpendiculartothearrayofcylinders. 32

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direction x or k 11 asafunctionofthedimensionlessquotientbetweentheradiusofacylinder, r ,andthedistancebetweenthem, d x -axis;andtheangle y -axisbetweencylindersand thehorizontalplanedenedinFigure1.3.Thevariable r takesonvaluesbetween0 : 025and 0 : 125andtheangle takesonvaluesbetween26to90degrees. Next,weapproximateeachpermeabilitycomponentbyafourth-orderpolynomialofthe form a 1 x 4 + a 2 x 3 y + a 3 x 3 + a 4 x 2 y 2 + a 5 x 2 y + a 6 x 2 + a 7 xy 3 + a 8 xy 2 + a 9 xy + a 10 x + a 11 y 4 + a 12 y 3 + a 13 y 2 + a 14 y + a 15 ,wherethecoecients a i ;i =1 ; 2 ;:::; 15areprovidedinTables3.4 and3.5.FortherighttopofFigure3.8,weestimatethenumericalpermeabilitytensorusing thefourth-orderpolynomials.Theabsoluteerrorsandthetwo-normareexpressedonthe bottomone.Similargraphscanbeobtainedfortheothercomponents.Figures3.9and3.10 showthenumericallygeneratedgraphsofthediagonalcomponents k 22 and k 33 andtheodiagonalcomponents k 12 , k 13 and k 23 ,respectively.TheL2-normforthedierencebetween thenumericalresultsandpolynomialapproximationsforeachcomponentispresentedin Table3.6. 3.5ComparisonwithKozeny-CarmanEquation TheKozeny-Carmanisafamousequationwhichcanindicateanempiricalrelationship betweentheporosityandtheisotropic,scalar,permeability[28], k = l 3 g k 1 S 2 ; .43 fromtherelationshipbetweenthesupercialvelocityandthepressuregradient v s = l 3 g k 1 S 2 4 p l ; .44 where k 1 isaconstantdependingonthegeometry; S isanareaofparticularsurfaceper unitvolumeofpackedspace,[ S ]=[1 =L ].Becausethisexpressionhasbeenwidelyused [15,27,35],wecompareournumericalresultswiththisrelationship,eventhoughtheisotropy assumptionforanarrayofparallelcylindersisclearlynotvalid.Fortheparallelcylinders, 33

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Figure3.8: Thetopleftgraphisthenumericallygeneratedpermeabilitycomponent k 11 whilethetoprightoneisthefourth-orderpolynomialapproximationofthetopleftgraph; thepointwiseabsoluteerrorareshownatthebottom. 34

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Figure3.9: Thediagonalcomponents k 22 and k 33 ofthepermeabilitytensor 35

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Figure3.10: Theo-diagonalcomponents k 12 ;k 13 and k 23 ofthepermeabilitytensor. 36

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Table3.4: Thefourth-orderpolynomialfunctions: a 1 x 4 + a 2 x 3 y + a 3 x 3 + a 4 x 2 y 2 + a 5 x 2 y + a 6 x 2 + a 7 xy 3 + a 8 xy 2 + a 9 xy + a 10 x + a 11 y 4 + a 12 y 3 + a 13 y 2 + a 14 y + a 15 approximating k 11 ;k 22 and k 33 . Coecients k 11 k 22 k 33 a 1 1 : 0198 e +000984 : 1116 e )]TJ/F16 11.9552 Tf 11.955 0 Td [(0031 : 3062 e +000 a 2 )]TJ/F16 11.9552 Tf 9.298 0 Td [(255 : 7726 e )]TJ/F16 11.9552 Tf 11.955 0 Td [(006 )]TJ/F16 11.9552 Tf 9.298 0 Td [(2 : 5282 e )]TJ/F16 11.9552 Tf 11.955 0 Td [(003 )]TJ/F16 11.9552 Tf 9.298 0 Td [(8 : 1104 e )]TJ/F16 11.9552 Tf 11.955 0 Td [(003 a 3 )]TJ/F16 11.9552 Tf 9.299 0 Td [(1 : 3507 e +000 )]TJ/F16 11.9552 Tf 9.299 0 Td [(1 : 0963 e +000 )]TJ/F16 11.9552 Tf 9.299 0 Td [(1 : 2944 e +000 a 4 )]TJ/F16 11.9552 Tf 9.298 0 Td [(18 : 5001 e )]TJ/F16 11.9552 Tf 11.955 0 Td [(006 )]TJ/F16 11.9552 Tf 9.298 0 Td [(12 : 9402 e )]TJ/F16 11.9552 Tf 11.955 0 Td [(006 )]TJ/F16 11.9552 Tf 9.298 0 Td [(24 : 0304 e )]TJ/F16 11.9552 Tf 11.955 0 Td [(006 a 5 2 : 4220 e )]TJ/F16 11.9552 Tf 11.955 0 Td [(0034 : 4395 e )]TJ/F16 11.9552 Tf 11.955 0 Td [(00311 : 9560 e )]TJ/F16 11.9552 Tf 11.955 0 Td [(003 a 6 671 : 5718 e )]TJ/F16 11.9552 Tf 11.955 0 Td [(003425 : 4400 e )]TJ/F16 11.9552 Tf 11.955 0 Td [(003365 : 6415 e )]TJ/F16 11.9552 Tf 11.955 0 Td [(003 a 7 )]TJ/F16 11.9552 Tf 9.298 0 Td [(151 : 3900 e )]TJ/F16 11.9552 Tf 11.955 0 Td [(0093 : 2985 e )]TJ/F16 11.9552 Tf 11.955 0 Td [(009279 : 1924 e )]TJ/F16 11.9552 Tf 11.955 0 Td [(009 a 8 44 : 3630 e )]TJ/F16 11.9552 Tf 11.955 0 Td [(00612 : 5010 e )]TJ/F16 11.9552 Tf 11.955 0 Td [(006 )]TJ/F16 11.9552 Tf 9.299 0 Td [(29 : 5497 e )]TJ/F16 11.9552 Tf 11.955 0 Td [(006 a 9 )]TJ/F16 11.9552 Tf 9.298 0 Td [(3 : 8002 e )]TJ/F16 11.9552 Tf 11.955 0 Td [(003 )]TJ/F16 11.9552 Tf 9.298 0 Td [(2 : 8409 e )]TJ/F16 11.9552 Tf 11.955 0 Td [(003 )]TJ/F16 11.9552 Tf 9.298 0 Td [(3 : 4598 e )]TJ/F16 11.9552 Tf 11.955 0 Td [(003 a 10 )]TJ/F16 11.9552 Tf 9.298 0 Td [(112 : 1466 e )]TJ/F16 11.9552 Tf 11.955 0 Td [(003 )]TJ/F16 11.9552 Tf 9.298 0 Td [(50 : 8733 e )]TJ/F16 11.9552 Tf 11.955 0 Td [(003 )]TJ/F16 11.9552 Tf 9.299 0 Td [(34 : 9274 e )]TJ/F16 11.9552 Tf 11.955 0 Td [(003 a 11 258 : 6547 e )]TJ/F16 11.9552 Tf 11.955 0 Td [(012 )]TJ/F16 11.9552 Tf 9.298 0 Td [(2 : 4185 e )]TJ/F16 11.9552 Tf 11.955 0 Td [(012176 : 7506 e )]TJ/F16 11.9552 Tf 11.955 0 Td [(012 a 12 )]TJ/F16 11.9552 Tf 9.298 0 Td [(10 : 3378 e )]TJ/F16 11.9552 Tf 11.955 0 Td [(009 )]TJ/F16 11.9552 Tf 9.298 0 Td [(5 : 9015 e )]TJ/F16 11.9552 Tf 11.955 0 Td [(009 )]TJ/F16 11.9552 Tf 9.299 0 Td [(170 : 4837 e )]TJ/F16 11.9552 Tf 11.955 0 Td [(009 a 13 )]TJ/F16 11.9552 Tf 9.298 0 Td [(8 : 0786 e )]TJ/F16 11.9552 Tf 11.955 0 Td [(006 )]TJ/F16 11.9552 Tf 9.298 0 Td [(2 : 1735 e )]TJ/F16 11.9552 Tf 11.955 0 Td [(00622 : 2997 e )]TJ/F16 11.9552 Tf 11.955 0 Td [(006 a 14 905 : 0796 e )]TJ/F16 11.9552 Tf 11.955 0 Td [(006566 : 7804 e )]TJ/F16 11.9552 Tf 11.955 0 Td [(006 )]TJ/F16 11.9552 Tf 9.299 0 Td [(274 : 0899 e )]TJ/F16 11.9552 Tf 11.955 0 Td [(006 a 15 804 : 5324 e )]TJ/F16 11.9552 Tf 11.955 0 Td [(006 )]TJ/F16 11.9552 Tf 9.298 0 Td [(2 : 7328 e )]TJ/F16 11.9552 Tf 11.955 0 Td [(0038 : 2408 e )]TJ/F16 11.9552 Tf 11.955 0 Td [(003 where a i ;i =1 ; 2 ;:::; 15arethecoecients; x = r=d ; y = whichisdenedinFigure1.3; r 2 [0 : 025 ; 0 : 125]and 2 [arctan.5,arctaninf],about26to90degree. 37

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Table3.5: Thefourth-orderpolynomialfunctions: a 1 x 4 + a 2 x 3 y + a 3 x 3 + a 4 x 2 y 2 + a 5 x 2 y + a 6 x 2 + a 7 xy 3 + a 8 xy 2 + a 9 xy + a 10 x + a 11 y 4 + a 12 y 3 + a 13 y 2 + a 14 y + a 15 approximating k 12 ;k 13 and k 23 . Coecients k 12 k 13 k 23 a 1 )]TJ/F16 11.9552 Tf 9.299 0 Td [(499 : 8635 e )]TJ/F16 11.9552 Tf 11.955 0 Td [(006297 : 6621 e )]TJ/F16 11.9552 Tf 11.955 0 Td [(003428 : 9162 e )]TJ/F16 11.9552 Tf 11.955 0 Td [(006 a 2 )]TJ/F16 11.9552 Tf 9.298 0 Td [(9 : 2789 e )]TJ/F16 11.9552 Tf 11.955 0 Td [(0061 : 2256 e )]TJ/F16 11.9552 Tf 11.955 0 Td [(003 )]TJ/F16 11.9552 Tf 9.299 0 Td [(1 : 4217 e )]TJ/F16 11.9552 Tf 11.955 0 Td [(006 a 3 933 : 9637 e )]TJ/F16 11.9552 Tf 11.955 0 Td [(006 )]TJ/F16 11.9552 Tf 9.299 0 Td [(459 : 2711 e )]TJ/F16 11.9552 Tf 11.955 0 Td [(003 )]TJ/F16 11.9552 Tf 9.299 0 Td [(371 : 3372 e )]TJ/F16 11.9552 Tf 11.955 0 Td [(006 a 4 80 : 3350 e )]TJ/F16 11.9552 Tf 11.955 0 Td [(009 )]TJ/F16 11.9552 Tf 9.299 0 Td [(36 : 4347 e )]TJ/F16 11.9552 Tf 11.955 0 Td [(006 )]TJ/F16 11.9552 Tf 9.299 0 Td [(5 : 9070 e )]TJ/F16 11.9552 Tf 11.955 0 Td [(009 a 5 )]TJ/F16 11.9552 Tf 9.298 0 Td [(1 : 1085 e )]TJ/F16 11.9552 Tf 11.955 0 Td [(0062 : 9797 e )]TJ/F16 11.9552 Tf 11.955 0 Td [(0031 : 9071 e )]TJ/F16 11.9552 Tf 11.955 0 Td [(006 a 6 )]TJ/F16 11.9552 Tf 9.298 0 Td [(305 : 3777 e )]TJ/F16 11.9552 Tf 11.955 0 Td [(006167 : 6087 e )]TJ/F16 11.9552 Tf 11.955 0 Td [(00376 : 7183 e )]TJ/F16 11.9552 Tf 11.955 0 Td [(006 a 7 )]TJ/F16 11.9552 Tf 9.298 0 Td [(393 : 8843 e )]TJ/F16 11.9552 Tf 11.955 0 Td [(012 )]TJ/F16 11.9552 Tf 9.299 0 Td [(10 : 4710 e )]TJ/F16 11.9552 Tf 11.955 0 Td [(009 )]TJ/F16 11.9552 Tf 9.299 0 Td [(78 : 2267 e )]TJ/F16 11.9552 Tf 11.955 0 Td [(012 a 8 18 : 3621 e )]TJ/F16 11.9552 Tf 11.955 0 Td [(00931 : 5771 e )]TJ/F16 11.9552 Tf 11.955 0 Td [(00617 : 8581 e )]TJ/F16 11.9552 Tf 11.955 0 Td [(009 a 9 )]TJ/F16 11.9552 Tf 9.298 0 Td [(251 : 6399 e )]TJ/F16 11.9552 Tf 11.955 0 Td [(009 )]TJ/F16 11.9552 Tf 9.298 0 Td [(3 : 1336 e )]TJ/F16 11.9552 Tf 11.955 0 Td [(003 )]TJ/F16 11.9552 Tf 9.299 0 Td [(1 : 5587 e )]TJ/F16 11.9552 Tf 11.955 0 Td [(006 a 10 39 : 8263 e )]TJ/F16 11.9552 Tf 11.955 0 Td [(00611 : 6868 e )]TJ/F16 11.9552 Tf 11.955 0 Td [(00320 : 6851 e )]TJ/F16 11.9552 Tf 11.955 0 Td [(006 a 11 754 : 8963 e )]TJ/F16 11.9552 Tf 11.955 0 Td [(015580 : 3406 e )]TJ/F16 11.9552 Tf 11.955 0 Td [(01216 : 5870 e )]TJ/F16 11.9552 Tf 11.955 0 Td [(015 a 12 )]TJ/F16 11.9552 Tf 9.298 0 Td [(33 : 8628 e )]TJ/F16 11.9552 Tf 11.955 0 Td [(012 )]TJ/F16 11.9552 Tf 9.299 0 Td [(140 : 4834 e )]TJ/F16 11.9552 Tf 11.955 0 Td [(00921 : 7867 e )]TJ/F16 11.9552 Tf 11.955 0 Td [(012 a 13 )]TJ/F16 11.9552 Tf 9.298 0 Td [(2 : 2598 e )]TJ/F16 11.9552 Tf 11.955 0 Td [(0095 : 7977 e )]TJ/F16 11.9552 Tf 11.955 0 Td [(006 )]TJ/F16 11.9552 Tf 9.299 0 Td [(4 : 9221 e )]TJ/F16 11.9552 Tf 11.955 0 Td [(009 a 14 147 : 4440 e )]TJ/F16 11.9552 Tf 11.955 0 Td [(009251 : 2534 e )]TJ/F16 11.9552 Tf 11.955 0 Td [(006349 : 0144 e )]TJ/F16 11.9552 Tf 11.955 0 Td [(009 a 15 )]TJ/F16 11.9552 Tf 9.298 0 Td [(3 : 5492 e )]TJ/F16 11.9552 Tf 11.955 0 Td [(006 )]TJ/F16 11.9552 Tf 9.298 0 Td [(4 : 1563 e )]TJ/F16 11.9552 Tf 11.955 0 Td [(003 )]TJ/F16 11.9552 Tf 9.298 0 Td [(6 : 8813 e )]TJ/F16 11.9552 Tf 11.955 0 Td [(006 where a i ;i =1 ; 2 ;:::; 15arethecoecients; x = r=d ; y = whichisdenedinFigure1.3; r 2 [0 : 025 ; 0 : 125]and 2 [arctan.5,arctaninf],about26to90degree. 38

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Table3.6: L2-normerrorsofthecomponentsofthepermeability k componentsL2-normerror k 11 2 : 54 e )]TJ/F16 11.9552 Tf 11.955 0 Td [(04 k 22 1 : 32 e )]TJ/F16 11.9552 Tf 11.955 0 Td [(04 k 33 2 : 48 e )]TJ/F16 11.9552 Tf 11.955 0 Td [(04 k 12 2 : 25 e )]TJ/F16 11.9552 Tf 11.955 0 Td [(06 k 13 2 : 11 e )]TJ/F16 11.9552 Tf 11.955 0 Td [(04 k 23 1 : 10 e )]TJ/F16 11.9552 Tf 11.955 0 Td [(06 thereisaone-to-onerelationshipbetweentheangleandporositythatcanbedetermined analytically. Mostapplicationsareusingtheporositytoimplementtotheirproblemsasaprimary variabletondthedependentvalueofthepermeability.Kozeny-Carmanequationindicating therelationshipbetweenvelocityandgradientofpressureprovidesaspecicformofthepermeability.Geosciencecommunitiessometimeusethisequationtocalculatethepermeability fortheirporousmediaproblems.Therefore,inthissection,aformulaofthepermeability functionsassociatedwiththeKozeny-Carmanequationareexpressedandcomparedwith ournumericalsolutions. ToderivetheformulaofthepermeabilityusingKozeny-Carmanequation,itisnecessary toknowthevolumeofthecylinderschangeddependingontheanglethattheparallelarray ofcylindersmakeswiththehorizontalplaneintheperiodiccell.Figure3.11,ontheleft, showsthecellarrayofcylinderswitheachcellconsistingof2ellipsoidalcylinders:onein themiddleandanotheronefromthesidesafterbondingthemtogether.Thevolumeisused todeterminethetotalvolumeofthecellwhichisappliedtoattainthevariable S inthe 39

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Figure3.11: Geometry:theleftgureshowsthecellarrayofcylinderswhentheangle betweenthecylindersandhorizontalplaneis75degrees;therightoneshowseachcellconsists of2ellipsoidalcylinders. formula.43: k = l 3 g k 1 S 2 = C k l 3 S 2 ; .45 where C k = g=k 1 isaconstantand S = surfaceareaofsolidstayinginuidphase totalvolumeofthecell = S A V t : .46 Becausethesurfaceareaofanellipsoidalcylinderis perimeterofshapeA L +2areaofshapeA ; .47 wherethelengthofthecylinder L andthecross-sectionalarea A aredenedinFigure3.12 andtheuidispastonlyonthesideoftheellipsoidalcylinders. S A =2 L perimeterofshapeA ; .48 whereonlytwoellipsoidalcylindersareinthecell.Fortheperimeterofanellipse,thereare manyformulascanbeused.Inthiswork,weemployonlytwodierentones: 3 r + r 1 )]TJ/F25 9.9626 Tf 11.955 12.599 Td [(q r + r 1 r +3 r 1 ; .49 40

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Figure3.12: Figureshowstheradiiofanellipseinbothmajorandminoraxisesand is theanglebetweenthehorizontalplaneandthesideofthecylinderwhile h istheheightof theperiodiccellwhichisperpendiculartothehorizontalplane. and 2 s r 2 + r 2 1 2 ; .50 where r and r 1 aretheradiioftheellipse A ,denedinFigure3.12and r alsotheradiusof thecylinder.Theequation.50issimplerbut r 1 shouldnotmorethanthreetimeslonger than r .However,equation.49presentedbyIndianmathematicianRamanujanprovides betterapproximation.Substituting.49into.48,wehave S A =2 L 3 r + r 1 )]TJ/F25 9.9626 Tf 11.955 12.599 Td [(q r + r 1 r +3 r 1 .51 whilethesecondformulagives S A =4 L s r 2 + r 2 1 2 : .52 FromFigure3.12,it'sobviousthattheradiusoftheellipse r 1 = r sin and L = h sin : .53 Thereforethevolumeofthe2cylindersinthecellis V s =2 rr 1 h = 2 sin r 2 h .54 41

