Numerical modelling of fibre suspensions in newtonian and non newtonian fluids

NUMERICAL MODELLING OF FIBRE SUSPENSIONS IN NEWTONIAN AND NON-NEWTONIAN FLUIDS DUONG-HONG DUC A THESIS SUBMITTED FOR THE DEGREE OF DOCTOR OF PHILOSOPHY DEPARTMENT OF MECHANICAL ENGINEERING NATIONAL UNIVERSITY OF SINGAPORE 2005 Acknowledgements It is a great pleasure to thank my mentor Professor Phan-Thien Nhan for recommending me to the SMA and NUS scholarship programmes, and then for introducing me this interesting and exciting area of research, as well as for guiding me to the scientific research over the years in the alma mater. I am deeply grateful for his great dedication in the supervision. I am also pleased to thank my supervisor Professor Yeo Khoon Seng for his continual support and guidance over the past three years. I would also thank Dr. Fan Xi-Jun for his original software as well as his significant help at the starting point of this work. I am also grateful Professor Khoo Boo Cheong for his ceaseless assistance whenever needed and particularly for allocating the resource of SMA’s clusters. I would also thank Dr. Chen Shuo, Professor E. Burdet, Dr. G. Chaidron, Dr. Le Minh Thinh for many interesting discussions and for their work in the preparation of some publications. This work has been supported by the Mechanical Engineering Department in the Engineering Faculty of the National University of Singapore, as well as the grant for the International Rheology Congress 15 in Korea. This work would not have been possible if it were not for the continuing encouragement and support of my parents and my siblings, as well as my dear friend Chieu Minh, for their unfailing belief in my ability. Last but not least I would like to thank my friends for all their continual support, particularly Mr. i Acknowledgements Daniel Wong for his helpful assistance in furnishing useful facilities in the computational lab as well as for familiarizing me with Singapore. ii Table of Contents Summary .v List of Tables .vi List of Figures vii Chapter 1: Introduction .1 Chapter 2: Literature reviews .12 2.1 Theory .12 2.1.1 Jeffery’s model .12 2.1.2 Fokker-Planck equation and equation of change .15 2.1.3 Folgar-Tucker Model .17 2.1.4 Closure approximations .17 2.1.5 Constitutive models for suspensions 19 2.1.6 Dilute suspensions: Transversely Isotropic Fluid (TIF) 19 2.1.7 Semi-concentrated suspensions: Dinh and Armstrong model .20 2.1.8 Concentrated suspensions: Phan-Thien – Graham Model. 22 2.2 Experimental results and numerical methods .23 2.2.1 Single particle systems .24 2.2.2 Multi-particle systems and boundary effects .25 2.2.3 Rheological predictions of fibre suspensions 31 2.2.4 Effects of non-Newtonian suspending fluids .34 2.3 Summary for Chapter .38 Chapter 3: DPD Method .40 3.1 Governing equations .43 3.2 Simulation procedure 46 3.2.1 Groot and Warren Algorithm .46 3.2.2 Rheological properties measurement .48 3.3 No-slip boundary conditions .49 3.3.1 SLLOD algorithm 50 3.3.2 Double layer wall and sliding wall method .51 3.4 Implementations 53 3.4.1 Serial programme .53 3.4.2 Parallel programme 55 3.5 Simulations of a Newtonian fluid and results .58 3.5.1 Simulations of Couette flow - SLLOD algorithm 58 iii Table of Contents 3.5.2 Couette flow - sliding wall method 60 3.5.3 Poiseuille flow with single layer wall 62 3.5.4 Poiseuille flow with double layer wall .64 3.5.5 Flow through a contraction and expansion channel .67 3.6 Concluding remarks 68 Chapter 4: Models in DPD and a model prediction for fibre suspensions .70 4.1 A model of fibre in DPD .70 4.2 VNADPD for modelling viscoelastic fluids .78 4.3 A prediction model .81 4.4 Summary for chapter 84 Chapter 5: Fibre suspensions in Newtonian and viscoelastic fluids .86 5.1 Fibre suspensions in a Newtonian fluid 86 5.2 Viscoelastic fluids with VNADPD .88 5.3 Fibre suspensions in viscoelastic fluids 89 5.3.1 Fibre suspensions in viscoelastic fluid I 90 5.3.2 Fibre suspensions in viscoelastic fluid II .94 5.4 Concluding remarks 98 Chapter 6: Other applications .99 6.1 Neutro-probe entering Brain Tissue 99 6.1.1 Introduction 99 6.1.2 Experiments .100 6.1.3 Simulations 101 6.2 Single DNA chains .102 6.2.1 Introduction 102 6.2.2 Mechanism of the model 104 6.2.3 Extensions of a single polymer chain in shear flows .106 6.3 Conclusions .109 Chapter 7: Conclusions and future work 111 Appendixes .116 Bibliographies .125 iv Numerical Modelling of Fibre Suspensions in Newtonian and non-Newtonian Fluids Summary In this thesis, Dissipative Particle Dynamics (DPD) models of fibre suspensions in Newtonian and non-Newtonian fluids are developed and presented. The results are validated with other experimental data and numerical