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 DOCTOR OF PHILOSOPHY OF ENGINEERING DEPARTMENT OF MECHANICAL & PRODUCTION ENGINEERING NATIONAL UNIVERSITY OF SINGAPORE 2007 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. ii 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 iii Nomenclature Symbols: ar aspect ratio b body force per unit mass CI interaction coefficient C configuration tensor Cb orbital constant d diameter of the fibre D deformation tensor Dr diffusivity F force F (b ) Brownian motion G Green function G single layer kernel I unit tensor J surface moment K double layer kernel Η operator symbol L length of the fibre L velocity gradient tensor M periodic summation n number of density n normal unit vector iv N normal stress difference O origin p pressure p unit vector field along the axis of a fibre P radial basis function x Cartesian coordinate R rotlet S surface area S stresslet t time t traction T period of the fibre’s orbit T torque u velocity U translational velocity V volume W vorticity tensor x average value of x Greek symbols: σ Cauchy stress tensor ϕ density function γ shear deformation γ shear rate (rate of deformation) v η viscosity ρ density φ volume fraction ψ normal stress coefficient ν Poisson ratio ω rotational velocity τ viscoelastic stress λ relaxation time ∆t time step T traction of the double layer kernel Superscripts: general convective derivative ∇ upper convected derivative ∆ lower convected derivative v viscoelastic component N Newtonian H homogeneous solution P particular solution * adjoint ∞ ambient Subscripts: initial position, time 1, 2,3 spatial coordinate s solvent viscosity vi v relative viscosity of the viscoelastic fluid sp specific viscosity f relative viscosity of the fibre suspensions Acronyms: BEM Boundary Element Method CDLBEM Completed Double Layer Boundary Element Method DLM Distributed Lagrange Multiplier DPD Dissipative Particle Dynamics FD Fictitious Domain Method FDM Finite Difference Method FEM Finite Element Method GCR Generalized Conjugate Residual method GMRES Generalized Minimal Residual method MD Molecular Dynamics MPI Message Passing Interface PVM Parallel Virtual Machine SPH Smoothed Particle Hydrodynamics vii TABLE OF CONTENTS 1. Introduction ------------------------------------------------------------------------------------- 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 viii 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 ix 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 x 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 xi 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: α = 300 and 600 .--------------------------------------------------------------------- 85 Figure 5.16: The rate of decay of the orbital constants in viscoelastic shear flow for ar = , β = / , γ = 1.0 . ------------------------------------------------------------------------- 89 xii Figure 5.17: The comparison of the orbits obtained from the numerical simulation and from the modelling with c = 0.018 for ar = , β = / , γ = 1.0 . ------------------------- 90 Figure 5.18: The comparison of the projections of the orbits into the flow-velocity gradient plane for ar = , β = / , γ = 1.0 .--------------------------------------------------- 90 Figure 5.19: The angle α changes with time, predicted by the present model, ar = , β = / , c = 0.018 and two different initial orientations: α = 300 , α = 600 . ---------- 91 Figure 6.1: Specific shear viscosity as a function of shear deformation in viscoelastic fluid (ηv = 1.15 , λ = 0.7 ) (a) ar = , φ = 5% ; (b) ar = , φ = 10% ; (c) ar = , φ = 1% ; (d) ar = , 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 , ar = , φ = 5% , γ = ). - 110 Figure 6.4: Jeffery orbits at different values of the orbit constant Cb . -------------------- 113 Figure 6.5: Orbital constant of a periodic fibre suspension in viscoelastic fluid ( ar = , 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 ar = of periodic fibre suspension 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 ar = of periodic fibre 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 xiii [...]... different values of the orbit constant Cb 11 3 Figure 6.5: Orbital constant of a periodic fibre suspension in viscoelastic fluid ( ar = 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% - 11 4 Figure 6.6: Orbits of a generic fibre with aspect ratio ar = 3 of periodic fibre suspension in viscoelastic. .. viscoelastic fluid at different volume fraction ( φ = 10 .8%, 27%, 43 .1% ) 11 5 Figure 6.7: (a) Orbits of a generic fibre with aspect ratio ar = 3 of periodic fibre suspension at volume fraction φ = 10 .8% , θ = 80o for various ηv (ηv = 1. 0 01, 1. 3, 1. 7 ), (b) A view along the vorticity axis - 11 6 Figure 6.8: Transient specific shear viscosity of periodic fibre suspension in viscoelastic. .. gradient plane for ar = 2 , β = 3 / 7 , γ = 1. 0 - 90 Figure 5 .19 : The angle α changes with time, predicted by the present model, ar = 2 , β = 3 / 7 , c = 0. 018 and two different initial orientations: α 0 = 300 , α 0 = 600 91 Figure 6 .1: Specific shear viscosity as a function of shear deformation in viscoelastic fluid (ηv = 1. 15 , λ = 0.7 ) (a) ar = 2 , φ = 5% ; (b) ar = 2... rate of decay of the orbital constants in viscoelastic shear flow for ar = 2 , β = 3 / 7 , γ = 1. 0 - 89 xii Figure 5 .17 : The comparison of the orbits obtained from the numerical simulation and from the modelling with c = 0. 018 for ar = 2 , β = 3 / 7 , γ = 1. 0 - 90 Figure 5 .18 : The comparison of the projections of the orbits into the flow-velocity gradient... suspensions using Ewald summation 10 2 6.3 Numerical procedure 10 4 6.4 Numerical examples and discussion 10 6 6.4 .1 Fibres are initially aligned along the vorticity axis - 10 7 6.4.2 Fibres are initially located on the flow-vorticity plane 11 2 6.5 Discussion 11 8 7 Conclusions ... 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 ... 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: α 0 = 300 and 600 - 85 Figure 5 .16 : The...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... 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% - 11 7 Figure 6.9: A comparison of the numerical effective shear viscosities (open symbols) with experiment results by Ganani and Powell (19 86) (solid symbols) 11 8 xiii ... 2 , φ = 10 % ; (c) ar = 4 , φ = 1% ; (d) ar = 5 , φ = 5% - 10 8 Figure 6.2: A comparison of the numerical effective shear viscosities (open symbols) with experiment results by M Sepehr et al (2004) (solid symbols) 10 9 Figure 6.3: Effect of viscoelastic suspending fluids ( λ = 0.7 , ar = 5 , φ = 5% , γ = 1 ) - 11 0 Figure 6.4: Jeffery orbits at different . Passing Interface PVM Parallel Virtual Machine SPH Smoothed Particle Hydrodynamics viii TABLE OF CONTENTS 1. Introduction 1 2. Governing equations and Formulation 11 2 .1. Kinematics. 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. 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