Periodic fibre suspension in a viscoelastic fluid a simulation by a boundery element method 1

PERIODIC FIBRE SUSPENSION IN A VISCOELASTIC FLUID – A SIMULATION BY A BOUNDARY ELEMENT METHOD By NGUYEN HOANG HUY B.Eng Ho Chi Minh University A THESIS SUBMITTED FOR THE DEGREE OF DO

PERIODIC FIBRE SUSPENSION IN A

VISCOELASTIC FLUID – A SIMULATION BY

A BOUNDARY ELEMENT METHOD

By NGUYEN HOANG HUY

(B.Eng Ho Chi Minh University)

A THESIS SUBMITTED FOR THE DEGREE OF DOCTOR OF PHILOSOPHY OF ENGINEERING DEPARTMENT OF MECHANICAL & PRODUCTION ENGINEERING

NATIONAL UNIVERSITY OF SINGAPORE

Acknowledgements

I would like to express my deep and sincere gratitude to my supervisors Professor Nhan Phan-Thien and Professor Khoo Boo Cheong for their constant encouragement, guidance and help during my research

I gratefully acknowledge the help of the National University of Singapore for giving

me the opportunity to work on a PhD thesis

My special thank is due to my families for their loving support Without their support, encouragement and understanding it would have been impossible for me to finish this thesis

Summary

This thesis presents some rheological problems of fibre suspensions in viscoelastic fluids which can be found in polymer processing industry and bioengineering Due to the difficulties in theoretical studies on fibre suspensions in viscoelastic fluid, numerical methods become an alternative approach to predict the fibre orientation in viscoelastic flow field and to study the effect of fibre structure on the macroscopic rheological properties of the fibre suspensions The completed double layer boundary element method was developed to simulate fibre suspensions in viscoelastic fluids A point-wise solver was applied to solve the viscoelastic constitutive equation The fixed least square method was employed in numerical fitting and differentiation without the need of volume meshing For dilute regime, a prolate spheroid rotating in shear flow of an Oldroyd-B fluid was simulated Based on the simulated orbit of a prolate spheroid in viscoelastic shear flow, a constitutive model for the weakly viscoelastic fibre suspensions was proposed and its predictions were compared with some available numerical and experimental results For non-dilute regime, formulation for periodic fibre suspensions in viscoelastic fluid was proposed It is demonstrated that these equations are applicable for the simulation of multiple fibres suspended in viscoelastic shear flow All simulated results are well compared to available experimental observations This indicates that these numerical methods can be used in rheological research to provide useful information on the behaviour of fibre suspensions in viscoelastic fluids

Keywords:

Fibre suspension, Viscoelastic matrix, Periodic, Viscosity, CDLBEM

Nomenclature Symbols:

r

I

b

r

( )b

N normal stress difference

p unit vector field along the axis of a fibre

Greek symbols:

γ
shear rate (rate of deformation)

η viscosity

t

Superscripts:

Subscripts:

1, 2,3 spatial coordinate

v relative viscosity of the viscoelastic fluid

f relative viscosity of the fibre suspensions

Acronyms:

