FLOW DYNAMICS OF COMPLEX FLUIDS USING NUMERICAL MODELS

Transcription

1 DEPARTMENT OF PHYSICS UNIVERSITY OF JYVÄSKYLÄ RESEARCH REPORT No. 8/3 FLOW DYNAMICS OF COMPLEX FLUIDS USING NUMERICAL MODELS BY Amir Shakib-Manesh Academic Dissertation or the Degree o Doctor o Philosophy To be presented, by permission o the Faculty o Mathematics and Natural Sciences o the University o Jyväskylä, or public Eamination in Auditorium FYS-1 o the University o Jyväskylä on December, 3 at 1 o clock noon Jyväskylä, Finland November, 3

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3 PREFACE This work has been carried out during the years mainly in the Department o Physics, University o Jyväskylä. I wish to thank my supervisors Pro. Jussi Timonen and Pro. Markku Kataja or guidance and support. Special thanks are due to Dr Jan Åström, Dr Antti Koponen, Dr Juha Merikoski, Dr. Jussi Maunuksela, Mr. Pasi Raiskinmäki and Mr. Ari Jäsberg or their valuable collaboration and advice. I would also like to thank Pro. Piroz Zamankhan who invited me to Finland, was my supervisor in the irst publication o this Thesis, and thereby introduced me to the world o physics. Financial support rom the Finnish Center o Technology (TEKES) and the Academy o Finland are grateully acknowledged. I would like to thank my parents to whose support I owe my successes, the people o the Department o Physics, and all my near riends. Jyväskylä, November 3 Amir Shakib-Manesh i

4 Amir Shakib-Manesh Flow dynamics o comple luids using numerical models Jyväskylä: University o Jyväskylä, 3, 137 p. (Research report/department o Physics, University o Jyväskylä, ISSN X; 8/3) ISBN Diss. ii

5 ABSTRACT In this thesis several multiphase low problems with comple boundaries have been analyzed numerically: granular lows, capillary rise and rheological properties o particulate suspensions. We analyze the long-time diusion coeicient in the shear low o inelastic, rough, hard granular spheres by molecular dynamics simulations, and observe a transition rom a disordered to a layered state at solid volume raction φ s =.565. This transition causes a sharp drop in the long-time transverse sel-diusion coeicient o particles. We veriy that the two-phase lattice-boltzmann method applied here is capable o modelling the capillary rise phenomenon. We also demonstrate the dependence o the dynamic contact angle on the capillary number, and the applicability o the Washburn equation when changing contact angle is properly included. Several benchmark studies are carried out to validate the lattice-boltzmann code developed or liquid-particle suspensions in the non-brownian regime. The main emphasis in this comparison is on the dependence o the viscosity o the suspension on the concentration o particles. We then analyze in more detail the dependence o viscosity on shear rate in a Couette low geometry. The onset o shear thickening is ound to be related to a maimum in the layering o particles near the channel walls. A comprehensive study o the various momentum transer mechanisms that contribute to the total shear stress indicates that both these phenomena are related to enhanced relative solid-phase stress. The apparent viscosity o the suspension is ound to be completely given by the ratio o the solid-phase stress to the total stress, and this dependence takes the orm o a simple rational unction. We inally analyze the low Reynolds number low o D liquid-particle suspensions, in which suspended particles may attach on channel walls, and deposited particles are detached i the hydrodynamic orce on them eceeds a given threshold. The behaviour o this system is controlled by concentration o particles and detachment threshold. At low values o these parameters there is no deposition on the walls, ollowed by a irstorder like transition to a phase in which deposition layers eist in the stationary state. Close to the transition, these layers display a meandering pattern whose wavelength is directly proportional to suspension velocity. The meandering phase is ollowed by a necking phase in which deposition layers still do not block the low channel. Finally a transition to a ully blocked low channel takes place. This last transition is driven by equilibrium luctuations in the deposition layers. Keywords: Multiphase lows, lattice Boltzmann, suspension, rheology, pipe clogging. iii

6 LIST OF PUBLICATIONS I. P. Zamankhan, W. Polashenski Jr., H. Vahedi Tareshi, A. Shakib-Manesh, P. J. Sarkomaa, Shear-Induced particle diusion in inelastic hard sphere luids, Phys. Rev. E 58, R537 (1998). II. P. Raiskinmäki, A. Shakib-Manesh, A. Koponen, A. Jäsberg, J. Merikoski and J. Timonen, Lattice-Boltzmann simulation o capillary rise dynamics, J. Stat. Phys. 17, 147 (). III. P. Raiskinmäki, A. Shakib-Manesh, A. Koponen, A. Jäsberg, M. Kataja and J. Timonen, Simulation o non-spherical particles suspended in a shear low, Comp. Phys. Comm. 19, 185 (). IV. A. Shakib-Manesh, P. Raiskinmäki, A. Koponen, M. Kataja and J. Timonen, Shear stress in a Couette low o liquid-particle suspensions, J. Stat. Phys. 17, 67 (). V. A. Shakib-Manesh, J.A. Åström, A. Koponen, P. Raiskinmäki and J. Timonen, Fouling dynamics in suspension lows, Eur. Phys. J. E 9, 97 (). The author o the thesis has participated in the ormulation o the problem, numerical analysis o the results and preparation o the article [I]. The author has written the irst drat o the articles IV,V, and also written selected parts o the articles II,III. The author has perormed almost all o the numerical computations needed, and participated in the ormulation o the problems, and the development o the analytial models. iv

7 CONTENTS Preace Abstract 1 Introduction... 1 Numerical methods Why computer simulations? Numerical simulations at dierent relevant length scales The molecular dynamics method Motion control Collision procedures Structural inormation Diusion The lattice-boltzmann method LB method or luid low Surace tension in the LB method LB method or suspension simulations Momentum transer in suspensions MD simulations o a granular medium Introduction Results and discussion Capillary rise The hydrodynamics o capillary rise Results and discussion Some rheological properties o liquid-particle suspensions Background Benchmark studies or the LB method Viscosity o liquid-particle suspensions Microstructure o dilatant suspensions Suspension microstructure in shear low Shear thickening Momentum transer analysis Suspension clogging Introduction Results and discussion Conclusions Reerences Part II. Published papers. v

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9 1. Introduction 1 1 INTRODUCTION Multiphase luids or so-called comple luids are common in daily lie. The term multiphase luids, which has been used or less than hal a century [1], stands or systems with dierent substances or dierent phases o matter (solid, liquid or gas) in coeistence. Multiphase low regimes can appear as dispersed regimes (i.e. not materially connected) such as particle, droplet or bubbly lows. Other orms o dispersed multiphase low regimes are slug low (large bubbles in a continuous luid), annular low (continuous luid along walls, gas in the center), and stratiied low (immiscible luids separated by identiiable interaces) []. There are also nondispersed multiphase lows such as luid low through a porous material. Multiphase luids have more complicated microstructure than single luids and their low dynamics depend on many physical parameters. Study o comple luids is important or several reasons. Flow processes o comple luids are encountered in numerous applications in many industries and in nature. Design and operation o multiphase processes call or understanding the basic mechanisms involved. This understanding is seldom available simply because o the complicated behaviour o multiphase materials. Thereore, studies o simple multiphase processes are needed as irst steps towards handling o realistic problems. So ar, perhaps apart rom turbulence, the dynamics o single-phase luids is airly well understood. However, there is a large number o unsolved undamental problems related to the macroscopic properties o multiphase materials. A primary diiculty in many multiphase low problems involves the prediction o the shape and position o the interace between dierent phases. Eperimental techniques are oten not designed or multiphase systems and have diiculties to accurately probe the low properties. In practice, the underlying multiphase character is oten neglected, and the system is treated ormally as a single-phase rheological or non-newtonian luid. In many applications with low volume raction o particles, droplets, or bubbles, the system may indeed behave like a single-phase luid. However, i concentration o the dispersed phase increases, signiicant changes may appear in the physical behaviour. For eample, in a single-phase luid the transer o momentum simply results rom internal motion or interaction o luid particles, while in multiphase

10 1. Introduction luids such as suspensions, viscosity, the resistance o the luid to low, arises rom many mechanisms that involve interactions between the dierent phases. This thesis addresses several mesoscopic multiphase problems and is based on molecular and mesoscopic luid dynamics simulations. Two major problems are discussed. On the one hand, the eect o shear on the dynamical and equilibrium properties, the microstructure and the rheological behaviour o monodisperse granular media and liquid-particle suspensions are discussed. First we investigate in detail the three-dimensional bounded and sheared granular low by using direct numerical simulations. The main emphasis o this study is on the diusion and microstructural behaviour o such materials under shear deormation. Concentrated suspensions behave similarly to these dense granular media, but with more comple microstructure and dynamics, due to the eistence o luid between the suspended particles. One o the major questions in the rheological behaviour o sheared particulate suspensions is whether they display a dilatant rheological behaviour. Industrial applications that involve this problem include mier blade damage or bowing o roller in milling operations, slurry ractures, and rupture o coating material in paper industry. It would also be important to better understand the operation o common viscometers. We have thus studied the rheological behaviour o sheared two- or three-dimensional non-brownian liquid-particle suspensions using numerical techniques. This study includes the eect o shear deormation on the microstructure o the suspension and on the transport coeicients such as viscosity. We also analyze various momentum transer mechanisms, and their contribution in the ormation o viscosity. On the other hand, Poiseuille low o luids is eamined without and with suspended colloidal particles. The ormer case, the pipe low o a luid, is related e.g. to low in porous materials, as narrow pores behave like capillary pipes. In a capillary pipe containing two immiscible luids, pressure dierence across the interace between these luids (capillary pressure) makes the denser luid rise inside the capillary even i there is an opposing eternal orce such as gravity. Using numerical simulations o a two-phase luid, we study this capillary phenomenon, e.g. the dynamics o the luid-solid contact angle. By studying the capillary rise, we can also benchmark our numerical techniques or modelling imbibition o luid in a gas-illed porous medium. We also study small-reynolds-number pipe low o a particle suspension, where the particles can deposit on channel walls, and deposited particles are detached i the hydrodynamic orce on them is above a threshold value. The behaviour o this system is determined by two parameters: the concentration o the suspension and the detachment threshold, and it has a rich phase diagram. Possible applications o the problem etend rom clogging o tiny blood veins in the human body, and

11 1. Introduction 3 blockage o narrow capillaries in coating colour slurries, to ouling o pipes in petroleum industry. In the analysis o the problems described above, two numerical methods were used. The molecular dynamics method was irst used to simulate the dynamics o hard spheres [3], but since then it has been etended to many other more complicated systems [4,5,6]. This method is a powerul tool in many physical problems, and it has been used to analyze e.g. rapid lows in a high Reynolds number regime o granular media [7], dilatancy phenomena [8] and luidization o grains [9]. We use this method to analyze sheared granular low. For modelling comple luids with moving internal boundaries, mesoscale methods like coarse-grained particle-based schemes (such as DPD) [1] or lattice-based methods (such as the lattice-boltzmann method [11,1,13,14] (LBM) and lattice-gas cellular automata [15,16,17,18]), are usually better than the conventional CFD methods [19]. LBM can also be seen as a special inite-dierence scheme or the kinetic equation o a discrete velocity distribution unction. It has evolved in recent years into an alternative and promising scheme or simulating various luid lows [13, 19,,1], and we use this method etensively in this thesis.

12 . Numerical methods 4 NUMERICAL METHODS.1 WHY COMPUTER SIMULATIONS? Modelling can be done by means o theory, laboratory prototypes, or numerical techniques. The theoretical models oten use, especially in complicated many-body problems, macroscopic equations to describe the physical phenomenon at hand, thereby neglecting many microscopic details o the system. Numerical methods usually lack this problem, and they are normally inepensive in comparison with laboratory eperiments. Numerical methods also provide eicient benchmarks or theoretical models. They can address details o comple phenomena that can be diicult or theory as well as or eperiments. Eperimental diiculties may arise rom compleity, epenses, danger or hazardous condition o the phenomenon. For eample, in a dense suspension low, measuring o the local instantaneous low quantities such as velocities and stresses, separately or both phases, is not easible. Such inormation can however be attained rom numerical simulations o the suspension low. Also, numerical techniques are versatile enough to combine or distinguish dierent physical phenomena, which have made them popular tools o analysis. The results o numerical techniques are however subject to uncertainties that arise rom e.g. the low model and the numerical techniques used. Care must thereore be taken when epressing the results o numerical simulations, and they should be supported by theoretical and/or eperimental benchmarks. In this thesis mainly two numerical techniques have been used or simulation o low problems, especially multiphase phenomena, starting rom microscale molecular dynamics o spherical particles up to mesoscale simulations o capillary rise phenomena, shear or Poiseuille low o suspension, and clogging o suspension low in channels. In each case, eperimental or theoretical evidence have been used or benchmarking the method.

