Dissipative particle dynamics simulation of micro concentric eccentric annular flows

DISSIPATIVE PARTICLE DYNAMICS SIMULATION OF MICRO-CONCENTRIC/ECCENTRIC ANNULAR FLOWS PENGFEI CHEN A THESIS SUBMITTED FOR THE DEGREE OF MASTER OF ENGINEERING DEPARTMENT OF MECHANICAL ENGINEERING NATIONAL UNIVERSITY OF SINGAPORE 2004 ACKNOWLEDGEMENTS I am very grateful to my supervisors, Professor Nhan Phan-Thien and Associate Professor Yeo Khoon Seng for giving me the great opportunity to study in such an interesting area. I would also like to express my sincere gratitude to them for their constant guidance and encouragement throughout the course of this work. I would like to thank Professor Fan Xijun, Dr. Dou Huashu, Dr. Chen Shuo, Dr. Lu Zhumin, Dr. Shi Xing, Mr. Wu Tao, Ms. Luo Chunshan and Ms. Zhao Xijing for their assistance and friendship. My entire family deserves a special gratitude for their unlimited support, encouragement and love throughout my stay in NUS. Finally, I would like to give my acknowledgement to the National University of Singapore for their Research Scholarship. Thanks are also due to all others who have helped me in one way or another in this effort. I Table of Contents Acknowledgement..........................................................................................................I Summary......................................................................................................................IV Nomenclature................................................................................................................V List of Figures...........................................................................................................VIII List of Tables..............................................................................................................XII Chapter 1 Introduction.......................................................................................1 1.1 Background...............................................................................……..1 1.2 Literature review.......................................................................……..4 1.3 Objective of this research project..............................................……..6 Chapter 2 Methodology.......................................................................................9 2.1 Basic equations of DPD method................................................……..9 2.2 Numerical Scheme..................................................................……...14 2.3 Initial Particle Models in concentric/eccentric flow field....................................................................................................16 Chapter 3 Steady Concentric/eccentric flows of simple DPD fluids at finite Reynolds numbers..................................................................……..20 3.1 Implementation of non-slip boundary condition in DPD...................................................................................................21 3.1.1 Boundary condition used in this thesis…………………....24 3.1.2 Comparison of boundary conditions: Poisseuille flow…....26 3.1.3 Comparison of boundary conditions: Concentric rotating cylinders flow……………………………………………..30 II 3.2 Effects of some DPD Parameters on simulation…………………...32 3.2.1 Effect of fluid particle density…………………….………33 3.2.2 Effect of conservative force factor between DPD particles………………………………………….………...36 3.3 Steady Circular Couette flow and eccentric flow of simple DPD fluids at finite Reynolds numbers………………..…………………….......43 3.3.1 Circular Couette flow of simple DPD fluids at finite Reynolds numbers…………………………………….…..43 3.3.2 Eccentric flow ( ω out / ω in = 0 ) of simple DPD fluids at finite Reynolds numbers…………………………………….…..48 Chapter 4 Steady Circular/eccentric flows of FENE Chain Suspension at finite Reynolds numbers……………………………….…………60 4.1 Circular Couette flow of FENE chain suspension at finite Reynolds numbers………………………………………………………….....62 4.2 Eccentric flow ( ω out / ω in = 0 ) of FENE chain suspension at finite Reynolds numbers………………..………………………………...70 Chapter 5 Conclusion and future works..........................................……..….80 5.1 Conclusion…………..…………………………………………...…80 5.2 Future works…………………………….………………………….81 Appendix A…………………………………………….………………………...…...84 Appendix B……………………………………………………………………...…....87 Reference…………………………………………………………………………88-92 III Summary Dissipative Particle Dynamics (DPD) is a fairly new method for simulating complex fluid flows and other colloidal phenomena. It is a mesoscopic method and offers the possibility of capturing some degree of molecular-level detail while conforming to continuum hydrodynamics at larger length scales. In this thesis, a new implementation of the no slip boundary condition in the modeling of solid boundaries is studied. This boundary is implemented to simulate the planar Poisseuille and circular Couette flow, and the results compare excellently with similar results derived by more traditional CFD methods. The effects of two important DPD parameters (particle density and conservative coefficient) are studied. It is shown that these two parameters affect the simulation accuracy considerably and should be carefully set. Furthermore, to confirm the ability of DPD method to provide numerically accurate results in simulating complex flow with rather complicated boundary conditions, the DPD method is employed to study the flow behavior of three dimensional microscopic concentric/eccentric flows at finite Reynolds numbers. A simple DPD fluid (made up of simple DPD particles) and then bio-molecular suspensions (FENE chains are used to model DNA macromolecules) are studied in detail respectively. IV Nomenclature It is not practical to list all the symbols that have been used. Below the author list the more important ones. Some of the symbols are defined as they are used. There are also occasions where the same symbol is assigned a different meaning in a different context, but the meaning should normally be clear from the usage. English alphabets: D Diffusion constant fij Interparticle force on particle i by particle j Fe External force exerting on particle i FijC Conservative force on particle i by particle j FijD Dissipative force on particle i by particle j FijR Random force on particle i by particle j H Spring constant of FENE chain model kB Boltzmann constant Lc Length of one segment of a FENE chain n Particle density N1, N2 The first and second normal stress Nb Number of beads in the FENE chain model p Constitutive pressure Re Reynolds number, ρ Rin2 ω η V Rin Radius of inner cylinder Rout Radius