models. First, the DPD method is studied and further developed to enhance its performance with regard to algorithms and no-slip boundary conditions. A novel no-slip boundary is proposed and successfully applied to different flows i.e. Poiseuille, Couette and complex flows. The algorithm is efficiently parallelized to speed up the computation. Secondly, a novel DPD model for fibre and a Versatile Network Approach DPD model for viscoelastic fluids are developed in order to simulate efficiently fibre suspensions in Newtonian and non-Newtonian fluids. The models are validated by comparing the numerical results with available theoretical solutions or experimental data. The rheological properties of fibre suspensions and the orientation of fibres under Couette flows are then investigated for the effects of different solvents, volume fractions, and shear rates. Those results will help to enhance our understanding of the flows of fibre suspensions and moreover the simulation can then be used to compute the rheological properties. On top of that, a modified version of the Folgar-Tucker’s constant is proposed to deal with viscoelastic suspensions. Coupled with this, a predictive model for rheological properties is suggested and good agreement with simulated data lends some confidence to its use for Newtonian and viscoelastic fibre suspensions. Lastly, the models are further extended to deal with several different applications. v List of Tables Table 1: Asymptotic values of Ai, i = to 20 Table 2: The viscosities of Newtonian solvent for different shear rates .61 vi List of Figures Fig. 1. The coordinate systems used to characterize the orientation of a single fibre14 Fig. 2. The structure of the double layer 52 Fig. 3. Face-centred cubic lattice .54 Fig. 4. DPD particles with fibre suspensions .56 Fig. 5. The interaction forces between particles within rc .56 Fig. 6. The communication between sub-domains 57 Fig. 7. The speed up and efficiency of parallel algorithm .57 Fig. 8.The relative velocity profile, temperature and density of Couette flow – SLLOD algorithm 58 Fig. 9. The shear stress and the normal stress differences – SLLOD algorithm 59 Fig. 10. The relative velocity profile, temperature and density of Couette flow .60 Fig. 11. The shear stress and the first and second normal stress differences .61 Fig. 12. Shear stresses versus shear rates in Newtonian fluid .62 Fig. 13. The fully developed velocity and the Navier-Stoke solution .63 Fig. 14. The temperature and density .63 Fig. 15. The shear stress and the analytical solution 63 Fig. 16. The first and second normal stress difference 64 Fig. 17. The fully developed velocity and the Navier-Stoke solution .65 Fig. 18. The temperature and density .65 Fig. 19. The normal stress differences .66 Fig. 20. The shear stress and the analytical solution 66 Fig. 21. The geometry of the contraction and diffusion channel .67 Fig. 22. The longitudinal velocity profile at x = -40.25 .68 Fig. 23. Temperature and density profile at x = -40.25 .68 Fig. 24. The osculating multi-bead rod model .71 Fig. 25. The relative drag coefficients versus relative? .76 Fig. 26. A network of particles containing particular particles (black round) at two different times, each particle can have a maximum of three links .79 Fig. 27. A comparison of the relative viscosity versus volume fraction between the vii Table of figures simulation results with experimental results of Ganani and Powell (1986), and with the Dinh and Amstrong’s model (1984) as well as our suggested model Eqs. (4.15) and (4.16) .86 Fig. 28. Shear rate dependent viscosity for fluid I and II 89 Fig. 29. Shear rate dependent viscosity of fibre suspensions in fluid I. 91 Fig. 30. Ci and det(pp) depend on shear rate .91 Fig. 31. The first normal stress difference of fibre suspensions in fluid I .92 Fig. 32. Minus second normal stress difference of fibre suspensions in fluid I 92 Fig. 33. The orientation of fibres suspended in fluid I and II 94 Fig. 34. The Ci depends on shear rate and volume fraction .94 Fig. 35. Shear rate dependent viscosities of fibre suspensions in fluid II 96 Fig. 36. First normal stress difference of fibre suspensions in fluid II 97 Fig. 37. Minus second normal stress difference of fibre suspensions in fluid II. 97 Fig. 38.Relative viscosities depend on volume fraction (fibre suspensions in fluid II) 98 Fig. 39. A neuroprobe and a Singapore dollar. 100 Fig. 40. Comparison of Viscosity dependence with shear rate for a pig brain and the VNADPD virtual fluid. The VNADPD particles can have a maximum of links and the FENE spring force H is equal to 20. 