13 . Numerical methods 5. NUMERICAL SIMULATIONS AT DIFFERENT RELEVANT LENGTH SCALES In colloidal suspensions, or eample, there are three important length scales, the microscopic size o the relevant molecules, the mesoscopic scale related to the size o colloidal particles, and the macroscopic scale which is that o the eective luid motion. In particulate suspensions the size o the suspended particles can be in the macroscopic scale. As the relevant length scales in suspensions dier by several orders o magnitude, it is not possible to simulate them by including details at all these length scales. All numerical methods are based on one basic length scale. For macroscopic modelling the suspension must be regarded as a continuum medium, and the description is in terms o the variations o the macroscopic velocity, density and pressure (and temperature) with space and time. The macroscopic scale simulations, which usually go under the general name o computational luid dynamics [,3] (CFD), use macroscopic continuum equations like the Navier-Stokes equation, whose solutions are very sensitive to the boundary conditions. Various CFD methods oer successul simulation tools or dierent single-phase luid dynamics problems within many branches o science and engineering [4]. For suspensions however, these methods are diicult to apply due to the moving boundaries o the suspended particles. In recent inite-element simulations it has been possible to analyse only cases with small numbers o particles [5,6], or simulations were limited to simple and regular particle geometries [7,8]. An attempt to circumvent the problems with moving or otherwise comple boundaries is provided by lattice-based methods which are based on a mesoscopic basic unit o the system, typically o the order o a micrometer, but depending o course on the details o the system. There are a ew popular methods, which use this approach, and they include the lattice-gas automata (LGA), its alteration the lattice- Boltzmann method (LBM) and the method o dissipative particle dynamics (DPD). Each o these methods has its advantages and limitations. Use o mesoscale simulation techniques can allow the eact treatment o small-scale macroscopic systems, and can be used to eectively study e.g. the rheology o colloidal suspensions and the dynamics o interaces. Simulations at the microscopic level have also been attempted in order to understand the truly microscopic behaviour o liquids. As Monte Carlo simulations are not suitable to describe the dynamic properties [9], the prevalent microscopic simulation method is that o molecular dynamics (MD) [3]. MD methods that apply dierent numerical techniques have thus been developed to study the dynamical behaviour o a large number o applications, see e.g. [9,3,31,3,33]. MD simulations are computationally very demanding, and obtaining bulk properties o, or eample, suspensions, would mean using billions o particles [34].

14 . Numerical methods 6 We review below in more detail the MD method and the LB method used in this work..3 THE MOLECULAR DYNAMICS METHOD Molecular dynamics (MD) provides a reasonable physical description o granular materials including their macroscopic averaged properties, hydrodynamics, and all luctuations, using only linear dynamics. A (granular) system o spheres is initialised by irst randomly placing the spheres in a bo. Then overlapping spheres are randomly moved until no overlaps remain. Thereater the system evolves in time according to Newton s equations o motion under chosen initial and boundary conditions, and applied orces and ields. Four major steps are needed to enrol a hard-sphere molecular dynamics simulation: motion control, determination o collision times, collision dynamics, and calculation o the particle properties..3.1 Motion control The motion o a rigid body o mass m and acceleration a= F / m, due to an eternal orce F, ollows Newton's Second Law. The total kinetic energy and momentum o the system must be conserved, as well as its total angular momentum J = m r i v i + I ω, i (.1) i i where r is the position, v the velocity and ω the angular velocity o particle i. I is i i the moment o inertia o the particles, which is mr / 5or a sphere and mr / or a disk, with R the respective radius o the particle. Verlet algorithm For the positions r the most popular time integration algorithm is the so-called Verlet algorithm. Forward and backward Taylor epansions o the position as a unction o time can be written in the orms ( t+ δt) = () t + δt () t + ( ) δt 3 ( t) + O( δt ) ( t δt) = () t δt () t + ( ) δt 3 () t + O( δt ) r r v 1/ a, r r v 1/ a. Adding the above two epressions gives the new position ater time δ t, ( t+ δ t) = ( t) ( t δt) + δt ( t) + O( δt 3 ) i (.) r r r a. (.3) There are several methods to make Eq. (.3) eplicit, the most common o these being the so-called hal-step leap-rog scheme [31], r t+ δt = r t + δtv t+ δt. (.4) ( ) ( ) ( ) So the velocity o the mid-step may be written as ( t) v( t + δ t ) = v ( t δ t ) + F m δt. (.5)

15 . Numerical methods 7 Similarly or the angle (orientation) o a particle (in the case o non-spherical particles) θ t+ δt = θ t + δtω t+ δt, (.6) and or its angular velocity ( ) ( ) ( ) ω( t+ δ t ) = ω ( t δt ) + T δt. (.7) I Later during the step, the current velocities are determined rom 1 v() t = ( v( t+ δt ) + v ( t δt ) ), (.8) and similarly or the angular velocities..3. Collision procedures ( t) Collision time Consider two smooth hard spheres with relative locations r ij = r i r j and velocities vij = vi v j (see Figure.1). j vj r(t) ij v i i j r (t) ij v (t) t ij ij i r ij (t+t ij ) Figure.1 Smooth hard-sphere collision: situation beore and at the collision moment. At the collision moment, the particle-particle distance should be equivalent to the particle diameter [9], ( + ) = () + () = r t t r t v t t R. (.9) ij ij ij ij ij Rearranging this epression gives a quadratic equation or the time to collision, with b r v ij ij ij t ij ij ij t ij vij vij ( ) b r R + + = (.1). One can see that Eq. (.1) is very important in collision detection: bij > ; no collision takes place, bij < and ( bij vij ( rij 4R )) < ; no collision takes place, bij < and ( bij vij ( rij 4R )) ; collision takes place. The solution o Eq. (.1) gives the minimum collision time [9], ij ( 4 ) (.11) b ij ± bij vij rij R tij =. (.1) v

16 . Numerical methods 8 O the two roots o solution (.1), the smaller one obviously gives the minimum time to collision between the two particles. Collision dynamics At the moment o collision, the total momentum should conserve, i.e., n v = v δ P / m, Here or smooth particles, the vector r, i.e., ij i i ij v = v + δ P n j j ij n ij / m. (.13) δ P is the projection o relative momentum m v ij along n rij vij δ Pij / m = r ij. (.14) 4R For smooth particles (with rictionless surace) the post-collision components o velocities perpendicular to the vector remain unchanged as well as their angular r ij velocities. However, or rough particles (see deinition below Eq. (.17)) one can derive [ 9] 1 I δ Pij / m = ( ij ) ( ij ) v + v n mr + I t. (.15) In the case o rough particles, the particles have a no-slip contact [35] and thereore the post-collisional velocities in the normal and tangential directions are v n = evn, (.16) v = β v. where e and t t β are the normal and tangential coeicients o restitution. β can be epressed in terms o e and the riction coeicient µ such that ( )( ) β = 1+ µ 1+ e 1 + mr / I v n / v t. (.17) The post-collision velocities and angular velocities can now be derived in a similar way to Eq. (.13). While e varies between and 1, corresponding to change rom inelastic to ully elastic material, β varies between 1 to +1 corresponding to change rom a perectly smooth to a perectly rough particle. For more details concerning the collision event, geometry o the system and simulation algorithm, see [9,36]. The angular velocities or rough particles are determined rom ω' i = ωi + rij δ Pij / I, (.18) ω' = ω + r δ P / I. j j ij ij Note that in the case the particles are not spherical, the vector tangential plane at the contact point between the two colliding particles..3.3 Structural inormation r ij is normal to the When analysing systems o particles that undergo Newtonian dynamics, it is important to iner also structural eatures o the system. This is important in particular when systems with high solid volume raction are considered, as in these

17 . Numerical methods 9 systems phase transition to ordered states may appear. We thereore eplain briely the main tools or the analysis o structural order. Radial distribution unction The radial distribution unction (pair correlation unction) measures correlation in the density o particles. It can be deined as the probability o inding another particle at distant r rom a particle [ 37], and can be epressed in the orm N N V g( r) = δ ( r() t r ij () t ) (.19) 4π rn, i i j where V is the volume o the system, δ the Dirac delta unction and the angular brackets denote conigurational average. Translational order Translational order appears as periodicity in the particle density, and can thus be ρ k t o the mass density [38], described by Fourier components ( ) N 1 ρk() t = ρr() t cos( k r i ), (.) N i= 1 Here wave vector k is such that vector or a regular lattice { r }. Bond orientational order 13 k π V, which becomes a reciprocal lattice Orientational order can best be parameterized with spherical harmonics, and we thus consider the quantity Q lm ( r) Ylm ( θ ( r), φ( r) ) [ 39], where Y lm ( θ ( r), φ( r) ) is a spherical harmonics with θ ( r ) and φ ( r ) the polar and azimuthal angle, respectively, and { r } is the set o midpoints o bonds between all nearest-neighbour (nn) particles. An order parameter which measures orientational order, can thus be deined as Nbond 1 where Q Q ( ) sample. lm bond α = 1 4π Ql = Q l + 1 l 1 lm, (.1) m=-1 lm r α, and the average is taken over all nn bonds o the N l SC (7-atom) BCC (15-atom) FCC (13-atom) HCP (13-atom) ICOS (13-atom).6.4 Table.1 Approimate values or Q [ l 39].

18 . Numerical methods 1 Bond-angle correlation unctions Bond-angle correlation unctions ( r) G measure the leibility with which a bond l can have local orientations in a given state. It can be deined as l 4π G l( r) = ( Wlm( r) Wlm( r + r )), (.) l + 1 N bond where W lm ( ) w Q ( ) α = 1 m= l r α lm r α is again a sum over all nn bonds o point r. Here are Wigner symbols [4]..3.4 Diusion Several deinitions can be used to calculate the sel-diusion coeicient o particles in the considered system [41]: the diusion equation, the velocity autocorrelation unction and the mean-square displacements o the particles. Analysis o the random Brownian motion o an isolated sphere by the diusion equation leads to the Stokes-Einstein relation [4] or the sel-diusion coeicient, D= lim 1 ( r( t+ τ ) r( t )). (.3) τ dτ Here t is the starting time, r is the position vector o the particle, d is the dimension o the space, and the brackets denote average over observation time intervals τ or a single particle or over many particles o an ensemble. This relation is obviously valid only i the mean square displacements o particles are asymptotically linear in time. This result can be generalized to that or a diusion tensor. Another way to determine the sel-diusion coeicient is to use the Green-Kubo ormula [ 43]..4 THE LATTICE-BOLTZMANN METHOD A general statistical mechanics description o luid motion is given by the Boltzmann equation (BE), without any prior assumptions about e.g. hydrodynamical stresses [44]. BE describes the evolution o particle distributions so that its level o description o a given system is somewhere between that o macroscopic hydrodynamics and that o microscopic molecular dynamics. The dierential orm o BE in the relaation time approimation is 1 eq t ( rv,, t ) + v ( rv,, t ) = ( ( rv,, t ) ( rv,, t )), (.4) τ rv t is the Mawell-Boltzmann distribution unction o a particle at where (,, ) location r with velocity v at time t, and τ is the relaation time due to collisions. The irst order solution O (1 τ ) o the BE by using the Chapman-Enskog epansion method gives the Euler equation [45]. This is done by substituting the Mawell- Boltzmann particle distribution in the steady state and by neglecting the disturbances. The Navier-Stokes equation is a second-order solution O ( τ ) o BE by D ij w α

19 . Numerical methods 11 the Chapman-Enskog method [45]. The continuum description o luid with second order accuracy in space and time is thus given by the continuity and Navier-Stokes (momentum) equations [46], ρ + ( ρu ) =, (.5) t ( ρu) T + ( ρ uu) = p + µ ( + ( ) ) +, t u u F (.6) respectively. Here µ is coeicient o viscosity in the viscous stress term. In the nearly incompressible limit the time derivative o density in Eq. (.5) is small. The terms on the right hand side o Eq. (.6) are the induced momentum rom the pressure gradient, viscous stress and the eternal orce F. The key idea behind the LB method [14,47,48,49], is to solve the Boltzmann equation on a regular lattice [11]. At every time step, each luid particle propagates to a neighbouring lattice point and undergoes local collisions, in which the momenta are redistributed. I the lattice spacing is much smaller than changes in the solid boundaries, e.g., o suspended particles, bulk-like behaviour can be recovered. Fluid particles cannot however be considered as being o microscopic size, they represent a coarse-grained scale which already can support hydrodynamic behaviour, i.e., are o mesoscopic size. One can rigorously derive the LB method rom kinetic theory [5], and there has been signiicant progress in the development o this method [11, 1,51,5]. Quite a lot o benchmarking has been carried out [53,54] in order to show that the LB method to second order accuracy provides a reliable, accurate and eicient algorithm or simulating low Reynolds number incompressible lows with comple boundaries. There are several dierent lattice-boltzmann models or incompressible Newtonian luids [,55,56,57,58]. The lattice-boltzmann (LB) method is just one o the methods that are based on solving a discretized orm o the Boltzmann equation, other techniques include, e.g., the lu splitting method [59]..4.1 LB method or luid low We use here the so-called lattice-bgk [6] (lattice-bhatnagar-gross-krook) algorithm with 9 links per lattice sites (DQ9) in two dimensions, and with 19 links per lattice sites (D3Q19) in three dimensions [, 55]. These algorithms have been successully tested, and the results o several benchmark tests or pure luid can be ound e.g. in [III,IV,61,6,63]. The LBGK method is based on the discrete kinetic equation 1 (, 1 ) (, ) ( eq i r+ ci t+ i r t = i ( r, t) i( r, t) ) + F. (.7) τ Here i(,) r t i(, r v= ci,) t is a probability distribution unction or luid particles at link i, i =,..., N, moving with discrete speeds c on a lattice { r }, τ is the BGK i

20 . Numerical methods 1 relaation parameter, and eq ( r, t is the local equilibrium distribution towards i ) which the particle populations are relaed. The set o discrete speeds are chosen such that conservation o mass and momentum as well as Galilean invariance are satisied. We choose c =, c1 = 1, c = N i= 1 C i ( ) =, or the rest particles, and motions along the nearest neighbour and net-nearest neighbour links, respectively. A orce term F is included in the LB method to model a pressure gradient needed e.g. in channel low. This body orce term adds a constant amount o momentum at every time step on each lattice point, and is equivalent to having a constant body orce (or gravity), F, in the Navier-Stokes equation Eq. (.6) [ 16]. The LB equation must satisy mass and momentum conservation, i.e., eq i N cici i= 1 ( ) =, (.8) 1 eq where C (,..., i 1 N) ( i ( r, t) i( r, t) ) is a single relaation time collision τ operator in the LBGK method, and represents the change in particle populations due to collisions. Based on kinetic theory, in the equilibrium state where all C. eq populations o dierent particle velocities are in equilibrium, ( ) i = The local equilibrium can be chosen in the orm o a quadric epansion o a local Mawellian distribution [5]: where the w i ci u ( ci u) u (,, r t u ) = ρ(, r t) w i 1+ +, 4 (.9) c c c w = = 4/9, w1 1/9, w = 1/36 or DQ9, and w = 1/3, w 1 = 1/18, w = 1/36 or (see Figure.) [ 13]. s s are a set o weight actors which can be chosen such that, e.g., s DQ (DQ9) (D3Q19) Figure. The lattice velocity vectors in the 9-link DQ9 (let) and 19-link D3Q19 models used in the LB method. The links to nearest neighbours have black arrows and those to net-nearest neighbours have white arrows.