of outer cylinder ri Position vector of particle i rm Maximum length of one chain segment of FENE chain model Sc Schmidt number T Temperature of the system ui Peculiar velocity of particle i vi Velocity vector of particle i • v i (t ) Acceleration of the particle i at the instant t v i (t ) Prediction of the velocity of the particle i at the instant t Greek alphabets: α ij Maximum repulsion force between particles i and j ω D (r ) r-dependent weighting function of dissipative force ω R (r ) r-dependent weighting function of random force γ Coefficient controlling the strengths of the dissipative force σ Coefficient controlling the strengths of random forces θ ij Gaussian variable α f α f w Maximum repulsive forces between fluid particles and wall particles α ww Maximum repulsive forces between wall particles f Maximum repulsive forces between fluid particles VI ρ Fluid density η Viscosity δ Displacement of inner cylinder axis from outer cylinder axis κ Radius ratio, Rin/Rout c Mean annular gap width, Rout-Rin ε Eccentricity ratio, δ / c ω Rotating angular velocity of the cylinder ζ ζ -coordinate of bipolar coordinate system θ θ -coordinate of bipolar coordinate system Φ Volume fraction of a dilute suspension VII List of Figures Figure 1 Illustration of generating fluid particles in an annular area………….…18 Figure 2 Illustration of generating wall particles in angular direction……….…..19 Figure 3 Illustration of generating wall particles in radial direction………….….19 Figure 4 Illustration of the boundary reflection………………………….……….26 Figure 5 The fully developed velocity profile compares with the Navior-Stokes solution in Poiseuille flow……………………………………….……...28 Figure 6 The temperature profile of a simple DPD fluid in Poiseuille flow….…..28 Figure 7 Shear stress distribution of a simple DPD fluid in Poiseuille flow….….29 Figure 8 Profiles of the first (N1) and second (N2) normal stress differences of a simple DPD fluid in Poiseuille flow……………………….……….…...29 Figure 9 The Comparison plot of density profiles in Poiseuille flow…………….30 Figure 10 The comparison of pressure profiles in Poiseuille flow………………...30 Figure 11 The comparison of velocity profiles of a simple DPD fluid in circular Couette flow………………………………………………………….…32 Figure 12 Illustration of the effect of fluid particle density on velocity of a simple DPD fluid in circular Couette flow………………………………….….34 Figure 13 Comparison of temperature profiles of a simple DPD fluid in circular Couette flow with different conservative force coefficient α f f …….…38 Figure 14 Comparison of density profiles of a simple DPD fluid in circular Couette flow with different conservative force coefficient α f f ………………...38 VIII Figure 15 Comparison of velocity profiles of a simple DPD fluid in circular Couette flow with different conservative force coefficient α f f ……………..…39 Figure 16 Illustration of the effect of α ww on density of a simple DPD fluid in circular Couette flow……………………………………….……….…..41 Figure 17 Comparison of temperature profiles of a simple DPD fluid in circular Couette flow with different conservative force coefficient α ww …….…42 Figure 18 Comparison of velocity profiles of a simple DPD fluid in circular Couette flow with different conservative force coefficient α ww …………..……42 Figure 19 Geometry for Circular Couette flow……………………………….…....43 Figure 20 Simulated Vx, Vz velocity and streamline contours of a simple DPD fluid in circular Couette flow………………………………………………....45 Figure 21 The fully developed tangential velocity profile of a simple DPD fluid in circular Couette flow………………………………………….………...46 Figure 22 Density and temperature profiles of a simple DPD fluid in circular Couette flow…………………………………………………………………..…46 Figure 23 Pressure profile of a simple DPD fluid in circular Couette flow…….…47 Figure 24 Shear stress distribution of a simple DPD fluid in circular Couette flow……………………………………………………………………..47 Figure 25 Profiles of the first and second normal stress differences of a simple DPD fluid in circular Couette flow………………………………………..….48 Figure 26 Bipolar coordinate system and geometric parameters of the eccentric annular region…………………………………………………………...50 IX Figure 27 Streamline patterns of a simple DPD fluid in eccentric flow ( ω out / ω in = 0 ) with different Reynolds numbers…………………………….………….58 Figure 28 Comparison of polar angles of separation and reattachment points of simple DPD fluids in eccentric flow ( ω out / ω in = 0 )…………….……....58 Figure 29 Comparison of tangential velocity profiles in the wide gap ( θ = π ) of simple DPD fluids in eccentric flow ( ω out / ω in = 0 , Re=37.2)…….…….59 Figure 30 Comparison of tangential velocity profiles in the narrow gap ( θ = 0 ) of simple DPD fluids in eccentric flow ( ω out / ω in = 0 , Re=37.2)……….….59 Figure 31 Comparison of velocity profiles of FENE chain suspensions of different concentrations in the Circular Couette flow……………………….…….64 Figure 32 Comparison of pressure profiles of FENE chain suspensions of different concentrations in the Circular Couette flow…………………….……….65 Figure 33 Comparison of Shear stress profiles of FENE chain suspensions of different concentrations in the Circular Couette flow…………….……..65 Figure 34 The conformation of some typical FENE chains in circular Couette flow withΦ=0.03883………………………………………………….……….67 Figure 35 The conformation of some typical FENE chains in circular Couette flow withΦ=0.11650…………………………………………………….…….68 Figure 36 The conformation of some typical FENE chains in circular Couette flow withΦ=0.34951……………………………………………….………….69 Figure 37 Streamline patterns of FENE chain suspension in eccentric flow X ( ω out / ω in = 0 ) with different volume fractions……………….