MS Thesis - DiVA portalkth.diva-portal.org/smash/get/diva2:435356/FULLTEXT01.pdf · 2011. 8....

Royal Institute of Technology

MS Thesis

Investigation of the effect of bending ratio on the rheology of a densesuspension of flexible fibers using lattice Boltzmann method

Author: Supervisor:Mubashar Khan Dr. Minh do QuangCo-supervisors:Prof. Gustav Amberg1 Prof. Cyrus K. Aidun

School of Engineering Sciences - Department of Mechanics

Stockholm, SE 100 44, Sweden

Dedication

I dedicate this thesis to my parents and family who supported me throughout my academiccareer.

i

Acknowledgement

I’m thankful to all the people who have supported me during MS Thesis. I express my profoundgratitude to my supervisor and co-advisors for their guidance. I must acknowledge that I learneda lot from Prof. Cyrus about microstructure of the fibers and the rheology of the suspensions.It has been a great experience to work with him. Prof. Gustav Amberg helped me understandthe underlying physics of non-Newtonoian fluids and phenomena occuring at micro-structuraland micro-fluidic scales. Dr. Minh guided me about practical issues related to numericaland simulation aspects. Dr. Jingshu Wu deserves special thanks for his remote assistance incompilation and understanding of his simulation code.

ii

Preface

This document is a thesis as a partial fulfillment of the requirements for the Degree of Mastersin Engineering Mechanics at Department Mechanics - Royal Institute of Technology (KTH),Stockholm-Sweden. The research conducted with the aim to categorically establish the mostinfluential parameter in the rheology of the flexible fiber suspensions. This goal has beenachieved and fiber stiffness has been found to have principal effect. Thus one question relatedto rheology of suspensions has been answered. The results have been presented already atDFD-10 of American Physical Society on Nov. 23, 2010. These results will also be publishedformally in Physics of fluid letters.

iii

Contents

Acknowledgement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ii

Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iii

Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . v

List of tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vi

List of figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viii

List of symbols and abbreviations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ix

1 Introduction 1

1.1 Introduction and overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.2 History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.3 Objective . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

1.4 General constitutive model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

1.4.1 Constutitive model for dense suspension of spheres . . . . . . . . . . . . . . 3

1.4.2 Constutitive model for flexible fiber suspension . . . . . . . . . . . . . . . . 4

1.5 Characteristics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

1.5.1 Bending ratio . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

1.5.2 Relative viscosity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

1.5.3 Normal stress difference . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

1.6 Fiber orientation and its distribution . . . . . . . . . . . . . . . . . . . . . . . . . . 8

2 LBM 9

2.1 Boltzmann-Maxwell to lattice Boltzmann equation . . . . . . . . . . . . . . . . . . 9

2.1.1 Boundary conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

2.1.2 Flexible fiber model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

2.2 Unit systems: physical and lattice . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

2.3 Validation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

2.4 Computational performance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

2.4.1 Hardware resources . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

2.4.2 Efficiency and speed-up . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

3 Results and discussion 17

3.1 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

3.2 Input and output parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

3.2.1 Input parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

3.2.2 Output parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

3.3 Fiber orientation distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

3.4 Relative viscosity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

3.5 First normal stress difference . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

3.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

3.7 Future recommendations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

iv

Appendices i

A Relative viscosity i

A.1 Graphs for relative viscosity ηr . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iA.1.1 Flexible fibers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iA.1.2 Rigid fibers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iii

B First normal stress difference v

B.1 Graphs for first NSD . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vB.1.1 Flexible fibers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vB.1.2 Rigid fibers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viii

C Second normal stress difference xi

C.1 Graphs for second NSD . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiC.1.1 Flexible fibers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiC.1.2 Rigid fibers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiii

v

List of Tables

2.1 Simulation start/end time and number of time-steps/min with increasing numberof processors on Key . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

3.1 Summary of the selected cases performed to determine the effect of BR againstshear rate γ̇ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

vi

List of Figures

1.1 Fiber orientation in a pure shear flow . . . . . . . . . . . . . . . . . . . . . . . . . . 8

2.1 D3Q19 velocity model (Courtesy of De (2009)) . . . . . . . . . . . . . . . . . . . . . 102.2 Boltzmann and Maxwell . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112.3 D2Q9 velocity model and the mid-node standard bounce back boundary condition 122.4 Fiber model in the code . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122.5 Validation of the simulation on Platon . . . . . . . . . . . . . . . . . . . . . . . . . 132.6 Efficiency of the code on the key and machines . . . . . . . . . . . . . . . . . . . . 152.7 Speed up of the code on the three machines . . . . . . . . . . . . . . . . . . . . . . 16

3.1 Relative viscosity ratio vs γ̇ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193.2 Relative viscosities of flexible and rigid fibers vs γ̇ . . . . . . . . . . . . . . . . . . . 203.3 Relative viscosity ratio vs γ̇ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203.4 First normal stress difference for flexible and rigid fibers vs γ̇ . . . . . . . . . . . . 213.5 Screen shots from animation of a single fiber in a shear flow with EY = 3000 lu . . 223.6 Screen shots from animation of a single fiber in a shear flow with EY = 1500 lu . . 223.7 Screen shots from animation of flexible fibers suspension in a shear flow at γ̇ = 50 s−1 233.8 Screen shots from animation of flexible fibers suspension in a shear flow at γ̇ =

500 s−1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

A.1 Variation of the relative viscosity ηr vs γ̇T at shear rate γ̇ = 50 s−1 . . . . . . . iA.2 Variation of the relative viscosity ηr vs γ̇T at shear rate γ̇ = 100 s−1 . . . . . . . iiA.3 Variation of the relative viscosity ηr vs γ̇T at shear rate γ̇ = 100 s−1 . . . . . . . iiiA.4 Variation of the relative viscosity ηr vs γ̇T at shear rate γ̇ = 200 s−1 . . . . . . . iiiA.5 Variation of the relative viscosity ηr vs γ̇T at shear rate γ̇ = 500 s−1 . . . . . . . iv

B.1 Variation of the first NSD N1 vs γ̇T at shear rate γ̇ = 50 s−1 . . . . . . . . . . . vB.2 Variation of the first NSD N1 vs γ̇T at shear rate γ̇ = 200 s−1 . . . . . . . . . . . viB.3 Variation of the first NSD N1 vs γ̇T at shear rate γ̇ = 250 s−1 . . . . . . . . . . . viB.4 Variation of the first NSD N1 vs γ̇T at shear rate γ̇ = 500 s−1 . . . . . . . . . . . viiB.5 Variation of the first NSD N1 vs γ̇T at shear rate γ̇ = 50 s−1 . . . . . . . . . . . viiiB.6 Variation of the first NSD N1 vs γ̇T at shear rate γ̇ = 100 s−1 . . . . . . . . . . . viiiB.7 Variation of the first NSD N1 vs γ̇T at shear rate γ̇ = 200 s−1 . . . . . . . . . . . ixB.8 Variation of the first NSD N1 vs γ̇T at shear rate γ̇ = 250 s−1 . . . . . . . . . . . ixB.9 Variation of the first NSD N1 vs γ̇T at shear rate γ̇ = 500 s−1 . . . . . . . . . . . x