21 . Numerical methods 13 The basic hydrodynamic variables are obtained in the LB method rom velocity moments in analogy with the kinetic theory o gases. The density ρ, the low velocity u and the momentum lu i= 1 Π ρ( r, t) = ( r, t ), N ρ ( r, t) u( r, t) = ci i( r, t ), N i= 1 N i i= 1 Π = cc ( r, t), i i i o the luid are given by respectively, where N = 9 or DQ9 and N = 19 or DQ (.3) With these choices Eq. (.7) can be shown [53] to lead in the continuum limit to the continuity and Navier-Stokes [57,64] equations with sound speed, c s 1 c I cc, (.31) N s = wi i i d i = 1 where d is the dimension o the space. For the DQ9 and DQ 3 19 models we use, Eq. (.31) gives c = 1 3 or the sound speed. The kinematic viscosity o the s simulated luid is given by ν = (τ 1) / 6 (in lattice units) [55]. The luid pressure can be epressed in the orm ( ) ( ) ( ) ( ) p r, t = c ρ r, t = c ρ r, t ρ, (.3) s s where ρ is the average density o the luid. Notice that the main collision equation o the LB method is valid only or a low Mach number low, or in the incompressible limit o the Navier-Stokes equation [65]. This ollows rom the low-velocity polynomial epansion made to obtain the equilibrium distribution unction eq (see Eq.(.9)), which approimates the Mawellian with second order accuracy [66,67]..4. Surace tension in the LB method In the LB method surace tension γ results rom interaction between luid particles and is not given as a known macroscopic parameter. It should satisy Laplace s law, which describes the balance o orces due to pressure dierence p across an interace, γ = p dv da, with dv and da small changes in the control volume and area o the interace, respectively. In the case o a spherical droplet with radius R, these changes are dv = 4π R dr and da = 8π RdR. One thus gets the Laplace law in the orm γ = p R. (.33) A two-phase LB model was developed by Shan and Chen [68] through a relative orcing scheme, 1 eq i( r+ ci, t+ 1 ) = i( r, t) + ( i ( r, t) i( r, t) ) + FG, i( r, t) + FW, i ( r, t ), (.34) τ

22 . Numerical methods 14 where F is the surace tension orce and F is the adhesive orce at each time step G t. The orce term F is an attractive short-range orce between neighbouring luid G particles [ 69] o the orm F () r = τψ () r Gψ ( r+ c ) c, where ψ () r = 1 ep[ ρ()] r G with the total particle density (,t) i W i i i ρ r deined in Eq. (.3), and G =, G1 = G, G = G (or zero link, nearest neighbour links and net-nearest neighbour links, respectively). Thus a single parameter G < controls the surace tension. Adhesive orces between the luid and solid phases were introduced via [69] F () r = τψ () r Ws( r+ c ) c, where W =, W = W, W =, and s =,1 or luid W i i i i 1 W and solid nodes, respectively. The interaction strength W is positive or a nonwetting and negative or a wetting luid. The algorithm can be divided into our major steps. In the irst step, the ields at time t, such as the densities and the velocities, are derived. In the second step, the collisions o the total populations are calculated. In the third step, surace tension is introduced and is combined with the total populations. Finally, propagation o the populations is achieved..4.3 LB method or suspension simulations Several lattice-boltzmann models or suspensions have already been developed [7,71,7,73]. In order to model the luid phase in liquid-particle suspensions, we used in Res. [III,IV] the lattice-boltzmann model o Re. [71]. This model was chosen because o computational convenience. It could be eiciently implemented on parallel processors, and all internal and eternal parameters o the system could easily be varied. All hydrodynamic orces acting on particles and stresses in the luid phase can also be evaluated without reerence to the averaged luid-dynamical quantities such as the viscous stress tensor [57]. The suspended solid particles obeyed Newtonian dynamics, and their motions were determined by the molecular dynamics method described in Section.3. Particle solid boundary nod Particle luid core node luid node Figure.3 Schematic picture o a disk particle in a regular lattice with solid, internal and eternal luid boundary nodes.

23 . Numerical methods 15 In the model used here, a solid particle consists o a solid matri, and o an interior luid that is called the luid core o the particle. There are two kinds o lattice points inside a particle, the interior points and the boundary points. Each boundary point has at least one link pointing to the luid phase. The lattice-boltzmann collision operation (Eq. (.7)) is applied at every lattice node including the boundary and interior points o the particles. The no-slip boundary condition at solid-luid interaces is usually realised in lattice- Boltzmann simulations through a simple bounce-back condition, where the momenta o the luid particles are reversed at the boundary points [7, 71]. The bounce-back condition can be generalised to moving boundaries, whereby the particle distributions are modiied at boundary points such that [7, 71] ( r+ c, t+ 1) = ( r+ c, t ) + ρ B( u c ). (.35) Here t + i i i i + i w indicates the time right ater the collision and i i denotes the bounce-back link. Also, coeicients B =, B = 1/ 3 and B = 1/1 apply or DQ9 in twodimensions, and B =, B = 1/ 6 and B = 1/1 or DQ 3 19 in three-dimensions, or the rest particles, and motions along the nearest neighbour and net-nearest neighbour links, respectively. Finally u is the wall velocity at point. It is determined by the particle center o mass position r and velocity velocity ω, such that 1 1 ( ) w w c w c r w U, and angular u = U+ r r ω. (.36) The last term in Eq. (.35) accounts or the momentum transer between the luid and the moving solid wall. Note that Eq. (.35) is also used to implement the moving solid boundary walls in Couette low with the channel wall velocity. The orces and torques on a solid particle are calculated as sums o momentum changes over the particle surace nodes, Fp() t = ρ uw, s (.37) Tp() t = rw ρ u w. The mass s s M o a solid matri is uniormly distributed in the interior and boundary points giving it an eective density ρ s. However, the solid matri interacts with the luid phase and with the core luid only at the boundary points. The total hydrodynamic orce and torque acting on the solid matri can be obtained at each time step by summing the eect o Eq. (.35) at all boundary points. The solid matri can then be moved according to normal Newtonian dynamics with particle properties such as location, velocity, angular velocity, orce and torque..4.4 Momentum transer in suspensions When analysing the rheological behaviour o particle suspensions, it is important to understand the underlying mesoscopic mechanisms that contribute to the total u w

24 . Numerical methods 16 Π observable shear stress (and apparent viscosity). To this end we need to calculate the stresses o dierent phases and momentum lues. The luid momentum tensor can be obtained rom the LB pre-collision and postcollision populations i and i (Eq. (.34)) such that N N αβ 1 1 Π ( r, t) = ciαciβ i( t) ci αciβ i ( t) r, + r, i= 1, (.38) i= 1 where α, β denote the spatial components o tensor. The convection tensor or each solid or luid phase ( v= s, ), Cv = ρvuu, (.39) can be directly calculated rom density ρ and velocity (o the node) u (Eq. (.3)), and the viscous stress tensor Σ is given by Σ = Π C Π. (.4) The total momentum lu F through any surace S which intersects the system is given by where Π where C F = Π ds, (.41) S is the total momentum tensor. In the present case it can be written as Π = C + C + Σ + Σ, (.4) and s s C s are the convective momentum tensors, and Σ and Σ s are the internal stress tensors or the luid and the solid phase, respectively. A schematic illustration o the simulation setup together with a snapshot o an actual solution or the Couette low o the suspension is shown later in Figure 6.1. In this igure the surace S = S ( y) is a plane perpendicular to the y ais. The total shear stress acting on this plane is deined by where y y y y τ = σ + σ + τ + τ C + C + Σ + Σ, (.43) T s s s s denotes averaging over space, time and ensemble. Here we calculate all averaged quantities in a macroscopically stationary state and, assuming ergodicity, perorm averaging over a long period o time, over volume or surace, and over a number o macroscopically identical systems. Stresses σ and σ s contain the stresses due to pseudo-turbulent motion o the two phases, τ contains the viscous stress o the luid phase and τ s contains the internal stress o particles (and corresponds to the elastic stress o physical solid particles). Stresses σ, τ and σ can all be directly calculated or each lattice point using Eqs. s (.38)-(.4). Notice that in σ s both the convection o the solid matri and the convection o the luid core have to be included. Notice also that the convection and stress o luid at the boundary points o the particles are included in σ and τ, respectively. Evaluation o the actual stress distribution inside solid particles would require solving, separately or each suspended particle, time-dependent elastic continuum

25 . Numerical methods 17 equations with boundary conditions given by the hydrodynamic orces due to the surrounding luid, and the impulsive orces due to particle-particle collisions. Fortunately, it is not necessary to carry out here this ormidable numerical eort. The stress τ s can be written in the orm τs = τc + τpp + τ p, (.44) where τ is the stress in the core luid while τ and τ are the stresses in the solid c matri caused by particle-particle collisions and by interactions between the luid and the particle, respectively. The stress can be obtained directly rom Eq. (.4). τ c In order to calculate stresses and τ, we consider an impulsive orce F due to τ pp particle-particle collisions, and a orce F due to the hydrodynamic interactions A p p that also acts on a surace ormed by the intersection o surace S with the particle (see the insets in Figure 6.1). Since the collisions are assumed rictionless, and the particles are circular and smooth, the angular velocity does not change in a collision. It is then easy to show that the collisional orce on can be written as where t d pp =, M s t p pp m p F (.45) is the collision time (1 in lattice units), p A p M s is the total mass o the solid p is the matri o the particle, m is the mass o the lower part o the particle, and d total momentum change o the particle due to collision (see inset (b) in Figure 6.1). In simulations, mass m can simply be calculated by counting the number o lattice d points that are located below the plane S inside the particle. While deriving the orce F, the eect o luid phase and luid core on the solid p matri must both be included. The equation o motion or the lower part o the particle, as shown in inset (a) o Figure 6.1, gives F = F F + m a+ m α r + m ω r e y. (.46) F id, p i, d o, d d d d d d Here and are the hydrodynamic orces due to the luid core and the luid phase out o the core on the solid matri (which can be obtained rom Eq.(.37)), is the mass o the lower part o the particle, and a, α and ω are the acceleration, angular acceleration and angular velocity o the whole solid matri, respectively. Finally, is a vector pointing rom the center o the particle to the center-o-mass o r d F od, the lower part o the particle. The contribution o collisions and hydrodynamic interactions to the total shear stress is now given by τ = F / A. p p p pp pp pp p m d τ = F / A and More details o the lattice-boltzmann suspension code we use, together with benchmark results, are given in Res. [III,IV].

26 3. MD simulations o a granular medium 18 3 MD SIMULATIONS OF A GRANULAR MEDIUM 3.1 INTRODUCTION Even though luids can be adequately modelled on the mesoscopic or macroscopic scale using the Navier-Stokes equations, it is nevertheless interesting to study microscopic scale phenomena like diusion and phase transitions. Many natural phenomena like avalanches, landslides, and soil luidisation, can be related to granular lows that involve diusion processes. Broadly speaking the individual molecules o a luid build up a granular medium through interparticle contacts [74,75,76]. In these systems miing is an important phenomenon which occurs due to the diusive motion o the particles. This diusion has been analyzed by kinetic theory o rapid granular lows [77,78], laboratory eperiments [79,8] and numerical simulations [81,8,83]. In Campbell s [8] review o results prior to 199 concerning diusion in molecular dynamics simulations o dense granular media, lack o diusion or solid volume ractions over.56 is reported. This means that diusive motion o particles may be suppressed by high densities. Campbell s conclusion was based on a series o computer simulations or unbounded granular shear lows. Such lack o diusion was, however, not observed in later computer simulations on bounded rapid granular lows [84]. Hence, it was important to urther eamine this issue, and we studied it by simulating a Couette low o inelastic, rough, hard spheres. Our simulations were carried out in a cell under periodic boundary conditions in the y and directions, with a velocity gradient imposed in the z direction. Two massive walls, with the same properties as the interior particles, were ied to move in opposite directions parallel to the direction (see Figure 3.1).