………….73 Figure 38 Comparison of tangential velocity profiles in the wide gap ( θ = π ) and narrow gap ( θ = 0 ) of FENE chain suspension in eccentric flow ( ω out / ω in = 0 )……………………………………………………………74 Figure 39 Illustration plot of the development of the back flow of FENE chain suspension ( φ = 0.11650 ) in the eccentric flow ( ω out / ω in = 0 )………...76 Figure 40 Snapshots showing the detailed backflow of one FENE chain in eccentric flow……………………………………………………………………...79 XI List of Tables Table 1 Angles of separation, reattachment and eddy center of different volume fractions………………………………………………………………...73 XII Chapter 1 Introduction 1.1 Background The numerical simulation of hydrodynamic interactions between suspended particles and the surrounding fluid phase is of interest in many engineering applications associated with particle transport, such as colloids, polymers, aerosols and physiological system [1]. The properties of these systems are often determined by their mesoscale structures, i.e. between the atomistic scale and the macroscopic scale, thus endowing a complex fluid with unique and interesting features [2]. Dynamic simulation of these systems presents unique problems that are difficult to address with established methods. On the one hand, the time and length scales of interest make simulation by molecular dynamics (MD) impractical, a limitation that is not likely to be solved in the near future, even with the rapid increases in computational speed that are expected to develop. On the other hand, colloidal length scales are often almost the same order as the flow domain size. (The ratio of the colloidal length to the characteristic length of flow field can be taken as the equivalent Knudsen number Kn of the flow. When Kn = O(1) , the flow may not be treated as a continuum). So purely continuum approaches, such as conventional computational fluid dynamics (CFD), are unacceptable, requiring as they do that one leaves out so many details at the molecular level. Moreover, people still cannot make the full connection from atomistic length-scale to the macroscopic world. Hence to obtain a better understanding of the phenomena that occur in mesocale, some intermediate 1 simulation techniques are developed that are aimed at a length-scale larger than the atomistic scale, but smaller than the macroscopic scale. Dissipative particle dynamics (DPD) is a so-called mesoscopic simulation method that provides one possible means of bridging the gap between purely molecular and continuum-level treatments. Dissipative particle dynamics is a stochastic simulation technique introduced by Hoogerbrugge and Koelman [3] in 1992 to simulate complex fluid dynamical phenomena. DPD combines features from molecular dynamics and lattice-gas automata (LGA) by introducing a LGA-type of time-stepping into MD schemes. In contrast to molecular dynamics simulations, the particles are supposed to represent the fluid on a mesoscopic level rather than a molecular level. For the simulation of macroscopic fluid dynamic phenomena, this implies an advantage of computational effort [4]. Though Brownian Dynamics Simulation (BDS), LGA and Lattice Boltzmann (LB) are mesoscale simulation methods, it is difficult for BDS to deal with complex flow field and for LGA and LB to cope with complex fluids and DPD has the advantage of more flexibility. The basic unit in DPD system is a set of discrete momentum carriers called particles that move in continuous space and discrete time-steps. The momentum carriers are coarse grained entities which are no longer regarded as molecules in a fluid but rather representing the collective dynamic behavior of a large number of molecules (a fluid “particle”). When first introduced by Hoogerbrugge and Hoelman [3], Dissipative 2 Particle Dynamics is such a method for simulation of the motion of this kind of “fluid particles” often referred to in continuum mechanics. Conceptually this method amounts to dividing up the molecules of a flow field into groups or packets, which have dimensions many times larger than the mean free path of an individual molecule. The mass m of each packet is localized to a point. As the points move, their interaction is ‘soft’ as a result of the fact that the packets are deformable, and this deformation is accompanied by a dissipation of energy. In addition, because of their small colloid-like dimensions, the thermal motion of the molecules in the packets gives rise to a random or Brownian contribution to their motion. A time step in a DPD simulation therefore consists of summing interactions that consist of three terms: a conservative repulsion that accounts for steric and energetic interactions between the molecules of interacting packets and this repulsion force prevents particles from overlapping [5], a dissipative interaction that accounts for energy that is lost due to internal friction or viscosity within a packet, and a random step that arises from the collective thermal motion of the molecules within a packet. The interactions are summed over all pairs of particles, in a way that guarantees that linear and angular momentum are conserved. Unless an explicit connection to molecular-level interactions is needed, the parameters that govern these interactions are chosen in any way that reproduces the continuum-level dimensionless groups that determine the behavior of a particular system. The last two terms, which account for dissipation and random motion, are necessarily 3 coupled by a fluctuation-dissipation theorem and the principle of equipartition of energy. These two terms combine to create a continuous pseudofluid in which the particles are suspended and free to interact hydrodynamically. The original algorithm of Hoogerbrugge and Koelman [3] did not satisfy this requirement, leaving in doubt whether a simulation could reach a true equilibrium, even at long times and in the absence of bulk motion. A slight revision of the original algorithm, produced by Espanol and Warren [6], remedied this problem, and has been used in most applications since that time. The method has received considerable theoretical support in other areas as well. Marsh et al. [7] make explicit connections between DPD and the Navier-Stokes equations by deriving the Fokker-Planck-Boltzmann equation for the single-particle distribution function, and solving it by the Chapman-Enskog method. Their derivation includes expressions for transport properties such as the shear viscosity that are valid in the limit of strong damping, where conservative repulsive interactions are negligible. Flekkoy and Coveney [8] discussed creating DPD particles by grouping molecules in MD simulations together. 1.2 literature review Since DPD method is a fairly recent development and the method is still evolving, only the fundamental and representative papers will be reviewed here. Literature review about the flow field in concentric/eccentric flows will be noted in the section on results and discussion. 