C.1 Variation of second NSD N2 vs γ̇T at shear rate γ̇ = 50 s−1 . . . . . . . . . . . . xiC.2 Variation of second NSD N2 vs γ̇T at shear rate γ̇ = 100 s−1 . . . . . . . . . . . xiiC.3 Variation of second NSD N2 vs γ̇T at shear rate γ̇ = 50 s−1 . . . . . . . . . . . . xiiiC.4 Variation of second NSD N2 vs γ̇T at shear rate γ̇ = 100 s−1 . . . . . . . . . . . xiiiC.5 Variation of second NSD N2 vs γ̇T at shear rate γ̇ = 200 s−1 . . . . . . . . . . . xiv

vii

C.6 Variation of second NSD N2 vs γ̇T at shear rate γ̇ = 250 s−1 . . . . . . . . . . . xivC.7 Variation of second NSD N2 vs γ̇T at shear rate γ̇ = 500 s−1 . . . . . . . . . . . xv

viii

List of symbols and abbreviations

N1 First normal stress difference SBB Standard bounce back BC’sNB1 First normal stress difference from Batchelor’s relation 〈〉 Average quantity within bracketsN2 Second normal stress difference φ0 Phase angleµ Coefficient of absolute or dynamic viscosity ∆tLBM Time step in LBMµeff Effective shear viscosity ∆x

LBM Unit gird size in LBMµfiber Dynamic viscosity coefficient in Batchelor’s relation kB Boltzmann constantη Shear viscosity D Fiber diameterηr Relative shear viscosity C Jeffery’s orbit constantηB Relative shear viscosity from Batchelor’s theory L Fiber lengthρ, ρ0 Fluid density θ ] between fiber axis and z-axisρf Fiber density φ ] between fiber axis and y-axisEY Young’s modulus n Number of fibers per unit volumeBR Bending ratio fi LBM distribution functionFEM Finite element method feqi LBM equilibrium distribution functionLBM Lattice Boltzmann method p(φ) Orientation distribution function for φFVM Finite volume method IBB Interpolated bounce backrP Fiber aspect ratio IBM Immersed boundary methodre Effective aspect ratio φvol Volume fractionσ Stress tensor cvf Coefficient of volume fractionτ Shear stress ρP Particle densityMi Mass of the i

th fiber ρr Particle to fluid density ratioΩi Angular speed of the i

th fiber a′ Semi-major axis of ellipsoidu Fluid velocity b′ Semi-minor axis of ellipsoidReγ̇ Particle Reynolds number ζ Coordinate in transverse directionPeγ̇ Péclet number I Moment of inertiaEBF External boundary force σxx Normal stress in x-directionNC1 First normal stress difference from Carter’s relation σyy Normal stress in y-directionηCr Relative shear viscosity from Carter’s relation i Indexwi Weightage factor along i index τBGK Relaxation time constant in BGK modelcs Pseudo speed of sound in LBM ν Kinematic viscosityei Discrete velocity in i direction ηp, comp Computational efficiencySp Speed up T

s1 Shortest time for the best serial program

ηrigid Shear viscosity of rigid fiber suspension Torbit Time period for Jeffery’s orbit rotation

Kc Proportionality constant in Carter’s relation for NC1 Tp Execution time for p-node computations

p Total number of processing cores fext External boundary forceJ, Ji Inter-particle collision term (discrete) ∇ Gradient operator

ix

Chapter 1

Introduction

This report briefly discusses the physical aspects of the the rheology of suspensions, thelattice Boltzmann method and the results obtained in the study.

1.1 Introduction and overview

Rheology of the particle suspensions has been a subject of great interest due to an un-precedented increase in research in the fields of biofluidics, micro/nano-fluidics, polymers,environmental sciences and many process industries. Blood and many bodily fluids aretypes of suspensions. Fibers are added in various proportions to vary the characteristicsof the suspension to achieve desired properties. But our knowledge about suspension isstill in infancy. Many rheological aspects need more thorough investigation. One of thecomplex aspects of suspensions is the variation of the properties with range of shear rateγ̇. Effect of deformability of the particles on the rheology of the suspensions has notbeen well understood and is subject to further research. Suspensions exhibit non-linearbehavior due to the involvement of many parameters. Rheology of dilute and semi-dilutesuspensions has been better understood than dense or concentrated suspensions in whichnon-hydrodynamic forces can not be ignored.

Advent of high performance machines with multicore architectures has also facilitated thelarge scale simulations and determine the parameters that characterize the suspensionbehavior or rheology. Rheology, a term used by Prof. Eugene C. Bingham, is the studyof deformation or flow of matter. This branch of science is, however, mainly focused onstudying and characterizing the so-called non-Newtonian fluids. Suspensions with highervolume fraction display deviation from Newton’s law of viscosity. This law states thatthe shear stress applied to the fluid is directly proportional to the gradient of velocity as,

τ = µ∂u

∂y(1.1)

where∂u

∂yis the velocity gradient and µ is constant of the proportionality known as

coefficient of absolute or dynamic viscosity. But non-Newtonian fluids exhibit a nonlinearcurve for viscosity due to setting in of the shear-thinning, shear-thickening, rheopexy orthixotripsy. The first or second normal stress differences are also non-zero. This impliesmicrostructure of the particle has become anisotropic under the conditions of shear flow.

1

1.2. HISTORY CHAPTER 1. INTRODUCTION

The restoring forces thus give rise to non-zero stress differences N1 ≡ σxx − σyy 6= 0and/or N2 ≡ σyy − σzz 6= 0. This will be discussed in detail in §1.5.This report have been organized into three main chapters. Chapter 1 covers the intro-duction and background to the rheology, overview, historical background, objectives ofthis study, general constitutive model and theoretical background of the parameters (i.e.bending ratio, relative viscosity, normal stress differences, fiber orientation and its dis-tribution) investigated in this study in terms of physical explanation and mathematicalrelations. Methodology is presented in chapter 2. It also describes the the theoreticalbackground and development of the method along with the boundary conditions, modelof the fibers , validation of the code and its measured computational performance. Thelast chapter is analysis and discussion of the results in relation with the established the-ories and fiber orientation distribution and finally the conclusion of the study. In theend, appendices have been provided for the graphs of selected simulation cases and thebibliography.