101 Fig. 41. Shear stress field before probe ceases motion. .102 Fig. 42. Shear stress field after probe stops .102 Fig. 43. The DPD chains model .104 Fig. 44. Probability distribution for polymer extension (projected in flow-vorticity plane) 108 Fig. 45. The comparison distribution extension between the experimental end-to-end (Sim. 1) and the projected extension in flow-vorticity plane (Sim.) .109 viii Chapter 1: Introduction Rod-like particle suspensions can be found in many important and diverse applications: short DNA separations, pulp suspensions, carbon nanotubes, and short-fibre reinforced composites, to name a few. The latter applications are becoming increasingly important in consumer goods as well as in industries such as high-quality sport’s equipment manufacturing and aerospace, where desirable properties such as strength, stiffness, toughness and light weight are necessary. While conventional materials such as metals and their alloys are strong and tough, they are also heavy. Fibre reinforced composites on the other hand possess all the mentioned desirable properties. These composites have been extensively developed and successfully deployed across many applications and industries over the last decades. There are two main types of fibre reinforced composites: the continuous-fibre composites (CFCs) and the short-fibre composites (SFCs). The continuous-fibre composites contain full length of reinforced particles over the dimension of the parts; whereas the short-fibre composites are reinforced by particles that are typically slender, and whose lengths are small compared to the overall dimension of the components. Therefore SFCs can be used in mass productions using techniques that have been developed for processing pure polymers, such as injection moulding, extrusion, and shear moulding compound, etc. [De and White (1996)]. In applications with intricate geometries, SFCs are preferred to continuous fibre composites, which often require costly and labour-intensive Appendices For re  Pe −1/  , re = ( 8π aR2 / 3E ln aR ) 0.5 (A.19) ηr = + ⎡⎣ + (15 / 4)( D −1 − 1) K − KPe −1/ ⎤⎦ φ (A.20) K = 0.822 K + 5.388( D −1 − 1) K (A.21) where Finally, for weak Brownian motion, Pe1/  re  : ⎪⎧ ⎤ ⎪⎫ ⎛ 15 ⎞ ⎡ ⎛ 1.792 ⎞ ⎛ K ⎞ ⎛ 3.052 ⎞ ⎛ aR ⎞ K 1 3K + K ) ⎥ ⎬ φ (A.22) − − − − ⎟ ⎜ ⎟ ⎜ ⎟⎢ ⎜ ⎟⎜ ⎟( re ⎠ ⎝ D ⎠ ⎝ re ⎠ ⎝ DPe ⎠ ⎝ ⎠⎣ ⎝ ⎦ ⎪⎭ ηr = + ⎨2 − ⎜ ⎪⎩ Doi and Edwards (1978): ηr = + (nl )3 if nl  (A.23) Bibbo et al. (1985): ηr = 1 − ( 4φ / π ) (A.24) 0.5 Berry and Russel (1987): ηr = + [η ]φ + K [η ] φ (A.25) 8aR2 − 0.020 Pe ) [η ] = ( 45ln 2aR (A.26) with: K = 0.4(1 − 0.0142 Pe ) 120 Appendices Appendix B The Schmidt number is usually defined: υ Sc = (A.27) D Where υ is the kinematic viscosity and D is the diffusion constant. Assuming the uniform density, the radial distribution function g(r) = 1.0, Groot and Warren (1997), the dissipative contribution to viscosity can be calculated by: ηD = 2πγρ 15 ∫ ∞ dr r w D ( r ) (A.28) Whereas the kinetic contribution to viscosity is defined: η K = ρν K = D / 2, (A.29) where D is the diffusion constant and calculated as: D= ∞ dt v i (0) ⋅ v i (t ) = τ k BT ∫0 τ = 4πγρ ∫ ∞ (A.30) dr r w D ( r ) (A.31) The viscosity can now be calculated: (sum of kinetic and dissipative viscosities) η =ηK +ηD (A.32) The Schmidt number is calculated: Sc = ν D = η ρD (A.33) With weight function wD (r ) = − r , the viscosity and the Schmidt number can be calculated by: η= 315k BT 512πγρ rc5 + 128πγ rc3 51975 (A.34) Sc = η (2πγρ rc4 ) ≈ + ( ρ D ) 1999k BT (A.35) 121 Appendices It is important to note that the viscosity of DPD fluid is a resulting property of a specific set of DPD parameters rather than the input parameter, thus the Eq. (A.34) is used for roughly predicting the viscosity of the simple DPD fluid. It is very useful for adjusting the DPD parameters to meet the viscosity of real fluid. To render the equations dimensionless in DPD, a specific scaling has to be applied to meet the standard units. Like other methods, the scaling law requires to choose three independent units, in DPD the length unit [σ], the mass unit [m] and the energy unit [ε] are often chosen corresponding to the characteristic length, mass and energy of the simulating fluid. The time unit [t] directly follows from the above and is given by ( mσ / ε ). Particularly, for the length unit, it is chosen based on the characteristic length of specific fluid, for instant in case modelling polymer chain the contour length of polymer chain should be scaled with the length of the modelled chain, while modelling fibre the length of fibre may be used for defining length unit. 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Multiphase Flow, 16(4):639, (1990) Zirnsak M. A., Hur D.U., Boger D.V., Normal stresses in fibre suspensions, J. Non-Newtonian Fluid Mech., 54:153, (1994) 135 [...]