A Modified Distributed Lagrange Multiplier/Fictitious Domain Method for Particulate Flows with Collisions

Transcription

1 A Modified Distributed Lagrange Multiplier/Fictitious Domain Metod for Particulate Flows wit Collisions P. Sing Department of Mecanical Engineering New Jersey Institute of Tecnology University Heigts Newark, NJ 070 T.I. Hesla and D.D. Josep Department of Aerospace Engineering and Mecanics University of Minnesota Minneapolis, MN A modified distributed Lagrange multiplier/fictitious domain metod (DLM) tat allows particles to undergo collisions is developed for particulate flows. In te earlier versions of te DLM metod for Newtonian and viscoelastic liquids described in [,] te particle surfaces were restricted to be at least one and alf times te velocity element size away from eac oter. A repulsive body force was applied to te particles wen te distance between tem was smaller tan tis critical value. Tis was necessary for ensuring tat conflicting rigid body motion constraints from two different particles are not imposed at te same velocity nodes. In te modified DLM metod te particles are allowed to come arbitrarily close to eac oter and even sligtly overlap eac oter. Wen conflicting rigid body motion constraints from two different particles are applicable on a velocity node, te constraint from te particle tat is closer to tat node is used and te oter constraint is dropped. An elastic repulsive force is applied wen te particles overlap eac oter. In our simulations, te particles are allowed to overlap as muc as one undredt of te velocity element size. Te modified DLM metod is implemented for bot Newtonian and viscoelastic liquids. Our simulations sow tat wen particles are dropped in a cannel, and te viscoelastic Mac number (M) is less tan one and te elasticity number (E) is greater tan one, te particles form a cain parallel to te flow direction. As in experiments, te new metod allows particles in te cain to touc eac oter. Te particles dropped in a Newtonian liquid, on te oter and, undergo caracteristic drafting kissing and tumbling. During te toucing pase, as in experiments, te two particles touc eac oter. Our results for te time dependent motion of a particle trown wit a given velocity towards te wall or anoter particle of te same size are in agreement wit asymptotic analytical results [3]. We also find good agreement between te analytical and simulations results for a particle sedimenting under gravity onto a orizontal surface. arcive/ddj/00/papers/moddlm-collisions/manuscript.doc

2 . Introduction Collisions or near-collisions between particles present severe difficulties in direct simulations of particulate flows. In point of fact, owever, smoot particles cannot actually contact eac oter in finite time in te continuous system, as sown in Hesla [3]. In ligt of tis, te term "collision" is somewat misleading at least for smoot particles. However, since te gap widt can become exceedingly small in certain situations, numerical truncation errors may allow actual contact (or even overlap) to occur in simulations. Even near-collisions can greatly increase te cost of a simulation, because in order to simulate te particle-particle interaction mecanism, te flow fields in te narrow gap between te converging particle surfaces must be accurately resolved. Te element size required for tis decreases wit te gap widt, leading to extremely small elements and greatly increased numbers of unknowns to be solved for. Actual overlap can lead to overdetermined systems of equations, and must tus be prevented at all costs. In numerical simulations to date [,,4-8], tis problem is avoided by introducing an artificial repulsive force between particles wic keeps te particle surfaces more tan one element apart from eac oter. In te standard distributed Lagrange multiplier/fictitious domain (DLM) metod [,,7-8], tis ensures tat te differing rigid-body motion constraints of te two particles are not simultaneously imposed at any one velocity node, wic would lead to a numerically overdetermined system of equations. In te arbitrary Lagrangian-Eulerian (ALE) metod [4-6], it ensures tat te element size doesn't become excessively small. Te ALE metod requires two layers of elements in te gap between te converging surfaces. In tis paper, we present a modification of te DLM finite-element sceme described in [,] wic allows particles to come arbitrarily close to eac oter, and even overlap sligtly. Te basic idea is to simply drop one of te conflicting rigid-body motion constraint terms in te equation of motion te one corresponding to te particle surface wic is farter away from te node in question. Tis trick avoids overdetermined systems of equations, but is applicable only as long as te overlap of te particles does not exceed one element. An elastic repulsive force is introduced to prevent particles from overlapping more tan tat. In actual experiments, one often (apparently) sees collisions. Tis may be attributed to one or more of te following: Te particle surfaces are not actually in contact on te microscopic scale. Te particle surfaces are not actually smoot on te microscopic scale. Some new pysics comes into play wen te gap is extremely narrow. However, tis is irrelevant for te present purposes, since we are simply solving te equations of motion of te fluid and particles, as given by continuum mecanics. arcive/ddj/00/papers/moddlm-collisions/manuscript.doc