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where h istheheightofthecellwhichisafunctionoftheangle ,denedinFigure3.12. Hence,thetotalvolumeoftheperiodiccell V t = V s V s =V t = V s s = V s 1 )]TJ/F20 11.9552 Tf 11.955 0 Td [( l = 2 sin )]TJ/F20 11.9552 Tf 11.955 0 Td [( l r 2 h ; .55 where l and s areuidandsolidvolumefractions,respectively.Substituting.51,.53, and.55into.46,wehave S = h = sin h 3 r + r 1 )]TJ/F25 9.9626 Tf 11.955 12.005 Td [(q r + r 1 r +3 r 1 i r 2 h = sin )]TJ/F20 11.9552 Tf 11.955 0 Td [( l .56 = 1 )]TJ/F20 11.9552 Tf 11.955 0 Td [( l r 2 " 3 r + r sin )]TJ/F25 9.9626 Tf 11.956 18.576 Td [(s 3 r + r sin r +3 r sin # .57 = )]TJ/F20 11.9552 Tf 11.955 0 Td [( l r sin 3sin +1 )]TJ/F25 9.9626 Tf 11.955 12.599 Td [(q sin +1sin +3 .58 Substituting.58into.45,wehave k = C k l 3 r 2 sin 2 )]TJ/F20 11.9552 Tf 11.955 0 Td [( l 2 h 3sin +1 )]TJ/F25 9.9626 Tf 11.955 12.005 Td [(q sin +1sin +3 i 2 : .59 Similarprocessisappliedto.52.Then S = h = sin q r 2 + r sin 2 = 2 = sin )]TJ/F20 11.9552 Tf 11.955 0 Td [( l r 2 h = p 2 )]TJ/F20 11.9552 Tf 11.955 0 Td [( l r sin q sin 2 +1.60 Substituting.60into.45,wehave k 1= C k 1 l 3 r 2 sin 2 2sin 2 +1 )]TJ/F20 11.9552 Tf 11.955 0 Td [( l 2 .61 where C k 1 isaconstant.Next,weapplythecurvettingmethodtotthebestcurveto thenumericaldata.Least-squaresregressionisonetechniquetoaccomplishthisobjective. Inthiswork,thecurvettingmethodfromMatlabisusedtodeterminetheconstantinthe linearrelationshipbetweenthenumericalresultsandtheformula3.59or3.61.Withsmall adjustmentinthedenominator,wehavethepermeability 1 3 tr k = k =1 : 6147 l 3 r 2 sin 2 )]TJ/F20 11.9552 Tf 11.955 0 Td [( l +0 : 6 2 h 3sin +1 )]TJ/F25 9.9626 Tf 11.955 12.005 Td [(q sin +1sin +3 i 2 +10 )]TJ/F18 7.9701 Tf 6.587 0 Td [(4 .62 42

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Figure3.13: Thegreenlinerepresentsthethesphericalpartofthepermeabilitytensorof thenumericalresultwhiletheblueandredonesarefromthefunctions.62and.63, respectively,wherex-axisistheporositydependingoneachangle. and 1 3 tr k = k 1= l 3 g k 1 S 2 =constant l 3 S 2 = 2 : 0128 l 3 r 2 sin 2 2 )]TJ/F20 11.9552 Tf 11.955 0 Td [( l +0 : 7 2 sin 2 +1 +10 )]TJ/F18 7.9701 Tf 6.587 0 Td [(4 ; .63 whichareshowninFigure3.13whentheradius r ofthecylindersis0 : 1.Aswecansee thereisexcellentagreementbetweentheKozeny-Carmanexpressionand1 = 3traceofthe permeability. 43

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Figure4.1: Thecartoonpictureisshowingthexedboundarywhiletheciliaismoving forwardandbackwardmakingtheangle withthehorizontalplaneinthePCL. 4.MODELINGTHEPCL ThepurposeofthisChapteristodevelopthegoverningequationsthatwillbeused tomodelthePCL,thelayercontainingbothciliaandalowviscosityuid.Weassume thePCLheightisxedandconsideratwo-dimensionaldomain.Figure4.1illustratesthe xedheightPCLandmucouslayer.Theredarrowsindicatetheciliabeatingbackwardand forwardgeneratingmetachronalwavesthatpropelthemucusoutofthelungwhilethegreen arrowsdemonstatetheuidowinthePCL.Forallanglesexcepttheonescloseto90 , thereisalayerofPCLuidbetweenthetipoftheciliaandthemucouslayer. NotethatthedensitiesofthePCLuidandthemucusaredierent.Ifthedierencein densitiesofthecontiguousuidsislarge,thefreeboundary,aninterfacewhoselocationisa prioriunknown,occursatthePCL/mucusinterfaceandmodelingthiswillbediscussedin Chapter6.AtpresentweassumethatthePCLheightisxed. Inthismodel,wedecomposethePCLintotwodomains, 1 and 2 .Thedomain 1 is denedtobetheregionthathasnociliawhile 2 consistsofcilia,seeFigure1.2.Because 1 isnotaporousmedium,thereisnopermeabilitytensorinthisregion.In 2 ,the permeabilityhasanitevalue.Typically,theStokesorNavier-Stokesequationsareusedin 1 andDarcy'sLawwiththeBeavers-Josephconditionin 2 ,[3,47,62].Inthisstudy,we 44

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usetheStokes-Brinkmanequationswhichareemployedbyseveralauthors,[57,67,68,75]. Next,webrieyclarifythedierencebetweenDarcy'sLawandtheBrinkmanequation andwhytheBrinkmanequationisemployedforthisproblem.Darcy'slawistypically employedwhereviscosityandinertialeectsarenegligible,andwheretheuidowing throughtheporousmediaisconsideredslow.Darcy'sLaw[37]is v D = )]TJ/F32 11.9552 Tf 10.494 8.088 Td [(k [ r p )]TJ/F20 11.9552 Tf 11.956 0 Td [( g ] ; .1 where v D isDarcy'svelocity; istheuiddensityand g isthegravity.Brinkman[25]claimed thatinsomecasestheviscousshearingstressesactingontheuidinDarcy'slawarenot negligiblesothatanadditionaltermshouldbeincluded.Torigorouslydeterminetheform ofDarcy'sLawincorporatingviscoussheeringstress,weuseHybridMixtureTheoryHMT, [10,12,36,103],andnondimensionalizationtoobtainformoftheBrinkmanequation,see Section4.4: k )]TJ/F18 7.9701 Tf 6.586 0 Td [(1 l v l )]TJ/F20 11.9552 Tf 11.955 0 Td [( l v s + r p )]TJ/F20 11.9552 Tf 13.553 8.088 Td [( l r l d l = g ; .2 whichistheequationusedindomain 2 .Forthedivergent-freecontinuityequation,4.2 isconsistentwithequationin[106]exceptthat.2includestheterm, l v s ,onthe left-handsidewhereasin[106]itisassumedthevelocityofthesolidphaseiszero.Theterm = l r l d l comesfromtheliquidphasestresstensorandwiththistermthegeneralized Darcy'sLawiscalledtheBrinkmanequation.Thisextratermhelpstomatchthetangential stressactingontheuidatthefree-uid/porous-mediuminterface. ToseetheconnectionwiththeStokesequation,ifwelet k )]TJ/F18 7.9701 Tf 6.586 0 Td [(1 gotozeroindomain 1 , .2canberewrittenas r p )]TJ/F20 11.9552 Tf 13.553 8.088 Td [( l r l d l = g : .3 If l isaconstantinspace,wehave r l d l = l v l i;j + v l j;i ;j = l v l i;jj + v l j;ij = l v l + l r r v l : .4 45

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Sinceweassumethattheuidisincompressible, r v l =0.5 indomain 1 .Therefore.3becomes r p )]TJ/F20 11.9552 Tf 11.955 0 Td [( g = l l v l ; .6 whichistheStokesequation.ThetypicalderivationoftheStokesequationisshownin Section4.5bybeginningwiththeNavier-Stokesequationwiththeporosity l =1. CouplingtheBrinkmanequation.2withthecontinuityequationgives[104] _ l + )]TJ/F20 11.9552 Tf 11.955 0 Td [( l r l v l )]TJ/F32 11.9552 Tf 11.955 0 Td [(v s =0 ; .7 andwenowhavethesystemofequationsintheporousmediumdomain 2 ,while.5and .6aretheequationsin 1 .Equation.7isderivedinSection4.2,where_ l isthematerial timederivativeoftheporositywithrespecttothesolidphase,_ l = @ l =@t + v s r l .In Section4.1,wederivethegeneralformofthemacroscalemomentumequationbyaveraging. Thecontinuityequation.7isderivedinSection4.2.Darcy'sLawandtheBrinkmanequationwhicharespecicformsofthemomentumequation,arederivedinSections4.3and4.4, respectively.TheStokesequationisdevelopedinSection4.5.Finally,thewell-posednessof theStokes-BrinkmansystemofequationsisshowninSection4.6forasecond-orderpermeabilitytensor.Previouslythishadbeenshownonlyforascalarcoecient. 4.1GeneralFormoftheMomentumEquationsusingHMT HybridMixtureTheoryHMTusesaveragingtheoremtoderivemacroscopiceldequationsandthenexploitstheentropyinequalitytoderiveconstitutiveequations.Totransfera microscalevariabletoamacroscalevariableisdenedintermoftheintrinsicphaseaverage. Thatistheaverageofthemicroscalevariableweightedbythevolumeofthephase.Inthis section,webrieydescribethederivationofthemacroscaleconservationofmassandmomentumbalancesforeachphaseusingHMT[9,12,74,103].Westartwiththeconservation 46

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ofmassatthemicroscale, @ @t + r v =0 : .8 Denetheindicatorfunction r ;t = 8 > > > < > > > : 1if r 2 V 0if r 2 V ; 6 = ; .9 and V astherepresentativeelementaryvolumeREV[6],and V denotestheportionof V inthe )]TJ/F16 11.9552 Tf 9.298 0 Td [(phaseand V istheportionof V inthe )]TJ/F16 11.9552 Tf 9.298 0 Td [(phase.Multiplying.8by , integrating.8withrespectto V anddividingbythevolume j V j ,wehave 1 j V j Z V @ @t dv + 1 j V j Z V r v dv =0 ; .10 Theaveragingtheorem[12,38]tellsushowtointerchangethepartialderivativesandthe integral, 1 j V j Z V @f @t dv = @ @t " 1 j V j Z V f dv # )]TJ/F25 9.9626 Tf 13.354 9.963 Td [(X 6 = 1 j V j Z A f w n da .11 1 j V j Z V r x f dv = r x " 1 j V j Z V f dv # + X 6 = 1 j V j Z A f n da ; .12 where f isthequantitiesintheeldequations,and A , w and n aretheportionof the interfacewithin V ,themicroscopicvelocityofinterface ,andtheoutwardunit normalvectorto V ,respectively.Applyingthetheoremtoeachtermin.10gives 1 j V j Z V @ @t dv = @ @t " 1 j V j Z V dv # )]TJ/F25 9.9626 Tf 13.353 9.963 Td [(X 6 = 1 j V j Z A w n da; .13 1 j V j Z V r v dv = r " 1 j V j Z V v dv # + X 6 = 1 j V j Z A v n da: .14 Therefore.10becomes @ @t " 1 j V j Z V dv # + r " 1 j V j Z V v dv # = X 6 = 1 j V j Z A w n da )]TJ/F25 9.9626 Tf 13.353 9.963 Td [(X 6 = 1 j V j Z A v n da: .15 47

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Dene = j V j j V j and X =1 ; .16 = 1 j V j Z V dv; .17 v = 1 j V j Z V v dv .18 ^ e = 1 j V j Z A [ w )]TJ/F32 11.9552 Tf 11.955 0 Td [(v ] n da; .19 sothatthemacroscaleequationoftheconservationofmass.15canberewrittenas @ @t + r v = X 6 = ^ e : .20 Next,weapplythesameprocesstotheconservationofmomentumatthemicroscale, @ v @t + r vv )-222(r t )]TJ/F20 11.9552 Tf 11.956 0 Td [( g = 0 ; .21 where t istheCauchystresstensor.Multiplying.21by ,integrating.21withrespect to V anddividingbythevolume j V j ,wehave 1 j V j Z V @ v @t dv )]TJ/F16 11.9552 Tf 21.074 8.088 Td [(1 j V j Z V r t dv + 1 j V j Z V r vv dv )]TJ/F16 11.9552 Tf 21.074 8.088 Td [(1 j V j Z V g dv =0 : .22 Applyingtheaveragingtheoremtoeachtermin.22,wehave 1 j V j Z V @ v @t dv = @ @t " 1 j V j Z V v dv # )]TJ/F25 9.9626 Tf 13.353 9.963 Td [(X 6 = 1 j V j Z A vw n da; .23 1 j V j Z V r t dv = r " 1 j V j Z V t dv # + X 6 = 1 j V j Z A tn da; .24 1 j V j Z V r vv dv = r " 1 j V j Z V vv dv # + X 6 = 1 j V j Z A vvn da: .25 Therefore.22becomes @ @t " 1 j V j Z V v dv # )-222(r " 1 j V j Z V t dv # + r " 1 j V j Z V vv dv # 48

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)]TJ/F16 11.9552 Tf 18.417 8.088 Td [(1 j V j Z V g dv = X 6 = 1 j V j Z A vw n da + X 6 = 1 j V j Z A tn da )]TJ/F25 9.9626 Tf 12.689 9.963 Td [(X 6 = 1 j V j Z A vvn da: .26 Dening D Dt = @ @t + v r .27 g = 1 j V j Z V g dv; .28 t = 1 j V j Z V t dv )]TJ/F16 11.9552 Tf 22.742 8.087 Td [(1 j V j Z V vv dv + v v .29 ^ T = 1 j V j Z A [ t + v w )]TJ/F32 11.9552 Tf 11.955 0 Td [(v ] n da )]TJ/F32 11.9552 Tf 11.955 0 Td [(v ^ e ; .30 equation.26canberewrittenas @ @t v + r v v )-222(r t )]TJ/F20 11.9552 Tf 11.955 0 Td [( g = X 6 = ^ T +^ e v ; .31 where X 6 = ^ T +^ e v = X 6 = 1 j V j Z A [ t + v w )]TJ/F32 11.9552 Tf 11.955 0 Td [(v ] n da: .32 Tosimplifythemacroscopicscaleofthelinearmomentumequation.31,webeginby applyingthechainruletothersttwotermof.31andthenemployingthemacroscale continuityequation.20toreplacethetimederivativeandadvectionterms, @ =@t + r v ,bytheinteractivequantityattheinterface , P 6 = ^ e ,asfollows. X 6 = ^ T +^ e v = @ @t v + r v v )-222(r t )]TJ/F20 11.9552 Tf 11.955 0 Td [( g = @ @t v + @ @t v + r v v + v r v )-222(r t )]TJ/F20 11.9552 Tf 11.955 0 Td [( g = " @ @t + r v # v + @ @t v + v r v )-222(r t )]TJ/F20 11.9552 Tf 11.955 0 Td [( g = X 6 = ^ e v + @ @t v + v r v )-222(r t )]TJ/F20 11.9552 Tf 11.955 0 Td [( g : .33 Canceling X 6 = ^ e v frombothsidesof.33andapplyingthematerialtimederivative denition.27,yields D v @t )-222(r t )]TJ/F20 11.9552 Tf 11.956 0 Td [( g = X 6 = ^ T .34 49

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whichisthegeneralformofthemacroscopicscaleofthemomentumequation. 4.2TheContinuityEquation Thecontinuityequationistheconservationofmass,andthegoalofthissectionisto deriveequation.7byusingsimplifyingassumptions.Werstbeginwiththemacroscale continuityequation @ @t + r v = X 6 = ^ e : .35 Applyingthechainruletothesecondtermof.35,weobtain @ @t + v r + r v = X 6 = ^ e : .36 Usingthematerialtimederivativedenition.27to.36,wehave D @t + r v = X 6 = ^ e : .37 Ourrstassumptionisthatthereisnochemicalreactionbetweentheliquidandsolidphases, sothattheright-handsideof.37iszeroanditbecomes D @t + r v =0 : .38 Wenextassumeboththeliquidandsolidphasesareincompressiblesothat D =@t =0. Then.38writtenforbothliquidandsolidphasesbecomes D l l Dt + l r v l =0 ; .39 D s s Dt + s r v s =0 : .40 Sincethevolumeofthesolidcanbewrittenintermsoftheliquidvolumefraction, s =1 )]TJ/F20 11.9552 Tf 10.093 0 Td [( l , wecaneliminate s from.40 )]TJ/F20 11.9552 Tf 10.494 8.088 Td [(D s l Dt + )]TJ/F20 11.9552 Tf 11.955 0 Td [( l r v s =0 : .41 50

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Adding.39and.41,weobtain D l l Dt )]TJ/F20 11.9552 Tf 13.151 8.087 Td [(D s l @t + l r v l )-222(r v s + r v s =0 : .42 Fromthedenitionofthematerialtimederivative: D l Dt = @ @t + v l r and )]TJ/F20 11.9552 Tf 13.15 8.088 Td [(D s Dt = )]TJ/F20 11.9552 Tf 12.607 8.088 Td [(@ @t )]TJ/F32 11.9552 Tf 11.955 0 Td [(v s r ; .43 wehave D l l Dt = @ l @t + v l r l and )]TJ/F20 11.9552 Tf 13.151 8.088 Td [(D s l Dt = )]TJ/F20 11.9552 Tf 10.494 8.088 Td [(@ l @t )]TJ/F32 11.9552 Tf 11.955 0 Td [(v s r l : .44 Addingthetwoequationsof.44,weget D l l Dt )]TJ/F20 11.9552 Tf 13.151 8.088 Td [(D s l Dt = v l )]TJ/F32 11.9552 Tf 11.955 0 Td [(v s r l : .45 Eliminatingtheterm D l l Dt )]TJ/F20 11.9552 Tf 13.15 8.087 Td [(D s l Dt bycombining.45and.42togethergives v l )]TJ/F32 11.9552 Tf 11.955 0 Td [(v s r l + l r v l )]TJ/F32 11.9552 Tf 11.955 0 Td [(v s + r v s =0 ; .46 or r l v l )]TJ/F32 11.9552 Tf 11.955 0 Td [(v s + r v s =0 : .47 Sowehave r v s = r l v l )]TJ/F32 11.9552 Tf 11.955 0 Td [(v s ; .48 andsubstituting.48into.41,weget D s l Dt + )]TJ/F20 11.9552 Tf 11.955 0 Td [( l r l v l )]TJ/F32 11.9552 Tf 11.955 0 Td [(v s =0 ; .49 whichistheformofthecontinuityequationforanincompressiblesolidandliquidphases thatwewilluse. 51

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4.3Darcy'sLaw Darcy'sLawisawidelyusedconceptusedtomodelowthroughaporousmedium.In thissection,wederiveageneralizedformofDarcy'sLawusingHMT.Weassumewehave onlytwophases,solidandliquid.Thusequation.34canbewrittenas l l D l v l Dt )-222(r l t l )]TJ/F20 11.9552 Tf 11.955 0 Td [( l l g l = l l ^ T l s ; .50 where ^ T l s istherateatwhichmomentumisexchangedfromthesolidphasetotheliquid phase.Notethatatthispointwehaven'tincorporatedanyconstitutiveequations.By exploitingtheentropyinequalityandlinearizing ^ T l s aboutthevariable v l )]TJ/F32 11.9552 Tf 12.023 0 Td [(v s where l and s standforliquidandsolidphases[103,109],weobtain l l ^ T l s = p r l )]TJ/F20 11.9552 Tf 11.955 0 Td [( l R v l )]TJ/F32 11.9552 Tf 11.955 0 Td [(v s : .51 where R issecond-ordertensorresultingfromlinearization.Now,weconsiderthemacroscale Cauchystresstensorontheleft-handsideof.50.Theequationforthestresstensor.29 involvesmicroscalevariables.Darcy'sLawisamacroscaleequationsowecannotuse.29 toobtainDarcy'sLaw.Byexploitingtheentropyequationweobtainthattheconstitutive equationfortheliquidstresstensorisgivenby t l = )]TJ/F20 11.9552 Tf 9.299 0 Td [(p I +2 d l ; .52 where p and d l arethepressureandtherateofdeformationfortheliquidphase.Substituting .51and.52into.50,wehave l l D l v l Dt )-222(r )]TJ/F20 11.9552 Tf 9.299 0 Td [( l p I + l 2 d l )]TJ/F20 11.9552 Tf 11.955 0 Td [( l l g l = p r l )]TJ/F20 11.9552 Tf 11.955 0 Td [( l R v l )]TJ/F32 11.9552 Tf 11.955 0 Td [(v s : .53 Notethat r l p I = l r p I + p I r l = l r p + p r l : .54 52