4 A key and attractive feature of DPD is its ability to reproduce continuum-level fluid mechanics over large enough length scales, even in the presence of inertia. However, in spite of the growing literature on the applications of DPD to various problems, there are still relatively few direct, quantitative comparisons between calculations done with DPD and well-established analytical and numerical results. Those comparisons that do exist pertain exclusively to low Reynolds numbers. Simulation results obtained by DPD have also been reported for several problems of interests. In their original paper, Hoogerbrugge and Koelman [3] calculated the flow-induced drag on a cylinder in a periodic array, and compared it with result reported by Sangani and Acrivos [9]. This comparison was made in the limit of low Reynolds number, where inertial effects in the flow are negligible. Boek et al. [10, 11] studied the rheological properties of colloidal suspensions of spheres and rods using dissipative particle dynamics, and measured the viscosity as a function of shear rate and volume fraction of the suspended particles. Furthermore Boek and van der Schoot [12] used DPD to study fluid flow through a periodic array of cylinders as a model for fluid filtration through a porous medium, and discussed the resolution effects in DPD. Both the calculations of Hoogerbrugge and Koelman [3] and those of Boek et al. [10, 11, 12] were done with the original algorithm, without the modification proposed by Espanol and Warren [6]; Boek et al [12] argued that the modification does not significantly alter the calculations of suspension rheology. Model polymers have been 5 constructed by linking DPD particles together with springs, and polymer dynamics and associated parameters have been studied by Kong et al. [13]. In addition, polymer-surfactant aggregation and micelle formation have been simulated by Groot [14], and an extension of DPD to non-Newtonian flows (Bosch [4]) has been proposed. More recently, Fan et al. [15] presented their simulation results for macromolecular suspension flows through microchannels. They also studied the Poiseuille flow of simple DPD fluids and FENE (Finitely Extendable Nonlinear Elastic) chains suspension and found that simple DPD fluids behave just like a Newtonian fluid while FENE chains suspension can be fitted by dilute suspensions. 1.3 Objective of this research project As noted above, DPD has its own advantages to numerically simulate the hydrodynamic interactions of various complex systems: such as polymer suspensions [13, 16, 17, 18], colloids [10, 11, 19] and multiphase fluids [20, 21, 22]. Although it is a promising technique for complex fluids, there are very few microfluidic applications of DPD in complex systems reported. The most recent literature that combines such a microscopic application of DPD in complex system as well as giving a quantitative simulation is noted in the paper of Fan et al. [15], but what is studied in their paper is about the simple Poiseuille flow. In this thesis, a more complex flow problem is studied: steady microscopic concentric/eccentric flow at finite Reynolds numbers. This work is also stimulated by the recent innovations in MEMS devices, especially in micro bearing applications of MEMS. This area is not fully investigated and 6 understood today. Unlike the Poiseuille flow, firstly, the boundary condition of concentric/eccentric flow is more complicated because it is a closed flow field and the implementation of the boundary conditions will have a more significant impact on the simulation results. Secondly, the flow patterns in concentric/eccentric flows are more intricate compared with that of Poiseuille flow at low Reynolds numbers, especially for eccentric flow (there is a clear recirculation region present in the flow field). Same with what are presented in the paper of Fan et al., two kinds of fluids are studied here: simple DPD fluids and FENE chains suspension. Some results on the deformation and migration of FENE are also reported. It should be noted that the purpose of this research program is not to propose DPD as an instrument for use in computational fluid mechanics, but rather to test its ability to capture continuum fluid mechanical effects in the presence of significant inertia. Even for colloidal systems, in which length scales are shorter than 10 µ m , the ability to capture inertial effects accurately is important. In their study of the Brownian motion of interacting particles, for example, Hinch and Nitsche [23] incorporated O(Re) corrections to the equations of motion for each frequency of oscillation. They found a nonlinear force of interaction between the particles that, even for very small Reynolds number, is of the order O(kT / R) . Here kT is the product of Boltzmann’s constant and temperature, and R is the particle radius. Such an interaction has an effect on the distribution of the particles, which in turn affects the thermodynamics and rheological 7 properties of a colloidal suspension. Since fluid simulation by DPD is a fairly new development, the results presented in this thesis are focused on a number of test problems. These problems include: improving the dynamic behavior of DPD method, studying the effects of basic DPD parameters ( α and n) on simulation results, and accessing the ability of DPD to provide numerically accurate results in simulating complicated flows with complicated boundary conditions. The simulation results presented in this work confirm the ability of DPD to provide quantitatively accurate results for complicated flows, provided conditions are such that the compression of the DPD fluid is not significant. In the present work, the DPD method is used to calculate three-dimensional flows at finite Reynolds numbers, and examine conditions under which Newtonian fluid and non-Newtonian fluid behaviour is reproduced quantitatively. First, the DPD method is outlined. This is followed by the description of a new implementation of DPD boundary condition, and then the effects of DPD parameters are studied, with some simulation details presented. The simulation details for concentric/eccentric flows with simple DPD fluids and FENE chains suspensions are presented next. 8 Chapter 2 Methodology 2.1 Basic equations of DPD method The DPD system consists of a set of interacting “particles”, whose time evolution is governed by Newton’s equation of motion. For a simple DPD particle i , dri = vi , dt dv i = ∑ fij + Fe , dt j ≠i (1) where ri and v i are the position and velocity vectors of particle i , and the unit of mass is taken to be the mass of a particle, so that the force acting on a particle is equal to its acceleration. Fe is the external force. fij is the interparticle force on particle i by particle j , which is assumed to be pairwise additive. The dynamic interactions between the particles are composed of two parts, dissipative and stochastic, complementing each other to ensure a constant value for the mean kinetic energy, kBT , of the system. The force fij consists of three parts, a conservative force, FijC , a dissipative force, FijD , and a random force, FijR : f ij = ∑ FijC + FijD + FijR . (2) i≠ j