1.2 History

The dependence of the relative viscosity of the suspension on volume concentration wasestablished by Einstein (1906). He worked on dilute suspension of spherical particles.He found that the energy dissipated by the suspension was equivalent to the sum of ki-netic energies of the particles in the suspension and the relative viscosity of the particleswas the function of volume fraction. Jeffery (1922) extended Einstein’s theory to ellip-tical particles suspended in Newtonian fluid. He also determined the stress and shearviscosity for a dilute suspension. The effect of Brownian motion was modeled as a diffu-sion term by Hinch and Leal (1972). Two-body interaction term was incorporated intoJeffery’s equation by Lee and Springer (1982). Anczurowski and Mason (1967) and Ok-agawa and Mason (1973) found Jeffery’s theory invalid in semi-dilute and concentratedregimes. Batchelor (1970, 1971) derived a general constitutive model for a suspensionin a Newtonian fluid. He also determined the stress field for particles with rP > 1 ina concentrated suspension. A method of extending the results to higher volume con-centrations from dilute theory was evolved by Doi and Edwards (1978a,b). Mechanicalinteraction between fibers in semi-dilute and non-Brownian suspension was developedby Dinh and Armstrong (1984). They included this term in a constitutive equation ofstate for the rheological behavior of the semi-concentrated suspensions. Fiber orienta-tion and its distribution was studied by Folgar and Tucker (1984). They modeled thefiber-fiber interaction by a dispersion term. Shaqfeh and Koch (1988) described how theparticles align their axis with the direction of the bulk flow and influence of the aspectratio on the alignment. Shaqfeh and Koch (1990) presented a kinetic theory that pre-dicts the dispersion of particle orientation as a result of hydrodynamic interactions only,about the principal axis of extension in uniaxial and planar extensional flows. Stoveret al. (1992) performed experimental work to measure the fiber-orientations suspensionin semi-dilute regime using cylindrical Couette device and investigated the contemporarytheories. Claeys and Brady (1993) developed a new simulation method for low Re flowproblems involving elongated particles in an unbounded fluid. Their technique extendsthe principles of Stokesian dynamics to ellipsoidal particle shapes. Forgacs and Mason(1959); Blakeney (1966); Goto et al. (1986) proved that slight variation in the shape of

2

CHAPTER 1. INTRODUCTION 1.3. OBJECTIVE

the particles changes the period of rotation and the drag on the the fibers and resultsin different shear viscosity of the suspension. Yamamoto and Matsuoka (1993); Jounget al. (2001) developed simulation of flexible fibers. They modeled the fibers as chainsconsisting of spring linked spheres and spheroids. Ross and Klingenberg (1997); Qi (2007)performed particle level simulations and studied the dynamics of flexible and rigid sus-pensions. They showed that the method can reproduce known dynamical behavior ofisolated rigid and flexible fibers. Schmid et al. (2000) and Lindström and Uesaka (2008)also studied the dynamics of particle suspensions and performed single fiber simulationsfor rigid and flexible fibers with high aspect ratios. Their flexible fiber model consistedof chain of rods. Wu and Aidun (2010a) developed a simulation for flexible fibers usingexternal boundary force.

1.3 Objective

Deformability of the particles/fibers in a suspension plays an important role in the rheol-ogy of suspension at higher volume concentrations. The aim of this study was to quantifythe relation of flexibility of fibers in terms of bending ratio BR with the relative viscosityηr and normal stress difference of a dense suspension in a Newtonian fluid. All theseterms are defined in subsequent sections.

1.4 General constitutive model

The general consitutive models for rigid particles/spheroids and flexible fibers with highaspect ratios are presented here. The models help qualitatively determine the importantphysical parameters and provide a top-down approach for understanding the complexphenomena of the rheology of the particle/fiber suspension. Simplified models can bederived by applying the suitable assumptions. These models have been subjected to non-dimensionalization with suitable scales which resulted in the mathematical definition andderivation of the important flow parameters like Reγ̇, Peγ̇, ηr and BR.

1.4.1 Constutitive model for dense suspension of spheres

Stickel and Powell (2005) proposed a general constitutive model for dense suspensions.It states that shear viscosity η of a concentrated suspension is a function of independent8 parameters.

η = f(D, ρP , n, µ, ρ0, kT, γ̇ or τ, t) (1.2)

Non-dimesionalizing the equation (1.2) results in 6 non-dimensional numbers

ηr = f(φvol, ρr, P eγ̇, Reγ̇, tr) (1.3)

where ηr, φvol, ρr, Peγ̇, Reγ̇, and tr are relative viscosity, volume fraction, density ratio,Péclet number, Reynolds number, and thermal

3

1.5. CHARACTERISTICS CHAPTER 1. INTRODUCTION

ηr =η

µ(1.4a)

φvol =4π

3nD3 (1.4b)

ρr =ρPρ0

(1.4c)

Peγ̇ =6π µD3 γ̇

kT(1.4d)

Reγ̇ =ρ0D

2 γ̇

µ(1.4e)

tr =t kT

µD3(1.4f)

However this model is valid for spheroid s or particles with aspect ratio of 1. It excludesdeformability of the particles.

1.4.2 Constutitive model for flexible fiber suspension

I propose the general constitutive model that includes the fibers with aspect ratios rP > 1and also the deformability by including Young’s modulus EY and length L of the fiber orrod. The total number of parameters becomes 11

η = f(D,L, ρr, n, µ, ρ0, kT, γ̇ or τ, t, EY ) (1.5)

and in non-dimesional form;

ηr = f(cvf , ρr, P eγ̇, Reγ̇, tr, BR) (1.6)

Equation (1.6) introduces two additional non-dimensional numbers rP and BR. φvol isreplaced by cvf as defined below. Reγ̇ is redefined in equation (1.7c) to include theaspect ratio. Similarly other non-dimensional numbers can be redefined.

rP =L

Dor

AsecD2

(1.7a)

cvf =π

4n rP D

3 (1.7b)

Reγ̇ =ρ0D

2 rP γ̇

µ(1.7c)

BR =EY (ln (2 re)− 1.5)

2µγ̇ r4P(1.7d)

Genesis of the bending ratio is discussed in §1.5.1,

1.5 Characteristics

1.5.1 Bending ratio

is a material property which describes the stiffness of the material against bending de-formation in elastic region. It is a dimensional parameter. Forgacs and Mason (1959)

4

CHAPTER 1. INTRODUCTION 1.5. CHARACTERISTICS

and Mason and Goldsmith (1967) introduced bending ratio for flexible fibers. This non-dimensional parameter relates Young’s modulus EY , shear rate γ̇ and aspect ratio rp (alsoreferred to as particle axis ratio in literature) with the absolute viscosity µ of ambientfluid. Physically, the of ellipsoidal particles results in orbits different than predicted byJeffery (1922)’s equation. He theoretically found that the major axis of a prolate spheroidrotates in a orbit with an effective aspect ratio re in equations (1.8) and (1.9).

tan θ =C re

(r2e cos2 φ+ sin2 φ)2

(1.8)

where C is Jeffery’s orbit constant, re is equivalent ellipsoidal axis ratio or effective aspectratio, θ, φ are colatitudinal and azimuthal orientation angles respectively. Thus the timeperiod is also changed

T orbit =2πreγ̇

for re � 1 (1.9)

Shear can induce buckling in fibers. theory predicts axial forces which is based on as-sumption that there is no-slip at fiber-fluid interface.