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Thus,.53canberewrittenas l l D l v l Dt + l r p + p r l )-222(r l 2 d l )]TJ/F20 11.9552 Tf 11.956 0 Td [( l l g l = p r l )]TJ/F20 11.9552 Tf 11.955 0 Td [( l R v l )]TJ/F32 11.9552 Tf 11.956 0 Td [(v s : .55 Canceling p l r l frombothsidesandneglectingtheinertialandviscosityterms,wehave l r p )]TJ/F20 11.9552 Tf 11.955 0 Td [( l l g l = )]TJ/F20 11.9552 Tf 9.298 0 Td [( l R v l )]TJ/F32 11.9552 Tf 11.955 0 Td [(v s ; .56 where g l isthegravityvector g = ; 0 ; )]TJ/F20 11.9552 Tf 9.299 0 Td [(g .Dividingbothsidesby l andletting R = l k )]TJ/F18 7.9701 Tf 6.587 0 Td [(1 ,wehaveDarcy'slaw: l v l )]TJ/F32 11.9552 Tf 11.955 0 Td [(v s = )]TJ/F32 11.9552 Tf 10.494 8.088 Td [(k r p )]TJ/F20 11.9552 Tf 11.955 0 Td [( g ; .57 wherethesupersript l isdroppedfromthegravityterminordertobeconsistentwiththe notationintheprevioussections. Byemployingthegeneralformofthemacroscopicscaleofthemomentumequation .34andlinearizingtherate, ^ T l s ,atwhichmomentumisexchangedfromthesolidphase totheliquidphase,wenowhavethewidelyusedmodel,Darcy'sLaw.Next,weshowthat Darcy'slawcanbeextendedtotheBrinkmanequation. 4.4DerivationoftheBrinkmanequation Inthissection,wederivetheBrinkmanequationbyusingHMTandnondimensionalization.WebeginwiththeversionofDarcy'sLawgivenby.55: l l D l v l Dt + l r p + p r l )-222(r l 2 d l )]TJ/F20 11.9552 Tf 11.956 0 Td [( l l g l = p r l )]TJ/F20 11.9552 Tf 11.955 0 Td [( l R v l )]TJ/F32 11.9552 Tf 11.956 0 Td [(v s : .58 Subtracting p r l frombothsides,dividingbothsidesby l andletting l k )]TJ/F18 7.9701 Tf 6.587 0 Td [(1 = R ,wehave D l v l Dt + r p )]TJ/F20 11.9552 Tf 13.552 8.087 Td [( l r l d l )]TJ/F20 11.9552 Tf 11.955 0 Td [( g = )]TJ/F20 11.9552 Tf 9.299 0 Td [( l k )]TJ/F18 7.9701 Tf 6.586 0 Td [(1 v l )]TJ/F32 11.9552 Tf 11.955 0 Td [(v s ; .59 orbyrearrangingtheterms, @ v l @t + v l r v l ! + k )]TJ/F18 7.9701 Tf 6.587 0 Td [(1 l v l )]TJ/F20 11.9552 Tf 11.955 0 Td [( l v s + r p )]TJ/F20 11.9552 Tf 13.552 8.088 Td [( l r l d l = g .60 53

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ScalingParameterDescriptionPrimaryDimensions L characteristiclength[ L ] f characteristicfrequency[1 =t ] v 0 characteristicspeed[ L=t ] p 0 referencepressure[ M= Lt 2 ] g 0 gravitationalacceleration[ L=t 2 ] Table4.1: Characteristicparameters where D l Dt = @ @t + v l r isthematerialtimederivative.Next,wenormalizeequation.60 andshowthatsometermsintheequationcanbeneglectedwithrespecttoourproblem. Tonormalize.60,wechoosescalingparametersdenedinTable4.1.Denethefollowing dimensionlessvariables: ~ v l = v l v 0 ; ~ v s = v s v 0 ; ~ p = p p 0 ; ~ g = g g 0 ; .61 ~ r = L r ; ~ t = ft; ~ = L 2 : .62 Substituting.61{.62into.60,weget fv 0 @ ~ v l @ ~ t + v 2 0 L ~ v l ~ r ~ v l + v 0 k )]TJ/F18 7.9701 Tf 6.586 0 Td [(1 l ~ v l )]TJ/F20 11.9552 Tf 11.955 0 Td [( l ~ v s + p 0 L ~ r ~ p )]TJ/F16 11.9552 Tf 14.148 8.088 Td [(1 l v 0 L 2 ~ r l ~ d l = g 0 ~ g : .63 where ~ d l = 1 2 ~ r ~ v T + ~ r ~ v .Multiplying.63by k =v 0 ,wehave l ~ v l )]TJ/F20 11.9552 Tf 11.955 0 Td [( l ~ v s = )]TJ/F32 11.9552 Tf 10.494 8.088 Td [(k [ f @ ~ v l @ ~ t + v 0 L ~ v l ~ r ~ v l + p 0 Lv 0 ~ r ~ p )]TJ/F20 11.9552 Tf 13.151 8.088 Td [(g 0 v 0 ~ g )]TJ/F16 11.9552 Tf 14.147 8.088 Td [(1 l L 2 ~ r l ~ d l ] : .64 Forourslowowproblem,wechoosethereferencetime t tobethetimeittakestheciliato gothroughonecycle, L tobetheheightofthecilia, and tobethedensityanddynamic viscosityofwaterattemperature40 Cand g 0 tobetheEarth'sgravity.Then t 1 : 029s ;L 7 10 )]TJ/F18 7.9701 Tf 6.586 0 Td [(6 m ; 992 : 2kg/m 3 g 0 =9 : 81m/s 2 ; .65 54

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0 : 653 10 )]TJ/F18 7.9701 Tf 6.587 0 Td [(3 kg/m s ;p 0 =10 )]TJ/F18 7.9701 Tf 6.587 0 Td [(1 kg/m s 2 ; .66 wheretheunitsareinInternationalSystemofUnits,SI.Thevaluesoftime t andlength L arefrom[77]and[20],respectively.Thenthecharacteristicspeedatthetipoftheciliais v 0 = L t = 10 )]TJ/F18 7.9701 Tf 6.587 0 Td [(6 = 4 = 1 : 029 5 : 34 10 )]TJ/F18 7.9701 Tf 6.586 0 Td [(6 m/s ; .67 where istheanglebetweentheciliaandthehorizontalplane.Forthepermeabilityand porosity,weemploythemaximumvaluesfromournumericalresults.Thereforethescalar permeabilityandporosityare k =10 )]TJ/F18 7.9701 Tf 6.586 0 Td [(14 m 2 ; l =1 ; .68 respectively.Forourconveniencetodeterminethevaluesofeachcoecient,werewrite .64as C 1 l ~ v l )]TJ/F20 11.9552 Tf 11.955 0 Td [( l ~ v s = )]TJ/F20 11.9552 Tf 9.299 0 Td [(C 2 @ ~ v l @ ~ t )]TJ/F20 11.9552 Tf 11.955 0 Td [(C 3 ~ v l ~ r ~ v l )]TJ/F20 11.9552 Tf 11.955 0 Td [(C 4 ~ r ~ p + C 5 ~ g + C 6 ~ r l ~ d l ; .69 wherethecoecientsaftersubstitutedbythevaluesfrom.65,.66,.67and.68 are C 1 =1.70 C 2 = kf =1 : 47 10 )]TJ/F18 7.9701 Tf 6.586 0 Td [(8 .71 C 3 = kv 0 L =1 : 15 10 )]TJ/F18 7.9701 Tf 6.586 0 Td [(8 .72 C 4 = kp 0 v 0 L =4 : 09 10 )]TJ/F18 7.9701 Tf 6.587 0 Td [(2 .73 C 5 = kg 0 v 0 =2 : 79 10 )]TJ/F18 7.9701 Tf 6.586 0 Td [(2 .74 C 6 = k l L 2 =2 : 04 10 )]TJ/F18 7.9701 Tf 6.586 0 Td [(4 : .75 Thecoecients C 2 and C 3 arerelativelysmallincomparisonwiththeothers.Neglecting theseterms,.60becomes k )]TJ/F18 7.9701 Tf 6.586 0 Td [(1 l v l )]TJ/F20 11.9552 Tf 11.956 0 Td [( l v s + r p )]TJ/F20 11.9552 Tf 13.553 8.088 Td [( l r l d l = g : .76 55

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whichistheBrinkmanequation. 4.5DerivationofStokesEquation Inthissection,wenondimensionalizetheNavier-Stokesequationtoshowthatsome termsinthisequationarenegligibleforourproblemindomain 1 .Webeginwiththe Navier-Stokesequationwhichisthedierentialequationforconservationofmomentumfor auidwithstressgivenbyaconstitutiveequationforaNewtonianuid.Forincompressible ow,itcanbewrittenintheform @ v @t + v r v ! = r p + g + v .77 UsingthesamecharacteristicparametersasgiveninTable4.1,thenondimensionalvariables aredenedtobe ~ v = v v 0 ; ~ p = p p 0 ; ~ g = g g 0 ; .78 ~ r = L r ; ~ t = ft; ~ = L 2 : .79 Substituting.78{.79into.77,wehave fv 0 @ ~ v @ ~ t ! + v 2 0 L ~ v ~ r ~ v = )]TJ/F20 11.9552 Tf 10.494 8.088 Td [(p 0 L ~ r ~ p + g 0 ~ g + v 0 L 2 ~ ~ v : .80 Multiplying.80by L 2 =v 0 ,weobtain fL 2 @ ~ v @ ~ t ! + Lv 0 ~ v ~ r ~ v = )]TJ/F20 11.9552 Tf 10.494 8.088 Td [(p 0 L v 0 ~ r ~ p + g 0 L 2 v 0 ~ g + ~ ~ v : .81 Notethatthecoecientinfrontof ~ v ~ r ~ v istheReynoldsnumber.Usingthevaluesfrom .65-.67,weobtainthecoecients: fL 2 =7 : 23 10 )]TJ/F18 7.9701 Tf 6.586 0 Td [(5 ; .82 Lv 0 =5 : 67 10 )]TJ/F18 7.9701 Tf 6.586 0 Td [(5 ; .83 p 0 L v 0 =2 : 00 10 2 ; .84 56

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g 0 L 2 v 0 =1 : 36 10 2 : .85 Neglectingthetermsassociatedwithcoecients.82-.83,equation.77becomes r p )]TJ/F20 11.9552 Tf 11.956 0 Td [( g = v ; .86 whichistheStokesequation. 4.6Well-posednessStokes-Brinkman Ingram[54]provedthewell-posednessoftheStokes-Brinkmansystemofequationsfora positiveconstantpermeability.Hereweshowtheexistenceanduniquenessforapermeability tensor k .RecalltheBrinkmanandcontinuityequationsare k )]TJ/F18 7.9701 Tf 6.586 0 Td [(1 l v l )]TJ/F20 11.9552 Tf 11.955 0 Td [( l v s + r p )]TJ/F20 11.9552 Tf 13.553 8.087 Td [( l r l d l = g ; in.87a _ l + )]TJ/F20 11.9552 Tf 11.955 0 Td [( l r l v l )]TJ/F32 11.9552 Tf 11.956 0 Td [(v s =0in4.87b where d l =1 = 2 r v l + r v l T istherateofdeformationandthesuperscript T isthe transpose.Thevelocityoftheliquid v l andthepressure p areunknown.Sincethesolid velocity v s ,dynamicviscosity ,porosity l andinverseofthepermeability k )]TJ/F18 7.9701 Tf 6.587 0 Td [(1 areknown, wemove k )]TJ/F18 7.9701 Tf 6.586 0 Td [(1 l v s totheright-handsideandequation.87abecomes k )]TJ/F18 7.9701 Tf 6.586 0 Td [(1 l v l + r p )]TJ/F20 11.9552 Tf 13.553 8.088 Td [( l r l d l = g + k )]TJ/F18 7.9701 Tf 6.587 0 Td [(1 l v s ; .88 whichistheStokesequationwhen k )]TJ/F18 7.9701 Tf 6.587 0 Td [(1 iszero.Denethespaces L 2 0 = f q 2 L 2 : Z qd =0 g .89 H 1 0 = f w 2 H 1 : w j @ =0 g .90 H 1 s = f w 2 H 1 : w j @ = s g .91 H )]TJ/F18 7.9701 Tf 6.587 0 Td [(1 = H 1 0 0 ; thedualof H 1 0 .92 V = f w 2 H 1 : w j @ =0and r w =0 g .93 V ? = f w ? 2 H 1 0 : Z w ? w =0 8 w 2 V g .94 57

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V 0 = f w 0 2 H )]TJ/F18 7.9701 Tf 6.587 0 Td [(1 : h w 0 ; w i H )]TJ/F19 5.9776 Tf 5.756 0 Td [(1 H 1 0 =0 8 w 2 V g .95 where V ? denotestheorthogonalof V in H 1 0 associatedwiththe H 1 seminorm jj H 1 ; V 0 isthepolarsetof V ; h ; i H )]TJ/F19 5.9776 Tf 5.756 0 Td [(1 H 1 0 representsthedualitypairingbetween H )]TJ/F18 7.9701 Tf 6.587 0 Td [(1 and H 1 0 ,andthetracetheorem4.5,below,ensurestheexistenceofthefunction s 2 H 1 = 2 @ usedinthedenitionofseminorm.Notethatforathree-dimensionaldomain, w 2 H 1 3 and r w 2 H 1 3 3 .However,forsimplicity,wewrite w 2 H 1 inanycaseandthe meaningfollowsfromthecontext.Letthevector f 1 = g + k )]TJ/F18 7.9701 Tf 6.587 0 Td [(1 l v s ,where v s isa boundedcontinuousfunction.Let f 1 2 H )]TJ/F18 7.9701 Tf 6.587 0 Td [(1 withthefollowingnorm: k f 1 k H )]TJ/F19 5.9776 Tf 5.756 0 Td [(1 =sup w 2 H 1 0 ; w 6 =0 h f 1 ; w i H )]TJ/F19 5.9776 Tf 5.756 0 Td [(1 H 1 0 k w k H 1 .96 where kk H 1 representsthestandardnormfor H 1 .Notethat r l d l = l v l + r r l v l : .97 Assume l isxedinspaceanddene v = l v l : .98 Thenequations.88and.87bcanberewrittenas k )]TJ/F18 7.9701 Tf 6.586 0 Td [(1 v + r p )]TJ/F20 11.9552 Tf 13.553 8.088 Td [( l v = f 1 + l r r v ; .99a r v = )]TJ/F16 11.9552 Tf 26.667 8.088 Td [(_ l )]TJ/F20 11.9552 Tf 11.956 0 Td [( l + r l v s ; .99b wheretheright-handsideof.99bisknown.Using.99btoeliminatethelasttermin .99aandletting f = )]TJ/F16 11.9552 Tf 10.686 0 Td [(_ l = )]TJ/F20 11.9552 Tf 9.733 0 Td [( l + r l v s and f = f 1 + = l r f ,wegettheStokes-Brinkman equationsinthefollowingform k )]TJ/F18 7.9701 Tf 6.586 0 Td [(1 v + r p )]TJ/F20 11.9552 Tf 13.553 8.088 Td [( l v = f .100a r v = f: .100b 58

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Denethelinearandbilinearfunctionals a v ; w = Z l r v : r w + Z k )]TJ/F18 7.9701 Tf 6.586 0 Td [(1 v w ; .101 b v ;q = )]TJ/F25 9.9626 Tf 11.291 14.058 Td [(Z q r v ; .102 c 1 w = h f 1 ; w i H )]TJ/F19 5.9776 Tf 5.756 0 Td [(1 H 1 0 )]TJ/F25 9.9626 Tf 11.955 14.058 Td [(Z l f r w ; .103 c 2 q = )]TJ/F25 9.9626 Tf 11.291 14.059 Td [(Z fq: .104 Then,theweakformulationof.100canbeexpressedas Problem4.1 WeakStokes-BrinkmanFind v 2 H 1 s andp 2 L 2 0 suchthat 8 w 2 H 1 0 ;a v ; w + b w ;p = c 1 w ; .105a 8 q 2 L 2 0 ;b v ;q = c 2 q : .105b BeforeweprovetheexistenceanduniquenessoftheStokes-Brinkmanequation,werst introducethefollowingnorms,seminormsandtheorems,[23]. Denition4.2 WeakderivativeWesaythatagivenfunction v 2 L 1 loc hasa weak derivative , D w v ,providedthereexistsafunction w 2 L 1 loc suchthat Z w x x dx = )]TJ/F16 11.9552 Tf 9.298 0 Td [(1 j j Z v x x dx 8 2 C 1 0 .106 where C 1 0 isthesetoffunctionswithcompactsupportin ; L 1 loc = f v : v 2 L 1 K 8 compact K interior g .Ifsucha w exists,wedene D w v = w . Denition4.3 SobolevnormLet k beanon-negativeinteger,andlet f 2 L 1 loc .Supposethattheweakderivatives D w f existsforall j j 6 k .Denethe Sobolevnorm k f k W k p := 0 @ X k j 6 k k D w f k p L p 1 A 1 =p .107 inthecase 1 6 p< 1 ,andinthecase p = 1 k f k W k 1 :=max j j 6 k k D w f k L 1 : .108 59

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Ineithercase,wedenethe Sobolevspaces via W k p := n f 2 L 1 loc : k f k W k p < 1 o : .109 Denition4.4 SeminormFor k anon-negativeintegerand f 2 W k p ,let j f j W k p = 0 @ X j j = k k D w f k p L p 1 A 1 =p .110 inthecase 1 6 p< 1 ,andinthecase p = 1 j f j W k 1 =max j j = k k D w f k L 1 : .111 Thespace W m 2 @ = H m @ canbeintroducedfortheboundary @ .Incaseof m =1 = 2, thespace H m @ isequippedwithanorm[40] k s k 2 H 1 = 2 @ = k s k 2 L 2 @ + j s j 2 1 = 2 ;@ .112 where j s j 2 1 = 2 ;@ = Z @ j x )]TJ/F20 11.9552 Tf 11.955 0 Td [(y j )]TJ/F18 7.9701 Tf 6.586 0 Td [( n +1 j s x )]TJ/F32 11.9552 Tf 11.955 0 Td [(s y j 2 dxdy .113 and n isthedimensionalnumber. Thefollowingtheoremsareimportanttoprovetheexistenceanduniquenessoftheweak solutions.Theproofsareinthecitationsreferredbelow.Thersttheoremisreferredtoas theinversetracetheoremanditensuresthatif s 2 H 1 = 2 @ ,thenthereexists v s 2 H 1 suchthatthetraceof v s on @ is s ,[40]. Theorem4.5 DirectandInverseTraceTheoremfor H 1 Thereexistpositiveconstants K and K 0 suchthat,foreach w 2 H 1 ,itstraceon @ belongsto H 1 = 2 @ and k w k H 1 = 2 @ 6 K k w k H 1 .Conversely,foreachgivenfunction s 2 H 1 = 2 @ ,there existsafunction v s 2 H 1 suchthatitstraceon @ coincideswith s and k v s k H 1 6 K 0 k s k H 1 = 2 @ : .114 60

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Thenexttheoremstatesthatthedivergenceoperatorisanisomorphismbetween L 2 0 and V ? [44],andtheLadyzhenskaya-Babuska-Brezzi,LBB,condition,whichisneededforthe stabilityofthemixedniteelementmethod,ismentioned[24]. Theorem4.6 Let beconnected.Then 1theoperator grad isanisomorphismof L 2 0 onto V 0 2theoperator div isanisomorphismof V ? onto L 2 0 Moreover,thereexists > 0 suchthat inf q 2 L 2 0 sup w 2 H 1 0 b w ;q k w k H 1 k q k L 2 > > 0.115 andforany q 2 L 2 0 ,thereexistsaunique v 2 V ? H 1 0 satisfying k v k H 1 6 )]TJ/F18 7.9701 Tf 6.586 0 Td [(1 k q k L 2 : .116 Toprovetheexistenceanduniqueness,Lax-Milgramisthemaintheoremusedinthe proof.Webeginwithsomedenitions[23]. Denition4.7 Abilinearform a ; onanormedlinearspace H issaidtobe bounded or continuous if 9 C< 1 suchthat j a v;w j 6 C k v k H k w k H 8 v;w 2 H .117 and coercive on E H if 9 > 0 suchthat a v;v > k v k 2 H 8 v 2 E: .118 InordertoprovetheexistenceanduniquenessofProblem.1,werequireoperatornotations.Recallthat L 2 0 0 = L 2 0 and H 1 0 0 = H )]TJ/F18 7.9701 Tf 6.586 0 Td [(1 where L 2 0 0 and H 1 0 0 arethedualspacesof L 2 0 and H )]TJ/F18 7.9701 Tf 6.586 0 Td [(1 ,respectively. Denition4.8 Let v ; w 2 H 1 and q 2 L 2 0 .Denelinearoperators A : H 1 0 ! H )]TJ/F18 7.9701 Tf 6.586 0 Td [(1 or A 2L H 1 0 ; H )]TJ/F18 7.9701 Tf 6.586 0 Td [(1 and B 2L H 1 0 ; L 2 0 by h A v ; w i H 1 0 H )]TJ/F19 5.9776 Tf 5.756 0 Td [(1 := a v ; w 8 v ; w 2 H 1 0 .119 61