Here the sum runs over all other particles within a certain cutoff radius rc , which is taken as the unit of length, i.e. rc =1 (see Appendix A). Since the time average of the dissipative and fluctuation forces is zero, they do not feature in the equilibrium behavior of the system, which is governed solely by conservative forces. The conservative force, FijC , is a soft repulsion force acting 9 along the line of centers and is given by α (1 − r / r )rˆ ij c ij  ij F = ij 0 C where (r < r ) ij c , (r ≥ r ) ij c (3) α ij is the maximum repulsion force between particles i and j ; and rij = ri − r j , rij = rij , rˆij = rij / rij is the unit vector directed along j to i . The dissipative force or drag force, FijD , on particle i by particle j , is given by FijD = −γ wD (rij )(rˆij ⋅ v ij )rˆij , (4) where ω D (r ) is an r-dependent weighting function that vanishes if r ≥ rc = 1 , vij = vi − vj . γ is a coefficient controlling the strengths of the dissipative force and characterizing the extent of dissipation in a single simulation step. The negative sign in front of γ indicates that the dissipative force is opposite to the relative velocity v ij . The random force or stochastic force, FijR , on particle i by particle j , is given by FijR = σ wR (rij )θij rˆij , (5) where ω R (r ) is also an r-dependent weight function that vanishes for r ≥ rc = 1 . σ is the coefficient controlling the strengths of random forces, and θ ij is a Gaussian variable with zero mean and variance equal to δ t −1 , where δ t is the time step: θ ij (t ) = 0 and θ ij (t )θ kl (t ' ) = (δ ikδ jl + δ ilδ jk )δ (t − t ' ) , (6) with i ≠ j and k ≠ l . The strength of the random force is the integral correlation 10 over a time scale considerably larger than its correlation time scale: 2 σω R (rij )  = +∞ ∫ FijR (t ) ⋅ FijR (t + τ ) dτ . (7) −∞ The detailed balance condition, similar to a Fluctation-Dissipation theorem relating the strength of the random force to the mobility of a Brownian particle, requires that ω (r ) = ω R (r )  D 2 and γ = σ2 2kBT , (8) where kB is the Boltzmann constant and T the temperature of the system. This will ensure that the temperature remains constant. We use the following weight function to improve on the Schmidt number for the system (Fan et al. [15]), instead of the quadratic function (1 − r / rc ) 2 that is usually adopted, 2  1 − r / rc 0 ω D (r ) = ω R (r )  =  ( r < rc ) . ( r ≥ rc ) (9) This weight function yields a stronger dissipative force between particles than that from the standard quadratic force for a given configuration of particles and interaction strength. It should be noted that in DPD, the random force between two particles, FijR , represents the results of thermal motion of all molecules contained in particles i and j . It tends to “heat up” the system. The dissipative force, FijD , on the other hand, reduces the relative velocity of two particles and removes kinetic energy from their mass centre to cool the system down. When the detailed balance is reached, the 11 system temperature will approach the given value. The dissipative and random forces act like the thermostat in molecular dynamics (MD). When simulating complex fluids and flows, simple DPD particles, described above, are used to model solvent or suspending fluid. The solid walls can be modeled by frozen DPD particles. The Finitely Extendable Nonlinear Elastic (FENE) chain is a model commonly used to model flexible polymer molecules in rheology; it is used in this thesis to model bio-molecules. Beads of the polymer chain are replaced by DPD particles. The intermolecular forces will act on these particles and should be added to the right hand side of Eq. (1). In the FENE chain [24], the force on bead i due to bead is j is FijS = − Hrij , 1 − (rij / rm) 2 (10) where H is the spring constant, and rm is the maximum length of one chain segment. The spring force increases drastically with rij / rm and becomes infinity as rij / rm =1.0. This model can capture the finite extensibility of the molecules and predict a shear-rate dependent viscosity and finite elongational viscosity. The mass of the beads is assumed to be unity, which is as same as that of other simple DPD fluid particles. 12 The time constant is important in characterizing molecular motion and can be formed from model parameter. Two constants with the time dimension can be obtained for the FENE spring [24], λH = ζ 4H , (11) where ζ is the friction coefficient of a bead, and λQ = ζ rm2 12kBT . (12) The FENE parameter, b, is the ratio of these two constans b= 3λ Q λH = Hrm2 . kBT (13) Chain models usually have a spectrum of relaxation times [24]. There is no closed form expression for the relaxation time spectrum for FENE chains; however, a modification of FENE chain, called the FENE-PM chain, has the same spectrum as the Rouse chain (bead and Hookean spring chain) [24]. A time constant can be defined for FENE-PM chain as [25]: N b2 − 1 b λ fene = λH , 3 b+3 (14) where Nb is the number of beads in the chain. The contour length is an appropriate parameter to represent the molecular size. If Lc represents the length of one segment of a molecular chain, the contour length of the molecular chain is simply ( Nb − 1) Lc . We may use the equilibrium length of a segment as an estimation of Lc . For the FENE spring [24], 13 3rm2 . Lc = b+5 (15) 2.2 Numerical Scheme The initial configuration of fluid and wall particles are generated separately by a pre-processing program and read in as input data. The total number of particles depends on the size and geometry of the flow domain, the densities of the fluid and wall materials. The initial velocities of fluid particles are set randomly according to the given temperature but the wall particles are frozen. At the beginning of the simulation the particles are allowed to move without applying the external force and rotation until the thermodynamic equilibrium state is reached. Then the rotating velocity is applied to the inner or outer cylinder wall particles and the non-equilibrium simulation starts. The forces computation and time updates are conducted with Cartesian coordinate system and the samples averaging work are carried out with polar coordinate system for concentric flow and double-polar coordinate system for eccentric flow. Since the dissipative force is dependent on the velocity, a modified version of the velocity-Verlet algorithm [26] is used. This algorithm can be described as follows: 14 1 ri (t + ∆t ) = ri (t ) + ∆tv i (t ) + ∆t 2 v i (t ) , 2 (16) v i (t + ∆t ) = v i (t ) + λ∆tv i (t ) , (17) v i (t + ∆t ) = v i (ri (t + ∆t ), v i (t + ∆t )) , 