F ' π γ̇ µ a′2M

ln

(2a′

b′

)− 1.75

(1.10)

where γ̇ is the shear rate, µ is the viscosity of suspending medium, a′, b′ are semi-span and

semi-diameter of the fiber respectively, M =tan2 θ sinφ cosφ

tan2 θ + 1. This implies that maximal

compressive force arises at φ = −45, θ = 0, ⇒ C = ∞,⇒ M = 1/2. Substituting thesevalues in equation (1.10), then calculating the least force required for buckling of thefiber,

Fmax 'π γ̇ µ a′2

2 ln

(2a′

b′

)− 1.75

(1.11)

Fmax =fmax a

′2

2(1.12)

rP =a′

b′(1.13)

fmax 'π γ̇ µ

ln (2 rP )− 1.75(1.14)

and combining with Euler’s equation for beam deflection

EY Id2ζ

dx2= −F ζ (1.15)

fmax =2EY I

a′4(1.16)

where EY is bending modulus, I is the moment of inertia and for cylindrical fiber it isπ b′4

4, F is the total compressive force, substituting the values and comparing equations

(1.14) and (1.16), we get the following relation

(γ̇ µ)crit =EY (ln (2 rP )− 1.75)

2 r4P(1.17)

5

1.5. CHARACTERISTICS CHAPTER 1. INTRODUCTION

As ln (2 rP ) − 1.75 = ln (2 re) − 1.5 ⇒rPre

= 0.78, However, Cox (1971) found the

relationrerP

=1.24√ln(rP )

by matching the experimental and the theoretical data obtained

by Anczurowski and Mason (1967). Substituting these relations in equation (1.17),

(γ̇ µ)crit =EY (ln (2 re)− 1.5)

2 r4P(1.18)

Defining bending ratio as BR =(γ̇ µ)critγ̇ µ

and substituting it in equation (1.18), we get

BR =EY (ln (2 re)− 1.5)

2µγ̇ r4P(1.19)

The equation (1.19) is valid for ambient fluid, no Brownian motion, aspect ratio rP � 1,no effects of inertia, no slip at the surface of the particle. determines the critical valuefor buckling of the fiber for a given rP , shear rate γ̇, and modulus of elasticity. Thusrelation given by equation (1.19), can be used to determine the critical value of any ofthe aforementioned parameters e.g. rP = rcrit. This relation gives the first mode of .

1.5.2 Relative viscosity

Relative viscosity ηr =µsusµ

is the most important characteristic of the particle suspen-

sion. Variation in the concentration cvf , material property of particles EY and suspendingmedia µ, aspect ratio rP , temperature and shear rate γ̇ are reflected in the correspondingchange in the relative viscosity. To establish the comparative importance of shear rateγ̇ and bending ratio BR on the relative viscosity ηr of the fiber suspension, simulationsare performed for both flexible and rigid fibers at different shear rates at cvf = 0.053.An empirical relation for relative viscosity of flexible fiber suspension has been given byJoung et al. (2001).

η = ηrigid

(1 +

A0

1 + eBR/A1

)(1.20)

where A0 and A1 are the constants determined by curve fitting to simulated data. A0depends on aspect ratio rP and coefficient of volume fraction cvf and A1 depends onlyon effective aspect ratio re. For understanding the relative viscosity of the suspensionand it is relation with fiber orientation distribution, Batchelor’s theory is used. HoweverBatchelor’s theory is valid for dilute regime only. Thus it neglects both hydrodynamicand non-hydrodynamic interatcions between particles fibers. The suspension in our studyis concentrated according to the classification by Doi and Edwards (1978a). In theirmethod nL3 value is used to categorize the suspension as dilute if nL3 < 1, semi-diluteif 1 < nL3 < L

Dand concentrated if nL3 > L

D. In this study nL3 = 17.3 and rP = 16.

ηB = 1 +µfiberµ〈p2x p2y〉 = 1 +

µfiber4µ

〈sin4 θ sin2 2φ〉 (1.21)

Equation (1.21), describes the depedence of relative viscosity on the fiber orientationangle. The relative viscosity of suspension is the maximum at for fiber orientation angle

φ =π

4or

3π

4and minimum for φ = 0,

π

2or π. This means that as fiber orientation

distribution becomes broader due to decreased bending ratio, a large number of fibers

6

CHAPTER 1. INTRODUCTION 1.5. CHARACTERISTICS

are oriented at various angles. This results in the increased relative viscosity. At randomorientation, the non-hydrodynamic interaction between fibers results in even higher rel-ative viscosity. However in case of the rigid fibers, the orientation distribution is narrowimplying most of the fibers are in the range of 0.4 to 0.6 [see figure 3.1].

1.5.3 Normal stress difference

Normal stress difference is generated due to asymmetric stresses arising as a result ofanisotropy in the microstructure. Primary or first Normal stress difference N1 is givenby the equation (1.22).

N1 = σxx − σyy (1.22)Experimental work by Petrich et al. (2000) and Carter (1967). Their results highlightthe importance of the non-hydrodynamic interaction and its influence on the rheologyof the suspensions. Batchelor’s theory underestimates first normal stress difference asit does not account for the non-hydrodynamic or mechanical interaction between fibers.This renders Batchelor’s theory invalid in semi-dilute and concentrated regimes. Non-hydrodynamic or mechanical interaction may also be responsible for the marked differencebetween behavior of flexible and rigid fiber suspensions. Petrich calculated the value ofN1 using Batchelor’s equation (1.26) and found that the experimental values were higherthan the theoretical prediction. This explains the significance of the non-hydrodynamicinteraction. A flexible fiber bends more than the relatively rigid fiber and it dependson the difference of the bending ratios. This difference in the response of rigid and flex-bile fibers to non-hydrodynamic interaction consequently appears in different rheologicalbehavior.The effect becomes more pronounced due to non-hydrodynamic interaction. This inter-action increases with fiber concentration. But the behavior of the fiber depends on itsdeformability. Thus bending ratio distinguishes the behavior of the suspension. Carter(1967) derived relation for the first normal stress difference,

NC1µγ̇∝ cvf r

2P

ln (2 re)− 1.8〈sin 2φ〉 (1.23)

and he suggests that,〈sin 2φ〉 ∝

√1/rP (1.24)

substituting equation (1.24) in equation (1.23), we get

NC1µγ̇

= Kccvf r

3/2P

ln (2 rP )− 1.8(1.25)

where Kc is the proportionality constant and determined experimentally. Equation (1.25)is independent of fiber orientation and describes the dependence of N1 on the volumeconcentration cvf and aspect ratio rP . Batchelor (1970) proposed the theory for slenderparticles and found relations for relative viscosity and first normal stress difference N1.