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h B v ;q i H 1 0 L 2 0 := b v ;q 8 v 2 H 1 0 ; 8 q 2 L 2 0 .120 Let B 0 2L L 2 0 ; H )]TJ/F18 7.9701 Tf 6.586 0 Td [(1 bethedualoperatorof B ,i.e. h B 0 q; v i = h q;B v i := b v ;q 8 v 2 H 1 0 ; 8 q 2 L 2 0 : .121 Withtheseoperators,Problem.1isequivalentlywrittenintheform: Problem4.9 Find v 2 H 1 s ;p 2 L 2 0 suchthat A v + B 0 p = f in H )]TJ/F18 7.9701 Tf 6.586 0 Td [(1 .122a B v = f in L 2 0 : .122b Next,weintroduceLax-Milgramtheoremwhichisoneofthemostimportanttheoremsused toprovetheexistenceanduniquenessoftheweaksolution. Theorem4.10 Lax-MilgramGivenaHilbertspace H; ; acontinuous,coercivebilinearform a ; andacontinuouslinearfunctional F 2 H 0 ,thereexistsaunique v 2 H such that a v;w = F w 8 w 2 H: .123 Thefollowinginequalitiesareappliedintheproofofcontinuitiesandcoercivity,[44,23] Theorem4.11 PoincareinequalityIf isconnectedandboundedatleastinonedirection, thenforeachinteger m > 0 ,thereexistsaconstant K = K m; > 0 suchthat k w k H m 6 K j w j H m 8 w 2 H m 0 ; .124 orforspace H 1 0 k w k L 2 6 K kr w k L 2 8 w 2 H 1 0 : .125 Theorem4.12 Holder'sInequalityFor 1 6 p;q 6 1 suchthat 1=1 =p +1 =q ,if f 2 L p and g 2 L q ,then fg 2 L 1 and k fg k L 1 6 k f k L p k g k L q : .126 62

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Theorem4.13 Cauchy-Schwarz'sInequalityThisissimplyHolder'sinequalityinthespecialcase p = q =2 .If f;g 2 L 2 ,then fg 2 L 1 and Z j f x g x j dx 6 k f k L 2 k g k L 2 : .127 forvectors v ; w 2 L 2 ,wealsohave Z j v w j 6 k v k L 2 k w k L 2 : .128 Wenextshowthatthe L 2 -normofdivergenceofafunctionin H 1 islessthanor equaltoamultiplicationofaconstantandthe H 1 -seminormofthefunctionandalso providetheproof.ThistheoremwillbeusedtoproveTheorem4.15. Theorem4.14 Let R n and w 2 H 1 .Then kr w k L 2 6 p n j w j H 1 : .129 Proof: Let u i = @w i =@x i where w = w 1 ;w 2 ;:::;w n .Since j u 1 + u 2 + :::;u n j 2 6 n u 2 1 + u 2 2 + ::: + u 2 n ,integratingandtakingsquarerootbothsides,wehave s Z j u 1 + u 2 + ::: + u n j 2 6 p n s Z u 2 1 + u 2 2 + ::: + u 2 n : .130 Then kr w k L 2 6 p n j w j H 1 : Thefollowingtheoremsshowthatthelinearfunctionals c 1 w , c 2 q andbilinearfunctionals a ; , b ; arecontinuousand a ; iscoercive. Theorem4.15 Thelinearfunctionals c 1 w , c 2 q andbilinearfunctionals a ; , b ; are continuousand a ; iscoercive,i.e., a w ; w > C c k w k 2 H 1 .131 where C c =min f =;C k g ; C k isapositivenumber.Inparticular, c 1 w 6 k f 1 k H )]TJ/F19 5.9776 Tf 5.756 0 Td [(1 + p n k f k L 2 k w k H 1 ; 8 w 2 H 1 ; .132 63

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c 2 q 6 k f k L 2 k q k L 2 ; 8 q 2 L 2 4.133 b v ;q 6 p n j v j H 1 k q k L 2 ; 8 v 2 H 1 ; 8 q 2 L 2 .134 a v ; w 6 C a k v k H 1 k w k H 1 ; 8 v 2 H 1 ; 8 w 2 H 1 ; .135 where n isthedimensionalnumber; C a =max f = l ; p 6 max 1 6 i;j 6 2 j k )]TJ/F18 7.9701 Tf 6.587 0 Td [(1 ij jg . Proof: Thelinearityof c 1 w and c 2 q andbilinearityof a v ; w and b v ;q areobvious. Nextweshowtheyarecontinuous.Let v ; w 2 H 1 and q 2 L 2 ,then j c 1 w j = jh f 1 ; w i H )]TJ/F19 5.9776 Tf 5.756 0 Td [(1 H 1 0 )]TJ/F25 9.9626 Tf 11.955 14.058 Td [(Z l f r w j .136 6 jh f 1 ; w i H )]TJ/F19 5.9776 Tf 5.757 0 Td [(1 H 1 0 j + Z l f r w .137 = h f 1 ; w i H )]TJ/F19 5.9776 Tf 5.756 0 Td [(1 H 1 0 k w k H 1 k w k H 1 + l Z f r w .138 6 k f 1 k H )]TJ/F19 5.9776 Tf 5.756 0 Td [(1 k w k H 1 + l k f k L 2 kr w k L 2 .139 6 k f 1 k H )]TJ/F19 5.9776 Tf 5.756 0 Td [(1 k w k H 1 + p n l k f k L 2 j w j H 1 .140 6 k f 1 k H )]TJ/F19 5.9776 Tf 5.756 0 Td [(1 + p n l k f k L 2 k w k H 1 ; .141 whereweapplythedenitionof H )]TJ/F18 7.9701 Tf 6.587 0 Td [(1 -norm,.96,andCauchy-Schwarz'sinequality, Theorem4.13,forthethirdinequality.Forthefourthinequalityweusethefactthat kr w k L 2 6 p n j w j H 1 where jj istheseminormand n isthedimensionalnumber.Forthe continuityof c 2 q ,itisobviousthat j c 2 q j = Z fq 6 k f k L 2 k q k L 2 ; .142 whereCauchy-Schwarz'sinequalityhasbeenapplied.Toshowcontinuityof b v ;q ,wehave j b v ;q j = Z q r v 6 k q k L 2 kr v k L 2 6 p n j v j H 1 k q k L 2 ; .143 whereCauchy-Schwarz'sinequalityandthefactthat kr w k L 2 6 p n j w j H 1 areapplied forthelasttwoinequalities; n denotesthedimensionalnumber.Toprovethecontinuityof a v ; w foratwo-dimensionaldomain,let v = v 1 ;v 2 .Werstconsider k k )]TJ/F18 7.9701 Tf 6.586 0 Td [(1 v k 2 L 2 = Z k )]TJ/F18 7.9701 Tf 6.587 0 Td [(1 11 v 1 + k )]TJ/F18 7.9701 Tf 6.587 0 Td [(1 12 v 2 2 + Z k )]TJ/F18 7.9701 Tf 6.587 0 Td [(1 21 v 1 + k )]TJ/F18 7.9701 Tf 6.586 0 Td [(1 22 v 2 2 64

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6 Z k )]TJ/F18 7.9701 Tf 6.586 0 Td [(1 11 v 1 2 + k )]TJ/F18 7.9701 Tf 6.586 0 Td [(1 12 v 2 2 + k )]TJ/F18 7.9701 Tf 6.587 0 Td [(1 21 v 1 2 + k )]TJ/F18 7.9701 Tf 6.587 0 Td [(1 22 v 2 2 + j 2 k )]TJ/F18 7.9701 Tf 6.586 0 Td [(1 11 k )]TJ/F18 7.9701 Tf 6.586 0 Td [(1 12 v 1 v 2 j + j 2 k )]TJ/F18 7.9701 Tf 6.587 0 Td [(1 21 k )]TJ/F18 7.9701 Tf 6.587 0 Td [(1 22 v 1 v 2 j d 6 2max 1 6 i;j 6 2 j k )]TJ/F18 7.9701 Tf 6.587 0 Td [(1 ij j 2 Z 2 v 2 1 +2 v 2 2 +2 j v 1 v 2 j 6 4max 1 6 i;j 6 2 j k )]TJ/F18 7.9701 Tf 6.587 0 Td [(1 ij j 2 Z v 2 1 + v 2 2 + j v 1 j 2 2 + j v 2 j 2 2 ! =2max 1 6 i;j 6 2 j k )]TJ/F18 7.9701 Tf 6.586 0 Td [(1 ij j 2 Z 3 v 2 1 +3 v 2 2 =6max 1 6 i;j 6 2 j k )]TJ/F18 7.9701 Tf 6.586 0 Td [(1 ij j 2 Z v 2 1 + v 2 2 =6max 1 6 i;j 6 2 j k )]TJ/F18 7.9701 Tf 6.586 0 Td [(1 ij j 2 k v k 2 L 2 ; .144 whereYoung'sinequality ab 6 a p =p + b q =q where a;b > 0 ;p;q> 0and1 =p +1 =q =1is appliedatthethirdinequality.Theproofthat a v ; w iscontinuousiscompletedwiththe following: j a v ; w j = j Z l r v : r w + Z k )]TJ/F18 7.9701 Tf 6.586 0 Td [(1 v w j .145 6 Z l r v : r w + Z k )]TJ/F18 7.9701 Tf 6.586 0 Td [(1 v w .146 6 l kr v k L 2 kr w k L 2 + k k )]TJ/F18 7.9701 Tf 6.587 0 Td [(1 v k L 2 k w k L 2 .147 6 l kr v k L 2 kr w k L 2 + p 6 max 1 6 i;j 6 2 j k )]TJ/F18 7.9701 Tf 6.587 0 Td [(1 ij jk v k L 2 k w k L 2 .148 6 C a k v k H 1 k w k H 1 .149 where C a =max f = l ; p 6 max 1 6 i;j 6 2 j k )]TJ/F18 7.9701 Tf 6.587 0 Td [(1 ij jg and.144isemployedtothethirdinequality. Beforeprovingthecoercivityofthebilinearform a w ; w ,weconsider,fortwo-dimensions, k )]TJ/F18 7.9701 Tf 6.586 0 Td [(1 w w = k )]TJ/F18 7.9701 Tf 6.586 0 Td [(1 11 w 2 1 + k )]TJ/F18 7.9701 Tf 6.586 0 Td [(1 22 w 2 2 +2 k )]TJ/F18 7.9701 Tf 6.587 0 Td [(1 12 w 1 w 2 .150 where w = w 1 ;w 2 and k )]TJ/F18 7.9701 Tf 6.587 0 Td [(1 issymmetric.Notethat k )]TJ/F18 7.9701 Tf 6.586 0 Td [(1 w w = w T k )]TJ/F18 7.9701 Tf 6.587 0 Td [(1 w asmatrix multiplicationwhere T isthetranspose.Forourproblem k )]TJ/F18 7.9701 Tf 6.586 0 Td [(1 isapositivedenitematrix anditsdiagonalentriesarepositivenumbers.Then k )]TJ/F18 7.9701 Tf 6.586 0 Td [(1 11 w 2 1 + k )]TJ/F18 7.9701 Tf 6.587 0 Td [(1 22 w 2 2 > 0when w > 0 .We nextneedtofocuson2 k )]TJ/F18 7.9701 Tf 6.587 0 Td [(1 12 w 1 w 2 .If2 k )]TJ/F18 7.9701 Tf 6.586 0 Td [(1 12 w 1 w 2 > 0,it'seasytoseethat k )]TJ/F18 7.9701 Tf 6.587 0 Td [(1 w w > k )]TJ/F18 7.9701 Tf 6.586 0 Td [(1 11 w 2 1 + k )]TJ/F18 7.9701 Tf 6.587 0 Td [(1 22 w 2 2 > min f k )]TJ/F18 7.9701 Tf 6.587 0 Td [(1 11 ;k )]TJ/F18 7.9701 Tf 6.586 0 Td [(1 22 g w 2 1 + w 2 2 = C k w 2 1 + w 2 2 where C k > 0.If2 k )]TJ/F18 7.9701 Tf 6.587 0 Td [(1 12 w 1 w 2 < 0, 65

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wehavetwocases: Case1: k )]TJ/F18 7.9701 Tf 6.587 0 Td [(1 12 > 0and w 1 w 2 < 0. Thus, k )]TJ/F18 7.9701 Tf 6.586 0 Td [(1 w w = k )]TJ/F18 7.9701 Tf 6.586 0 Td [(1 11 w 2 1 + k )]TJ/F18 7.9701 Tf 6.586 0 Td [(1 22 w 2 2 )]TJ/F16 11.9552 Tf 11.955 0 Td [(2 k )]TJ/F18 7.9701 Tf 6.586 0 Td [(1 12 j w 1 w 2 j . WenowapplyYoung'sinequality: ab 0. Then j w 1 w 2 j = j w 1 jj w 2 j < j w 1 j 2 = 2+ j w 2 j 2 = 2. Thus, k )]TJ/F18 7.9701 Tf 6.586 0 Td [(1 11 w 2 1 + k )]TJ/F18 7.9701 Tf 6.586 0 Td [(1 22 w 2 2 )]TJ/F16 11.9552 Tf 11.955 0 Td [(2 k )]TJ/F18 7.9701 Tf 6.586 0 Td [(1 12 j w 1 w 2 j . >k )]TJ/F18 7.9701 Tf 6.587 0 Td [(1 11 w 2 1 + k )]TJ/F18 7.9701 Tf 6.586 0 Td [(1 22 w 2 2 )]TJ/F16 11.9552 Tf 11.955 0 Td [(2 k )]TJ/F18 7.9701 Tf 6.586 0 Td [(1 12 j w 1 j 2 = 2+ j w 2 j 2 = 2 = k )]TJ/F18 7.9701 Tf 6.586 0 Td [(1 11 w 2 1 + k )]TJ/F18 7.9701 Tf 6.586 0 Td [(1 22 w 2 2 )]TJ/F20 11.9552 Tf 11.955 0 Td [(k )]TJ/F18 7.9701 Tf 6.586 0 Td [(1 12 j w 1 j 2 + j w 2 j 2 = k )]TJ/F18 7.9701 Tf 6.586 0 Td [(1 11 )]TJ/F20 11.9552 Tf 11.955 0 Td [(k )]TJ/F18 7.9701 Tf 6.586 0 Td [(1 12 w 2 1 + k )]TJ/F18 7.9701 Tf 6.586 0 Td [(1 22 )]TJ/F20 11.9552 Tf 11.955 0 Td [(k )]TJ/F18 7.9701 Tf 6.586 0 Td [(1 12 w 2 2 . Forourproblem, k )]TJ/F18 7.9701 Tf 6.586 0 Td [(1 isdiagonallydominant. Then, C k =min f k )]TJ/F18 7.9701 Tf 6.586 0 Td [(1 11 )]TJ/F20 11.9552 Tf 11.955 0 Td [(k )]TJ/F18 7.9701 Tf 6.586 0 Td [(1 12 ; k )]TJ/F18 7.9701 Tf 6.586 0 Td [(1 22 )]TJ/F20 11.9552 Tf 11.955 0 Td [(k )]TJ/F18 7.9701 Tf 6.586 0 Td [(1 12 g > 0. Therefore, k )]TJ/F18 7.9701 Tf 6.587 0 Td [(1 w w >C k w 2 1 + w 2 2 . Case2: k )]TJ/F18 7.9701 Tf 6.586 0 Td [(1 12 < 0and w 1 w 2 > 0. Since k )]TJ/F18 7.9701 Tf 6.587 0 Td [(1 isbounded, 9 C b suchthat k )]TJ/F18 7.9701 Tf 6.587 0 Td [(1 12 = )]TJ/F20 11.9552 Tf 9.299 0 Td [(C b where C b > 0. Thus, k )]TJ/F18 7.9701 Tf 6.587 0 Td [(1 w w = k )]TJ/F18 7.9701 Tf 6.587 0 Td [(1 11 w 2 1 + k )]TJ/F18 7.9701 Tf 6.587 0 Td [(1 22 w 2 2 )]TJ/F16 11.9552 Tf 11.955 0 Td [(2 C b w 1 w 2 = k )]TJ/F18 7.9701 Tf 6.587 0 Td [(1 11 w 2 1 + k )]TJ/F18 7.9701 Tf 6.586 0 Td [(1 22 w 2 2 )]TJ/F16 11.9552 Tf 11.955 0 Td [(2 C b j w 1 w 2 j = k )]TJ/F18 7.9701 Tf 6.586 0 Td [(1 11 w 2 1 + k )]TJ/F18 7.9701 Tf 6.586 0 Td [(1 22 w 2 2 )]TJ/F16 11.9552 Tf 11.955 0 Td [(2 C b j w 1 jj w 2 j WenowapplyYoung'sinequality: j w 1 jj w 2 j < j w 1 j 2 = 2+ j w 2 j 2 = 2 Therefore, k )]TJ/F18 7.9701 Tf 6.587 0 Td [(1 11 w 2 1 + k )]TJ/F18 7.9701 Tf 6.587 0 Td [(1 22 w 2 2 )]TJ/F16 11.9552 Tf 11.955 0 Td [(2 C b j w 1 jj w 2 j >k )]TJ/F18 7.9701 Tf 6.587 0 Td [(1 11 w 2 1 + k )]TJ/F18 7.9701 Tf 6.587 0 Td [(1 22 w 2 2 )]TJ/F16 11.9552 Tf 11.956 0 Td [(2 C b j w 1 j 2 = 2+ j w 2 j 2 = 2 = k )]TJ/F18 7.9701 Tf 6.587 0 Td [(1 11 w 2 1 + k )]TJ/F18 7.9701 Tf 6.587 0 Td [(1 22 w 2 2 )]TJ/F20 11.9552 Tf 11.955 0 Td [(C b j w 1 j 2 + j w 2 j 2 = k )]TJ/F18 7.9701 Tf 6.586 0 Td [(1 11 )]TJ/F20 11.9552 Tf 11.955 0 Td [(C b w 2 1 + k )]TJ/F18 7.9701 Tf 6.586 0 Td [(1 22 )]TJ/F20 11.9552 Tf 11.955 0 Td [(C b w 2 2 > min f k )]TJ/F18 7.9701 Tf 6.586 0 Td [(1 11 )]TJ/F20 11.9552 Tf 11.955 0 Td [(C b ; k )]TJ/F18 7.9701 Tf 6.587 0 Td [(1 22 )]TJ/F20 11.9552 Tf 11.955 0 Td [(C b g w 2 1 + w 2 2 Forourproblem, C b islessthanthediagonalentriesof k )]TJ/F18 7.9701 Tf 6.586 0 Td [(1 . Thus, C k =min f k )]TJ/F18 7.9701 Tf 6.586 0 Td [(1 11 )]TJ/F20 11.9552 Tf 11.956 0 Td [(C b ; k )]TJ/F18 7.9701 Tf 6.586 0 Td [(1 22 )]TJ/F20 11.9552 Tf 11.955 0 Td [(C b g > 0 Hence, 9 C k > 0 ; k )]TJ/F18 7.9701 Tf 6.587 0 Td [(1 w w >C k w 2 1 + w 2 2 Fromallofthecasesabove,wehave 9 C k > 0 ; k )]TJ/F18 7.9701 Tf 6.586 0 Td [(1 w w > C k w 2 1 + w 2 2 .151 66