27 Z 3. MD simulations o a granular medium 19.5 X Y Z Y X -.5 Figure 3.1 A schematic picture o the initial coniguration o dark and light particles in the computational bo which includes the wall particles. The number o particles in the interior and in the walls varied rom 496 to 484, and rom 4 to 65, respectively. This meant that or concentration ranged rom.5 to.6. All the simulations began with an initial set o random overlapping spheres using the method o Re. [85]. Then the individual spheres were moved randomly until the overlaps were removed. On one side o the plane Z = the particles were marked dark, whereas on the other side the particles were marked white (see Figure 3.1). 1 Granular shear low with average shear rate rom zero to γ = u H 4 s was then studied using the molecular dynamics method (see chapter ): w w u w is the wall velocity and H is the gap between the walls. The particles were modelled as inelastic, rough, hard spheres with radius R or which H R 1. In order to simulate a dense granular low, Lun and Bent [86] suggested certain values or the dissipation parameters. Using their values (i.e., the coeicient o restitution e =.93, the surace riction coeicient µ =.13, and the phenomenological constant or sticking contacts β =.4 the value reported previously [8]. 3. RESULTS AND DISCUSSION ), the collision requency was 6 khz, which was close to From the simulations, stress luctuations, diusion coeicient tensors, velocity distributions, density proiles, radial distribution unctions, and correlation unctions were determined.

28 3. MD simulations o a granular medium (a) (b)..1 z z. z -.. z (c). z. z y -. Figure 3. Snapshots o (a) the initial coniguration o white and black particles, and at (b) t * = 59 and (c) t * = 5, projected to the z (let) and yz planes (right). The arrows indicate the shear low direction. Snapshots o a system with solid volume raction φ =.565 and shear rate γ w 4 s are shown in Figure 3.. The snapshots are projections o particle locations to planes parallel and normal to the shear low, respectively. From Figures 3. (b) and (c) a diusive evolution o the system can be seen. This is in contrast with what Campbell [8] reported on the absence o diusion. The lack o sel-diusion may have been induced by an ordered initial coniguration in their simulations. The ordered layers in the z plane rom a close to heagonal pattern in the yz plane (the pattern is also close to icosahedral). g(r*) (a) r* G 6 (r*) Z * [Z/R] 5 <T kin > (b) <T> V <φ z > Z * [Z/R] r* V -5 Figure 3.3 (a) Radial distribution unction in the equilibrium situation 1 * * ( φ =.56, γ w = 4 s, t 5 ) as a unction o dimensionless radial distance r =r R. (b) Velocity proile and kinetic granular temperature at t * = 15, with 1 * φ ~.5 and γ w = 4 s, as unctions o dimensionless channel height ( Z = Z R). The inset shows the average density proile across the sample.

29 3. MD simulations o a granular medium 1 The radial distribution unction shown in Figure 3.3a has a peak at r * =r R 3, indicating localised ordering around the particles. This means that the initially disordered system has evolved to an ordered state in the presence o shear low. The degree o translational order ( ρ k ) was tested according to Eq. (.) by choosing k=( π l ) z, where z is a unit vector in the z direction. We ound that is.8 or l H 11 (see also Figure 3.3b). This could indicate a layered state with eleven layers. We also ound that ρ k in the direction (, +, ) y z, thus veriying the absence o cubic symmetry (the sliding cc phase). We Q r as investigated thereore a set o orientational order parameters, denoted by ( ) described in Eq.(.1) and Table.1, associated with each bond with 1 nearest neighbours. Nonzero value o this order parameter indicates an ordered structure. We ound that Q.4 6 suggesting an icosahedral lattice. However, the value Q8.4 suggested that the symmetry o the bond-oriented states is not perectly icosahedral and, as can be seen rom the inal coniguration in Figure 3., the structure is rather close to that o a simple heagonal lattice. More inormation regarding possible types o orientational order o the above-mentioned system can be obtained rom the bond-angle correlation unctions G (see Eq. (.)) [39]. The * value o 6 ( ) G r at large r*, shown in the inset o Figure 3.3a, also indicates that the system is close to an icosahedrally oriented liquid, possessing a degree o symmetry intermediate between those o a crystal and a liquid. In conclusion, or a high average shear rate, the system o inelastic, rough, hard spheres displays the symmetry o imperect icosahedral or heagonal liquid or solid volume ractions larger than φ =.565. In d dimensions, the average translational kinetic energy o particles whose N 1 d velocities satisy the Mawell-Boltzmann distribution, is < mu i i >= kt. The i= 1 kinetic granular temperature T kin is given by where ρik, T kin 1 d k = 1 Nk d i= 1 ρ ( u u ) ( u u ) i, k i k i k Nk i= 1 ρ ik,, l lm (3.1) is the volume o slice k covered by particle i, and u is the particle velocity k k with the average uk ρi, ku i ρ i, k. N N i= 1 i= 1 In Figure 3.3b we show the velocity and density proiles, and the local kinetic granular temperature. A layered structure in the yz plane is evident rom the density proile, and an S-shaped velocity proile and a high granular temperature near the walls can also be observed. We ound that the granular temperature is i

30 3. MD simulations o a granular medium proportional to the transverse sel-diusion coeicient, D * T, which can be understood rom dimensional analysis (Eq. (.3)). The amplitude luctuations o the dimensionless normal stress on the walls * P t = P ρp 4R γ w with t * =tuw H the dimensionless time and ρ p the particle ( ( ) density [87]), was ound to obey a strongly asymmetric distribution similar to those 1 observed in recent eperiments [79]. For a system with φ =.565 and γ w = 4 s at * t, the amplitude o stress luctuations on the wall increased with increasing coeicient o restitution (rom e =.84 to e=.93), and decreased with surace riction (rom µ =.41 down to µ =.13) as shown in Figure 3.4a. For e =.84 and µ =.41 at t * >, the luctuations were much smaller than those observed by Savage and Sayed [ 87]. In the opposite case, by increasing the coeicient o restitution to e =.93, and decreasing the surace riction coeicient to µ =.13, the result was closer to those in the annular shear cell tests o Re. [87]. However, all the results obtained were much smaller than those or gravity-driven channel lows [79]. The decay o the absolute value o the mean dimensionless normal stress, P *, appears to be almost eponential beore is, however, an additional decrease o * t, as evidenced by Figure 3.4a. There * * 7 P at about t (~5 1 collisions), which might indicate a phase transition. Other evidence or a transition could be obtained rom translational and bond orientational parameters [ I, 87], and rom the radial distribution unction. The stress luctuations on the walls increased with the solid volume raction when the latter was increased rom.56 to.58 (c.. Figure 3.4a). Meanwhile the sel-diusion coeicient (measured rom the slope o the mean square displacement curve or dimensionless time intervals τ * > 1), decayed urther approaching a value close to those o recent eperimental observations [79, 83]. The increased stress luctuations may be the result o higher dissipation at.58 leading to the decrease in the transverse sel-diusion coeicients. At φ =.58, diusion decays rapidly with * -5 time (at t = 45 ) to D* 1 ( the diamonds in Figure 3.4b). This also is an evidence o a phase transition to an ordered state or to a structural arrest. In an ordered system luctuations induced changes in the geometry o the cage ormed by the nearest neighbours around a particle become inrequent. This could result in a dramatic decay o the long-time transverse sel-diusion coeicient.

31 3. MD simulations o a granular medium 3 P * [P/(4ρ p R γ w )] (a) φ s =.565, e=.84, µ=.41 φ s =.565, e=.93, µ=.13 φ s =.58, e=.93, µ= t * 1 - < Z * >[< Z /4R >] 1 (b) D * =81-4 φ s =.565 at t*=159 φ s =.565 at t*=45 D * =1-5 φ s =.58, at t*= 1 3 τ * [τ u w /H] D * =1-4 Figure 3.4 (a) Dimensionless normal stress, eerted by the particles on the bottom wall as a unction o dimensionless time t *, or three indicated sets o parameters. (b) Dimensionless mean square transverse displacement < Z * > as a unction o dimensionless time τ *, or three sets o parameters that include the starting time o the * analysis t. Solid lines are linear its to the data or τ * > 1. Also shown in Figure 3.4b are the sel-diusion coeicients or the cases * 3 φ =.565, t = 159, 45. The coeicients decay rom ~ 1 up to 1 4 * when t increases rom 159 to 45. This is consistent with the result presented in Figure 3.4a. In order to test that the behaviour we ound above or rough particles is not due to roughness only, we did some o the analyses or systems o smooth particles. For a solid volume raction o.565 and a suiciently long starting time o t* 4, our results show that the calculated dimensionless long-time transverse sel-diusion - coeicient or the system comprised o smooth particles is D * =1. 1, which is an * -3 order o magnitude higher than that or the system o rough particles D =1.5 1 (not shown). The above-mentioned value or the system o smooth particles is close to those reported in Re. [ 79] or moderate shear rates. It appears that shearing o particles with rough suraces generates lower normal stress than a similar shearing o smooth particles. This may be interpreted such that rough particles tend to have more rotational energy than smooth particles. Moreover, particles with rough suraces lack transverse diusional movements. This observation supports the Menon and Durian results [8] in that the dynamics o grains in a dense granular low are dominated by collisions rather than sliding contacts. A comparison between diusion coeicients or dierent volume ractions in granular media revealed that or dilute systems, the velocity autocorrelation unction also decays eponentially as predicted by kinetic theory [88]. Results are shown in Figure 3.5 or a dilute and a dense system. For the higher volume raction, φ ~.51, the velocity autocorrelation decays aster, and correspondingly there is a smaller diusion coeicient.

32 3. MD simulations o a granular medium (a) 1. (b) <v () v (τ)>.3..1 φ s 3% φ s 51% < X * >, < Z * >.8.4 4D * 4D * zz 4D * τ * 4D * zz 1 τ * Figure 3.5 (a) Velocity autocorrelation unction as a unction o dimensionless time or φ ~ 3% (black symbols) and φ ~ 51% (white symbols). (b) Dimensionless mean square transverse displacements < Z * > o a system o smooth particles as a unction o dimensionless time τ *. Solid lines are linear its to the data. Filled symbols denote 3% solid volume raction and white symbols 51% solid volume raction. Our results also show that in the stream direction the sel-diusion coeicient was much higher than in the transverse directions or both smooth and rough particles, in agreement with the eperimental results o Hsiau and Shieh [89]. In conclusion, our numerical simulations o a sheared, dense, monosized granular material indicate there is a phase transition at long run times or systems o rough particles, which causes a sharp drop in the dimensionless transverse long-time seldiusion coeicient o the particles. However, the system is diusive at solid volume actions even higher than.56. The structure o the ordered state was ound to be closer to that o a simple heagonal lattice rather than icosahedrally oriented liquid. A similar behaviour could be seen in system o smooth particles so that the sel-diusion coeicient decreased by increasing concentration. In this case, the seldiusion coeicient was much higher in the stream direction than in the transverse directions, and much smaller than that or rough particles.

33 4. Capillary rise 5 4 CAPILLARY RISE In this section we review our results or simulations on single- and two-phase luids. The purpose o these simulations was to better understand the solid-liquid (wetting) and gas-liquid (surace tension) interactions. It would also be interesting to see how well the LB method is capable o simulating the well-known capillary rise phenomenon, and the method was thus tested on an uprising luid in a narrow capillary pipe [II]. 4.1 THE HYDRODYNAMICS OF CAPILLARY RISE The capillary rise phenomenon has been o wide theoretical and practical interest in the recent years [9,91,9,93,94], with applications ranging rom simple capillary rise and imbibition (penetration o liquid (droplet) into a porous material) to droplet spreading and other phenomena related to wetting. The basic analytical theories or capillary rise were developed a long time ago [95], but recent eperimental and numerical techniques have provided new insight into this problem: on the numerical side irst the LGA models [16, 14,96,97,98] and then the more recent LB method [II, 55, 96,99]. In the classical analysis by Washburn [1], the motion o an incompressible luid is treated as a Poiseuille low [11]. When we consider the rise o an incompressible liquid in a capillary pipe, we can thus start rom the Hagen-Poiseuille equation or a ully developed pipe low under a pressure drop o P, 4 dq π Pr =, (4.1) dt 8 h+ h where dq dt π r dh d ( ) µ + = t is the volumetric low rate, h is the height o the rising liquid column rom the level o the liquid surace outside the pipe, o the capillary pipe immersed in the liquid, r h + is the length is the radius o the pipe, and is the viscosity o the liquid. The total pressure drop may be epressed as a sum o capillary pressure and the static pressure eerted by gravity: γ cosθd P= ρ gh, (4.) r µ

34 4. Capillary rise 6 where luid density, surace tension and dynamic contact angle are denoted by ρ, γ and θ d, respectively. In the classical capillary rise theory the contact angle is constant, but in the phenomenologically corrected version [1] o this theory contact angle varies with velocity, and is thus called the dynamic contact angle. Combining Eqs. (4.1) and (4.) one can write dh 8 µ ( h+ h+ ) = rγ cosθd r ρ gh. (4.3) dt By including inertial and entrance eects [11], and by rearranging the terms, Eq. (4.3) becomes 1 d h 8 µ ( h+ h+ ) dh 1 dh cos θd = ρrh + + ρ ( ) r + ρ rgh. (4.4) γ dt r dt 4 dt This is the Washburn equation whose solution we will use to compare with the simulated column height, with the dynamical contact angle irst determined by simulations. Inertial orces can usually be ignored in the overdamped limit 3 ( 3µ γ cosθ ρ ) 15 r < r = g c d [1], which condition is satisied by the parameter values in our system. Asymptotically and at long times ( t ), when dh dt, one inds that cos h = γ θ, (4.5) ρ gr where θ = limθ d. Due to the rough discretization and the steep density variation t especially or small r, it is diicult to accurately measure θ d directly rom the simulated density ield. For an indirect determination o θ d, we irst apply the onedimensional Reynolds transport theorem in a control volume to estimate the rate o change o the momentum o the system. The upper control surace moves with the meniscus o the liquid column (no outlow rom the control volume) and the lower control surace is ied at the lower end o the pipe. Then the rate o change o the total momentum inside the control volume is d d ( m ) syst [ d V] ( )d dt v = ρ ρ dt v + CV v v n A, (4.6) CS where m= m() t is the total mass o the luid inside the control volume, v is the luid velocity, and CV and CS denote the volume and the surace area o the control volume, respectively. By integration we ind that d dv dh d ( mv) cv [ d V] r h dt vρ = ρ π CV + v =, dt dt dt (4.7) vρ( v n)d A= v( ρ πr v). CS