NB1 = µfiberγ̇(〈p3x py〉 − 〈px p3y〉

)= −µfiberγ̇

4〈sin4 θ sin 4φ〉 (1.26)

Equation (1.26) is valid only in the dilute regime. But it is useful as it explains thedependence of the first normal stress difference on the fiber orientation angles φ and θ.

7

1.6. FIBER ORIENTATION AND ITS DISTRIBUTION CHAPTER 1. INTRODUCTION

As we will see later in the discussion that bending ratio BR has a strong influence on thefiber orientation and it affects rheological characteristics of the suspension.

1.6 Fiber orientation and its distribution

Orientation of the fiber with respect to flow affects its mircrostructure and interatctionwith the ambient fluid and non-hydrodynamic interaction with other fibers. In the case offiber suspensions, the orientation distribution is of interest as it explains the rheologicalphenomena in conjunction with other theories [see §3.3 for details]. In pure shear flow,Fibers tend to orient their extensional axis along the direction of the flow.

Figure 1.1: Fiber orientation in a pure shear flow

Figure 1.1 shows the orientation of a fiber in spherical coordinate system. φ and θ are theorientation angles and x-axis points in the direction of the flow. The fiber experiencesvarying stresses at different orientations in the flow as discussed in §1.5.1.

8

Chapter 2

Lattice Boltzmann method (LBM)and fiber model

The simulation code used in this study is based on lattice Boltzmann equation devel-oped by Wu and Aidun (2010a). In this section, we overview the methodology and itsdevelopment.

2.1 Boltzmann-Maxwell to lattice Boltzmann equation

Navier Stoke’s equations (NSEs) have been traditionally used as the governing rule forflow

∇ · u = 0∂ui∂t

+ u · ∇ui = −1

ρ

∂p

∂xi+ ν∇2 · u+ f (2.1)

Though NSEs are the most general flow governing equations and valid in the continuumregion. These are applicable with certain modifications at micro-scales. But alternativemethods have been sought to at micro-scales which provide better perspective aboutmicrofluidic and microstructural phenomena.The simulation code used in this study is based on lattice Boltzmann method. LatticeBoltzmann has been derived from Boltzmann-Maxwell equation (2.2). Lattice Boltzmannequation is equivalent to incompressible NSE [see Chapman (1916); Enskog (1917)]

(∂

∂t+ e · ∇r + a · ∇e

)f(e, r, t) = J (2.2)

f(e, r, t) in equation (2.2) represents the probability distribution function for single parti-cle. Where ∇r and ∇e denote the spatial and velocity gradients respectively. J representsthe interparticle-collision term. This conservation equation is non-linear in nature. Fromthis general equation the relation for finite discrete velocity can be derived as

∂tfi(r, t) + ei · ∇fi(r, t) = Ji(f), i = 0, ......., Q (2.3)

where the index i denotes the discrete velocity directions. Lattice Boltzmann (LB) equa-tion is derived as;

9

2.1. BOLTZMANN-MAXWELL TO LATTICE BOLTZMANN EQUATION CHAPTER 2. LBM

Figure 2.1: D3Q19 velocity model (Courtesy of De (2009))

fi(xe + ei, t+ 1) = fi(x

e, t) +1

τBGK[f eqi (x

e, t)− fi(xe, t)] (2.4)

where fi(xe, t) and f eqi (x

e, t) are distribution and equilibrium distribution functions re-spectively. LB equation (2.4) is an algebraic expression. It is also known as BGK modelBhatnagar et al. (1954). It is also evident that LBE uses regular stepping in spatialand temporal coordinates. The relations between macroscopic (ρ, u) and discrete (fi, ei)paraemters are as follows,

ρ(xe, t) =∑i

fi(xe, t) (2.5)

ρ(xe, t) u(xe, t) =∑i

fi(xe, t) ei (2.6)

The equilibrium distribution function f eqi (xe, t) is calculated by the following relation;

f eqi = wi ρ [1 + 3 (ei · u) +9

2(ei · u)2 −

3

2u2] (2.7)

where wi is the weight function and its values depend on the velocity model used. D2Q9is mostly used for two dimensional simulations. This model has 9 discrete directionsi.e. i = 0, 1, 2, · · · , 8 [see figure 2.3a]. The weight functions are w0 = 49 , w1−4 =

19

andw5−8 =

136

. Relaxation time τ and ν are related ν = 2τ−16

and pseudo speed of sound

cs =√

1/3. Few more velocity models are FHP, D2Q21, for 2-D, D3Q15 and D3Q19 for3-D. In our simulation code, D3Q19 velocity model has been implemented as shown infigure 2.1.The weight functions for D3Q19 are w0 =

13

for the rest or origin, w1−6 =118

in orthogonaldirections and w7−18 =

136

in diagonal directions.LB equation (2.4) can be split into two subequations i.e. (1) streaming equation and(2) collision equation. In streaming the particles translates from one node to the nextnode. In collision exchange of the momentum between the two colliding particles takesplace. For further reading about LBE and its derivation consult the article by Aidunand Clausen (2010) and books by Wolf-Gladrow (2000); Succi (2001); Sukop and Thorne(2006).

2.1.1 Boundary conditions

A brief overview of BC is presented in this section. For details standard references shouldbe consulted. Boundary conditions are very important in fluid dynamics as these reflect

10

CHAPTER 2. LBM 2.1. BOLTZMANN-MAXWELL TO LATTICE BOLTZMANN EQUATION

(a) Boltzmann (b) Maxwell

Figure 2.2: They are the pioneers and their work forms the basis for the LBM

the influence of the ambient conditions on the flow domain. LBM has an advantageover other flow simulation techniques due to its compatibility with a large number ofversatile boundary conditions. However implementing a boundary condition in LBMcan be a tricky task. History of research on BC’s for LBM starts with Ziegler (1993)and this continues as latest being the external boundary force EBF proposed by Wuand Aidun (2009). There are numerous BC’s for LBM with a matching number of theirclassifications. Few of the most common boundary conditions are list below without anyclassification

Periodic This BC is very simple and easy to code. It can be used where surface effectsare negligible and only the physics of the bulk flow itself is of interest. Basically thisis implemented by setting the “incoming” BC equal to the corresponding “outgoing”BC and/or vice-versa i.e. f in = f out.