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Hence, Z k )]TJ/F18 7.9701 Tf 6.586 0 Td [(1 w w > C k Z w 2 1 + w 2 2 = C k k w k 2 L 2 : .152 Wearenowreadytoshowcoercivity.Webeginwiththeformula.101.Then a w ; w = Z l r w : r w + Z k )]TJ/F18 7.9701 Tf 6.586 0 Td [(1 w w .153 > l kr w k 2 L 2 + C k k w k 2 L 2 .154 > C c k w k 2 H 1 ; .155 where C c =min f =;C k g . Nextitisnecessarytoknowthefollowingformulationsothatitwillbeusedinthe proofofthenexttheorem.Assumethat f 1 2 H )]TJ/F18 7.9701 Tf 6.586 0 Td [(1 ; s 2 H 1 = 2 @ and f 2 L 2 .Bythe inversetracetheorem4.5,wehave 9 v s 2 H 1 suchthat v s j @ = s ; .156 and 9 K s > 0suchthat k v s k H 1 6 K s k s k H 1 = 2 @ : .157 Therefore,wehavethefollowingequalities, Z f = Z r v = Z @ v n = Z @ s n = Z @ v s n = Z r v s ; .158 whereweappliedthedivergencetheoremtothesecondandlastequalitiesand n istheunit outwardpointingnormaloftheboundary @ .Theequalitiesshowthat Z f )-222(r v s =0.159 whichimpliesthat f )-222(r v s 2 L 2 0 .FromTheorem4.6,then 9 ! v 0 2 V ? H 1 0 suchthat r v 0 = f )-222(r v s : .160 WenowcanprovetheexistenceanduniquenessoftheStokes-Brinkmanequations. 67

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Theorem4.16 Well-posednessoftheStokes-BrinkmanequationsAssumethat f 1 2 H )]TJ/F18 7.9701 Tf 6.586 0 Td [(1 , f ;f 2 L 2 and s 2 H 1 = 2 @ .Thereexistsaunique v 2 H 1 s ;p 2 L 2 0 satisfyingProblem4.1,equations.105.Moreover, k v k H 1 6 1 C c k f 1 k H )]TJ/F19 5.9776 Tf 5.757 0 Td [(1 + p n k f k L 2 + C a C c +1 k ^ v k H 1 .161 where ^ v = v s + v 0 and k p k L 2 6 1 k f 1 k H )]TJ/F19 5.9776 Tf 5.756 0 Td [(1 + p n l k f k L 2 + 1 C a k v k H 1 : .162 where istheconstantin.115. Proof: Let f 1 2 H )]TJ/F18 7.9701 Tf 6.586 0 Td [(1 , f ;f 2 L 2 and s 2 H 1 = 2 @ .Let ^ v = v s + v 0 .Forany w 2 V ,let F w = c 1 w )]TJ/F20 11.9552 Tf 12.532 0 Td [(a ^ v ; w .FromTheorem4.15, c 1 and a ; arelinearand bilinearrespectively.Then F islinear.Moreover,thecontinuitiesof c 1 and a ; impliesthat F iscontinuous.ByapplyingtheLax-MilgramTheorem,thereexistsa unique ~ v 2 V H 1 0 suchthat a ~ v ; w = F w .Let v = ~ v + ^ v = ~ v + v s + v 0 . Therefore, v j @ = ~ v j @ + v s j @ + v 0 j @ =0+ s +0= s because v 0 2 V ? H 1 0 ,and ~ v 2 V H 1 0 .Moreover, r v = r ~ v + r v s + r v 0 =0+ r v s + f )-243(r v s = f since ~ v 2 V and r v 0 = f )-268(r v s ,.160.Therefore, v isin H 1 s andsatisesthe continuityequation.Toshowthat v isunique,weapplythecoercivityof a ; .Since a ~ v ; w = F w = c 1 w )]TJ/F20 11.9552 Tf 12.134 0 Td [(a ^ v ; w and v = ~ v + ^ v , a v ; w = c 1 w .Let v 1 and v 2 satisfy a v 1 ; w = c 1 w and a v 2 ; w = c 1 w .Then a v 1 )]TJ/F32 11.9552 Tf 12.564 0 Td [(v 2 ; w =0forany w 2 V .Thus, a v 1 )]TJ/F32 11.9552 Tf 12.01 0 Td [(v 2 ; v 1 )]TJ/F32 11.9552 Tf 12.009 0 Td [(v 2 =0.Therefore,0= a v 1 )]TJ/F32 11.9552 Tf 12.01 0 Td [(v 2 ; v 1 )]TJ/F32 11.9552 Tf 12.01 0 Td [(v 2 > C c k v 1 )]TJ/F32 11.9552 Tf 12.01 0 Td [(v 2 k 2 H 1 > 0.Since, C c > 0, k v 1 )]TJ/F32 11.9552 Tf 11.603 0 Td [(v 2 k H 1 =0.Then v 1 = v 2 inthe H 1 -norm.Hence,thereexistsaunique v suchthat a v ; w = c 1 w forall w 2 V . Next,weshowthatthereexists p 2 L 2 0 satisfyingProblem4.1or4.9.Let h F 1 ; w i = c 1 w .Since a ~ v ; w = c 1 w )]TJ/F20 11.9552 Tf 10.1 0 Td [(a ^ v ; w , c 1 w )]TJ/F20 11.9552 Tf 10.1 0 Td [(a ~ v ; w )]TJ/F20 11.9552 Tf 10.1 0 Td [(a ^ v ; w =0orinoperatornotation F 1 )]TJ/F20 11.9552 Tf 11.533 0 Td [(A ~ v )]TJ/F20 11.9552 Tf 11.532 0 Td [(A ^ v =0,byDenition4.8.Then F 1 )]TJ/F20 11.9552 Tf 11.533 0 Td [(A ~ v )]TJ/F20 11.9552 Tf 11.532 0 Td [(A ^ v 2 V 0 .Let B 0 = r : L 2 0 ! V 0 . FromTheorem.6andtheisomorphicproperty,thereexistsaunique p 2 L 2 0 suchthat 68

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B 0 p = F 1 )]TJ/F20 11.9552 Tf 11.625 0 Td [(A ~ v )]TJ/F20 11.9552 Tf 11.625 0 Td [(A ^ v = F 1 )]TJ/F20 11.9552 Tf 11.625 0 Td [(A v or A v + B 0 p = F 1 .Hence,thereexistsaunique v 2 H 1 s and p 2 L 2 0 satisfyingProblem4.1. Inordertoprove.161,werstconsider k v 0 k H 1 .Using f )-272(r v s 2 L 2 0 and applyingequations.160and.116,wehave k v 0 k H 1 6 )]TJ/F18 7.9701 Tf 6.586 0 Td [(1 k f )-222(r v s k L 2 6 )]TJ/F18 7.9701 Tf 6.586 0 Td [(1 k f k L 2 + kr v s k L 2 6 )]TJ/F18 7.9701 Tf 6.586 0 Td [(1 k f k L 2 + p n k v s k H 1 6 )]TJ/F18 7.9701 Tf 6.586 0 Td [(1 k f k L 2 + p nC s k s k H 1 = 2 @ ; .163 whereTheorem4.14isappliedtothethirdinequalityandTheorem4.5isappliedtothe fourthinequality.Then k ^ v k H 1 = k v s + v 0 k H 1 6 C s k s k H 1 = 2 @ + )]TJ/F18 7.9701 Tf 6.586 0 Td [(1 k f k L 2 + p nC s k s k H 1 = 2 @ = )]TJ/F18 7.9701 Tf 6.586 0 Td [(1 k f k L 2 ++ p n )]TJ/F18 7.9701 Tf 6.587 0 Td [(1 C s k s k H 1 = 2 @ : .164 Next,weapplythecoercivityof a ; toobtainaboundfor ~ v : C c k ~ v k 2 H 1 6 a ~ v ; ~ v = c 1 ~ v )]TJ/F20 11.9552 Tf 11.956 0 Td [(a ^ v ; ~ v 6 k f 1 k H )]TJ/F19 5.9776 Tf 5.756 0 Td [(1 + p n k f k L 2 k ~ v k H 1 + C a k ^ v k H 1 k ~ v k H 1 ; .165 whereTheorem4.15isappliedtothesecondinequality.Dividingbothsidesby C c k ~ v k H 1 , wehave k ~ v k H 1 6 1 C c k f 1 k H )]TJ/F19 5.9776 Tf 5.756 0 Td [(1 + p n k f k L 2 + C a C c k ^ v k H 1 : Using.165,wehave k v k H 1 = k ~ v + ^ v k H 1 6 1 C c k f 1 k H )]TJ/F19 5.9776 Tf 5.756 0 Td [(1 + p n k f k L 2 + C a C c k ^ v k H 1 + k ^ v k H 1 69

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= 1 C c k f 1 k H )]TJ/F19 5.9776 Tf 5.757 0 Td [(1 + p n k f k L 2 + C a C c +1 k ^ v k H 1 : .166 Next,weemploycondition.115toobtainaboundforpressure p : k p k )]TJ/F18 7.9701 Tf 6.587 0 Td [(1 L 2 sup w 2 H 1 0 b w ;p k w k H 1 =sup w 2 H 1 0 b w ;p k w k H 1 k p k L 2 > > 0 : .167 Rearrangingwehave k p k L 2 6 )]TJ/F18 7.9701 Tf 6.587 0 Td [(1 sup w 2 H 1 0 b w ;p k w k H 1 : .168 Notethat b w ;p = c 1 w )]TJ/F20 11.9552 Tf 11.956 0 Td [(a v ; w 6 k f 1 k H )]TJ/F19 5.9776 Tf 5.756 0 Td [(1 + p n k f k L 2 k w k H 1 + C a k v k H 1 k w k H 1 : .169 Substituting.169into.168,weobtain k p k L 2 6 )]TJ/F18 7.9701 Tf 6.586 0 Td [(1 sup w 2 H 1 0 k f 1 k H )]TJ/F19 5.9776 Tf 5.756 0 Td [(1 + p n k f k L 2 + C a k v k H 1 = )]TJ/F18 7.9701 Tf 6.587 0 Td [(1 k f 1 k H )]TJ/F19 5.9776 Tf 5.756 0 Td [(1 + p n k f k L 2 + C a )]TJ/F18 7.9701 Tf 6.587 0 Td [(1 k v k H 1 ; .170 andwehavecompletetheproofofthewell-posednessoftheStokes-Brinkmanequations, Theorem4.16. Wenowhaveasystemofequations: k )]TJ/F18 7.9701 Tf 6.587 0 Td [(1 l v l )]TJ/F20 11.9552 Tf 11.955 0 Td [( l v s + r p )]TJ/F20 11.9552 Tf 13.553 8.088 Td [( l r l d l = g ; in.171 _ l + )]TJ/F20 11.9552 Tf 11.955 0 Td [( l r l v l )]TJ/F32 11.9552 Tf 11.955 0 Td [(v s =0in.172 v l = s on @ ; .173 whicharewell-posedforaxednumericaldomainandboundaryconditionsdenedby thefunction s 2 H 1 = 2 @ . 70

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t = t 0 t = t 1 Figure5.1: TheleftguresshowsthePCLwhen t = t 0 wheretheciliaisperpendicularto thehorizontalplanwhiletherightonedisplaysthePCLwhen t = t 1 wheretheciliamake anangle tothehorizontalplan,where islessthan90degrees. 5.NUMERICALRESULTSFORTHEFIXEDHEIGHTMODEL, TWO-DIMENSIONALMODEL InthisChapter,weapplyamixedniteelementmethodtothesystemoftheStokesBrinkmanequationsandthecontinuityequation: k )]TJ/F18 7.9701 Tf 6.587 0 Td [(1 l v l )]TJ/F20 11.9552 Tf 11.955 0 Td [( l v s + r p )]TJ/F20 11.9552 Tf 13.553 8.087 Td [( l r l d l = g ; in 1 [ 2 .1 _ l + )]TJ/F20 11.9552 Tf 11.955 0 Td [( l r l v l )]TJ/F32 11.9552 Tf 11.955 0 Td [(v s =0 : in 1 [ 2 .2 wheretheStokesequationisappliedindomain 1 t whiletheBrinkmanequationisemployedin 2 t .Figure5.1showsthedomains.Theleftonerepresentsthedomainwhen theciliaareperpendiculartothehorizontalplaneattime t = t 0 .Inthiscase,wehave onlyporousmedium, 2 ,inthePCLsoonlytheBrinkmanequationisapplied.Theright gureshowsthePCLwhentheciliamakeanangle tothehorizontalplaneattime t = t 1 . Inthisstudytheangle isbetweenarctan : 5toarctaninforabout26to90degrees. Whenevertheangle islessthan90 ,bothsubdomains, 1 and 2 ,havenonzeroareasand theStokes-Brinkmanequationsareapplied.Theboundaryconditionsforthedomainsare discussedinSection5.1.ThevariationalformulationandthemodeldiscretizationarepresentedinSection5.2,whilethenumericalresultsandthevalidationareprovidedinSection 5.3. 71

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5.1BoundaryConditions Inthissection,appropriatelyphysicallymeaningfulboundaryconditionsthatalsoguaranteetheexistenceofauniquesolutionarediscussed.Tobewell-posedeithervelocityor thetractionvector t n mustbeprescribedeverywhereontheboundary.Furthermore,for atwo-dimensionaldomain,weneedtwoscalarequationsateachboundaryforthevelocity [11].Forthepressure,weconsiderthebilinearfunctional.102: b w ;p = )]TJ/F25 9.9626 Tf 11.291 14.058 Td [(Z p r w .3 = Z w grad pd )]TJ/F25 9.9626 Tf 11.955 14.058 Td [(Z )]TJ/F32 11.9552 Tf 7.779 8.579 Td [(w p n d )]TJ -165.61 -29.35 Td [(= Z w grad pd .4 whereweuseGreen'srstidentitytogetthesecondequalityand w isin H 1 0 soitiszero ontheboundary.Thisimpliesthat b w ;p doesnotchangeifweaddaconstantfunction to p .Notethat.1onlydeterminespressureuptoanadditiveconstant.Thisconstantis usuallyxedbyenforcingthenormalization[21] Z pd =0 : .5 Wedividetheboundaryintovepieces:thefree-uid/porous-mediuminterface, 1 2 , thefree-uid/uidinterfacebetweenthePCLandmucouslayeratthetopof 1 [ 2 ,the leftandrightsidesof 1 [ 2 andthebottomofthePCL,tobeabletosolvethesystemsof equationsnumerically.Onthesides,weassumeperiodicityforbothvelocitiesandpressure. Thevelocitiesareassumedzeroatthebottom. Letusrstconsidertheboundaryconditionsatthefree-uid/porous-mediuminterface. Manyauthors[3,47,62]whousetheDarcyequationin 2 andtheStokesequationin 1 applytheBeavers-Josephcondition[8]whichstatesthatthevelocitycomponentparallelsto theinterfacecanslipaccordingto du v dx 2 = m p k u v slip )]TJ/F20 11.9552 Tf 11.955 0 Td [(u p ; .6 72

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where m isamaterialdependent,dimensionless,parameter; k isthepermeability; u v and u p arethevelocitiesoftheviscousandporousows,respectivelyand u v slip istheslipvelocity dependingonthestructureofpermeablematerialwithintheboundaryregion,heightofthe domainanduidviscosity.Thedierenceofthetangentialcomponentsofthevelocitiesin .6isthejumpduetoanominalboundarylayerattheinterface.RecallthatinDarcy's lawthereisnoviscousterm,2 d l =0,whichisthedeviatoricpartofthestresstensor.Asa resultthereisnotanaturalwaytobalancetheshearstresses.TheBeavers-Josephboundary conditionprovidesatransitionmodelfromtheviscoustotheporousows[43,46,75,88]. Iftheporosityisclosedtooneattheinterface,theviscousstressintheviscousowis completelytransferredtotheuidintheporousmedia. Alternately,ifoneusestheBrinkmanequationin 2 andtheStokesequationin 1 , thedeviatoricpartofthestresstensorscanbeconsidered.Forthiscase,bothcontinuous anddiscontinuousshearstressboundaryconditionshavebeenconsidered[39,68].Werst assumecontinuityofthenormalcomponentofstress,i.e. t l n j 1 = t l n j 2 ; .7 where t l = )]TJ/F20 11.9552 Tf 9.299 0 Td [(p I +2 d l .Wealsoneedtoconservemassacrosstheinterfacemassuxmust beequal[11]: v l n j 1 = v l n j 2 .8 wherethesubscripts1and2refertothedomain 1 and 2 ,respectivelyand n istheunit normalvectorpointingoutwardoftheboundary.Ifthenormalcomponentofthevelocityis continuousacrosstheinterface,weautomaticallyhave.8. Forthetopboundaryconditionwerstconsidertheshearstress.Insteadofmucus,air isrstassumedatthetopofPCL.Thereforetheboundaryconditiononthetopof 1 is t 12 = t 21 =2 d l = @v l 1 @x 2 + @v l 2 @x 1 =0.9 73

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Figure5.2: AtwodimensionalCartesiancoordinatesystemwithaxislines x 1 and x 2 and thecartoonpictureoftheciliainthePCLwithboundaryconditions. where v l = v l 1 ;v l 2 .Theequation.9canberewrittenas @v l 1 @x 2 = )]TJ/F20 11.9552 Tf 10.991 8.088 Td [(@v l 2 @x 1 : .10 Sinceweassumethattheverticalvelocityiszeroatthisinterface, @v l 1 @x 2 = @v l 2 @x 1 =0 : .11 Hence,ourinitialmodelasshowninFigure6.2is k )]TJ/F18 7.9701 Tf 6.587 0 Td [(1 l v l )]TJ/F20 11.9552 Tf 11.955 0 Td [( l v s + r p )]TJ/F20 11.9552 Tf 13.552 8.088 Td [( l r l d l = g ; in 1 [ 2 .12 _ l + )]TJ/F20 11.9552 Tf 11.955 0 Td [( l r l v l )]TJ/F32 11.9552 Tf 11.955 0 Td [(v s =0 : in 1 [ 2 .13 v l =0atthebottomof 2 .14 v l and p areperiodiconthesides.15 v l iscontinuousat 1 1 .16 dv l 1 dx 2 = dv l 2 dx 1 =0onthetopof 1 : .17 5.2ModelDiscretization Theappropriatediscretizationofthegoverningequationsistherststeptoobtaina numericalsolutionusingthemixedniteelementmethod.Webeginbyderivingtheweak formulationoftheStokes-Brinkmanequations.12-.13whichareequivalentto.1 74