35 4. Capillary rise 7 where v and n are liquid velocity and a unit vector normal to the inlet control surace, respectively. Assuming a constant velocity through the control surace, we obtain an epression or θ by substituting Eq. (4.4) into Eqs. (4.6) and (4.7), d d ( mv) 1 3 cos θ cv d = [ rv 8 ( h h) v rgh]. πrγ dt 4 ρ π + πµ ρ π (4.8) ht () This equation can be used to determine θ d rom the simulated. Notice that this is the correct orm or Eq. (8) in Re. [ II]. However, the results remain almost unchanged, especially at long times, because the rise velocity is very small. The height o the column ht, or the location o the gas-liquid interace, was () determined as the turning point o a least-squares it by a tanh unction o the density proile in the vertical direction. A number o empirical relationships or wetting have been discussed in the literature, all o which epress the dynamic contact angle θ d as a unction o the capillary number Ca = vµ γ during spreading such that θ = (Ca, θ ), where the d a + advancing contact angle θ a is that o a spreading liquid in the limit v. We adopted the epression generally used in studies o capillary rise [1], cosθ = cosθ Ca B, (4.9) d a A where A and B are constants. This dependence on θ a and simulations, and a reasonable agreement was ound. 4. RESULTS AND DISCUSSION Ca o θ d was tested by We irst simulated a three-dimensional two-phase luid system by using the LB method [II] with periodic boundaries in the lateral directions and without any pipe. The simulation code has already been tested or single-phase luid and low through porous media, and many benchmark results have been reported [6]. Ater a certain number o iterations, when the system reached the equilibrium state, the pipe was inserted, and the simulation was continued until saturation o the capillary rise (c.. the snapshots in the insets o Figures 4.3 and 4.4 or r=5 and, respectively). The simulations reported below were perormed or pipe radii r = 51,,, while the ull domain size was 5 5, and or r =. In all capillary rise simulations the relaation parameter was τ = 1, and unless stated otherwise, the adhesive parameters were W = 1. and G = 15.. This resulted in ρ ρ, with the bulk density o the liquid phase ρ = 5., viscosity µ =. 375, and the surace tension γ =. 85. Dimensional analysis or the group o variables (,,,,, ) ht g ρ µ γ was done or presenting the results in dimensionless units. We thus deined the characteristic 3 time, velocity and height as t = µ ρ γ 3., v = γ µ.3 and.74 h = µ ρ γ, respectively. In the presence o gravity, we also used the dimensionless parameter g = µ g ργ. 4 3 g

36 4. Capillary rise 8 An important issue is whether variation in the capillary radius changes the dynamics o the low in the two-phase low model. To address this question we calculated the velocity proile or pipes with dierent radii, and compared them with the analytical results [11] (see Figure 4.1). This velocity proile could be determined rom a steady low o the single- or two-phase luid at its equilibrium density between two ininite (periodic boundary conditions in the lateral directions) parallel plates. A constant body orce at each lattice point produced the low without any phase separation in the two-phase model. The well-known analytical results or Poiseuille low [11] are also shown in Figure 4.1 by the ull lines r= r=1 r=5 LBM, W=, G= Analytical result [11]..4.6 v Figure 4.1 A typical eample o the velocity proile o the ully developed low o a single-phase (W = G = g = ) luid, and o a two-phase (W = 1., G = 15., g = ) luid, between two parallel plates. A comparison with analytical results showed that in our simulations the velocity proile had the correct (Poiseuille) parabolic orm but it was shited orward. This relative deviation increased or decreasing r as epected rom the boundary eects. The reason or this deviation was a strong boundary-layer eect. The adhesive orce deined in Eq. (.34), generates a low-density liquid layer on the pipe surace. F G Due to very low resolution, we did not epect sensible results or r =, and or strong adhesion also r = 5 and r = 1 were too small. As seen in the density proiles o Figure 4., the narrow pipe with radius r = did not have the bulk liquid density anywhere in the pipe. The decay o density close to the inside walls o the pipe was however similar or all r = 51,,, at least or weak adhesion orces, e.g. W =.1 (see Figure 4.). The proportion o the liquid with bulk density decreased with decreasing radius. For increasing adhesion we observed that the simulated velocity proile approached the theoretical curve, and that the boundary eects decreased in the density proile (see the curves or r = in Figure 4.). At the same time, due to increased discrete-

37 4. Capillary rise 9 lattice eects at the gas-solid interace, values or cosθ d greater than unity were obtained or small r, which is clearly unphysical..5 ρ r= r=5 r=1 r=, w=.11 r=, w= Figure 4. The density proiles across the capillary pipe or radii r =,5, 1 and as unctions o distance rom the centerline o the pipe. With increasing adhesion (not shown), the density o the luid in the irst layer net to the wall increased, but the range o the wall eect (liquid layer with a thickness o a ew lattice spacing) in the density was the same. In Figures 4.3 and 4.4 the column height and the dynamic contact angle are shown or r = 5 and as unctions o simulation time. For comparison we also show the results with and without the presence o gravity, while W = 1. was kept constant. For increasing pipe radius there was a decrease in the column height which is epected rom Eq. (4.3). Without gravity, as epected rom Eq. (4.3), the column rose aster with increasing pipe radius. Moreover, there was an increase in the column height or increasing pipe radius, which is epected. The solid lines in Figures 4.3 and 4.4 show the corresponding numerical solutions o Eq. (4.4), where the dynamic contact angle θ d was determined rom Eq. (4.8).

38 4. Capillary rise π/ π/3 π/6 θ d h/h 1 θ d h/h No g θ d h/h g*=.146 Washburn 1 3 t/t Figure 4.3 The height o the column and the dynamic contact angle as unctions o simulation time or the pipe radius r = 5. The inset shows three snapshots o the capillary rise: initial, intermediate, and steady state (side view). 4 3 π/ π/3 π/6 θ d h/h 1 θ d h/h No g θ d g*=.75 h/h Washburn 1 3 t/t Figure 4.4 The height o the column and the dynamic contact angle as unctions o simulation time or the pipe radius r =. The inset shows three snapshots o the capillary rise: initial, intermediate, and steady state (side view). I the contact angle was kept constant, a satisactory agreement between the Washburn equation and the simulation data could only be obtained or a short time window up to t/ t 1. On the other hand, by using θ d in the Washburn equation as obtained rom Eq. (4.8), good agreement was ound or large r up to the latest data points, where some deviation appeared as a result o an increase in the interace velocity close to the end o the pipe. At the beginning o the simulations we had cosθ d =, or θd = π/, as can be seen rom Figures 4.3 and 4.4. Ater this the contact angle θ d decreased with increasing

39 4. Capillary rise 31 time or decreasing Ca. The steady state contact angle was quickly reached, especially in the presence o gravity, in a ew hundred time steps o the total simulation time o about 5 time steps. The increase o cosθ at small Ca results rom the liquid reaching the end o the pipe. For small capillary numbers and long times, i.e. or slow interace velocities, θ d approached a constant as epected. For increasing pipe radius there was an increase in the steady state contact angle, which is unphysical and is due to the diiculty o determining the precise angle rom simulation results. For stronger gravity the relative importance o the other resistive orces in Eq. (4.4) diminished, and the behaviour o θ d was more consistent. This supports the idea that the rictional orces in our simulation model are somewhat dierent rom those assumed in the hydrodynamical derivation o Eq. (4.8). We also eamined the speed o capillary rise as shown in Figure 4.5 as a unction o time or both zero and non-zero gravity. The column irst sharply speeded up and 1/ 3 then decayed towards the stationary state approimately as ~ t. More simulation time is however needed to properly i the eponent. While the sharp initial acceleration resulted rom the ast implantation o the pipe into the system, the decay o velocity or non-zero gravity was almost independent o capillary radius..3. v/v v/v, r=, zero gravity v/v,r=5,zerogravity d v/v t/t t/t v/v,r=,zerogravity v/v, r=5, zero gravity v/v, r=, g*=.75 v, r=5, g*=.146 Figure 4.5 Normalized velocity o the column as a unction o simulation time. In conclusion, our test results are promising: The capillary phenomenon can indeed be studied by LBM at the hydrodynamic level. However, due to discreteness eects one needs rather thick (in lattice units) pipes or realistic kinetics. For the parametrization used here, the diameter o the pipe should be at least 3 lattice units. Future work needs to be done to validate or modiy the theoretical velocity proile o a narrow capillary with strong adhesive orces at walls. For the static properties in the presence o gravity, somewhat thinner pipes are appropriate, while thicker pipes are needed in the case o strong adhesion.

40 5. Some rheological properties o liquid-particle suspensions 3 5 SOME RHEOLOGICAL PROPERTIES OF LIQUID- PARTICLE SUSPENSIONS In this chapter I eplain results or Couette low o liquid-particle suspensions. In the simulations lattice-boltzmann methods or the luid (Chapter 4) and those o molecular dynamics or the Newtonian motion o solid colloidal particles (Chapter 3) are combined. The main motivation or this part o the work was to provide benchmarking or subsequent simulations on liquid-particle suspensions. We clariy the role o particle shape, shear rate, and solid volume raction (area raction), i.e. concentration, in the rheological properties o 3D (D) suspensions. The viscosity o suspensions is analysed as a unction o shear rate and concentration. 5.1 BACKGROUND Suspensions o submicrometer-sized particles appear in many industrial processes e.g. in paper industry, in emulsion technology, in environmental processes, and in biological systems, to mention just a ew eamples [see e.g. 13]. There is thus wide interest in understanding the low dynamics and rheological properties o such liquid-particle suspensions. Due to their complicated microstructure, the rheology o suspensions may depend on many actors such as particle-volume raction, particle size, particle shape, ionization, suspending liquid, low rate and shear rate. Recently, it has also been shown that polydispersity may have a signiicant eect on the hydrodynamics o particle suspensions [14,15,16]. Attractive and repulsive interactions between the particles have urthermore been ound to have a signiicant eect on suspension s rheology [17]. One should understand better how e.g. particle interactions aect the rheological properties o the suspension or increasing concentration o particles, and the two-phase character o the suspension may give rise to complicated microstructural behaviour in dierent low regimes. The theoretical (both analytical and numerical) study o suspension rheology is diicult or primarily two reasons. First, the hydrodynamic interactions between the suspended particles involve several length and time scales. There are short-range lubrication orces, which are two-body interactions. There is an intermediate range in which many-body hydrodynamic interactions are important. There are also longrange interactions that must be properly renormalized. Second, the eact

41 5. Some rheological properties o liquid-particle suspensions 33 knowledge o the orces and stresses or a given coniguration o the particles is not suicient to determine the rheology, because an average over dierent particle conigurations is needed. These conigurations are themselves results o an interplay between the eternal driving orces (e.g. an imposed shear low or gravity) and the internal hydrodynamics, i.e. the inter-particle and Brownian orces. Thus, the system is completely coupled [18]. For these reasons, theoretical approaches have generally assumed highly idealized systems, such as dilute spherical suspensions at low shear rates. Eperimental knowledge is also limited as the operation o even the most commonly used viscometers is not always properly under control. Analytical and numerical studies have traditionally been based on various continuum models. However, these models always include several parameters o microscopic origin that are diicult to determine. Thus, simulation results or particulate suspensions would be most helpul and have immediate and wide applicability. On the computational side detailed analyses o microstructures have indeed become easible through direct simulation o the motions o individual particles. Nevertheless, such studies are still computationally heavy, and usually involve approimations. Viscosity o liquid-particle suspensions The viscosity o a suspension o monodisperse spheres can depend on several dimensionless parameters. The most important o these are the solid volume 3 raction φ = (4/ 3) πρnr and the Peclet number Pe = γ d / D. Here d is the particle diameter, ρ n the particle number density, R the radius o the particles, γ w the shear rate, and the diusion coeicient related to the Brownian motion o the particles. D The Peclet number epresses the ratio o the hydrodynamic shear orces and the diusive Brownian orces acting on the suspended particles. The Brownian orces tend to bring the suspended particles back to their equilibrium coniguration, which is continuously disturbed by the hydrodynamic shear orces acting on the particles. I the Brownian orces are omitted (as is the case here), the shear Reynolds number ( H) Reγ = ρ duw µ γ wd ν = is used to characterise the low conditions instead o the Peclet number. Here ν is the kinematic viscosity o the carrier luid, and ρ is its density. For D discs, e.g., the area raction is φ = πρ n R. The unctional dependence o viscosity on φ and w Pe (or Re γ ) is known theoretically only in a ew limiting cases. Einstein derived at zero Reynolds number his amous ormula or the relative apparent viscosity µ o the suspension ( µ r = µ a µ with µ a the apparent viscosity o the suspension and µ the viscosity o the pure luid), 5 µ r = 1+ φ. (5.1) r