No-slip This is the next simplest BC. It is implemented by specifying the zero flowvelocity at solid nodes. This boundary condition is implemented when boundarywalls are present and their influence on the flow can not be neglected. It can becoded by reversing the particle directions on a particular node in such a way that neteffect is zero. However there are two major ways to specify this BC. One famousno-slip BC is the standard bounce back (SBB). It is based on BC of lattice gas.Physically the particles at solid nodes are bounced back such that net momentumor velocity is zero.

1. On the node The physical wall lies exactly on the nodes

2. Mid node The boundary exists between two rows of nodes

SBB is not continuous as the boundary is defined between solid and fluid nodes.Thus oscillations in the solutions are observed. This has been resolved by usinginterpolated bounce back (IBB) BCs. These are based on interpolation (linear,quadratic, multi-reflection) between the the nodes.

Free-slip This BC holds when the boundary walls have negligible influence on the flowingparticles. Thus tangential component of the velocity is not zero i.e. no momentum-exchange between the wall and the particles.

Sliding walls This boundary condition is different in the sense it is not-stationery ormoving. This is implemented by setting the speed of the particles equal to that ofwall u(W ) = uw. It is only tangential component of the velocity that is not zero.

11

2.1. BOLTZMANN-MAXWELL TO LATTICE BOLTZMANN EQUATION CHAPTER 2. LBM

Immersed Boundary Method Peskin (1977) proposed this method. It requires twogrid systems. The solid nodes at boundary are connected through high stiffnesssprings. It is works as follows

• immersed solid nodes are streamed or moved with velocity calculated by inter-polation from the ambient fluid velocity

• calculation of the forces within the solid deformable particle

• feedback the forces to the ambient fluid through the same interpolation functionas in step 1

• and solve NS in the discrete form for calculating the new velocity

External Boundary Force This method was originally developed for NS equation byGoldstein et al. (1993). Wu and Aidun (2009) extended this concept to LBM byincluding a discrete external boundary force. Thus the equation (2.4) is modifiedto,

fi(xe + ei, t+ 1) = fi(x

e, t) +1

τBGK[f eqi (x

e, t)− fi(xe, t)] +3

2wi f

ext · ei (2.8)

where fext is external boundary force.

(a) D2Q9 (b) Half-way SBB

Figure 2.3: 2.3a shows D2Q9 a 2 dimensional velocity model with 9 discrete directions and 2.3bshows the mid node or half-way SBB schematics

2.1.2 Flexible fiber model

Fibers have been modeled as the chains of links and hinges as shown in figure 2.4a . Thelinks possess the degree of freedom to twist and bend and this makes them deformable.All the links have same length. The length of the fiber is then L = N l and aspect ratiorp = D/L. The length and diameter of the fiber sufficiently large and thus the influence ofBrownian motion can be ignored. This model is valid for a Newtonian and incompressibleambient fluid. There are 4 nodes on the circumference of each rod as shown in figure2.4b.

12

CHAPTER 2. LBM 2.2. UNIT SYSTEMS: PHYSICAL AND LATTICE

(a) Rods and hinges (b) Four nodes on the circum-ference of the fiber rod

Figure 2.4: Fiber model in the code

2.2 Unit systems: physical and lattice

Lattice Boltzmann method uses parameters in lattice units (lu). The lattice units andphysical units are interconvertible through non-dimensional numbers. Reynolds numberReγ̇ and bending ratio BR are the two important non-dimensional numbers that allowthe conversion of values from lattice units into physical units and vice versa. An examplecalculation is presented here to illustrate the process. Example calculation for the caseof γ̇ = 25 s−1 for both rigid and flexible fibers is presented here to demonstrate theconversion from lattice units to phyiscal units and vice-versa. The suspending fluid hasµ = 13 Pa · s , density ρ = 970 kg m−3 ⇒ ν = 0.0134 m2 s−1 , rP = 16, fiber diameterD = 0.12 mm or = 0.2 lu, where lu denotes the lattice unit. For rigid fiber BR = 2940,substituting these values in equations (1.7c) and (1.19), we get ⇒ Reγ̇ = 4.298 × 10−4,EY = 7.495× 1010 Pa . For flexible fibers, BR = 0.25, and we get EY = 6.373× 106 Pa

2.3 Validation

The lattice Boltzmann simulation code with external boundary force, used in this study,has been developed and validated by Jingshu Wu and C. K. Aidun. Complete details ofthe validation of the code are present in Wu (2010, p. 40). The validation results havebeen published [see Wu and Aidun (2009, 2010a)].

13

2.4. COMPUTATIONAL PERFORMANCE CHAPTER 2. LBM

Figure 2.5: Validation of the simulation on Platon

Figure 2.5, shows the graph taken from Wu (2010) and the two yellow circles representthe points reproduced through running the code on the Platon machine for flexible andrigid fibers with rP = 16, cvf = 0.053, nL

3 = 17.3 at BR = 0.25 and 2940. This impliesthat the simulations performed for this study match with the validated results. Hencethe results are reliable and reproducible on different machines.

2.4 Computational performance

The simulation code has been parallelized with OpenMP technique. Thus all the simu-lations were performed on SMP machines or nodes. To save time, many instances of thecode were run simultaneously.

2.4.1 Hardware resources

Following three machines have been used for execution of simulations.

Ferlin at PDC, KTH. It has 672 compute nodes. Each node consists of 2 x Quad coreIntel Xeon E5430 @ 2.66 GHz and 1333 FSB (Harpertown) processors. A total of5440 cores and 5.44 TB of main memory. The operating system on Ferlin is LinuxCentOS

Platon at LUNARC, Lund University. It has 216 nodes. Each node consists of 2 x Quadcore Intel Xeon E5520 @ 2.26 GHz. A total of 1728 processors. Memory per nodeis 24 GB (3 GB per core). The operating system on Platon is Linux CentOS

Key at PDC, KTH. SMP machine with 32 Intel Itanium IA64 cores @ 1.6 GHz with18 MB cache. The total main memory is 256 GB. The operating system on Key isLinux CentOS

From the description above, it is evident that the nodes of Ferlin and Platon are com-parable. Intel Xeon E5430 @ 2.66 GHz has higher frequency than Intel Xeon E5520 @

14

CHAPTER 2. LBM 2.4. COMPUTATIONAL PERFORMANCE

2.26 GHz. But latter features hyperthreading and it can handle up to 8 threads. Thusit would be interesting to compare the the computational performace of one of thesemachines versus Key.