3 Te modified DLM metod is verified by comparing te time dependent trajectories for two circular particles falling in a cannel for two different mes refinements, and for two different time steps. It is sown tat te results are independent of te mes resolution and te time step. It was discussed in [,,9,0] tat wen two or more particles are dropped in a cannel filled wit a viscoelastic liquid te orientation of particles relative to te direction of flow is determined by De Uλ te Mac number M= Re De and te elasticity number E=. Here De = r is te Debora Re D ρl UD number, and Re = is te Reynolds number, were λ r is te relaxation time of te fluid, η is te η zero sear viscosity of te fluid, U is te particle velocity, and D is te particle diameter. Specifically, wen M is less tan one and E is greater tan one, te particles align temselves parallel to te flow direction. Te distance between te particles of a cain may be very small. In fact, te particles may even approximately touc eac oter. On te oter and, wen tese conditions are not satisfied, or wen te fluid is Newtonian, te particles undergo drafting kissing and tumbling. Experiments sow tat during kissing te particles come very close to eac oter, and may even approximately touc eac oter. In te next section te strong and weak forms of governing equations as well as te modified DLM sceme tat allows particle-particle near collisions is presented. In te last section, we present several test cases to validate te sceme, and to demonstrate tat it performs as advertised.. Computational Sceme Te computational sceme we propose is a modification of te DLM finite-element sceme described in [,]. In tis sceme, te fluid flow equations are solved on te combined fluid-solid domain, and te motion inside te particle boundaries is forced to be rigid-body motion using a distributed Lagrange multiplier. Te fluid and particle equations of motion are combined into a single combined weak equation of motion, eliminating te ydrodynamic forces and torques, wic elps ensure stability of te time integration. Te time integration is performed using te Marcuk-Yanenko operator splitting metod, wic is first-order accurate. See [,] for furter details.. Governing Equations In tis paper we will present results for two-dimensional flows. Let us denote te domain containing te Newtonian or viscoelastic fluid and N particles by Ω, and te interior of te it particle by P i (t). For simplicity we will assume tat te domain is rectangular wit boundary Γ. Te four sides of te domain will be denoted by Γ, Γ, Γ 3, and Γ 4 (see Figure ), and Γ will be used to denote te arcive/ddj/00/papers/moddlm-collisions/manuscript.doc 3

4 upstream part of Γ. Te viscoelastic fluid is modeled by te Oldroyd-B model. Te governing equations for te fluid-particle system are: u c ρ L + u. u = ρ L g - p +.( t λ A) +.(η sd) in Ω\ P(t) () r. u = 0 in Ω\ P(t) () u = u L on Γ (3) u = U i + ω i x r i on (t), i=,,n, (4) wit te evolution of te configuration tensor A given by A + u. A = A. u+ u T.A - t λ A = A L r P i (A - I), (5) Here u is te velocity, p is te pressure, η s is te solvent viscosity, ρ L is te density, D is te symmetric part of te velocity gradient tensor, c is a measure of polymer concentration in terms of te zero sear viscosity, and were λ r is te relaxation time. Te zero sear viscosity η = η s + η p, were η p on Γ. = c η s is te polymer contribution to viscosity. Te fluid retardation time is equal to equations are solved wit te following initial conditions: u = u t= 0 t= 0 0 A = A 0 were u 0 and A 0 are te known initial values of te velocity and te configuration tensor. Te particle velocity U i and angular velocity ω i are governed by λ r. Te above + c dui M i = Mig + F i (8) dt dω dt i I i = i t= 0 T U = U i i,0 (6) (7) (9) (0) ω i t= 0 = ω i,0 () were M i and I i are te mass and moment of inertia of te it particle, and F i and T i are te ydrodynamic force and torque acting on te it particle. In tis investigation we will assume tat te particles are circular, and terefore we do not need to keep track of te particle orientation. Te particle positions are obtained from arcive/ddj/00/papers/moddlm-collisions/manuscript.doc 4