42 5. Some rheological properties o liquid-particle suspensions 34 Einstein omitted in his analysis the Brownian motion o the particles, and all interactions between them. For this reason Einstein s ormula is only valid or very small (smaller than about 5%) solid volume ractions φ. When the solid volume raction is increased, the eects o interactions between the suspended particles become signiicant. The inclusion o Brownian eects and hydrodynamic twoparticle interactions lead to a φ term in the epansion o the viscosity [19,11], 5 µ r( γ w) = 1 + φ + k( γ w) φ. (5.) Notice that µ r now also depends on the shear rate γ w, an eect which will become increasingly important or increasing volume ractions φ. This dependence is included in the actor k which has the ollowing limiting behaviour: (i) in the lowshear limit ( γ w ), where Brownian motion dominates (Region I o Figure 5.1), k = 6. [ 11]; (ii) in the high-shear limit ( γ w ), where hydrodynamic contributions dominate (Region III in Figure 5.1), k = 5. [19]. Unortunately, Eq. (5.) agrees with eperiments only up to volume ractions o order 15 %. For still higher concentrations many-particle interactions must be taken into account. This is where theoretical diiculties become more severe. µ r I II III γ w (1/s) Figure 5.1 Typical viscosity o a liquid-particle suspension as a unction o shear rate. When etending the dilute-suspension analyses to higher concentrations, diiculties arise in computing the many-body hydrodynamic interactions, and in determining the conigurations o the microstructure. A simple and oten used epression is the relative viscosity o the (semi-empirical) Krieger- Dougherty model [111], ηφ ma φ µ r = 1, (5.3) φma where η is the intrinsic viscosity o the suspension, which varies between.5 and.67 or rigid spheres depending on whether the particles are charged or not, and φ is the maimum packing raction at which viscosity diverges. Eperimental ma results o Van Der Wer and De Krui [11] suggest that ηφ ma is close to. The values o the maimum packing ratio φ ma which we use in scaling the volume

43 5. Some rheological properties o liquid-particle suspensions 35 raction are.785 (D) and.65 (3D), which correspond to lowing arrangements o simple cubic and heagonal packings, respectively, perpendicular to the plane o shear [15, 11]. 5. BENCHMARK STUDIES FOR THE LB METHOD We carried out numerous benchmark studies to ensure the correctness and accuracy o our results and our simulation codes. As shown in Chapter 4 or luid-low simulations by the LB method, good agreement was ound between the theoretical and simulated velocity proiles in a channel low (see Figure 4.1). For suspension simulations an obvious test is also to compare the hydrodynamic radius R o particles with their nominal radius R. The hydrodynamic radius H H R was derived by simulating luid low through an ininite array o discs in D or spheres in 3D. The simulation was perormed by placing a solid ied particle in an arbitrary location in a periodic cubic bo with side length L, and subject to luid stream with velocity U. Analytical epressions or the drag coeicient are known or both D and 3D systems [ 113,114], and the latter is given by FD = 6πµ RU H - + R - 3 H R 6 H RH L L L. (5.4) We showed in Re. [III], and show here in Table 5.1, that the hydrodynamic radii o discs in D were systematically smaller than R, but only by less than 1 a lattice point. R <RH> Table 5.1. Input radius R and the mean hydrodynamic radius <R H > or a regular array o disc-shaped (D) particles placed in dierent locations. The same test was carried out in 3D by locating a solid sphere in an arbitrary location in a periodic cubic bo, and the mean hydrodynamic radius was ound somewhat smaller than the nominal radius R (see Table 5.). The small discrepancy both in D and in 3D is mostly due to our realization o the no-slip boundary condition. R <RH> Table 5.. Input radius R and the mean hydrodynamic radius <R H > or a regular array o spherical particles (3D). Furthermore, or a disc-like particle moving under an eternal orce, such as gravity, in the middle o a channel, good agreement was ound in the orces induced on the particle and the walls, between our results and those calculated by FEM or the LB method o Re. [7] (see Table in Re. [III]).

44 5. Some rheological properties o liquid-particle suspensions 36 In another benchmark study, the hydrodynamic orces on a ied disc with orced counterclockwise rotation close to a moving wall were simulated, and compared with the results o a commercial inite-volume solver (Fluent). Simulations were carried out or several particle radii and or two dierent gaps between the centre o the particle and the moving wall, and the results are shown in Table 3 o Re. [III]. We also analysed the hydrodynamic lubrication orces between two discs scattering with a non-zero impact parameter, and in central collision [III]. The simulation results were compared with the analytical results o Re. [115]. In both cases the relaation parameter was τ = 1. In the scattering case, the lubrication orce is [115] FSc = µ Ut J J 1 s J O hs with ( ) 1 1 π s, h 1 and J tan ( ) s, (5.5) ( ) = = + =. In central collision the lubrication orce is given by [115] FC = µ Un 3 J + + s J + O + ( s ). (5.6) s h 4 4h 4h Here s= ( D1 R) /R is the relative spacing between the two particles o radius R and o centre-to-centre distance D 1 with 1 and denoting the particles, = ( U U ) t with U1 and U the particle velocities and t a unit vector in the U t 1 direction o motion o the particles, and U n = ( U1 U) n with n a unit vector normal to the direction o the line connecting the particles at their shortest mutual distance. The relative normalization o the analytical and simulated lubrication orces was done so that the orce on a particle by the other particle is asymptotically (or large relative distances) the same independent o the method. F * C (a) s t n F * Sc (b) s t n 1 Radius 5.4 Radius 1.8 Radius Dodd et.al. [115] s 1 Radius 5.4 Radius 1.8 Radius Dodd et.al. [115] s Figure 5. (a) Lubrication orce on two particles in central collisions as a unction o their dimensionless distance s. (b) Lubrication orce on two particles scattering with a non-zero impact parameter as a unction o the dimensionless impact parameter s. The ull curves are the analytical results or the respective cases[115]. It is evident rom Figure 5. that the simulation results are in rather good agreement with the analytical solutions. The accuracy was even better in the case o central collision and higher lattice resolutions. Similar behaviour was ound in Re. [61] or

45 5. Some rheological properties o liquid-particle suspensions 37 corresponding 3D simulations. As also shown in Re. [73], lubrication orces given by the LB method are typically smaller than the corresponding analytical predictions when particles are very close to each other (see Figure 5.b). However, they are strong enough to keep the particles separated most o the time. In order to test the inertial behaviour o the suspended solid particles, we also perormed the lateral migration tests o Aidun and Lu [7], and Feng et. al. [116] (see Figures 5.3 and 5.4). We ound good agreement with their results, when a particle initially placed near a wall migrated to the middle o the channel as shown in Figures 5.3 and 5.4 or three dierent ratios o partice density to luid density. In Figure 5.4, an overshoot rom the middle o the channel is also accurately reproduced by the present method..5 y/h.4.3. ρ s /ρ = ρ s /ρ =3 ρ s /ρ =5 Aidun & Lu [7] Feng et al. [116] Re γ = /L Figure 5.3 Comparison o the lateral migration o a particle in a vertical channel o length L ( direction) and width H (y direction) with the results o Aidun and Lu [7], and Feng et. al. [116], or Re 1. γ =.5 y/h.4.3. ρ s /ρ = ρ s /ρ =3 ρ s /ρ =5 Aidun & Lu [7] Feng et al. [116] Re γ = /L Figure 5.4 Comparison o the lateral migration o a particle in a vertical channel (the same as in Figure 5.3) with the results o Aidun and Lu [7], and Feng et. al. [116], or Re 3. γ =

46 5. Some rheological properties o liquid-particle suspensions VISCOSITY OF LIQUID-PARTICLE SUSPENSIONS In order to simulate the Couette low o a liquid-particle suspension, suspension was placed between two moving solid walls oriented in the direction and separated by a distance H. The walls moved with speed in opposite directions. Couette-low conditions are thus created with the mean shear rate u w γ = u H. Periodic boundary conditions were imposed in the direction. The simulation grid was rectangular and usually up to lattice points. The diameter o the particles was 1 lattice units. Volume raction (area raction) o the particles was varied between φ = 6% and φ = 55%. The shear Reynolds number ( Re = ρ d γ w µ ) was varied between.14 and A typical duration o the γ macroscopically transient states was -6 simulation steps depending on the system size. The necessary time-averaged quantities were calculated or the stationary states over 4 iteration steps, which even in the worst cases corresponded to several mean periods o a particle traversing the system. We computed the relative apparent viscosity µ r = µ a µ o the suspension. Here µ a, the apparent viscosity o the suspension, is given by µ a = τt / γ w with τ T the total shear stress on the moving walls. We irst analysed the eect o particle shape on the viscosity, and show in Figure 5.5 µ as a unction o φ or disk-shaped and star-shaped particles with Re 1.. γ It is evident that the viscosity in the case o star-shaped particles (a cross-shaped combination o two perpendicular ellipsoidal particles, with two dierent ais ratios a and major aes l : a= 4.3, l = 17. and a=.7, l = 8. ) increases much more rapidly with increasing solid volume raction than in the case o disc-shaped particles. Instead, the viscosity o ellipsoidal particles quite closely ollows that o disc-like particles (not shown). Also shown in Figure 5.5 is the D version o the semiemperical ormula by Krieger and Dougherty [ 111], Eq. (5.3) with φ ma =.785 and ηφ ma =. r w w

47 5. Some rheological properties o liquid-particle suspensions 39 µ r Krieger & Dougherty (D) [111] Aidun & Lu, D LB method [7] Brady & Bossis, D Stokesian [117] Disk particles Star particles a=4.3 Star particles a= φ [%] Figure 5.5 The simulated relative apparent viscosity µ r or disk shaped and star-shaped particles as a unction o the solid volume raction φ or Reγ 1.. Also shown are the semi-empirical ormula by Krieger and Dougherty [111] (see Eq.(5.3)), and the simulation results o Aidun and Lu [7] and Brady and Bossis [117]. The three-dimensional Couette low o monodispersed randomly distributed spherical particles was simulated in a luid, which was modelled using the threedimensional D3Q19 model with 19-links, introduced in Chapter. The simulation grid was 18 18, the diameter o the particles 1 lattice units, Re.1, and volume raction o the particles was varied between γ = φ = 6% and φ = 55%. A snapshot o these simulations is shown in Figure 5.6a. A simpliied version o the 3D simulation o Couette low o monodispersed spherical particles was also carried out such that the particles were restricted to move only in one layer along the y plane. In these monolayer simulations the shear Reynolds number was varied between.1 and 15, and φ between 6% and 5%. A snapshot o these simulations is shown in Figure 5.6b.

48 5. Some rheological properties o liquid-particle suspensions 4 (a) (b) Figure 5.6 (a) A snapshot o a shear low o suspended spherical particles as simulated by the three-dimensional LB method. (b) A snapshot o a shear low o a monolayer o suspended spherical particles as simulated by the 3D (m) LB method. We computed the relative apparent viscosity µ r o the suspension or both versions o the 3D simulations. In Figure 5.7 we show the relative apparent viscosities or Re γ =.1 as unctions o the scaled volume raction φ / φ together with the 3D Krieger ormula, or which ηφ ma = 1.6. ma 6 Krieger & Dougherty (3D) [111] 3D 3D (m) 4 µ r φ /φ ma Figure 5.7 Simulated relative apparent viscosity or small Reynolds numbers, in a 3D as well as in a 3D (m) (monolayer) system. Also shown is the 3D Krieger ormula [111]. As can be seen rom Figure 5.7, both 3D and 3D(m) methods very accurately ollow the Krieger ormula with less than % deviations below φ / φ.5. In Figure 5.8 we compare the simulated relative viscosities o D and 3D(m) suspensions or three Reynolds numbers.1, 1.5, and 11.7, plotted as unctions o the scaled volume raction φ / φ. Also the D Krieger ormula is shown or ma comparison, and it should be applicable at low Reynolds numbers (see Eq.(5.3)). ma

49 5. Some rheological properties o liquid-particle suspensions D Krieger & Dougherty (D) [111] Re γ µ r 6 4 3D (m) φ /φ ma Figure 5.8 The simulated relative apparent viscosity µ r or disk like (D) and spherical 3D (m) particles restricted to move in a monolayer as a unction o normalised solid volume raction φ / φma : φ ma =.785 in D and.65 in 3D. Results are shown or three values o Re γ. Also shown is the D version o Krieger ormula. As shown in Figure 5.8, the magnitude o relative apparent viscosity is ound to be higher in the D system than in a 3D monolayer o spheres or corresponding values o scaled concentration. For increasing shear rate the deviation rom Krieger ormula occurs at lower concentration, which is epected due to shear thickening. Note that the results or dilute 3D(m) systems ollow closely the Krieger ormula, even or relatively high Reynolds numbers. For increasing solid volume raction, the 3D relative viscosity seems to approach the corresponding D value. In Figure 5.9 we show the relative apparent viscosity or Re γ =. 1 as a unction o scaled volume raction φ / φ together with a number o previous eperimental ma (van der Wer and de Krui) [11] and numerical results, all obtained or Reynolds numbers Re γ < 1. The numerical results included in Figure 5.9 are obtained using Stokesian approimation [ 15, 117,118,119], lattice-gas simulations [1], or lattice- Boltzmann simulations [7], or both two-dimensional and three-dimensional systems. The values o the maimum packing ratio φ ma used in scaling the volume raction were.785 (D) and.65 (3D), as above. It is evident rom Figure 5.9 that the results obtained here agree very well with the previous results shown.

50 5. Some rheological properties o liquid-particle suspensions 4 µ r Aidun & Lu, D LB method [7] Brady & Bossis, D Stokesian [117] Chang & Powell, D Stokesian [15] Ladd et al., D lattice-gas [1] Present D result Krieger & Dougherty (3D) [111] van der Wer & de Krui, ep. [11] Brady, 3D Stokesian [119] Zhu et al., 3D Stokesian [118] Present 3D result φ/φ ma Figure 5.9 Comparison o the simulated relative apparent viscosity with previous eperimental and numerical results or small Reynolds numbers, in D as well as 3D systems. We also carried out several other benchmark studies such as stress distribution around a particle in shear low, and dierent relative motion o particles near a wall. In addition, benchmark tests reported in Res. [7, 71, 73] indicate that the method applied here is adequate or realistic suspension simulations. We can conclude, based on the above tests and benchmarks, that the undamental physical background o the model used here is correct.