1 v i (t + ∆t ) = v i (t ) + ∆t [ v i (t ) + v i (t + ∆t ) ] , 2 (18) (19) where v i (t ) denotes the acceleration (i.e., the total net force) of the particle i at the instant t and the position ri (t ) , v i (t + ∆t ) is the prediction of the velocity of the particle at the instant t + ∆t , and λ is an empirically introduced parameter, which account for some additional effects of the stochastic interactions. If the total force is velocity independent, the standard velocity-Verlet algorithm is recovered for λ =0.5. Groot and Warren [26] found that the optimum value of λ is 0.65. For this value of λ , the time step can be increased to ∆t =0.06 without loss of temperature control in simulating an equilibrium system with the fluid density ρ =3 and random force coefficient σ =3 [26]. The stress tensor is calculated using the Irving-Kirkwood model [27]: 1 V   rijfij   ∑i muiui + ∑i ∑ j >i  = −n ∑ mu u + ∑∑ r f S=− i i i ij ij i (20) , j >i where n is the number density of particles, ui = Vi − V ( X) is called the peculiar velocity of particle i (velocity relative to the local bulk velocity field), with Vi being the velocity of particle i and V ( X) being the stream velocity at position X , and ... denotes the ensemble average. The first term on the right side in the above 15 equation describes the contribution to the stress from the momentum transfer of DPD particles and the second term from the interparticle forces. For simple DPD particle i , the forces fij are calculated from Eqs. (2) to (5). If particle i denotes a bead of molecule chains, fij should include the total spring force on the bead. The constitutive pressure can be determined from the trace of the stress tensor S : 1 p = − trS . 3 (21) 2.3 Initial Particle Models in concentric/eccentric flow field Since we assume that the purpose of the simulation is to study the equilibrium fluid state, then the nature of the initial particle configuration should have no influence whatsoever on the outcome of the simulation. In choosing the initial coordinates, the usual method is to position the atoms at the sites of a lattice whose unit cell size is chosen to ensure uniform coverage of the simulation region. Typical lattices used in three dimensions are the face-centered cubic(FCC) and simple cubic, whereas in two dimensions the square and triangular lattices are used; if the goal is the study of the solid state, then this will dictate the lattice selection. There is little point in laboriously constructing a random arrangement of atoms, typically using a Monte Carlo procedure to avoid overlap, since the dynamics will produce the necessary randomization very quickly. An obvious way of reducing equilibration time is to base the initial state on the final state of a previous run. 16 The function of FCC arrangement (with the optional of unequal edges) is to generate four particles per unit cell. To generate more particles in one cell, there are other arrangement methods. For example, the diamond lattice, which can generate eight particles per unit cell, is a slightly more complicated form of the FCC code since the lattice is most readily defined as two staggered FCC lattices, one of which is offset along the diagonal by a quarter unit cell. Similarly, to distribute 12 particles in one unit cell, three staggered FCC lattices can be built up with offset of 0.125 unit cell. These three arrangements are used in this thesis to generate the required particle density respectively: n = 4 , n = 8 and n = 12 . (1) Fluid Particle disposition First, particles are distributed evenly in a prism whose square section is bigger than the outer circle; then deduct particles outside of the outer circle and particles within the inner circle, and the shade area shown in Figure 1 represents the needed flow domain. In this three-dimensional problem, the periodic boundary condition is applied to fluid boundaries of the axis direction, which is y direction of Figure 1. Particles that leave the simulation region of y direction immediately reenter the region through the opposite face. For this steady, low Reynolds number case, this application of periodic boundary condition is workable and feasible. 17 Z Y X Figure 1 Illustration of generating fluid particles in an annular area. (2) Wall Particle disposition Solid walls are modeled by frozen DPD particles; three wall layers with staggered distribution in angular (Fig.2) and radial (Fig. 3) directions. Due to the soft interaction between DPD particles, it is difficult to prevent particles from penetrating the wall if the wall particle density is too low compared with the fluid particle density; however, a higher density of the wall will produce a stronger repulsive force between the wall DPD particles and fluid DPD particles, and consequently a large density fluctuation appears near the wall. So normally the wall particle density is chosen to be of the same order of that of fluid particles. 18 On the other hand, the repulsive force between wall particles and fluid particles is also determined by the value of repulsive force coefficient between them. So the wall particle density should be set rationally, although there is no certain mode to distribute the wall particles for such a special wall boundary contour. Figure 2 Illustration of generating wall Figure 3 Illustration of generating wall particles in angular direction. particles in radial direction. 19 Chapter 3 Steady Concentric/eccentric flows of simple DPD fluids at finite Reynolds numbers Some of implementation details on DPD are briefly introduced here. The program is usually divided into 2 parts: a pre-processing program and a main one. In pre-processing program, the initial particle configurations for the fluid, wall and DNA chains (if any) are generated and read into the main program as input data. When the density of fluid and the geometry of flow domain are defined, the wall and DNA chains particles if any will be set, and then all fluid particles are initially located at the sites of a face-centered cubic (FCC) lattice. In the main program, the initial velocities of fluid particles are set randomly according to the given temperature. The velocities of wall particles are set at zero. At the beginning of simulation the fluid particles are allowed to move without applying any external forces until a thermodynamic equilibrium state is reached. Then the external force field is applied to fluid particles and the non-equilibrium simulation starts. In this kind of simulation, computing the interaction forces between particles takes the most computational time. To cope with this problem, here a cell sub-division and linked-list method described by Rapaport [28] is applied. In this method, firstly the computational domain is divided into a grid of cells. Then linked lists associate 20 particles with the cells in which they reside at any given instance. The linked list therefore associates a pointer with each data item and to provide a non-sequential path through the data. Thus it needs only a one dimension array to store the list and makes the search of neighboring particles more efficiently. Three parts will be discussed below: firstly the boundary conditions used in this thesis are outlined, afterwards the effects of two DPD parameters on the simulation are analyzed, then the simulation results of simple DPD particles in concentric/eccentric flows are presented. The simulation results of DNA polymer chains suspension in concentric/eccentric flows will be presented in the next chapter. 