2.4.2 Efficiency and speed-up

As specifications of the machines used in §2.4.1 have been described, an overview of thecomputational efficiency and speed up of the code is presented here. Computationalefficiency means the measured performance against the maximum or peak performanceof the application on that machine. Speed up means the performance gain due to theparallelization. Mathematical relations for speed up Sp and computational efficiency ηpare given below:

Sp =T s1Tp

(2.9)

where T s1 is the shortest time for the best serial program, Tp is the execution time forp-node computation

ηp =Spp

(2.10)

where p is the total number of processors.

Table 2.1: Simulation start/end time and number of time-steps/min with increasing number ofprocessors on Key

15

2.4. COMPUTATIONAL PERFORMANCE CHAPTER 2. LBM

Figure 2.6: Efficiency of the code on the key and machines

Figure 2.6, describes the computational efficiency of the code on the different machines.It is noticeable that Itanium based machine “Key“ is inefficient. In addition to thelower frequency of the Itanium CPU, the machine is shared. A dedicated machine ofthe same specification will perform better. It is possible on this machine to go upto 24cores, however, the efficiency plummets to below 30% as conflict with other applicationsincreases. The code is efficient on Platon machine. The machine has more than 80% ofthe efficiency while running on eight cores. Since the code was parallelized with OpenMP,it was of no benefit to use more than a single node. But multiple instances of the codewere run in parallel on dedicated nodes to save time. Compiling the code with Intelcompilers made the code run faster and more efficiently.

Figure 2.7: Speed up of the code on the three machines

Figure 2.7, shows the speed up of the code on the two machines. The code has nearlya linear speed up on Platon. This implies that the code is scalable. This is one of theprincipal merits of the LBM simulations. Scalability on Key is not as good as on Platonbut with increasing the cores from 8 to 24 shows some improvement.

16

Chapter 3

Results and discussion

3.1 Summary

Table 3.1 presents the selected cases for rigid and flexible fibers. In these cases, thesuspending fluid has µ = 13 Pa · s , and density ρ = 970 kg m−3 ⇒ ν = 0.0134 m2 s−1 ,aspect ratio rP = 16, and fiber diameter D = 0.12 mm or = 0.2 lu. Bending ratios are0.25 and 2940 for flexible and rigid fibers respectively. Simulation have been performedfor different shear rates, bending ratios, Reynolds number and fiber stiffness.

Table 3.1: Summary of the selected cases performed to determine the effect of BR against shearrate γ̇

These cases range from low to high shear rates γ̇. The Reynolds number Reγ̇ � 1 andPeγ̇ � 1 i.e. the system has no influence of inertia and Brownian motion.

3.2 Input and output parameters

3.2.1 Input parameters

Input parameters for the simulation code are in lattice units (lu). However the physicaland the lattice units are interconvertible through non-dimensional parameters. This hasbeen explained in §2.2. Following is the list of the important input parameters;

• Shear rate γ̇

17

3.3. FIBER ORIENTATION DISTRIBUTION CHAPTER 3. RESULTS AND DISCUSSION

• Aspect ratio rP

• Reynolds number Reγ̇

• Coefficient of volume fraction cvf

• Young’s modulus EY

• Bending ratio BR

3.2.2 Output parameters

Output parameters for the simulation code are normalized. Following is the list of theimportant output parameters;

• Time steps T

• Non-dimensional time γ̇T

• Primary normal stress difference N1

• Secondary normal stress difference N2

• Relative viscosity ηr,fiber

• Fiber orientation distribution function p(φ)

3.3 Fiber orientation distribution

Orientation of the fibers in the shear flow determines the magnitude of the stresses gen-erated in it. The microstructure may become anisotropic as a consequnece of the defor-mation. Thus the orientation affects important characteristics of the fibers. Distributionof orientations of the fibers statistically represents the picture of the suspension of fibers.It is interesting to note the dependence of the orientation distribution of the fibers on thebending ratio from figure 3.1. The higher the bending ratio the narrower the distribu-tion. This signifies that flexible fibers exhibit broader spread of orientations. Physicallyit translates into random and more hydrodynamic and mechanical interaction betweenthe flexible fibers than their rigid counterparts. Thus it is comprehensible that flexiblefiber suspension has higher relative viscosity ηr [see figure 3.2].

18

CHAPTER 3. RESULTS AND DISCUSSION 3.4. RELATIVE VISCOSITY

Figure 3.1: Relative viscosity ratio vs γ̇

It is interesting and intutive to explain the normal stress difference and the relative viscos-ity of the rigid and flexible fiber suspensions in relation with fiber orientation distribution.

3.4 Relative viscosity

To investigate the influence of the bending ratio on the relative viscosity of the rigid andflexible fiber suspensions at a given aspect ratio rP = 16, two sequences of simulationshave been performed i.e. at BR = 0.25 and 2940. Each sequence shows the variation ofrelative viscosity against the shear rate. The shear rate γ̇ varies from low to high i.e. 1,5, 25, ........, 750 s−1. As shown in figure 3.2, for BR = 0.25 i.e. flexible fibers, relativeviscosity of the suspension initially increases rapidly and then stabilizes at constant valueof 2. This is very significant result as it implies that there is no influence of the shearrate on the relative viscosity. Mathematically,

ηr,flex =

{f(γ̇, BR) if γ̇ ≤ 250 s−1,f(BR) if γ̇ > 250 s−1.

(3.1)

Above the shear rate of 250 s−1, rigid fibers show 30% − 40% lower relative viscositythan the flexible fibers. Forgacs and Mason (1959) made the similar observation. Theseresults can be illustrated with Batchelor’s equation (1.21). This relation is valid in di-lute suspensions only. However, it relates the relative viscosity with the fiber orientation.Fiber orientation distribution provides a good insight into the microstructure of the fibersand relates it to the macroscopic or bulk rheology of the suspension. The influence offiber-fiber mechanical interaction is essentially included in Batchelor’s relation due to itsdependence on the fiber orientation distribution.

19

3.5. FIRST NORMAL STRESS DIFFERENCE CHAPTER 3. RESULTS AND DISCUSSION

Figure 3.2: Relative viscosities of flexible and rigid fibers vs γ̇

Figure shows the curve for ratio of relative viscosities of flexible to rigid fibers. Thisprovides another way to view the difference in the behavior of flexible and rigid andhence the influence of bending stiffness on the rheology of the suspension.