CDLBEM Completed Double Layer Boundary Element Method

DLM Distributed Lagrange Multiplier

GCR Generalized Conjugate Residual method

GMRES Generalized Minimal Residual method

TABLE OF CONTENTS

1 Introduction - 1

2 Governing equations and Formulation -11

2.1 Kinematics -11

2.1.1 Velocity and Acceleration -12

2.1.2 Velocity Gradient and Rate of Deformation -12

2.2 Conservation Laws -13

2.2.1 Conservation of Mass -13

2.2.2 Conservation of Linear Momentum -13

2.2.3 Conservation of Angular Momentum -14

2.3 Constitutive Equations -14

2.3.1 Newtonian fluids -14

2.3.2 Viscoelastic fluids -14

2.3.2.1.Generalized Newtonian model - 17

2.3.2.2.Upper convected Maxwell model - 18

2.3.2.3.Oldroyd-B model - 18

2.3.2.4.Giesekus model - 19

2.3.2.5.Phan-Thien Tanner model - 20

3 Suspension theory -22

3.1 Jeffery’s orbit -22

3.2 Equation of change -26

3.3 Folgar-Tucker model -27

3.4 Transversely isotropic fluid models -27

3.5 Dinh-Armstrong model -28

3.6 Phan-Thien-Graham model -29

3.7 Bulk suspension properties -30

4 Boundary Element Method -34

4.1 The Boundary Element Method -34

4.2 Betti’s Reciprocal Theorem -35

4.3 Integral Representation -35

4.3.1 Kelvin state -35

4.3.2 Integral representation -37

4.4 Single and Double Layer Potentials -39

4.4.1 Single Layer -39

4.4.2 Double Layer -41

4.5 Boundary Integral Equations -44

4.5.1 Direct BEM -44

4.5.2 Indirect BEM -45

4.6 Completed Double Layer Boundary Element Method (CDLBEM) -47

4.6.1 Completed Double Layer BEM -48

4.6.1.1 Completion process - 49

4.6.1.2 Deflation process - 50

4.6.2 Completed Double Layer Traction Problem -52

5 Mobility problem of a single particle in viscoelastic fluid -55

5.1 Introduction -55

5.2 CDLBEM formulation for viscoelastic fluid -56

5.3 The Particular Solution -60

5.4 Numerical Implementation -63

5.4.1 Field points -63

5.4.2 Fixed least square method -64

5.4.3 Point-wise solver for constitutive equation -69

5.5 Numerical procedure -71

5.6 Prolate Spheroid in Viscoelastic Shear Flow -74

5.7 Modeling Single Fibre In Dilute Fibre Suspensions -85

5.7.1 Modeling -85

5.7.2 Results -89

5.8 Discussion -92

6 An extension of CDLBEM for periodic fibre suspensions -94

6.1 Introduction -94

6.2 Formulation -95

6.2.1 Boundary integral equation -95

6.2.2 Traction solution for periodic suspensions using Ewald summation -99

6.2.3 Velocity solution for periodic suspensions using Ewald summation 102

6.3 Numerical procedure - 104

6.4 Numerical examples and discussion - 106

6.4.1 Fibres are initially aligned along the vorticity axis - 107

6.4.2 Fibres are initially located on the flow-vorticity plane - 112

6.5 Discussion - 118

7 Conclusions - 119

8 References - 123

List of Figures

Figure 2.1: Steady simple shear flow - 15

Figure 5.1: Fixed and moving field points for a 2D case for illustration Open circles denote the surface element nodes, filled circles denote the moving points and the plus signs denote the fixed points. - 64

Figure 5.2: The coordinate systems used to characterize the orientation of the fibre 74

Figure 5.3: Illustration of Jeffery orbits for a fibre for various values of orbital constant C - 75

Figure 5.4: Effect of elasticity on the period of a prolate spheroid in shear flow - 77

Figure 5.5: Jeffery’s orbit compared to the numerical orbits. - 78

Figure 5.6: Jeffery’s orbit compared to the numerical orbits. - 79

Figure 5.7: Jeffery’s orbit compared to the numerical orbits. - 80

Figure 5.8: The angle α changes with time - 80

Figure 5.9: Jeffery’s orbit compared to numerical orbits with different relative viscosities - 81

Figure 5.10: Jeffery’s orbit compared to numerical orbits with different relative viscosities, as viewed along the vorticity axis. - 82

Figure 5.11: Jeffery’s orbit compared to numerical orbits at different initial configurations - 82

Figure 5.12: Jeffery’s orbit compared to numerical orbits at different initial orientations, as viewed along the vorticity axis - 83

Figure 5.13: Jeffery’s orbit compared to numerical orbits at different aspect ratios - 83

Figure 5.14: Jeffery’s orbit compared to numerical orbits at different aspect ratios, as viewed along the vorticity axis - 84

Figure 5.15: The angle α changes with time with two different initial orientations:α0 =300and 600. - 85

Figure 5.16: The rate of decay of the orbital constants in viscoelastic shear flow for 2 r a = , β =3 / 7, γ
=1.0 - 89

Figure 5.17: The comparison of the orbits obtained from the numerical simulation and from the modelling with c=0.018 for a r = , 2 β =3 / 7, γ
=1.0 - 90 Figure 5.18: The comparison of the projections of the orbits into the flow-velocity gradient plane for a r = , 2 β =3 / 7, γ
=1.0. - 90 Figure 5.19: The angle α changes with time, predicted by the present model, a r = , 2

3 / 7

β = , c=0.018 and two different initial orientations: 0

α = - 91

Figure 6.1: Specific shear viscosity as a function of shear deformation in viscoelastic fluid (ηv =1.15,λ=0.7) (a) a r = ,2 φ =5%; (b) a r = ,2 φ =10%; (c) a r = ,4 φ =1%; (d) a r = ,5 φ =5% - 108 Figure 6.2: A comparison of the numerical effective shear viscosities (open symbols) with experiment results by M Sepehr et al (2004) (solid symbols) - 109 Figure 6.3: Effect of viscoelastic suspending fluids (λ =0.7,a r = ,5 φ =5%,γ
=1) - 110 Figure 6.4: Jeffery orbits at different values of the orbit constant C b - 113 Figure 6.5: Orbital constant of a periodic fibre suspension in viscoelastic fluid (a r = , 2 Oldroyd-B fluid with ηv =1.3,λ =0.7) as a function of shear deformation (a) φ =0.1%, (b) φ =1.58%. - 114 Figure 6.6: Orbits of a generic fibre with aspect ratio a r = of periodic fibre suspension 3

in viscoelastic fluid at different volume fraction (φ =10.8%, 27%, 43.1%) - 115 Figure 6.7: (a) Orbits of a generic fibre with aspect ratio a r = of periodic fibre 3 suspension at volume fraction φ =10.8%,θ =80o for various ηv (ηv =1.001, 1.3, 1.7), (b) A view along the vorticity axis - 116 Figure 6.8: Transient specific shear viscosity of periodic fibre suspension in viscoelastic fluid (ηv =1.3,λ =1.6) at different fibre concentrations (a) φ =27%, (b) φ =43.1% - 117 Figure 6.9: A comparison of the numerical effective shear viscosities (open symbols) with experiment results by Ganani and Powell (1986) (solid symbols) - 118