3.1 Implementation of non-slip boundary condition in DPD As with any other methods in computational fluid dynamics, the issue of boundary conditions has to be addressed in DPD. Until now, the boundary conditions in DPD are usually treated in two ways: 1. In order to simulate a shear flow such as that generated in a Couette geometry, the Lees-Edward technique has been used [3, 6, 19], which is a way of avoiding the modelization of physical boundaries. 2. When considering the boundary conditions on the surface of solid objects as in the modelization of colloidal suspensions, the method of “freezing” some portions of the fluid described by DPD particles has been used [7, 29, 30, 31]. In 21 this model, those particles are fixed but still can interact with other free particles [32]. Due to the soft repulsion between DPD particles, it is difficult to prevent fluid particles from penetrating the wall particles. Near-wall particles may not be slowed down enough and slip may then occur. To avoid this, higher density of wall particles and larger repulsive forces have to be adopted to strengthen the wall effects. This, however, results in large density distortions in the flow field, which is similar to what happened in MD simulation. Special treatments have been proposed to implement no slip boundary condition in DPD simulation without using frozen wall particles. Revenga et al. [33, 34] used effective forces to represent the effect of wall on fluid particles instead of using wall particles. For planar wall, the effective forces can be obtained analytically. But these forces are not sufficient to prevent fluid particles from crossing the wall. When particles cross the wall, a wall reflection is used to reflect particles back to the fluid. However, a slip velocity appears at the wall if the wall density is equal to that of the fluid, and no slip results only if the wall has a much higher density than that of the fluid. To cope with this problem, Willemsen et al. [35] added an extra layer of particles outside of the simulation domain. The position and velocity of particles in this layer are determined by the particles inside the simulation domain near the wall, such that the mean velocity of a pair of particles inside and outside the wall satisfies the given boundary conditions. Furthermore, to reduce the density distortion near the wall they created an extra second set of layer by shifting all 22 the particles being located between rc and 2rc from the boundary into this layer, and considering the repulsive forces between those particles and free particles. Obviously, this method is quite expensive computationally and very difficult to use for the complex surface of solid objects such as a sphere or cylinder. Since the effective forces are not sufficient for keeping the fluid confined and one has to specify what happens when a DPD particle crosses the line that defines the position of the wall. Normally, there are three different possibilities: 1. Specular reflection: such that the parallel component of the momentum of the particles is conserved and the normal components is reversed. 2. Maxwellian reflection: where the particles are introduced back into the system according to a Maxwellian distribution of velocities centered at the velocity of the wall. 3. Bounce back reflection: in which both components of the velocity are reversed. In [34], Revenga et al. compared these three methods, and they found that for small dimensionless friction coefficient τ = γλ / dvT (where γ is the friction coefficient, λ is the average distance between particles, vT = kBT / m is the spatial dimension and is the thermal velocity), slip appears in specular and Maxwellian reflections and the bounce back produces stick boundary conditions for any values of τ . But for small τ , the bounce back scheme produces an anomalous temperature behavior. 23 Later, considering the merits and disadvantages of above methods, Fan et al. [15] present their model based on the above discussions and obtained good simulation results for Posieuille flow. They still used frozen particles to represent the wall, but near the wall a thin layer is assumed where the no slip boundary condition holds. To meet this boundary condition, they enforced a random velocity distribution in this layer with zero mean and corresponding to the given temperature. Similar to Revenga et al.’s reflection law, Fan et al. [15] further required that particles within this layer always leave the wall. The velocity of particle i in the layer is (n⋅ V ) − n⋅ V  ,  Vi = VR + n   → 2 → → R R (22) → where VR is the random vector and n the unit vector normal at the wall pointing into the fluid domain. The above equation means that the normal velocity component of fluid particles within the assumed thin layer will always move away from the wall and the parallel component of the momentum of the particle is conserved. In this section, a new boundary condition model of the frozen particles to represent the wall is presented which satisfies no-slip boundary condition and also has its own merits. Firstly, this boundary condition model is outlined, and then simulation results from this boundary condition are compared with those of Fan et al.’s model [15] in two cases: Poisseuille flow and circular Couette flow. 3.1.1 Boundary condition used in this thesis Whenever a particle attempts to cross one of these walls it is reflected back into the 24 interior; In order to produce a non slip effect, each particle colliding with the wall has all memory of its previous velocity erased, and is reflected back into the system with a new velocity having a random direction and fixed magnitude which is set to a value corresponding to the wall