Figure 3.3: Relative viscosity ratio vs γ̇

3.5 First normal stress difference

More than 20 simulations were performed to understand the phenomena of anisotropyin the microstructure of the fibers and its dependence on both the shear rate and thebending ratio. From figure 3.4, it can be seen that the bending stiffness affects the first

20

CHAPTER 3. RESULTS AND DISCUSSION 3.6. CONCLUSIONS

normal stress difference. Flexible fibers have higher degree of anisotropy than the rigidfibers and consequently show higher N1 values.

Figure 3.4: First normal stress difference for flexible and rigid fibers vs γ̇

Similar to the relative viscosity curve in figure 3.2, the flexible fibers have reached almost aconstant value forN1 and show imperceptible varriation with shear rate γ̇. It seems almostno or very weak dependence on the shear rate. Rigid fibers, on the other hand, show astromg dependence on the shear rate and their N1 curve shows appreciable variation withthe shear rate. N1 values for rigid fibers increase with the shear rate and consequentlyat γ̇ = 750 s−1 it has same value as the curve for flexible fibers. However, it would bepremature to conclude the crossover of the curves at shear rates γ̇ > 750 s−1.

3.6 Conclusions

From the discussion in §3.4 and 3.5, it is consequently shown that the fiber stiffnessplays the most significant role in the rheology of the flexible fiber suspensions. Ratherat higher shear rates bending ratio is the only parameter that governs the rheology ofthe suspension and only low shear rates have some effect on the relative viscosity [seeequation (3.1)]. In the case of relative viscosity ηr, bending ratio clearly dominates therheology and shear rate has no influence. In the case of the first normal stress differenceit is also a strong function of bending ratio but it weakly depends on the shear rate also.It has been observed in figure 3.4 that at higher shear rates the rigid fibers also reachthe same value as the flexible fibers. Further higher shear rates may result in rigid fibersdisplaying higher first normal stress difference than the flexible fibers. This impiles higherdegree of anisotropy in the microstructure. It can be attributed to the higher rigidityor bending stiffness causing the development of higher stress values at higher shear ratesand unlike flexible fibers that buckle to dissipate the energy, rigid fiber microstructure

21

3.6. CONCLUSIONS CHAPTER 3. RESULTS AND DISCUSSION

becomes highly anisotropic in extensional and transverse directions. The reasons are stillnot clear and it should be investigated further.

Figure 3.5: Screen shots from animation of a single fiber in a shear flow with EY = 3000 lu

Figure 3.5 and 3.6 show that higher bending stiffness EY value results in less tip deflectionfor an individual fiber.

Figure 3.6: Screen shots from animation of a single fiber in a shear flow with EY = 1500 lu

At low shear rate γ̇ = 50 s−1 the flexible fibers behave as rigid fibers and align theirextensional axis with the flow as evident from figure 3.7.

22

CHAPTER 3. RESULTS AND DISCUSSION 3.7. FUTURE RECOMMENDATIONS

Figure 3.7: Screen shots from animation of flexible fibers suspension in a shear flow at γ̇ = 50 s−1

However, at higher shear rates the fibers deform and entagle with each other and interactnon-hydrodynamically.

Figure 3.8: Screen shots from animation of flexible fibers suspension in a shear flow at γ̇ =500 s−1

Thus bending stiffness is the single most important variable and above γ̇ = 250 s−1, shearrate has no influence on the rheology and relative viscosity is the function of bending ratioηr = ηr(BR).

3.7 Future recommendations

This study can be extended to include the more aspect ratios. bending ratios and volumefractions. The simulation code can be modified to include heterogeneous multi aspect

23

3.7. FUTURE RECOMMENDATIONS CHAPTER 3. RESULTS AND DISCUSSION

ratio and multi form particles and fibers simultaneously. A comparative study of differentfiber models can also be performed to find the model that best matches the experimentalresults. Further different kind of electrical forces can be modeled to investigate theelectrostatic and Coulomb forces on a single fiber and suspension of fibers. Parallelizingthe code with MPI or hybrid OpenMPI approach will enable the code to run on thousandsof cores in distributed machines and hence very large simulations will be possible.

24

Appendices

25

Appendix A

Relative viscosity

For legend, the following colors are used

• The red curve is the moving average

• The blue curve is the the original data

A.1 Graphs for relative viscosity ηr

The simulation cases shown in the following figures are performed with parameter valuesdescribed in §2.2

A.1.1 Flexible fibers

Figure A.1: Variation of the relative viscosity ηr vs γ̇T at shear rate γ̇ = 50 s−1

i

A.1. GRAPHS FOR RELATIVE VISCOSITY ηR APPENDIX A. RELATIVE VISCOSITY


ii

APPENDIX A. RELATIVE VISCOSITY A.1. GRAPHS FOR RELATIVE VISCOSITY ηR

A.1.2 Rigid fibers



iii

A.1. GRAPHS FOR RELATIVE VISCOSITY ηR APPENDIX A. RELATIVE VISCOSITY


iv

Appendix B

First normal stress difference




B.1 Graphs for first normal stress difference N1

The simulation cases shown in the following figures are performed with parameter valuesdescribed in §2.2

B.1.1 Flexible fibers

Figure B.1: Variation of the first NSD N1 vs γ̇T at shear rate γ̇ = 50 s−1

v

B.1. GRAPHS FOR FIRST NSD APPENDIX B. FIRST NORMAL STRESS DIFFERENCE



vi

APPENDIX B. FIRST NORMAL STRESS DIFFERENCE B.1. GRAPHS FOR FIRST NSD


vii


B.1.2 Rigid fibers



viii

APPENDIX B. FIRST NORMAL STRESS DIFFERENCE B.1. GRAPHS FOR FIRST NSD



ix



x

Appendix C

Second normal stress difference




C.1 Graphs for second normal stress difference N2

Graphs for second normal stress difference are given for reference purpose only. Due tohigh noise to original signal ratio,N2 is not reliable and consequently not used to reachany conclusion. The simulation cases shown in the following figures are performed withparameter values described in §2.2

C.1.1 Flexible fibers

Figure C.1: Variation of second NSD N2 vs γ̇T at shear rate γ̇ = 50 s−1

xi

C.1. GRAPHS FOR SECOND NSD APPENDIX C. SECOND NORMAL STRESS DIFFERENCE


xii

APPENDIX C. SECOND NORMAL STRESS DIFFERENCE C.1. GRAPHS FOR SECOND NSD

C.1.2 Rigid fibers



xiii

C.1. GRAPHS FOR SECOND NSD APPENDIX C. SECOND NORMAL STRESS DIFFERENCE



xiv

APPENDIX C. SECOND NORMAL STRESS DIFFERENCE C.1. GRAPHS FOR SECOND NSD


xv

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