51 6. Microstructure o dilatant suspensions 43 6 MICROSTRUCTURE OF DILATANT SUSPENSIONS In this chapter we analyse the dilatant rheological behaviour o Couette low o liquid-particle suspensions. The purpose o this work is to better understand the cause o shear thickening and the eects on dilatancy o the microstructure o the suspension. We study the mechanisms o momentum transer and shear stress o liquid-particle suspensions in two dimensions using numerical simulations. The Couette low geometry (shear low) o liquid-particle suspension is relevant or many applications, and appears e.g. in common viscometers. Oten in such practical low problems, the underlying two-phase character is neglected, and the suspension is treated as a non-newtonian luid. Such an approach may be useul or a dilute, homogeneous, isotropic suspension with small particles o regular shape. However, in dense suspensions their two-phase nature cannot be neglected. This is due to the importance then o other mechanisms or momentum transport in addition to viscous stress in the luid phase. These mechanisms include particle-particle interactions and stress inside the particles. Moreover, depending on the scale o the details included, eects o boundaries or complicated microstructures such as clustering [11] and layering [1] o particles can make the two-phase low nonhomogeneous or non-isotropic. In order to understand the underlying mesoscopic mechanisms that contribute to the total observable shear stress and consequently the apparent viscosity o a liquidparticle suspension, we compute the stresses o dierent phases and momentum lues. A snapshot o shear stress carried by the luid can be seen in Figure 6.1, where higher shear stress areas are noticeable between solid particles. The coloured contours indicate the viscous shear stress in the luid phase. As we concentrate here on the basic mechanisms that are responsible o the solid volume raction and Reynolds number dependence o viscosity, we only consider the zero-gravity situation.

52 6. Microstructure o dilatant suspensions 44 (a) y F p m d r d F o,d A p F i,d u w (b) F pp 1 -F m d A p F τ y, S z -u w Figure 6.1 A snapshot o a two-dimensional Couette-low o liquid-particle suspension solved by the LB method. Colour coding indicates viscous shear stress in the luid phase. The two insets show the orces used in calculating the internal particle stress. The Couette low o the carrier liquid is solved using the lattice-boltzmann method while the motion o the cylindrical (in D) and spherical (in 3D(m)) particles o radius R suspended in the luid is governed by Newtonian mechanics. The numerical technique is presented in chapter and benchmarking o the simulations against results o other numerical techniques and eperiments was reported in the previous chapter. The suspension is placed between two moving solid walls oriented in the direction and separated by a distance H. The walls move with speed u in opposite directions. Couette-low conditions are thus created with the mean shear rate γ = u H. Periodic boundary conditions are applied in the direction. The w w simulation grid is rectangular and usually o size 1818 lattice points. In some cases lattices o size 5656 and even lattice points are used. The diameter o particles is 1 lattice units and their density (including the solid matri and the core luid) is about 3.5 times the density o the carrier luid. We chose this particular value or the density ratio as it is relevant or pigment suspensions used in paper w

53 6. Microstructure o dilatant suspensions 45 coating. As discussed in the previous chapter, the results we obtain are not very sensitive to this value. Volume raction (area raction) o the particles is varied between 1% to 5% and the shear Reynolds number between.14 and Simulations are started rom a randomly distributed particle coniguration (see Figure 6.1) with the luid at rest (only walls moving). 6.1 SUSPENSION MICROSTRUCTURE IN SHEAR FLOW In an equilibrium state, monodisperse suspensions show dierent structures depending on particle interactions, concentration and shear rate. For eample, it is known that long range repulsion between the particles tends to order the system while short range repulsive orces lead to a luid-like structure [17]. We irst consider the eect o concentration on the microstructure o the suspension. Figure 6. shows the proiles o the average velocity U o the suspension, and the local volume raction o the particles, between the moving plates in a macroscopically stationary low or Reγ = 1.5, and or two systems with dierent volume ractions o particles. For each value o coordinate y (c.. Figure 6.1), the average is taken over the surace S(y), over time, and over an ensemble o three macroscopically identical systems. The average velocity is calculated or the local velocity ield irrespective o the phase that occupies the location. The velocity U thus represents the miture velocity, i.e., the total volumetric lu o the suspension in the direction. For the relatively dilute suspension shown in Figure 6.a, the particles seem to concentrate near the center o the channel. Weak and somewhat diuse layering may be observed near the walls. The velocity proile slightly deviates rom linear and has a shallow S shape such that the shear rate is lower near the center. This can be understood on the basis o decreased particle concentration and, consequently, lower apparent viscosity near the walls. 1y/H φ y / φ (a) 1 φ y / φ (b) y/h U/u w U/u w Figure 6. The local volume raction o particles φ y and the mean velocity proile o the suspension U across the channel width H or a suspension with Reγ = 1.5 and mean solid volume ractions φ = 1% (a) and φ = 48% (b). Thick solid line indicates the volume raction, and thin solid line is the suspension velocity divided by the velocity o the wall ( u w ). Dashed line represents a linear velocity proile.

54 6. Microstructure o dilatant suspensions 46 For the denser suspension shown in Figure 6.b, the most spectacular phenomenon is the strong layering o particles near the walls. Weaker layering is visible even in the central region o the channel. Notice that such layering was also observed in light-diraction eperiments by Homan [13]. The shear rate oscillates strongly near the walls, and the velocity proile seems to have eatures o slippage at the walls. At the central part, the shear rate is clearly lower than the average value (indicated by the slope o the dashed line). It is evident rom Figure 6. that the mean low o the suspension is aected by the underlying two-phase nature o the luid. In order to ind the shear-rate dependence o layering, and the related velocity proiles, we then determined the velocity and the distribution proiles or a suspension with φ = 5%, which should display layering as indicated above, or three dierent shear rates. Figure 6.3a shows the local volume raction o particles, and Figure 6.3b the mean velocity proile o the suspension. In the low shear Reynolds number region ( Re = γ.1) the suspension is in the Newtonian low regime, and the velocity proile is quite linear. A strong layering o particles is observed, as epected, near the walls, and it appears or all three shear Reynolds numbers. Weaker layering is visible even in the central region o the channel. Between the irst layer and the wall there remains a narrow zone which contains almost entirely luid. Eperiments also oten show such a layer o pure luid in a gap less than a particle radius that decreases or increasing concentration [ 14]. At Re = γ 1. layering appears to be somewhat more pronounced near the channel walls in comparison with the two other cases. In this regime, the velocity proile starts to become nonlinear (Figure 6.3b). By increasing the shear rate beyond Re > 1. γ, the concentration proile in the middle o the channel appears to latten out (Figure 6.3a) (a) 1.75 (b) y/h.5 Re γ.1 Re γ 1 Re γ 1 y/h <φ y > Re γ.1 Re γ 1. Re γ U/u w Figure 6.3 Variation o the solid volume raction o particles (a), and the mean velocity proile o the suspension (b), across the channel width H or three dierent shear Reynolds numbers and or the mean solid volume raction φ = 5%. We would like to analyse in a more quantitative way the observed layering, and later also its possible relation to the apparent viscosity o the suspension. To this end

55 6. Microstructure o dilatant suspensions 47 we need to quantiy layering, and do this by deining a layering inde through a recursive procedure. A layered cluster parallel to a channel wall is composed o particles whose centres lie within a distance δ y = β R rom the line which goes through the center o mass o the cluster and is parallel to the wall, and whose nearest-neighbour distances (between the centres o the particles) are within δ = αλ in the direction o the wall (see Figure 6.4). Here λ = HL / n is the mean ree path o the particles, n the total number o particles, and α and β coeicients that we have chosen to be 3 and 1, respectively. This deinition is built into a recursive search routine which is continued until no new particles are ound that satisy the deinition. At irst a particle is chosen at random and the layered cluster it possibly belongs to is identiied. Then another (so ar unidentiied ) particle is randomly chosen and the process is repeated, until all particles are identiied. Layered cluster with n L =4 particles δy<βr R c.m. δ<αλ New layered cluster with n L =5 particles u w Figure 6.4 A schematic presentation o the search routine or layered clusters. A cluster o our particles is already identiied, and a new particle (shaded) is then ound that satisies the criteria or the maimum horizontal distance between the nearest-neighbour centres and or the maimum distance o the centre o the particle rom the horizontal line that goes through the centre o mass o the cluster. We use α = 3, β = 1 and λ = HL n. Here we consider only those layered clusters with n > L particles that are closest to the walls. We average over the simulation in the stationary situation (at every n L 5 th time step over time steps) and over both walls. This procedure produces a non-zero n even i all possible conigurations were random L distributions. We thus determine n L by averaging over independent random conigurations, and subtract it rom random distribution. A layering inde is then deined to be ( L L ) n L n L to ind layering in ecess o π R n n, (6.1) Lφ such that it is the ratio o the solid volume raction the averaged layered cluster closest to a wall occupies o a ull layer, and the solid volume raction φ o the

56 6. Microstructure o dilatant suspensions 48 suspension. The result o measuring this layering inde or si dierent concentrations is shown in Figure 6.5 as a unction o shear Reynolds number. layering inde φ 5% 48% 44% 36% 4% 1% Re γ Figure 6.5 Variation o layering inde as a unction o shear Reynolds number or si concentrations. The solid lines are guides or the eye. As can be seen rom Figure 6.5, or increasing shear Reynolds number the layering inde irst increases, but then, ater a maimum, it continuously decreases. This tendency is clear or concentrations above 3%, or which the maimum appears beore Re = γ 1. We might tentatively think that there are two competing eects in play here. On the one hand, an eect o walls is to suppress the rotation o nearby clusters so that they tend to align themselves along the walls. On the other hand, lubrication orces between the suspended particles tend to disorder and eventually break the clusters especially or increasing shear Reynolds numbers when even the single-phase luid low begins to become unstable. Whether this is indeed the relevant mechanism, we cannot tell at the moment. This kind o behaviour resembles the decrease o ordering reported by Dratler et. al. [15] in relation to shear thickening. This would also indicate that changes in the microstructure o the suspension are related to its shear thickening behaviour. It is thus instructive to also treat the suspension as a non-newtonian single-phase luid, and compute its apparent viscosity, as would be measured in an appropriate viscometric eperiment. 6. SHEAR THICKENING From the D simulations described above we also determined the time-averaged total shear stress on the channel walls as a unction o shear rate. From the total shear stress we get the relative apparent viscosity o the suspension, and this is shown in Figure 6.6 as a unction o shear Reynolds number or si concentrations. For solid volume ractions above about 3%, the relative viscosity begins to increase

57 6. Microstructure o dilatant suspensions 49 with increasing shear Reynolds number at about Reγ.5, while being more or less independent o Re γ beore that value. For lower concentrations the apparent viscosity begins to increase at higher shear rates. The dependence o apparent viscosity on Re γ appears to be a power law (not shown here). In the corresponding 3D(m) simulations o monolayer arrays o spheres we ound very similar behaviour (see Chapter 5). µ r φ Re 1 w 1 3 5% 48% 44% 36% 4% 1% Re γ Figure 6.6 Relative apparent viscosity µ r as a unction o shear Reynolds number Reγ = ρ duw µ H or si solid volume ractions φ. For more inormation, the corresponding values o the wall Reynolds number Re w = ρ uh w µ are also indicated. We can now at least partly address the question how layering is related to the relative apparent viscosity, and plot in Figure 6.7 the layering inde as a unction o relative apparent viscosity or the three highest solid volume ractions considered, or which layering inde was ound to have a maimum below Re. γ =1

58 6. Microstructure o dilatant suspensions 5 layering inde φ 5% 48% 44%.6 Figure 6.7 Variation o layering inde µ r or three concentrations. Solid lines are guides or the eye. as a unction o relative apparent viscosity The data points in Figure 6.7 are or dierent shear Reynolds numbers such that, or each concentration, Re γ increases rom let to right. It is evident rom this igure that below a certain threshold value layering inde is independent o µ r, and comparison with Figures 6.5 and 6.6 reveals that this threshold roughly coincides with the maimum in the layering inde and the onset o shear thickening. Above the threshold, i.e. in the shear thickening regime, layering inde continuously decreases with increasing µ. We might thus conclude that disappearance o r layering cannot be the cause o shear thickening, but rather that it results rom a mechanism which appears to be responsible also or increasing apparent viscosity. This leads us to study the mechanisms o stress and momentum transport that contribute to the ormation o viscosity. 6.3 MOMENTUM TRANSFER ANALYSIS Momentum transer in multi-phase systems such as particulate suspensions, is by nature a multi-scale phenomenon. We thus computed the stresses o dierent phases and momentum lues including the viscous stress in the luid phase, the structural stress inside the particles, and the inertial lues that arise rom the pseudo-turbulent luctuations o both phases. The mean shear stress inside each individual particle is calculated indirectly rom the hydrodynamic orces that act on the surace o the particle. Evaluation o the hydrodynamic orces acting on the particles and the stresses in the luid phase could be done without reerence to the averaged luid-dynamical quantities such as the viscous stress tensor, because the LB method is not based on the conventional continuity and Navier-Stokes equations, but on a discretized Boltzmann equation. The details o the momentum transer calculations are shown in chapter. A snapshot o the shear stress carried µ r

59 6. Microstructure o dilatant suspensions 51 by the luid can be seen in Figure 6.1, where higher shear stress areas are noticeable between solid particles. First we compare in Table 6.1 the dierent contributions to the total shear stress o the suspension. φ σ / τ T s τ T σ / τ / τ T τ s / τ T 1% % Table 6.1 Relative contributions o the dierent momentum transer mechanisms to the total shear stress τ T or two values o the solid volume raction. Stresses σ and σ s are the pseudo-turbulent stresses o the luid and solid phases, respectively, τ is the viscous stress o the luid phase, and τ s is the shear stress in the solid phase (internal shear stress in the suspended particles). It is evident that the convective (pseudo-turbulent) stresses are very small in comparison with the other stress terms, and need not be taken into account. Moreover, the contribution o direct collisions was also negligible (always less than.5% o the total stress). This is due to highly dissipative collisions between particles in our simulations. In Figure 6.8a we show the shear stress carried by the luid, τ τ T, and the shear stress carried by particles, τ s τ T, both scaled with the total shear stress τ T, or the solid volume raction φ = 1%. In Figure 6.8b the same quantities are shown or φ = 5%. It is evident rom Figures 6.8 that both shear stresses are nearly independent o Re γ at low Reynolds numbers, but begin to change when the Reynolds number eceeds a threshold value that depends on the solid volume raction. Above the threshold value, the shear stress o the solid phase increases with increasing Re γ, while the shear stress o the luid phase respectively decreases. The relative contribution to the total shear stress o the solid phase seems also to strongly depend on the solid volume raction o the suspension.