temperature. In addition, for the wall of the rotating cylinder the local (tangential) wall velocity is added to yield a new velocity vector. This mechanism is sufficient to drive the fluid rotation and dissipate the thermal energy generated by the sheared flow. The main differences from Fan et al’s model [15] lie in: 1. Only particles which intend to cross the wall boundary instead of all particles lying within the thin layer are imposed with a new velocity corresponding to the given temperature; 2. In addition, the updated positions of those reflected particles are traced accurately in the program rather than being given an approximate position as noted by Rapaport [28]. This idea can be shown clearly with Figure 4: Different with Maxwellian reflection scheme (noted above), in which the new velocity is distributed with the random directions, the boundary condition used here retains the parallel part of the old velocity and makes the vertical vector point into the interior area (see Equation (22)). 25 Position at time t + ∆t Position at time t (a) t 0 → t 0 + λ∆t (0 < λ < 1) with the (b) t 0 + λ∆t → t + ∆t (0 < λ < 1) with original velocity the new velocity Figure 4 Illustration of the boundary reflection. 3.1.2 Comparison of boundary conditions: Poisseuille flow Fan et al.’s boundary model [15] gives a rather good simulation results for simple Poisseuille flows, but in their article an obvious fluctuation of density and pressure in the region near the wall was noted, although this fluctuation is not as severe as what was predicted by Molecular Dynamics simulation. In this section, to make a convincing comparison, the same parameters are selected as noted in [15]. The density of n =4.0, with 21600 simple DPD particles placed in the channel and 2160 wall particles are located in three layers parallel to the (x, y) plane in each side. The fluid domain is given by -30 ≤ x ≤ 30, -1.5 ≤ y[...]... of two DPD parameters on the simulation are analyzed, then the simulation results of simple DPD particles in concentric/ eccentric flows are presented The simulation results of DNA polymer chains suspension in concentric/ eccentric flows will be presented in the next chapter 3.1 Implementation of non-slip boundary condition in DPD As with any other methods in computational fluid dynamics, the issue of. .. Re=37.2)…….…….59 Figure 30 Comparison of tangential velocity profiles in the narrow gap ( θ = 0 ) of simple DPD fluids in eccentric flow ( ω out / ω in = 0 , Re=37.2)……….….59 Figure 31 Comparison of velocity profiles of FENE chain suspensions of different concentrations in the Circular Couette flow……………………….…….64 Figure 32 Comparison of pressure profiles of FENE chain suspensions of different concentrations... density of fluid and the geometry of flow domain are defined, the wall and DNA chains particles if any will be set, and then all fluid particles are initially located at the sites of a face-centered cubic (FCC) lattice In the main program, the initial velocities of fluid particles are set randomly according to the given temperature The velocities of wall particles are set at zero At the beginning of simulation. .. and wall particles are generated separately by a pre-processing program and read in as input data The total number of particles depends on the size and geometry of the flow domain, the densities of the fluid and wall materials The initial velocities of fluid particles are set randomly according to the given temperature but the wall particles are frozen At the beginning of the simulation the particles... the wall if the wall particle density is too low compared with the fluid particle density; however, a higher density of the wall will produce a stronger repulsive force between the wall DPD particles and fluid DPD particles, and consequently a large density fluctuation appears near the wall So normally the wall particle density is chosen to be of the same order of that of fluid particles 18 On the other... velocity vectors of particle i , and the unit of mass is taken to be the mass of a particle, so that the force acting on a particle is equal to its acceleration Fe is the external force fij is the interparticle force on particle i by particle j , which is assumed to be pairwise additive The dynamic interactions between the particles are composed of two parts, dissipative and stochastic, complementing each... Streamline patterns of a simple DPD fluid in eccentric flow ( ω out / ω in = 0 ) with different Reynolds numbers…………………………….………….58 Figure 28 Comparison of polar angles of separation and reattachment points of simple DPD fluids in eccentric flow ( ω out / ω in = 0 )…………….…… 58 Figure 29 Comparison of tangential velocity profiles in the wide gap ( θ = π ) of simple DPD fluids in eccentric flow ( ω out... way of avoiding the modelization of physical boundaries 2 When considering the boundary conditions on the surface of solid objects as in the modelization of colloidal suspensions, the method of “freezing” some portions of the fluid described by DPD particles has been used [7, 29, 30, 31] In 21 this model, those particles are fixed but still can interact with other free particles [32] Due to the soft... distribution of the particles, which in turn affects the thermodynamics and rheological 7 properties of a colloidal suspension Since fluid simulation by DPD is a fairly new development, the results presented in this thesis are focused on a number of test problems These problems include: improving the dynamic behavior of DPD method, studying the effects of basic DPD parameters ( α and n) on simulation. .. concentric/ eccentric flows with simple DPD fluids and FENE chains suspensions are presented next 8 Chapter 2 Methodology 2.1 Basic equations of DPD method The DPD system consists of a set of interacting “particles”, whose time evolution is governed by Newton’s equation of motion For a simple DPD particle i , dri = vi , dt dv i = ∑ fij + Fe , dt j ≠i (1) where ri and v i are the position and velocity vectors of particle ... fij Interparticle force on particle i by particle j Fe External force exerting on particle i FijC Conservative force on particle i by particle j FijD Dissipative force on particle i by particle. .. length of one chain segment of FENE chain model Sc Schmidt number T Temperature of the system ui Peculiar velocity of particle i vi Velocity vector of particle i • v i (t ) Acceleration of the particle. .. behavior of a large number of molecules (a fluid particle ) When first introduced by Hoogerbrugge and Hoelman [3], Dissipative Particle Dynamics is such a method for simulation of the motion of this