5 dx i = dt i t= 0 U X = X i i,0 were X i, 0 is te position of te it particle at time t=0. () (3). Weak form of equations and finite-element discretization Te approac used for obtaining te weak form of te governing equations stated in te previous section was described in [,,7]. In obtaining tis weak form, te ydrodynamic forces and torques acting on te particles can be completely eliminated by combining te fluid and particle equations of motion into a single weak equation of motion for te combined fluid-particle system. For simplicity, in tis section we will assume tat tere is only one particle. Te extension to te manyparticle case is straigtforward. Te solution and variation are required to satisfy te strong form of te constraint of rigid body motion trougout P(t). In te distributed Lagrange multiplier metod tis constraint is removed from te velocity space and enforced weakly as a side constraint using a distributed Lagrange multiplier term. It was sown in [,7] tat te following weak formulation of te problem olds in te extended domain: For a.e. t>0, find u Ω ρ L ρ + ρ L d W uγ, A W A, p L ( Ω 0 ), λ Λ (t ) du g vdx dt Ω p. v dx + du dω ' M g V + I ξ F. V = dt dt Ω η D[ u]: D[ v]dx s, U R, and ω R, satisfying c v.. λ r Ω λ, v ( V + ξ r) P(t) A dx for all v W 0, V R, and ξ R, (4) q. udx = 0 for all q L ( Ω ), (5) Ω µ, u ( U + ω r) = 0 for all µ Λ (t), (6) P(t) A T + u A - A u - u A + ( A - I) s dx t = 0 for all s W A0, (7) Ω λ r u = u t = 0 o in Ω, (8) A = = A in Ω, (9) t 0 o arcive/ddj/00/papers/moddlm-collisions/manuscript.doc 5

6 as well as te kinematic equations and te initial conditions for te particle linear and angular velocities. Here F is te additional body force applied to te particles to limit te extent of overlap (see equation (7) below) and λ is te distributed Lagrange multiplier W uγ = { v H ( Ω) v = uγ(t) on Γ}, 0 W0 = H ( Ω), W A 3 = { A H ( Ω) A = A L (t) on Γ }, W A0 3 = { A H ( Ω) A = 0 on Γ }, L0( Ω) = {q L ( Ω) Ω q dx = 0 }, (0) and Λ (t) is L (P(t)), wit.,. denoting te L inner product over te particle, were P(t) Γ is te upstream part of Γ. In our simulations, since te velocity and µ are in L, we will use te following inner product ( µ. v ) x µ v = d. (), P(t) P(t) In order to solve te above problem numerically, we will discretize te domain using a regular finite element triangulation T for te velocity and configuration tensor, were is te mes size, and a regular triangulation T for te pressure. Te following finite dimensional spaces are defined for approximating W uγ, W 0, W A, A0 W, L ( Ω ) and L ( Ω 0 ) : W uγ, = T Γ, 0 { v C ( Ω) v P P for all T T, v = u on Γ}, W 0 = { v C ( Ω) v P P for all T T, v = 0 on }, () 0, T Γ L 0 = {q C ( Ω) q P for all T T }, T L 0, = {q L q dx = 0}, (3) Ω W A 0 3 = { s C ( Ω) s P P P for all T T, s = A T L, on Γ }, W A0, 0 3 = { s C ( Ω) s P P P for all T T, s = 0 on Γ }, (4) T arcive/ddj/00/papers/moddlm-collisions/manuscript.doc 6