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and.2:Find v ;p 2 H 1 L 2 0 suchthat k )]TJ/F18 7.9701 Tf 6.587 0 Td [(1 v + r p )]TJ/F20 11.9552 Tf 13.552 8.088 Td [( l v = g + k )]TJ/F18 7.9701 Tf 6.587 0 Td [(1 l v s + l r f .18 r v = f; .19 withtheboundaryconditions.14-.17where f = )]TJ/F16 11.9552 Tf 10.687 0 Td [(_ l = )]TJ/F20 11.9552 Tf 12.687 0 Td [( l + r l v s andisa computationaldomain.Notethateverytermontherighthandsideof.18isgivensince thevelocityoftheciliaorsolidphaseisknown,seee.g.[20].Writing.18asscalar equationsintwo-dimensionsgives: h k )]TJ/F18 7.9701 Tf 6.586 0 Td [(1 11 v 1 + k )]TJ/F18 7.9701 Tf 6.586 0 Td [(1 12 v 2 i )]TJ/F20 11.9552 Tf 13.553 8.088 Td [( l " @ 2 v 1 @x 2 1 + @ 2 v 1 @x 2 2 # + @p @x 1 = l h k )]TJ/F18 7.9701 Tf 6.587 0 Td [(1 11 v s 1 + k )]TJ/F18 7.9701 Tf 6.586 0 Td [(1 12 v s 2 i + l @f @x 1 ; .20 h k )]TJ/F18 7.9701 Tf 6.586 0 Td [(1 21 v 1 + k )]TJ/F18 7.9701 Tf 6.586 0 Td [(1 22 v 2 i )]TJ/F20 11.9552 Tf 13.553 8.088 Td [( l " @ 2 v 2 @x 2 1 + @ 2 v 2 @x 2 2 # + @p @x 2 = )]TJ/F20 11.9552 Tf 9.298 0 Td [(g + l h k )]TJ/F18 7.9701 Tf 6.586 0 Td [(1 21 v s 1 + k )]TJ/F18 7.9701 Tf 6.586 0 Td [(1 22 v s 2 i + l @f @x 2 ; .21 wheregravityisgivenby g = ; )]TJ/F20 11.9552 Tf 9.298 0 Td [(g .Tondtheweakformulation,wemultiply.20.21bytestfunctions w i 2 H 1 0 , i =1 ; 2andintegrateoverthedomain= 1 [ 2 . Thisyields Z h k )]TJ/F18 7.9701 Tf 6.587 0 Td [(1 11 v 1 + k )]TJ/F18 7.9701 Tf 6.587 0 Td [(1 12 v 2 i )]TJ/F20 11.9552 Tf 13.553 8.088 Td [( l " @ 2 v 1 @x 2 1 + @ 2 v 1 @x 2 2 # + @p @x 1 ! w 1 d = Z l h k )]TJ/F18 7.9701 Tf 6.587 0 Td [(1 11 v s 1 + k )]TJ/F18 7.9701 Tf 6.587 0 Td [(1 12 v s 2 i + l @f @x 1 ! w 1 d : .22 Z h k )]TJ/F18 7.9701 Tf 6.587 0 Td [(1 21 v 1 + k )]TJ/F18 7.9701 Tf 6.587 0 Td [(1 22 v 2 i )]TJ/F20 11.9552 Tf 13.553 8.088 Td [( l " @ 2 v 2 @x 2 1 + @ 2 v 2 @x 2 2 # + @p @x 2 ! w 2 d = Z )]TJ/F20 11.9552 Tf 9.298 0 Td [(g + l h k )]TJ/F18 7.9701 Tf 6.587 0 Td [(1 21 v s 1 + k )]TJ/F18 7.9701 Tf 6.586 0 Td [(1 22 v s 2 i + l @f @x 2 ! w 2 d : .23 Integratingbypartsthepressureterm,thesecond-ordertermandthesourceterm f ,we havetheweakformulation:Find v ;p 2 H 1 L 2 0 suchthat Z h k )]TJ/F18 7.9701 Tf 6.586 0 Td [(1 11 v 1 + k )]TJ/F18 7.9701 Tf 6.586 0 Td [(1 12 v 2 i w 1 d + l Z " @v 1 @x 1 @w 1 @x 1 + @v 1 @x 2 @w 1 @x 2 # d )]TJ/F25 9.9626 Tf 11.955 14.058 Td [(Z p @w 1 @x 1 d = Z l h k )]TJ/F18 7.9701 Tf 6.587 0 Td [(1 11 v s 1 + k )]TJ/F18 7.9701 Tf 6.586 0 Td [(1 12 v s 2 i w 1 d )]TJ/F20 11.9552 Tf 13.552 8.087 Td [( l Z f @w 1 @x 1 d )]TJ/F25 9.9626 Tf 11.955 14.058 Td [(Z )]TJ/F20 11.9552 Tf 7.779 8.579 Td [(pw 1 n 1 d )-222(+ l Z )]TJ/F25 9.9626 Tf 7.779 26.113 Td [(" @v 1 @x 1 n 1 + @v 1 @x 2 n 2 # w 1 d )-222(+ l Z )]TJ/F20 11.9552 Tf 7.779 8.579 Td [(fw 1 n 1 d )]TJ/F20 11.9552 Tf 7.314 0 Td [(; 8 w 1 2 H 1 0 .24 75

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Z h k )]TJ/F18 7.9701 Tf 6.586 0 Td [(1 21 v 1 + k )]TJ/F18 7.9701 Tf 6.586 0 Td [(1 22 v 2 i w 2 d + l Z " @v 2 @x 1 @w 2 @x 1 + @v 2 @x 2 @w 2 @x 2 # d )]TJ/F25 9.9626 Tf 11.955 14.058 Td [(Z p @w 2 @x 2 d = Z )]TJ/F20 11.9552 Tf 9.298 0 Td [(g + l h k )]TJ/F18 7.9701 Tf 6.586 0 Td [(1 21 v s 1 + k )]TJ/F18 7.9701 Tf 6.587 0 Td [(1 22 v s 2 i w 2 d )]TJ/F20 11.9552 Tf 13.553 8.088 Td [( l Z f @w 2 @x 2 d )]TJ/F25 9.9626 Tf 11.955 14.058 Td [(Z )]TJ/F20 11.9552 Tf 7.779 8.579 Td [(pw 2 n 2 d )-222(+ l Z )]TJ/F25 9.9626 Tf 7.779 26.113 Td [(" @v 2 @x 1 n 1 + @v 2 @x 2 n 2 # w 2 d )-222(+ l Z )]TJ/F20 11.9552 Tf 7.779 8.579 Td [(fw 2 n 2 d )]TJ/F20 11.9552 Tf 7.314 0 Td [(; 8 w 2 2 H 1 0 .25 where n 1 ;n 2 istheoutwardunitnormalvectorand)-326(istheboundaryofthedomain. Let T h beatriangulationofdomainand V h = f v 2 H 1 : v j K isquadratic ; 8 K 2 T h g .26 H h = f q 2 L 2 0 : q j K islinear ; 8 K 2 T h g : .27 benite-dimensionalsubspacesof H 1 andL 2 0 ,respectively.Inniteelementmethod weapproximatethesolutions v i ;p 2 V h H h byletting v i x = M X m =1 m x v m i = T V i ; .28 p x = L X l =1 l x p l = T P : .29 where m and l arecalledbasisfunctionswhileandaretheirvectorforms; V i and P arevectorsofthevelocitiesandpressure,respectively,andthenumbers M and L are determinedbytheinterpolationfunction.Forexample,foratetrahedralelement, M =10 forquadraticfunctionfor v i and L =4forlinearfunctionfor p .Substituting.28and .29into.24and.25andthebasisfunctioninto w 1 and w 2 ,wehave Z k )]TJ/F18 7.9701 Tf 6.587 0 Td [(1 11 T d V 1 + Z k )]TJ/F18 7.9701 Tf 6.586 0 Td [(1 12 T d V 2 + l " Z @ @x 1 @ T @x 1 d + Z @ @x 2 @ T @x 2 d ! V 1 # )]TJ/F25 9.9626 Tf 11.955 14.058 Td [(Z @ @x 1 T d P = l Z k )]TJ/F18 7.9701 Tf 6.587 0 Td [(1 11 v s 1 + k )]TJ/F18 7.9701 Tf 6.586 0 Td [(1 12 v s 2 d )]TJ/F20 11.9552 Tf 13.553 8.088 Td [( l Z f @ @x 1 d )]TJ/F25 9.9626 Tf 11.955 14.058 Td [(Z )]TJ/F16 11.9552 Tf 7.779 8.579 Td [( T n 1 d )]TJ/F32 11.9552 Tf 7.314 0 Td [(P + l " Z )]TJ/F16 11.9552 Tf 7.779 8.579 Td [( @ T @x 1 n 1 d )-222(+ Z )]TJ/F16 11.9552 Tf 7.779 8.579 Td [( @ T @x 2 n 2 d )]TJ/F25 9.9626 Tf 7.314 17.534 Td [(! V 1 # + l Z )]TJ/F20 11.9552 Tf 7.779 8.579 Td [(f n 1 d )]TJ/F20 11.9552 Tf 7.314 0 Td [(; .30 Z k )]TJ/F18 7.9701 Tf 6.587 0 Td [(1 21 T d V 1 + Z k )]TJ/F18 7.9701 Tf 6.587 0 Td [(1 22 T d V 2 + l " Z @ @x 1 @ T @x 1 d + Z @ @x 2 @ T @x 2 d ! V 2 # )]TJ/F25 9.9626 Tf 11.955 14.059 Td [(Z @ @x 2 T d P = Z )]TJ/F20 11.9552 Tf 9.299 0 Td [(g + l k )]TJ/F18 7.9701 Tf 6.586 0 Td [(1 21 v s 1 + k )]TJ/F18 7.9701 Tf 6.586 0 Td [(1 22 v s 2 d )]TJ/F20 11.9552 Tf 13.552 8.088 Td [( l Z f @ @x 2 d 76

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)]TJ/F25 9.9626 Tf 11.955 14.059 Td [(Z )]TJ/F16 11.9552 Tf 7.779 8.579 Td [( T n 2 d )]TJ/F32 11.9552 Tf 7.314 0 Td [(P + l " Z )]TJ/F16 11.9552 Tf 7.779 8.579 Td [( @ T @x 1 n 1 d )-222(+ Z )]TJ/F16 11.9552 Tf 7.779 8.579 Td [( @ T @x 2 n 2 d )]TJ/F25 9.9626 Tf 7.314 17.534 Td [(! V 2 # + l Z )]TJ/F20 11.9552 Tf 7.779 8.579 Td [(f n 2 d )]TJ/F20 11.9552 Tf 7.314 0 Td [(: .31 Toformanelementmatrixfortheniteelementmethodin 2 , ~ A = Z e 2 T d e 2 ; ~ K ij = Z e 2 @ @x i @ T @x j d e 2 ~ Q i = Z e 2 @ T @x i d e 2 ; .32 ~ F 1 = l " Z e 2 k )]TJ/F18 7.9701 Tf 6.587 0 Td [(1 11 v s 1 + k )]TJ/F18 7.9701 Tf 6.586 0 Td [(1 12 v s 2 d e 2 !# )]TJ/F20 11.9552 Tf 13.552 8.088 Td [( l Z e 2 f @ @x 1 d e 2 )]TJ/F25 9.9626 Tf 11.955 14.058 Td [(Z )]TJ/F22 5.9776 Tf 5.288 2.745 Td [(e 2 T n 1 d )]TJ/F21 7.9701 Tf 7.314 4.936 Td [(e 2 P + l " Z )]TJ/F22 5.9776 Tf 5.289 2.745 Td [(e 2 @ T @x 1 n 1 d )]TJ/F21 7.9701 Tf 7.314 4.936 Td [(e 2 + Z )]TJ/F22 5.9776 Tf 5.289 2.745 Td [(e 2 @ T @x 2 n 2 d )]TJ/F21 7.9701 Tf 7.314 4.936 Td [(e 2 ! V 1 # + l Z )]TJ/F22 5.9776 Tf 5.288 2.745 Td [(e 2 f n 1 d )]TJ/F21 7.9701 Tf 7.314 4.936 Td [(e 2 ; .33 ~ F 2 = Z e 2 )]TJ/F20 11.9552 Tf 9.298 0 Td [(g + l k )]TJ/F18 7.9701 Tf 6.587 0 Td [(1 21 v s 1 + k )]TJ/F18 7.9701 Tf 6.586 0 Td [(1 22 v s 2 d e 2 )]TJ/F20 11.9552 Tf 13.553 8.088 Td [( l Z e 2 f @ @x 2 d e 2 )]TJ/F25 9.9626 Tf 11.955 14.059 Td [(Z )]TJ/F22 5.9776 Tf 5.288 2.746 Td [(e 2 T n 2 d )]TJ/F21 7.9701 Tf 7.314 4.937 Td [(e 2 P + l " Z )]TJ/F22 5.9776 Tf 5.289 2.746 Td [(e 2 @ T @x 1 n 1 d )]TJ/F21 7.9701 Tf 7.314 4.937 Td [(e 2 + Z )]TJ/F22 5.9776 Tf 5.289 2.746 Td [(e 2 @ T @x 2 n 2 d )]TJ/F21 7.9701 Tf 7.314 4.937 Td [(e 2 ! V 2 # + l Z )]TJ/F22 5.9776 Tf 5.288 2.745 Td [(e 2 f n 2 d )]TJ/F21 7.9701 Tf 7.314 4.936 Td [(e 2 : .34 where e 2 istheelementdomainsuchthat 2 = [ e e 2 .Then.30and.31become k )]TJ/F18 7.9701 Tf 6.587 0 Td [(1 11 ~ AV 1 + k )]TJ/F18 7.9701 Tf 6.586 0 Td [(1 12 ~ AV 2 + = l ~ K 11 + ~ K 22 V 1 )]TJ/F16 11.9552 Tf 17.382 4.417 Td [(~ Q T 1 P = ~ F 1 ; .35 k )]TJ/F18 7.9701 Tf 6.587 0 Td [(1 21 ~ AV 1 + k )]TJ/F18 7.9701 Tf 6.586 0 Td [(1 22 ~ AV 2 + = l ~ K 11 + ~ K 22 V 2 )]TJ/F16 11.9552 Tf 17.382 4.417 Td [(~ Q T 2 P = ~ F 2 ; .36 wherethesuperscript T denotesthetranspose.Applyingthesameprocesstothecontinuity equation.19: @v 1 @x 1 + @v 2 @x 2 = )]TJ/F16 11.9552 Tf 20.554 8.088 Td [(_ l 1 )]TJ/F20 11.9552 Tf 11.955 0 Td [( + l @v s 1 @x 1 + l @v s 2 @x 2 ; .37 wehavetheweakform )]TJ/F25 9.9626 Tf 11.291 14.058 Td [(Z e 2 @ T @x 1 d e 2 V 1 )]TJ/F25 9.9626 Tf 11.955 14.058 Td [(Z e 2 @ T @x 2 d e 2 V 2 = )]TJ/F25 9.9626 Tf 11.291 14.058 Td [(Z e 2 )]TJ/F16 11.9552 Tf 22.115 8.088 Td [(_ l 1 )]TJ/F20 11.9552 Tf 11.955 0 Td [( l + l @v s 1 @x 1 + l @v s 2 @x 2 ! d e 2 : .38 Let ~ F 3 = )]TJ/F25 9.9626 Tf 11.291 14.059 Td [(Z e 2 )]TJ/F16 11.9552 Tf 22.114 8.088 Td [(_ l 1 )]TJ/F20 11.9552 Tf 11.955 0 Td [( l + l @v s 1 @x 1 + l @v s 2 @x 2 ! d e 2 : .39 77

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Then.38becomes )]TJ/F16 11.9552 Tf 13.789 2.905 Td [(~ Q 1 V 1 )]TJ/F16 11.9552 Tf 16.445 2.905 Td [(~ Q 2 V 2 = ~ F 3 : .40 Writingthesystemofequations.35,.36and.40inelementmatrixform,wehave 0 B B B B B B @ k )]TJ/F18 7.9701 Tf 6.586 0 Td [(1 11 ~ A + = l ~ K 11 + ~ K 22 k )]TJ/F18 7.9701 Tf 6.586 0 Td [(1 12 ~ A )]TJ/F16 11.9552 Tf 11.422 2.905 Td [(~ Q T 1 k )]TJ/F18 7.9701 Tf 6.586 0 Td [(1 21 ~ A k )]TJ/F18 7.9701 Tf 6.587 0 Td [(1 22 ~ A + = l ~ K 11 + ~ K 22 )]TJ/F16 11.9552 Tf 11.422 2.905 Td [(~ Q T 2 )]TJ/F16 11.9552 Tf 11.422 2.905 Td [(~ Q 1 )]TJ/F16 11.9552 Tf 11.422 2.905 Td [(~ Q 2 0 1 C C C C C C A 0 B B B B B B @ V 1 V 2 P 1 C C C C C C A = 0 B B B B B B @ ~ F 1 ~ F 2 ~ F 3 1 C C C C C C A : .41 Wenowhavethematrixformofthediscretesystemofequationsindomain 2 . Wenextndtheelementmatrixformfordomain 1 .Notethatthemomentumequations in 1 arethesameasthosein 2 excepttherearenovelocitytermsandtheporosityisone. Applyingthesameprocessasthatappliedtoobtain.41,wehave l " Z e 1 @ @x 1 @ T @x 1 d e 1 + Z e 1 @ @x 2 @ T @x 2 d e 1 ! V 1 # )]TJ/F25 9.9626 Tf 11.956 14.059 Td [(Z e 1 @ @x 1 T d e 1 P = )]TJ/F25 9.9626 Tf 11.291 14.058 Td [(Z )]TJ/F22 5.9776 Tf 5.288 2.745 Td [(e 1 T n 1 d )]TJ/F21 7.9701 Tf 7.314 4.936 Td [(e 1 P + l " Z )]TJ/F22 5.9776 Tf 5.289 2.745 Td [(e 1 @ T @x 1 n 1 d )]TJ/F21 7.9701 Tf 7.314 4.936 Td [(e 1 + Z )]TJ/F22 5.9776 Tf 5.289 2.745 Td [(e 1 @ T @x 2 n 2 d )]TJ/F21 7.9701 Tf 7.314 4.936 Td [(e 1 ! V 1 # ; .42 l " Z e 1 @ @x 1 @ T @x 1 d e 1 + Z e 1 @ @x 2 @ T @x 2 d e 1 ! V 2 # )]TJ/F25 9.9626 Tf 11.956 14.058 Td [(Z e 1 @ @x 2 T d e 1 P = )]TJ/F25 9.9626 Tf 11.291 14.058 Td [(Z )]TJ/F22 5.9776 Tf 5.288 2.746 Td [(e 1 T n 2 d )]TJ/F21 7.9701 Tf 7.314 4.936 Td [(e 1 P + l " Z )]TJ/F22 5.9776 Tf 5.289 2.746 Td [(e 1 @ T @x 1 n 1 d )]TJ/F21 7.9701 Tf 7.314 4.937 Td [(e 1 + Z )]TJ/F22 5.9776 Tf 5.289 2.746 Td [(e 1 @ T @x 2 n 2 d )]TJ/F21 7.9701 Tf 7.314 4.937 Td [(e 1 ! V 2 # ; .43 Let e 1 betheelementdomainsuchthat 1 = [ e e 1 .Writing.42,.43and.38in thematrixform,wehave 0 B B B B B B @ = l ~ K 11 + ~ K 22 0 )]TJ/F16 11.9552 Tf 11.422 2.905 Td [(~ Q T 1 0 = l ~ K 11 + ~ K 22 )]TJ/F16 11.9552 Tf 11.422 2.905 Td [(~ Q T 2 )]TJ/F16 11.9552 Tf 11.422 2.905 Td [(~ Q 1 )]TJ/F16 11.9552 Tf 11.423 2.905 Td [(~ Q 2 0 1 C C C C C C A 0 B B B B B B @ V 1 V 2 P 1 C C C C C C A = 0 B B B B B B @ ~ B 1 ~ B 2 ~ F 3 1 C C C C C C A .44 where ~ B 1 = )]TJ/F25 9.9626 Tf 11.291 14.059 Td [(Z )]TJ/F22 5.9776 Tf 5.289 2.745 Td [(e 1 T n 1 d )]TJ/F21 7.9701 Tf 7.314 4.936 Td [(e 1 P + l " Z )]TJ/F22 5.9776 Tf 5.288 2.745 Td [(e 1 @ T @x 1 n 1 d )]TJ/F21 7.9701 Tf 7.314 4.936 Td [(e 1 + Z )]TJ/F22 5.9776 Tf 5.289 2.745 Td [(e 1 @ T @x 2 n 2 d )]TJ/F21 7.9701 Tf 7.314 4.936 Td [(e 1 ! V 1 # ; .45 and ~ B 2 = )]TJ/F25 9.9626 Tf 11.291 14.058 Td [(Z )]TJ/F22 5.9776 Tf 5.289 2.745 Td [(e 1 T n 2 d )]TJ/F21 7.9701 Tf 7.314 4.936 Td [(e 1 P + l " Z )]TJ/F22 5.9776 Tf 5.288 2.745 Td [(e 1 @ T @x 1 n 1 d )]TJ/F21 7.9701 Tf 7.314 4.936 Td [(e 1 + Z )]TJ/F22 5.9776 Tf 5.289 2.745 Td [(e 1 @ T @x 2 n 2 d )]TJ/F21 7.9701 Tf 7.314 4.936 Td [(e 1 ! V 2 # : .46 78