... Understanding the rheology of fibre suspensions is the key in processing of shortfibre composites Historically, rheological models of fibre suspensions are constructed based on: 1 The motion of an individual fibre in a homogeneous media: this helps to understand the reciprocal influence between the kinematics of the suspending fluid and the orientation of the suspended particle or fibre 2 A suspension of. .. Chapter 1: Introduction prediction model for predicting rheological properties of fibre suspensions is suggested Up to this point, fibre suspensions in Newtonian or non -Newtonian fluids are simulated by integrating the fibre model with the DPD Newtonian fluid or the viscoelastic VNADPD fluids The implementations of these suspensions are presented in chapter 5 The rheological properties of fibre suspensions. .. significance in industries, the rheology of short -fibre suspensions has been studied intensively during the last few decades In general, fibre suspensions are characterized in terms of their volume faction, φ = nπd2l/4, and aspect ratio, aR = l/d, where n is the number of density, l is the fibre length and d is the diameter of the fibre [Doi and Edwards (1978a, b)] The concentration of fibre suspensions. .. A novel DPD model of fibre suspensions is proposed and presented in Chapter 4 The fibre- solvent interaction and the fibre- fibre interaction are thoroughly studied and investigated through the single-rod problem Besides, in DPD, polymeric liquids are usually modelled by suspending some polymer chains in Newtonian fluids [Kong, Manke, and Madden (1994, 1997); Schlijper, Hoogerbrugge, and Manke (1995)]... properties of fibre suspensions in both Newtonian and non -Newtonian fluids In Chapter 2, we review available models of fibre suspensions and their associated problems Since many contributions have been made in this field, we will focus attention only on several key models that are directly relevant to the particular subject of our study In Chapter 3, we will introduce the DPD method in detail The governing... et al., Fibre Suspensions in Newtonian and Non -Newtonian Fluids: DPD Simulations and a Model Prediction, 2005b - Duong-Hong et al., A DPD Model for Simulating Rheological Properties of Fibre Suspensions in Viscoelastic Media, 2005a 10 Chapter 1: Introduction - Duong-Hong et al., Numerical Simulation of Soft Solids by the Versatile Network Approach: Application to a Neuroprobe entering a brain tissue,... systems and boundary effects Fibre- fibre interaction is a crucial effect in non- dilute suspensions since it influences strongly the rheological properties of suspensions as well as the orientation of fibres Many authors have attempted to tackle this issue by both experiments and numerical simulations From an experimental view point, it is very difficult to observe directly the effects of fibre- fibre interactions... reported several experiments of semi-concentrated suspensions in Newtonian and non -Newtonian fluids and the results are compared with the model of Dinh and Armstrong However, the discrepancy between them is clearly observable Folgar and Tucker (1984) developed an evolution equation for concentrated fibre suspensions, where the particle-particle interaction is taken into account by adding a diffusion term to... Manke, and Madden (1994, 1997); Schlijper, Hoogerbrugge, and Manke (1995)]; colloids [Koelman and Hoogerbrugge (1993); 7 Chapter 1: Introduction Boek et al (1996, 1997)]; and multi-phase fluids [Coveney and Español (1997); Coveney and Novik (1996); Novik and Coveney (1997)] In this thesis, the DPD method is further developed to efficiently model fibre suspensions in both Newtonian and non -Newtonian fluids. .. V , or φ aR ≤ 1 In semi-concentrated regime, the 2 condition 1 < φ aR ≤ aR applies so that each fibre is confined in a volume of d 2l < V < dl 2 In this regime the fibres have only two rotating degrees of freedom since the average spacing between two neighbouring fibres is greater than the fibre diameter but less than the fibre length Finally, a suspension satisfying the condition of φ aR > 1 , is . Suspensions in Newtonian and non -Newtonian Fluids Summary In this thesis, Dissipative Particle Dynamics (DPD) models of fibre suspensions in Newtonian and non -Newtonian fluids are developed and. for predicting rheological properties of fibre suspensions is suggested. Up to this point, fibre suspensions in Newtonian or non -Newtonian fluids are simulated by integrating the fibre model. engineering models, which can qualitatively predict the rheological properties of fibre suspensions in both Newtonian and non -Newtonian fluids. In Chapter 2, we review available models of fibre