60 6. Microstructure o dilatant suspensions 5 1. (a) 1. (b).8.8 τ s /τ T, τ /τ T.6.4 τ /τ T, τ s /τ T Re γ Re γ Figure 6.8 The scaled viscous ( τ τ T, open squares) and solid ( τ s τ T, illed squares) stresses as unctions o the shear Reynolds number Reγ, or suspensions with φ = 1% (a), and φ = 5% (b). Solid lines are guides or the eye, and τ T is the total shear stress. To make plain the dependence on the solid volume raction o the shear stresses, we show in Figure 6.9 both shear stresses as unctions o the solid volume raction at a ied shear Reynolds number Re 3 γ. τ /τ T, τ s /τ T φ [%] Figure 6.9 The scaled viscous ( τ τ T, open squares) and solid ( τ s τ T, illed squares) stresses as unctions o the solid volume raction or Re γ 3 ; τ T is the total shear stress. At this shear Reynolds number, the contribution o the solid phase eceeds that o the luid phase at about φ = 5%, and at φ = 5% the contribution o the solid phase is almost 9%. Comparison o Figures 6.6 and 6.8 indicates that, as a unction o shear Reynolds number, the solid-phase contribution begins to increase roughly at the onset o shear thickening, which is also roughly where the layering inde has its maimum.

61 6. Microstructure o dilatant suspensions 53 As the shear stress o the solid phase appears to be an important variable, we show in Figure 6.1 the relative apparent viscosity o the suspension as a unction o τ τ, the ratio o the shear stress carried by the particles to the total shear stress. s T The data points in this igure correspond to shear Reynolds numbers that vary between Re =.1 and Re γ γ = µ r <φ s > 5 % 48 % 36 % 4 % 1 % 6% τ s /τ T Figure 6.1 The relative apparent viscosity as a unction o τ s τ T or si solid volume ractions. The solid line is a it by Eq. (6.) to the observed values. It is evident rom Figure 6.1 that all data points collapse on a single curve, which indicates that the apparent viscosity only depends on τ s τ T, and not directly on the solid volume raction or shear Reynolds number. From a it to the data points we ind urthermore that the scaling curve deined by the data collapse, very closely ollows a simple epression 1 µ = r 1 τ τ. (6.) s T So ar we have not ound any eplanation or this scaling orm. We can conclude by stating that the rheological behaviour o the liquid-particle suspension displays comple phenomena such as dilatancy, i.e. shear thickening, nonlinear velocity proile, layering o particles and apparent slip near the solid walls. A detailed study o various momentum transer mechanisms revealed that shear thickening is related to increasing solid-phase stress, accompanied by decreasing layering near the walls. The apparent relative viscosity was ound to only depend on the relative solid-phase stress, through a simple analytical epression.

62 7. Suspension clogging 54 7 SUSPENSION CLOGGING 7.1 INTRODUCTION Pipe or channel blockage is a common phenomenon that may occur in a wide range o physicochemical and biological processes such as, or eample, in pipeline transportation o oil or coal in a liquid [16], wheat transport in the pipes o silos, coating colour in capillary viscometer and cholesterol in human blood vessels [17]. There are two qualitatively dierent phenomena that may cause the blockage: jamming at high densities o suspended particles, and blockage due to growing deposited layers on channel walls. Particle deposition on channel walls narrows the low channel and nontrivial patterns like meandering or necking o the remaining low channel may appear. A typical eample o a meandering pattern is that o rivers made by sedimentation o soil particles on riverbanks o slow streams (see e.g. [18,19]). It would be o undamental interest to ind a common physical mechanism that drives the pattern ormation in the above-mentioned eamples as they display dierent patterns even though the involved physics may be rather similar. In order to analyse the basic mechanisms, we constructed a numerical model. This model should take as simple a orm as possible but still capture the mechanisms that are relevant in real systems. There are two competing mechanisms in our model: Particle deposition leads to changes in the local low velocities, which may in turn cause detachment o deposited particles. In the simulations we applied the simplest possible attachment and detachment rules, and did not consider more complicated mechanical interactions between the particles and the low. Large system sizes and long runs were required to obtain possible phase boundaries, and thereore we used two-dimensional simulations to get better accuracy. We used the lattice-boltzmann method eplained above to simulate the channel low o particle suspension [V]. When a particle touched a wall, it was assumed to get attached to it (see Figure 1.1). Mutual attachment o suspended particles were not allowed to happen. The reely loating particles should thus be understood as mesoscopic particle aggregates o the typical size o reely loating clusters. When

63 7. Suspension clogging 55 the total orce on an attached particle (i.e. an aggregate) increased beyond a certain threshold value τ P, the particle was detached and became again suspended in the liquid (c.. Figure 7.1). Free particle Attached particle (a) (b) Figure 7.1 A schematic presentation o attachment (a) and detachment (b) o a particle near a channel wall. Suspension was driven through a periodic channel o width H 3 Rand length L 15 R by a pressure gradient P, at low particle Reynolds numbers Re p. 1 (reers here to Re 4 = ρ v R H p µ, where v is the velocity, ρ density, and viscosity o the carrier luid.). The number o suspended particles with a diameter o R = 9 lattice units varied rom 5 up to 1 ( φ = NπR HL varied rom.5 to.55), and the particle to luid density ratio was / µ ρ ρ.5. These particles were non-buoyant spherical discs whose mutual collisions due to the hydrodynamic lubrication orces were inelastic. 6 The simulation times were chosen large enough O( ) p ~ 1 to be well in the stationary regime, where the velocity and concentration distributions used to iner the properties o the system are ully developed. A constant pressure gradient P was imposed between the inlet and the outlet by using a body orce, i.e., a certain momentum was added at each lattice point at each time step, including the lattice points that are inside the ree and the attached particles. A particle approaching a wall was attached to the wall i its distance was less than or equal to one lattice unit rom the wall or another already attached particle. As the (dimensionless) threshold parameter we used the total orce on a deposited particle normalized with the drag orce on a particle near a wall, * Ftotal τ P Fp = 4π R P H 4R/3. (7.1) Here the total orce F total ( ) includes the lit and drag orces, collision orces due to other particles, and the body orce. Normalization is o course arbitrary and this particular choice was made as it produces convenient values or the thresholds.

64 7. Suspension clogging RESULTS AND DISCUSSION For a ied number o particles, a channel low o suspension with deposition can be characterised by only two parameters: The solid volume raction φ and the threshold τ P [V]. I τ P is increased rom zero at a constant φ, we ind up to si distinct regimes in the behaviour o the suspension: ordinary pipe-low ( τ P τ d ), two types o deposition in the both meandering and necking regimes, and blocking τ > τ ) (see Figures 7. and 7.3). ( P b Deposition o particles began as P τ was increased beyond a threshold value ( ) τ φ. Above this threshold curve, dierent types o deposition pattern were ound on the channel walls. There was a non-zero region above τ ( φ ) in which the number o deposited particles was less than the number o particles needed to block the system. This means there is an equilibrium in which a luctuating interace ht (, ) at position and time t, separates the deposited layer rom the rest o the suspension on both channel walls. This interace was etracted by connecting the center points o attached particles, which were locally the most distant rom the wall. The resolution o the interace was thereore one particle diameter. d d Figure 7. Snapshots with color coded luid velocities (violet is zero velocity and red is high velocity). (a) Meandering or τ P = 18., φ = 6., and C s = 11.. (b) Necking or τ =. 67, φ = 46., and C = 13.. (c) Blockage or τ =. 95 and φ = 46.. P C is not deined because o blocking. s s By calculating the cross correlation between the upper and lower interace curves, we could eamine i they were correlated. This makes it possible to distinguish between a meandering type and a necking type o deposition. Cross correlation is deined as P

65 7. Suspension clogging 57 C s = (, ) Cov h h σσ 1 1 ( )( ( ) ()) 1 ( ) ( ) () Cov h, h = n h, t h t h, t h t, ( ( ) ()) σ = n h, t h t 1 α α α,, (7.) where σ is the standard deviation, n is the number o points at the interace curves, α = 1, reers to the upper and lower interaces, respectively, the bar denotes average over an interace, and the angular brackets denote ensemble average. For low paths just above the transition ( τ P τ d ), cross correlation was positive indicating a meandering type o deposition (Figure 7.a). There was an eception at φ 38., where the intermediate region between the onset o deposition and blocking was very narrow. As τ P approached the blocking limit τ P = τ b, cross correlation typically became negative, i.e. the interaces at opposite sides o the channel were anti-correlated. This indicates a necking behaviour o the low path. In this case luctuations may more probably drive the two interaces together, so that ull blocking o the low path appears (Figure 7.c). Because o large luctuations it is however diicult to determine the eact phase boundary between the meandering type and necking type o behaviours. We could however construct the phase diagram by numerically determining the approimate locations o three phase boundaries, and support them by semianalytical arguments [V]. The possible phase boundary between meandering and necking types o low paths was diicult to determine even numerically, but we give an estimate based on cross-correlations also or that boundary (assuming that it eists). The phase diagram together with theoretical estimates or three phase boundaries are shown in Figure 7.3.

66 7. Suspension clogging X X X X X X τ P X X X X ballistic no deposition φ X φ c X X X X X X X X X homogeneous τ b (φ) τ d (φ) Figure 7.3 A generic phase diagram that describes the location o si phases in the ( τ, φ space. ( ) Full blocking, ( ) deposition layers with necking, C <, and ( ) P ) deposition layers with meandering, C s. The dotted line is a simple spline it between negative and positive correlations. To the let o φ = φ c deposition is ballistic, and to the right o it deposition layers are homogeneous. The ull lines are phase boundaries given by the semi-analytical arguments described in the tet. In the phase with deposition layers at low φ and high τ P, there was no detachment o particles, and columns o particles typical o ballistic deposition [13,131] without relaation were ormed. This is illustrated in Figure 7.4a. At high φ and low τ P, the system was characterised by rapid attachment and detachment events, which, ater etensive rearrangements, resulted in more or less homogeneous and dense layers o deposited particles. Unlike in the case o ballistic deposition, density was higher in the deposited layer than in the suspension, and the thickness o the deposited layers grew until detachment and attachment events were in balance (Figure 7.4b). Finally, at high φ and high τ P, rapid deposition occurred until the channel became ully blocked (Figure 7.c). We ound that the concentration o the suspension and the deposited layers φ s and φ d, respectively, did not depend much on τ P within the used low regime, and that they crossed at φ = φ c 38., which thus marks the transition rom ballistic to homogeneous deposition (see Figure 7.4c). s

67 7. Suspension clogging (c) φ s,φ d φ d φ s φ Figure 7.4 Superimposed snapshots with grey scale coded particle velocities regardless o luid contours (white is high particle velocity and black is zero particle velocity). (a) Ballistic type o deposition, τ P = 14., φ = 3., and φd/ φs 76.. (b) Dense and homogeneous deposition layers or τ P =. 53, φ = 5., and φd/ φs (c) The average concentration o particles in the deposited layers φ and the concentration o suspended particles in the low channel φ s, as unctions o the solid volume raction φ or τ =.1. P The boundary or the onset o deposition ( τ d ) was estimated by N a = N det, where N a and N det are the attachment and detachment rates, respectively. Obviously when N det > N at, there is no deposition. N at is proportional to the rate o collisions between particles and walls, which can be determined numerically by preventing attachment in the suspension. On the other hand, N det is proportional to the rate at which the luctuating total orces eceed τ P. These orces were numerically evaluated by recording the collision orces on a single deposited particle. By equalling the two rates determined in this way, we ormed the curve τ ( φ ) shown in Figure 7.3. It is diicult to determine numerically the nature o the onset o the deposition transition, but i we ollow the asymptotic amount o net deposited particles N = lim N ( t ), across the transition line τ ( φ ), it looks like N has a inite jump t a there to a non-zero value (Figure 7.5b). Below the transition d d d increases initially due to transient deposition, but ater a relatively short time reaches a maimum, and decays eponentially to zero thereater. Above the transition the initial rise in N is very similar, but there is no maimum, and N keeps on increasing with a a very long saturation time. We ind in act that this saturation is algebraic in time. This can be seen in Figure 7.5a which shows the concentration o deposited particles or dierent values o φ. Below φ 38., i.e. in the case o ballistic deposition, saturation is very slow because o shadowing eects [13, 131]. It is so slow that it is not possible to determine the actual time dependence o with the statistics we can get rom the etremely long simulations needed in this case. On the other hand, a N a N a