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Notethatthevelocitiesandpressurein 1 and 2 arenotthesamebutwestillusethe samenotationforsimplicity.Sincethenormalandshearstressesarecontinuousacrossthe interface 1 2 wherecansharethenodesbetweenthepure-uidandporousmedium domains,thevelocityandpressuretermsinthesurfaceintegrals ~ B 1 and ~ F 1 areequalin magnitudebuthaveoppositesignswhichcanceleachotherinthenaldiscretizedequation, [43,89,99].Similarly,thevelocityandpressuretermsinthesurfaceintegrals ~ B 2 and ~ F 2 can becanceledinthenaldiscretizedequation.Therefore,byusingniteelementmethod,the thestress-continuityconditionisautomaticallysatisedandnospecialprocedureisneeded toimposesuchinterfacialcondition. 5.3ValidationoftheCodeandNumericalResults BeforeweprovidetheStokes-Brinkmannumericalresultsforourproblem,wevalidate ourresultsbycomparingwithanexactsolutionforwhichtheboundaryconditionsare simpleenoughtodeterminetheanalyticsolution.Theboundaryconditionimposedisthat at x 2 = H c 1 + H c 2 = H c ,seeFigure5.1,theliquidisdraggedwithaconstantvelocity u 0 by animpermeableplate.UsingtheBrinkmanequationtomodeluidowin 2 ,thegeneral formforthevelocityis[58] u x 2 = 8 > > > < > > > : C c + C k x 2 )]TJ/F20 11.9552 Tf 11.955 0 Td [(H c 2 if x 2 >H c 2 Ce x 2 )]TJ/F21 7.9701 Tf 6.587 0 Td [(H c 2 = p k if x 2
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0 @u @x 2 j x 2 = H )]TJ/F22 5.9776 Tf -0.614 -6.276 Td [(c 2 = @u @x 2 j x 2 = H + c 2 ; .50 where+and )]TJ/F16 11.9552 Tf 12.929 0 Td [(refertothefreeuidandporousmediumrespectivelyand 0 istheeective viscositywhichisaparametermatchingtheshearstressatthefree-uid/porous-medium interface[68],weobtainexpressionsforthecoecients C c ;C k and C : C c = u 0 1+ 0 = H c )]TJ/F20 11.9552 Tf 11.955 0 Td [(H c 2 = p k ; .51 C k = 0 C c p k ; .52 C = C c : .53 Eliminating C c ;C k and C from.47using.51-.53,wehave u x 2 = u 0 1+ 0 H c )]TJ/F20 11.9552 Tf 11.955 0 Td [(H c 2 p k 8 > > > > > < > > > > > : 1+ 0 x 2 )]TJ/F20 11.9552 Tf 11.955 0 Td [(H c 2 p k ! if x 2 >H c 2 e x 2 )]TJ/F20 11.9552 Tf 11.955 0 Td [(H c 2 p k if x 2
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Figure5.3: Velocityprolesinthe x 2 directionor y -axisoftheexactsolutionandour numericalresultwhere u 0 =1; =1; l =0 : 64457; H c =1and H c 2 =0 : 7071. Table5.1: L 2 -normerrorofthevelocityfortheStokes-Brinkmanequationswhere isthe anglebetweenthearrayofcylindersandthehorizontalplane. angle L 2 -normerrors 26.50.0466 30.91.0092 34.90.7471 38.60.7548 41.91.0390 45.01.5918 51.30.8077 56.30.9084 63.40.6294 75.90.9683 90.01.2073 81

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Figure5.4: Velocityprolesofthenumericalandexactsolutionsusingourpermeability resultswiththecorrespondingangle ;u0=1; =1; H c =1. 82

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Figure5.5: Convergenceofthevelocityprolesofthenumericalresultstotheexact solutionswhentheangle is45 83

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Table5.2: L 2 -normerrorsofthenumericalandexactsolutionsofStokes-Brinkmanequationswhenthenumbersofelementsareincreasingwhere#dofisthenumberofdegreesof freedomand x istheuniformlengthofeachelement. #ele#dof xL 2 -normerrors 20010030.11.8434 80038030.051.5918 3200148030.0251.1824 12800584030.01250.4385 angle =90 .Ourinitialnumericalresult,Figure5.6,iscalculatedwhentheshearstress atthefree-uid/porous-mediuminterfaceisassumedtobezero.Thismeansnoshearforce onthefreeuidandonlythetipofciliamovestheuid.Thelinesfromthelefttorightare theincreasingangle betweenthearrayofcylindersandthehorizontalplane.Ascanbe seenfromFigure5.6thevelocityintheporousmedium 2 increasesas increases.The velocityoftheStokesuidat 1 2 isthesameasthevelocityintheporousmediumdue tothecontinuityofnormalandshearstressesattheinterface. 84

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Figure5.6: Velocityproleinthe x 2 directionoftheciliamakingangles =26 ; 30 ;:::: 90 withthehorizontalplanewhentheshearstressiszeroatthefree-uid/porous-medium interface. 85

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Figure6.1: Thecartoonpictureisshowingthefreeboundaryhavingaunknowncurve s whiletheciliaismovingforwardandbackwardmakingtheangle withthehorizontalplane inthePCL. 6.FREEBOUNDARYTWO-DIMENSIONALMODEL ThepurposeofthisChapteristoproposeamodelforthefreeboundaryprobleminthe two-dimensionaldomain= 1 [ 2 .Figure6.1illustratesthedomainofinterestandthe freeboundarybetweenthePCLandmucouslayers,whoseheightisaprioriunknown.The curve s ,whichisafunctionof x and t ,isanunknownandchangesduetothemovementof thecilia. Thereexitmanyapproachesforsimulatingfreeboundaryproblemsincludingtheimmersedboundarymethod[72,81],thevolumeofuidVOFmethod[50,76]andthelevel setmethod[79].Forexample,HirtandNichols[50]usesthevolumeofuidmethodto treatcomplicatedfreeboundarycongurations,whichisproposedoriginallybyNicholset al.[76].Theideaoftheimmersedboundarymethodisprovidedin[72]and[81]whereitis usedtomodelsystemsofelasticstructuresormembranesdeformingandinteractingwith uidows.Inthisapproach,theuidisrepresentedinanEuleriancoordinateframeand thestructuresinaLagrangiancoordinateframe.Adisadvantageofthismethodisthat imposingoftheboundaryconditionsisnotstraightforwardcomparedtotheothers.Forthe levelsetmethod,thecurvesattheinterfacearetrackedonaxedCartesiangridEulerian approach.Themethodisagreattoolformodelingtime-varyingobjectssuchasadropof oiloatinginwater[79].Oneofadvantagesoflevelsetmethodistwotothreedimensions canbecodedquicklywhichistimecomsumingfortheVOFmethod.However,oneofthe 86

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considerableadvantagesoftheVOFmethodisthattheyconservemasswellwhilethelevel setmethoddoesnot.Forsteady-stateproblems,authorsin[82,110]employednewnite elementmethodstosolvefreeboundaryproblems.Moreover,SaavedraandScott[86]presentedtheoreticalerroranalysisofaniteelementmethodofafreeboundaryproblemfor viscousliquid.Forthesteady-stateLaplaceequation,ZhangandBabuska[110]solvedthe freeboundaryprobleminaxeddomainexactly,aswellasanexponentialrateofconvergence.Theyprovedthatthesequenceofsolutionsoffreeboundaryproblemsconvergesto thesolutionofthegivenfreeboundaryproblem.Inaddition,Petersonetal.[82]provides anewiterativemethodfortwo-dimensionalfree-surfaceproblemswitharbitraryinitialgeometries.Theyusedanelasticdeformationofthemeshtopreservethecontinuity.Next, weprovideourmodelandoutlinethemethodfrom[82]togetanideatobeabletoapply toourfreeboundaryproblem. 6.1ModelProblemandBoundaryConditions Wesummarizethemodelandboundaryconditionsinthissection.Weassumethat thethefreesurfaceatthefree-uid/porous-mediuminterfaceisrepresentedbyacurve s s;t = x s;t ;y s;t parametrizedbyarclength s anddependedontime t .Atthe freeboundary,curve s ,thekinematicboundaryconditionatwhichamaterialpointonthe boundaryremainsontheboundarycanbeexpressedas[82] n v = n @ s @t ; .1 where n istheoutwardfree-surfacenormalvectorand v = l v l .Theabsenceofamaterial derivativeinthesteady-statemomentumequation.1allowstheproblemtobesolved usingquasi-steady-statemethods.Thefree-surfacelocationcanbeupdatedexplicitlyusing thekinematicboundarycondition.Beforewementionthenumericalmethod,theprimary systemofequationsofourcomputationaldomains 1 and 2 is k )]TJ/F18 7.9701 Tf 6.587 0 Td [(1 l v l )]TJ/F20 11.9552 Tf 11.955 0 Td [( l v s + r p )]TJ/F20 11.9552 Tf 13.552 8.088 Td [( l r l d l = g ; in 1 [ 2 ;t> 0.2 87

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Figure6.2: AtwodimensionalCartesiancoordinatesystemwithaxislines x 1 and x 2 andthecartoonpictureofthemovingciliacreatingafreeboundarycurve s withboundary conditions. _ l + )]TJ/F20 11.9552 Tf 11.955 0 Td [( l r l v l )]TJ/F32 11.9552 Tf 11.955 0 Td [(v s =0 ; in 1 [ 2 ;t> 0.3 n v = n @ s @t at x 2 = s .4 v =0at x 2 =0 ; .5 v and p areperiodicat x 1 = s 0 and x 1 = s 1 : .6 6.2NumericalImplementation Inthissection,wesummarizethemethodfrom[82]tobeabletoadapttoourfreeboundaryproblem.Intheabsenceofextremecurvature,thesemi-explicitmethodisanattractive approach.Tosolvethefreeboundaryproblemviatheniteelementmethod,weconsidertwo mainsteps.First,theowoftheliquidisdeterminedusingaxedmesh.Second,ateach iteration,themeshismodiedecientlywithsomeformofcontinuousmeshdeformation [71,82].Boththeparticularformofcontinuousdeformationandthemeshtopologymust bedeterminedinadvance.Inordertoupdatethemesh,thekinematicboundarycondition isemployed.Frequently,conservationofmassisalsotakenintoconsideration.However,in thiswork,onlythekinematicconditionwillbeappliedateachiteration.Ifthechangeof massisnotconservedwithinanallowableerror,theformulaoftheconservationofmasswill beconsidered. 88

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Thefree-surfaceboundarylocationisupdatedusingarst-orderexplicitscheme.The kinematicboundaryconditionissolvedfor _ s using[41]withthetimestepbeingselected usingtheCFLcondition t 6 min 1 4 l min v max ; max .7 where v max isthelargestvelocityfoundfromthemostrecentsolution; l min denotesthe lengthoftheshortestedgeinthemeshand max isaprescribedmaximumtimestepandthe constant1 = 4arisesduetotheuseofthequadraticelements.Itissometimesnecessaryto employsmallervaluesoftheconstantinordertoensurestability. Oncethefreesurfaceislocated,theinteriornodesarerepositionedviathecontinuous deformationmodelwhichmaintainstheoriginalconnectivity.Ifthemeshqualityfallsbelow prescribedthresholds,themeshisregeneratedusingastandardDelaunaytriangulation algorithmandonlythelocationsoftheboundarynodesispreserved. Thesemi-explicitschemecanbeemployedwhenthetheequationsarelinearsothatthe matricesaresymmetricandthelocationofthefree-surfacenodesneednotbeconsideredas variablesduringtheowsolutionphase,thenreducingthecomputationalcost. Ifsurfacetensionisinvolved,thefree-surfaceboundaryconditionsdependonthepattern ofthefree-surface.Becausetheaccuracyofthefree-surfaceboundaryconditionsdepends directlyontheaccuracyofthefree-surfacerepresentation,theaccuracyofthesolutionis sensitivetotheresolutionofthemeshattheboundary.Foratwo-dimensionaldomain,if thenumberofboundaryunknownsincreaseslinearly,thenthenumberofinteriorunknowns mustincreasequadratically.Becausequadraticelementsallowustoobtainmoreaccuracy thanlinearelements,anerdiscretizationontheboundarythanintheinteriorofthedomain maybepossible.Whenthefreesurfacehasnon-zerosurfacetension,thegradientofthe stressatthefreesurfaceis ^ t = )]TJ/F20 11.9552 Tf 9.299 0 Td [( n .8 89

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where isthesurfacetension; isthecurvatureand n istheoutwardunitnormalvector. Next,weclarifyhowtoimplement.8intothestinessmatrix.Afterapplyingthevariationalformulatothemomentumequation,wehavethatthedivergenceofthestresstensor movestothetestfunction, Z r w i t d = )]TJ/F25 9.9626 Tf 11.291 14.059 Td [(Z @ w i ^ t ds = Z @ w i n ds .9 where w i isthetestfunctionand.8isappliedatthelastequality.Theequation.9is thestressboundaryconditionswhichcanbebroughttotheright-handsideofthemomentum equation.Toexplicitlycalculate.9,wemayemploy n = d ds ; .10 where isthetangentatthecurve s .Thenatafreesurfaceedge @ e ,wehave Z @ e w i n ds = Z @ e w i d ds ds = w i j @ e )]TJ/F25 9.9626 Tf 11.955 14.058 Td [(Z @ e dw i ds ds; .11 wherethetangent isgivenby = 3 X i =1 s i dw i ds ; .12 where i rangesoverthethreebasisfunctionsactiveonanyparticularedgeandthevariables s i ;i =1 ; 2 ; 3,arethepositionsoftherelevantboundarynodes.Theaboveprocedureis summarizedasfollows: 1.Chooseaninitialfreeboundarylocationandsetupthenodes. 2.Solveforthevelocityandpressureelds. 3.Ifevolutioniscompleted,thenstop. 4.Updatethefreeboundaryusingthevelocityeld. 5.Ifboundarynodesneedtobeadjusted, Mergeboundaryedges. 90

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Splitboundaryedges. Adjustboundaryedges. Remeshthedomain. Interpolatetheprevioussolutionsontonewmeshifnecessary. Goto2. 6.Solvetheelasticproblemandrelocateinteriornodes. 7.Goto2. Theproceduresofmerging,splittingandadjustingcanbefoundin[82]. Inthischapter,wesummarizeapossibleapproachtosolvethefreeboundaryproblem. Anattractiveofthisstrategyisbecauseourfreeboundaryprobleminvolveswiththeabsence ofextremecurvature.Moreover,oneofadvantagesofthisapproachistime-savingbecause thenewadjustmentproblemdependsonlyontheselectionofaninitialboundarymesh. 91

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7.CONCLUSION Inthisthesiswemodeledthepericiliarylayer,PCL,oflungs.ThePCLcontainscilia, hair-likeorganellesthatresideonthesurfaceoftheepitheliumandwhosepurposeisto sweepdebrisawayfromlungs,andthelowviscousliquidinwhichtheciliareside,PCLuid. ThePCLuidismodeledasanincompressibleuidundergoingStokesow,andtheciliais modeledasaperiodicarrayofcylindersthatrotateabouttheirbasewithheightvaryingas afunctionoftheangel.WedevelopedamathematicalmodelofthemovementofthePCL uidduetothemovementofthecilia. TheportionofthePCLwhichcontainstheciliawastreatedasaporousmedium.The permeabilityoftheporousmediumwasdeterminedbyusinghomogenizationtodeterminea systemofequationwhichwerethennumericallysolvedusinganiteelementmethod.This approachallowedustondanexpressionforthepermeabilityintermsofthegeometry ofthearrayofcylinderthatisnumericallyinexpensive.Theresultswerevalidatedby comparisonwithresultsavailableintheliteraturewithgoodagreement.Inaddition,the calculatedpermeabilitywasshowntobeasymmetricandpositivedenitematrix[90],asit theoreticallyshouldbe,andfordierentsizesofperiodiccells,thecalculatedpermeability matriceswerenumericallycorrespondingaswasexpected. Asisconsistentwiththephysicalmeaningofpermeabilityameasureoftheeasewith whichuidcanmovethroughtheporousmedium,thepermeabilitycalculationsindicated thediagonalcomponentsofthepermeabilitydecreaseastheanglebetweenthecylinders andthebasedecrease,seeFigure3.3,andforxedangles,theydecreaseforincreasingradii, seeFigure3.2.Further,wecanseefromtheresultsthatthepermeabilitytendstozeroif theporositytendstozero.Theseresultscanbeextendedtogeneralcasesbyvaryingthe anglesandradiiasinFigures3.8,3.9,3.10.Moreover,weprovidedexplicitfunctionsofthe componentsofthepermeabilitytensorasafunctionoftheratio r=d andtheangle ,see Tables3.4and3.5. 92

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BecausetheKozeny-Carmanequationisamongthemorepopularequationsusedto determinethescalarpermeabilityasafunctionofporosity,wecomparedtheKozeny-Carman permeabilityresultswithonethirdthetraceofournumericalpermeabilitytensor.Because periodicarrayofangledcylindersproduceanisotropicpermeabilitytensors,itwouldnot havebeenexpectedthatusingtheKozeny-Carmanequationwouldgivecomparableresults. Howeverwefoundtheoppositetobetrue-theywereinexcellentagreement.However, becauseoftheanisotropicnatureoftheproblem,theKozeny-Carmanequationscannotbe usedforourproblem. TomodelthePCL,weusedthepermeabilitytensorinasystemofequationsthatmodel thePCLatthemacroscale.ThismodeliscomposedoftheBrinkmanequation,whichcan bederivedusingHybridMixtureTheoryHMT,inthepartofthePCLcontainingthe cilia, 2 ,andtheStokesequationinthepartofthePCLabovethecilia, 1 .TheStokesBrinkmanequationswithstress-continuityconditionsattheporous-mediumandfree-uid interfacewereusedasthemathematicalmodeltodeterminethevelocityandpressureelds oftheuidinthePCL.Thisallowedustomatchthestresseldbetweenthetwodomains. WealsoshowedtheexistenceanduniquenessoftheStokes-Brinkmanequationswhenthe permeabilitycoecientisasecond-ordertensor.Numerically,thetwo-dimensionalmixed niteelementmethodwasappliedtotheregions 1 and 2 .Thecodewasvalidatedby comparingtheresultwithanexactsolutionwherethetopofthefreeuidisdraggedwitha constantvelocity.Numericalresultsareprovidedforthecaseofstresscontinuityatthefreeuid/porousmediuminterfaceandthevelocityofapendulumisappliedforthevelocityof thecilia.Finally,wealsoproposedamethodforthesolvingtheproblemasafreeboundary problemusingthesemi-explicitmethodof[82]. Ourproblemismotivatedbyabiologicalinterestandthemathematicalmodelisdriven bybiologicalinsight.Themodeldevelopedcouldbeusedtodevelopmedicationforlung diseases.Somelungdiseasescauseconditionsinwhichsomefunctionsofthelungsare adverselyaectedbecausethebodyproducesabnormallythickandstickymucuswhichcan 93

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blocktheairwaysandwhereciliahaveahardtimeremovingthethickmucus.Examples ofsuchdiseasesincludeCysticFibrosisCF,ChronicBronchitisCBandPrimaryCiliary DyskinesiaPCD.Forthegenetic-lung-disorderPCDdisease,theciliaareabnormalordo notmove.BetterunderstandingofhowtheuidphaseinthePCLarepropelledbycilia andhowtheuidpropertiesaectthemovementcanlendtomoreeectivetreatments.For example,physiciansareusingsaltaddedtothePCLuidtomakeiteasiertoexpelthick mucusoutoflungs.ThetreatmentindicatesthepropertyofPCLuidisimportantandits interactionwithcilianeedstobebetterunderstood.Havinganappropriatemodelcanlead tomoreeectivemedication. FutureworkincludesextendingthePCLmodeltomodelthree-dimensionsandincorporatingbetweenthePCLandmucus. 94

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APPENDIXA.NOMENCLATURE variablesdenitionsunitsequations a diameterofcylinders,2 r [ a ]= L Figure1.3 a a arithmeticaverageberdiameterofcylinder[ a a ]= L .7 a s surfaceaverageberdiameterofcylinder[ a s ]= L .7 a i i=1,....,15,coecientsofpolynomials[ a i ]=1Table3.4 a ; bilinearfunctional.101 A aellipticshapeFigure3.12 h A ; i alinearoperatorDenition.8 A ablockmatrixinthestinessmatrix.39 ~ A ablockmatrixinthestinessmatrix.32 h B ; i alinearoperatorDenition.8 h B 0 ; i thedualoperatorof B Denition.8 b ; bilinearfunctional.102 c 1 linearfunctional.103 c 2 linearfunctional.104 C aconstantparameter[ C ]= L T .5 C v unitconversionfactor.7 C d dragcoecient[ C d ]=1.3 C c apositiveconstant[ C c ]=1.131 C k aconstant[ C k ]=1.45 d distancebetweenciliainastraightdirection[ d ]= L Figure1.3 d 1 distancebetweenciliaindiagonaldirection[ d 1 ]= L Figure1.3 95

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variablesdenitionsunitsequations d l rateofdeformation[ d l ]= 1 T .2 D ablockmatrixinthestinessmatrix.39 D j ablockmatrixinthestinessmatrix.35 D w aweakderivativeDenition.2 D =Dt materialtimederivativeof phase.27 e i theunitvectorinthedirectionof y i axis[ e i ]=1.31 ^ e interactivequantityattheinterface [^ e ]= 1 T .19 E jj ablockmatrixinthestinessmatrix.36 E = E 11 + E 22 + E 33 .37 f asourceterm.100b f asourceterm.100a f s = f s 1 ;f s 2 ;f s 3 ,sourceterminStokeequation[ f s ]= M L 2 T 2 .2 F d dragforceoneachobject[ F d ]= M LT 2 .1 F d totaldragforce[ F d ]= M T 2 .5 F ij ablockmatrixinthestinessmatrix.36 F ablockmatrixinthestinessmatrix.39 ~ F 1 asourceterminelementmatrix.33 ~ F 2 asourceterminelementmatrix.34 ~ F 3 asourceterminelementmatrix.39 g gravity[ g ]= L T 2 .38 g = ; 0 ; )]TJ/F20 11.9552 Tf 9.298 0 Td [(g gravityvector[ g ]= L T 2 .1 g mass-averagedgravityof phase[ g ]= L T 2 .28 h piezometrichead[ h ]= L .38 h heightofellipsoidalcylinder[ h ]= L Figure3.12 H aHilbertspace.19 96

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variablesdenitionsunitsequations H c H c = H c 1 + H c 2 [ H c ]= L Figure1.2 H c 1 heightofdomain 1 [ H c 1 ]= L Figure1.2 H c 2 heightofdomain 2 [ H c 2 ]= L Figure1.2 H h nite-dimensionalsubspaceof L 2 0 .27 H 1 0 aSobolevsubspaceof H 1 .90 H 1 s aSobolevsubspaceof H 1 .91 H )]TJ/F18 7.9701 Tf 6.586 0 Td [(1 dualspaceof H 1 0 .92 H 1 = 2 @ aSobolev-Slobodeckijspacebelow.95 k ascalarpermeability[ k ]= L 2 .10 k = f u i ,permeabilitytensor[ k ]= L 2 .36 k 1ascalarpermeability[ k 1]= L 2 .61 k )]TJ/F18 7.9701 Tf 6.587 0 Td [(1 inverseofpermeabilitytensor[ k )]TJ/F18 7.9701 Tf 6.586 0 Td [(1 ]= 1 L 2 .2 k dimensionlessscalarpermeability[ k ]=1below.11 k dimensionlesspermeability[ k ]=1.43 k RM nondimensionalscalarpermeability[ k RM ]=1Table3.2 Rocha&Cruz k s nondimensionalpermeabilityofsphere[ k s ]=1Table3.2 k 1 aconstantdependingongeometry[ k 1 ]=1.43 K aconstantTheorem.5 K 0 aconstantTheorem.5 K hydroulicconductivity[ K ]= L T .38 ~ K ij ablockmatrixinthestinessmatrix.32 l thicknessofacell[ l ]= L .7 l min lengthoftheshortestedge[ l min = L ].7 L lengthofaellipsoidalcylinder[ L ]= L Figure3.12 97

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variablesdenitionsunitsequations L c lengthofdomain[ L c ]= L Figure1.2 L 2 0 aSobolevsubspaceof L 2 .89 L 1 loc thesetoflocallyintegrablefuncitonsbelow.106 n outwardunitnormalvector[ n ]=1.1 n outwardunitnormalvectorto V [ n ]=1.11 p pressure[ p ]= M L T 2 .2 p i asymptoticexpansionofpressure p [ p i ]= M L T 2 .3 P vectorof p l ;l =1 ; 2 ;::::;L .29 v i vectorof v m i ;m =1 ; 2 ;::::;M .28 q Darcy'svelocity[ q ]= L T .36 q i solutionofsystemofequations.30-2.32[ q i ]= L .30-2.32 q dimensionlessvariable q i [ q ]=1.40-.42 Q vectorof q j ;j =1 ; 2 ;:::;n .28 ~ Q i ablockmatrixinthestinessmatrix.32 r radiusofacylinder[ r ]= L Figure1.3 r 1 aradiusofaellipseFigure3.12 r p radiusofasphere[ r p ]= L .3 Re = aU= ,Reynoldsnumber[ Re ]=1below.5 s afunctiononboundary @ below.95 S areaofparticularsurfaceperunitvolume[ S ]= 1 L .43 S A surfaceareaofsolidstayinginuid[ S A ]= L 2 .46 t stresstensor[ t ]= M L T 2 .1 t stresstensorof phase[ t ]= M L T 2 .29 ^ t gradientofthestress[ ^ t ]= M L 2 T 2 .8 ^ T accelerationtermattheinterface [ ^ T ]= L T 2 .30 98

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variablesdenitionsunitsequations u exactsolutionofStokes-Brinkman[ u ]= L T .47 u 0 constantvelocityatthetopofthePCL[ u 0 ]= L T .48 u i solutionofsystemofequations.30-.32[ u i ]= L 2 .30-.32 u dimensionlessvariable u i [ u ]=1.40-.42 u p velocityoftheporousow[ u p ]= L T .6 u v velocityoftheviscousow[ u v ]= L T .6 u v slip slipvelocity[ u v slip ]= L T .6 f u i = k ,permeabilitytensor[ f u i ]= L 2 .33 u j vectorof u m j ;m =1 ; 2 ;::::;M .27 U upstreamvelocityofuid[ U ]= L T below.5 U ablockmatrixinthestinessmatrix.39 v = v 1 ;v 2 ;v 3 ,uidvelocityofStokesequation[ v ]= L T .1 alsousedinthemomentumequation[ v ]= L T .21 alsoused v = l v l inBrinkmanequation[ v ]= L T .98 v i i =1 ; 2 ; 3 ::: asymptoticexpansionofStokesvelocity v [ v i ]= L T .3 v l mass-velocityofliquidphase[ v l ]= L T .2 v s mass-velocityofsolidphase[ v s ]= L T .2 v mass-averagedvelocityofphase [ v ]= L T .18 f v 0 = q ,Darcy'svelocity[ f v 0 ]= L T .33 f v 1 averageof v 1 overdomain[ f v 1 ]= L T .33 v a averagevelocityofuidinacell[ v a ]= L T .6 v s supercialvelocityofuidinacell[ v s ]= L T .4 v s avelocitywhich v s j @ = s .114 v 0 r v 0 = f )-222(r v s .160 99

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variablesdenitionsunitsequations ^ v ^ v = v s + v 0 .161 v D supercial-velocityvectororDarcy'svelocity[ v s ]= L T .1 v max largestvelocity[ v max ]= L T .7 V aSobolevsubspaceof H 1 .93 V ? orthogonalofVin H 1 0 .94 V 0 polarsetofV.95 V h nite-dimensionalsubspaceof H 1 .26 V i vectorof v m i ;m =1 ; 2 ;::::;M .28 V s volumeofcylindersinacell[ V s ]= L 3 .54 V t totalvolumeofacell[ V t ]= L 3 .46 w velocityofinterface [ w ]= L T .11 x = x 1 ;x 2 ;x 3 ,macroscopicvariable[ x ]= L Figure2.2 y = y 1 ;y 2 ;y 3 ,microscopicvariable[ y ]= L Figure2.2 y dimensionlessvariable y [ y ]=1.40-.42 z z-axisinCartesiancoordinate[ z ]= L .38 anonzeroparameter[ ]=1Figure2.3 m materialdependentparameter[ m ]=1.6 anonzeroparameter[ ]=1Figure2.3 ij identitymatrix[ ij ]=1.2 t timestep[ t ]= T .7 max maximumtimestep[ max ]= T .7 A portionoftheinterface within V [ A ]= L 2 .12 k relativeerrorof k s withrespectto k RM [ k ]=1Table3.2 V representativeelementaryvolume[ V ]= L 3 .22 V portionof -phase[ V ]= L 3 .9 100

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variablesdenitionsunitsequations V portionof -phase[ V ]= L 3 .9 0 < 1,aparameter[ ]=1Figure2.1 l porosity[ l ]=1below.6 s solidvolumefraction[ s ]=1.7 thesurfacetension[ ]= M T 2 .8 )-4120()]TJ/F21 7.9701 Tf 63.883 -1.793 Td [(F [ )]TJ/F21 7.9701 Tf 7.314 -1.793 Td [(S []= L 2 Figure1.3 )]TJ/F21 7.9701 Tf 7.314 -1.793 Td [(F uidboundary[)]TJ/F21 7.9701 Tf 240.844 -1.793 Td [(F ]= L 2 Figure1.3 )]TJ/F21 7.9701 Tf 7.314 -1.793 Td [(S solidboundary[)]TJ/F21 7.9701 Tf 240.844 -1.793 Td [(S ]= L 2 Figure1.3 thecurvature[ ]= 1 L .8 dynamicviscosity[ ]= M L T .2 periciliarylayerPCL[]= L 2 Figure1.2 1 domainbetweenthetipofciliaandmucus[ 1 ]= L 2 Figure1.2 2 domaincontainingcilia[ 2 ]= L 2 Figure1.2 e anelementdomainbefore.27 F uiddomain[ F ]= L 3 Figure1.3 S soliddomain[ S ]= L 3 Figure1.3 i quadraticbasisfunctions.27 vectorofthequadraticbasisfunction3.27 i linearbasisfunction.28 vectorofthelinearbasisfunctions.28 densityofuid[ ]= M L 3 .38 mass-averageddensityof phase[ ]= M L 3 .17 thetangentatthecurve[ ]=1.10 anglebetweenciliaandhorizontalplane[ ]=1Figure1.2 101

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FigureB.1: Nodalnumberingforthe27-variable-number-of-nodeselementwheretheorigin isatthecenterofthebrick. APPENDIXB.BASISFUNCTIONS Sincesomebasisfunctionsarenotprovidedexplicitlyinbooks,inthisAppendix,we providebasisfunctionsoftwodierentelements:Brickandtetrahedralelementswhichare usedinthiswork.Thebasisfunctionsofthethree-dimensionalBrickelementortriquadratic 27-node,seeFigureB.1wherethetriquadraticcoordinatesare ; and ,canbederived fromquadraticone-dimensionalelement,seeFigureB.2.Eachshapefunctionhasaunit valueononenodeandzeroontherest.Thebasisfunctionsofthequadraticelementare [52]: N 1 = )]TJ/F20 11.9552 Tf 11.955 0 Td [( 2 )]TJ/F20 11.9552 Tf 11.955 0 Td [( 3 1 )]TJ/F20 11.9552 Tf 11.955 0 Td [( 2 1 )]TJ/F20 11.9552 Tf 11.956 0 Td [( 3 = 1 2 )]TJ/F16 11.9552 Tf 11.955 0 Td [(1 ; B.1 N 2 = )]TJ/F20 11.9552 Tf 11.955 0 Td [( 1 )]TJ/F20 11.9552 Tf 11.955 0 Td [( 3 2 )]TJ/F20 11.9552 Tf 11.955 0 Td [( 1 2 )]TJ/F20 11.9552 Tf 11.956 0 Td [( 3 =1 )]TJ/F20 11.9552 Tf 11.955 0 Td [( 2 ; B.2 N 3 = )]TJ/F20 11.9552 Tf 11.955 0 Td [( 1 )]TJ/F20 11.9552 Tf 11.955 0 Td [( 2 3 )]TJ/F20 11.9552 Tf 11.955 0 Td [( 1 3 )]TJ/F20 11.9552 Tf 11.956 0 Td [( 2 = 1 2 +1 : B.3 102

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FigureB.2: Nodalnumberingforthe3-variable-number-of-nodeselement Sincethecoordinateofthebasisfunction 1 inFigureB.1is )]TJ/F16 11.9552 Tf 9.298 0 Td [(1 ; )]TJ/F16 11.9552 Tf 9.299 0 Td [(1 ; )]TJ/F16 11.9552 Tf 9.298 0 Td [(1,thebasisfunctions ofthethree-dimensionalBrickelementatthatnodeis 1 ;; = N 1 N 1 N 1 = 1 8 )]TJ/F16 11.9552 Tf 11.955 0 Td [(1 )]TJ/F16 11.9552 Tf 11.955 0 Td [(1 )]TJ/F16 11.9552 Tf 11.955 0 Td [(1 : B.4 Similarly,wehave 2 ;; = N 3 N 1 N 1 = 1 8 +1 )]TJ/F16 11.9552 Tf 11.955 0 Td [(1 )]TJ/F16 11.9552 Tf 11.955 0 Td [(1 ; B.5 3 ;; = N 3 N 3 N 1 = 1 8 +1 +1 )]TJ/F16 11.9552 Tf 11.955 0 Td [(1 ; B.6 4 ;; = N 1 N 3 N 1 = 1 8 )]TJ/F16 11.9552 Tf 11.956 0 Td [(1 +1 )]TJ/F16 11.9552 Tf 11.955 0 Td [(1 ; B.7 5 ;; = N 1 N 1 N 3 = 1 8 )]TJ/F16 11.9552 Tf 11.956 0 Td [(1 )]TJ/F16 11.9552 Tf 11.955 0 Td [(1 +1 ; B.8 6 ;; = N 3 N 1 N 3 = 1 8 +1 )]TJ/F16 11.9552 Tf 11.955 0 Td [(1 +1 ; B.9 7 ;; = N 3 N 3 N 3 = 1 8 +1 +1 +1 ; B.10 8 ;; = N 1 N 3 N 3 = 1 8 )]TJ/F16 11.9552 Tf 11.956 0 Td [(1 +1 +1 ; B.11 9 ;; = N 3 N 2 N 1 = 1 4 +1 )]TJ/F20 11.9552 Tf 11.955 0 Td [( 2 )]TJ/F16 11.9552 Tf 11.955 0 Td [(1 ; B.12 10 ;; = N 2 N 3 N 1 = 1 4 )]TJ/F20 11.9552 Tf 11.955 0 Td [( 2 +1 )]TJ/F16 11.9552 Tf 11.955 0 Td [(1 ; B.13 11 ;; = N 1 N 2 N 1 = 1 4 )]TJ/F16 11.9552 Tf 11.955 0 Td [(1 )]TJ/F20 11.9552 Tf 11.956 0 Td [( 2 )]TJ/F16 11.9552 Tf 11.955 0 Td [(1 ; B.14 103

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12 ;; = N 2 N 1 N 1 = 1 4 )]TJ/F20 11.9552 Tf 11.955 0 Td [( 2 )]TJ/F16 11.9552 Tf 11.955 0 Td [(1 )]TJ/F16 11.9552 Tf 11.955 0 Td [(1 ; B.15 13 ;; = N 2 N 2 N 1 = 1 2 )]TJ/F20 11.9552 Tf 11.955 0 Td [( 2 )]TJ/F20 11.9552 Tf 11.955 0 Td [( 2 )]TJ/F16 11.9552 Tf 11.956 0 Td [(1 ; B.16 14 ;; = N 3 N 2 N 3 = 1 4 +1 )]TJ/F20 11.9552 Tf 11.955 0 Td [( 2 +1 ; B.17 15 ;; = N 2 N 3 N 3 = 1 4 )]TJ/F20 11.9552 Tf 11.955 0 Td [( 2 +1 +1 ; B.18 16 ;; = N 1 N 2 N 3 = 1 4 )]TJ/F16 11.9552 Tf 11.955 0 Td [(1 )]TJ/F20 11.9552 Tf 11.956 0 Td [( 2 +1 ; B.19 17 ;; = N 2 N 1 N 3 = 1 4 )]TJ/F20 11.9552 Tf 11.955 0 Td [( 2 )]TJ/F16 11.9552 Tf 11.955 0 Td [(1 +1 ; B.20 18 ;; = N 2 N 2 N 3 = 1 2 )]TJ/F20 11.9552 Tf 11.955 0 Td [( 2 )]TJ/F20 11.9552 Tf 11.955 0 Td [( 2 +1 ; B.21 19 ;; = N 1 N 1 N 2 = 1 4 )]TJ/F16 11.9552 Tf 11.955 0 Td [(1 )]TJ/F16 11.9552 Tf 11.955 0 Td [(1 )]TJ/F20 11.9552 Tf 11.956 0 Td [( 2 ; B.22 20 ;; = N 3 N 1 N 2 = 1 4 +1 )]TJ/F16 11.9552 Tf 11.956 0 Td [(1 )]TJ/F20 11.9552 Tf 11.955 0 Td [( 2 ; B.23 21 ;; = N 3 N 3 N 2 = 1 4 +1 +1 )]TJ/F20 11.9552 Tf 11.955 0 Td [( 2 ; B.24 22 ;; = N 1 N 3 N 2 = 1 4 )]TJ/F16 11.9552 Tf 11.955 0 Td [(1 +1 )]TJ/F20 11.9552 Tf 11.955 0 Td [( 2 ; B.25 23 ;; = N 3 N 2 N 2 = 1 2 +1 )]TJ/F20 11.9552 Tf 11.955 0 Td [( 2 )]TJ/F20 11.9552 Tf 11.955 0 Td [( 2 ; B.26 24 ;; = N 2 N 3 N 2 = 1 2 )]TJ/F20 11.9552 Tf 11.955 0 Td [( 2 +1 )]TJ/F20 11.9552 Tf 11.955 0 Td [( 2 ; B.27 25 ;; = N 1 N 2 N 2 = 1 2 )]TJ/F16 11.9552 Tf 11.955 0 Td [(1 )]TJ/F20 11.9552 Tf 11.955 0 Td [( 2 )]TJ/F20 11.9552 Tf 11.955 0 Td [( 2 ; B.28 26 ;; = N 2 N 1 N 2 = 1 2 )]TJ/F20 11.9552 Tf 11.955 0 Td [( 2 )]TJ/F16 11.9552 Tf 11.955 0 Td [(1 )]TJ/F20 11.9552 Tf 11.955 0 Td [( 2 ; B.29 27 ;; = N 2 N 2 N 2 =1 )]TJ/F20 11.9552 Tf 11.955 0 Td [( 2 )]TJ/F20 11.9552 Tf 11.955 0 Td [( 2 )]TJ/F20 11.9552 Tf 11.955 0 Td [( 2 : B.30 Nextwealsoprovidethebasisfunctionsof4-nodetetrahedronand10-nodequadratic tetrahedron.Forthelineartetrahedronelement,thebasisfunctionsare N 1 r;s;t = r; B.31 N 2 r;s;t = s; B.32 N 3 r;s;t = t; B.33 N 4 r;s;t =1 )]TJ/F20 11.9552 Tf 11.955 0 Td [(r )]TJ/F20 11.9552 Tf 11.955 0 Td [(s )]TJ/F20 11.9552 Tf 11.956 0 Td [(t: B.34 104

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FigureB.3: Nodalnumberingforthe4-variable-number-of-nodestetrahedronelement. FigureB.4: Nodalnumberingforthe10-variable-number-of-nodesquadratictetrahedron element. Forthequadratictetrahedronelement,thebasisfunctionsare N 1 r;s;t =2 r 2 )]TJ/F20 11.9552 Tf 11.955 0 Td [(r; B.35 N 2 r;s;t =2 s 2 )]TJ/F20 11.9552 Tf 11.956 0 Td [(s; B.36 N 3 r;s;t =2 t 2 )]TJ/F20 11.9552 Tf 11.955 0 Td [(t; B.37 N 4 r;s;t =2 )]TJ/F20 11.9552 Tf 11.955 0 Td [(r )]TJ/F20 11.9552 Tf 11.955 0 Td [(s )]TJ/F20 11.9552 Tf 11.955 0 Td [(t 2 )]TJ/F16 11.9552 Tf 11.955 0 Td [( )]TJ/F20 11.9552 Tf 11.956 0 Td [(r )]TJ/F20 11.9552 Tf 11.955 0 Td [(s )]TJ/F20 11.9552 Tf 11.955 0 Td [(t ; B.38 N 5 r;s;t =4 rs; B.39 N 6 r;s;t =4 st; B.40 N 7 r;s;t =4 t )]TJ/F20 11.9552 Tf 11.955 0 Td [(rt )]TJ/F20 11.9552 Tf 11.955 0 Td [(st )]TJ/F20 11.9552 Tf 11.955 0 Td [(t 2 ; B.41 N 8 r;s;t =4 r )]TJ/F20 11.9552 Tf 11.955 0 Td [(r 2 )]TJ/F20 11.9552 Tf 11.955 0 Td [(rs )]TJ/F20 11.9552 Tf 11.955 0 Td [(rt ; B.42 N 9 r;s;t =4 rt; B.43 N 10 r;s;t =4 s )]TJ/F20 11.9552 Tf 11.955 0 Td [(rs )]TJ/F20 11.9552 Tf 11.955 0 Td [(s 2 )]TJ/F20 11.9552 Tf 11.955 0 Td [(st : B.44 105

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