7 were Γ is te upstream part of Γ. Te particle inner product terms in [4] and [6] are discretized using a triangular mes similar to te one used in []..3 Collisions and Near-Collisions Wen tere is more tan one particle, te rigt-and side of Equation (4) becomes a sum of particle inner product terms, one for eac particle, and tere is a separate equation like (6) for eac particle. If two particles overlap, te problem becomes overconstrained in te overlap region. Te two "copies" of Equation (6) represent incompatible side constraints on u. And on te rigt-and side of Equation (4), te terms corresponding to te two overlapping particles represent two supposedly independent distributed Lagrange multipliers, associated wit te respective rigid-body motion constraints. But as just noted, tese are constraints are incompatible. In te finite-element discretized equations, te problem becomes overconstrained even before te particles actually overlap. Specifically, it becomes overconstrained wen two particles bot overlap a single velocity element. Te velocity field u is a single linear function on tis element, but te two discretized "copies" of Equation (6) are "trying" to force it to matc two different rigid motions (in te portions of te element wic overlap te respective particles). In [,], tis situation was avoided by imposing a particle-particle repulsive force strong enoug to ensure tat te particle-particle gap never becomes so small tat bot particles intersect a single velocity element. In te modified DLM sceme described in tis article we allow te particles to touc or sligtly overlap, and eliminate te inconsistency in te equations by te following strategy: if eac particle inner product is represented by a nonsymmetric matrix, wose rows correspond to te nodes of te particle mes, and wose columns correspond to te nodes of te velocity mes, ten for a given particle, we "zero out" any column wic corresponds to a velocity node wic is closer to (te center of) anoter particle. In oter words: () If v is a velocity test function corresponding to a given one of te tree vertices of a velocity element overlapped by two particles, we drop, from te rigt-and side of Equation (4), te particle inner product term corresponding to te particle wose center is farter from te given vertex. () In te "copy" of Equation (6) corresponding to a given particle, if µ is any particle test function for tat particle's mes and te velocity field u is expanded wit respect to te global velocity basis functions, ten we drop any nonzero terms corresponding to velocity nodes wic are closer to anoter particle. A little reflection will convince te reader tat te resulting equations are no longer overdetermined even wen two particles overlap as long as te overlap is not too large. arcive/ddj/00/papers/moddlm-collisions/manuscript.doc 7

8 In our code, we impose a repulsive force wen te particles overlap. In simulations, te repulsive force is assumed to be large enoug so tat te overlap between te particles and te walls is smaller tan one undredt of te velocity element size. Te additional body force wic is repulsive in nature is added to equation (8). Te particle-particle repulsive force is given by were F P i,j = 0 for R d i, j > k( Xi X j)(r di, j), for d i, j < R (5) d i, j is te distance between te centers of te it and jt particles, R is te particle radius wic is assumed to be te same for all particles, X i and X j are te position vectors of te particle centers, and k is te stiffness parameter. Te repulsive force between te particles and te wall is given by F W i,j = 0 for d i > R k w ( X X )(R d ), for d i < R (6) i j i were d i is te distance between te centers of te it particle and te imaginary particle on te oter side of te wall Γ j, and k w is te stiffness parameter for particle wall collision (see Figure ). Te above particle-particle repulsive forces and te particle-wall repulsive forces are added to equation (8) to obtain were dui Mi dt = M g + F i i ' i + F F ' i N = F j= j i P i, j + 4 F W j i, j= (7) is te repulsive force exerted on te it particle by te oter particles and te walls. Te repulsive force acts only wen te particles overlap eac oter. 3. Results We present te results of several simulations, to demonstrate tat te sceme works correctly, and tat it reproduces interesting dynamical beavior. According to an analysis by Hesla [3], actual collisions between smoot rigid particles cannot occur in finite time. Hesla gives a rigorous matematical proof of tis fact for Newtonian fluids in bot two and tree dimensions. He also proves, in te usual case were te "colliding" particles ave te same density, tat te gap widt (t) between teir surfaces will never become smaller tan a certain (positive) minimum value, wic depends on te initial conditions of te problem. Tese results are exact tat is, tey are obtained directly arcive/ddj/00/papers/moddlm-collisions/manuscript.doc 8

9 from te full Navier-Stokes equations (in conjunction wit Newton's equations for te particles), wit no approximations of any kind. Altoug (t) will never become zero in te continuous system, it can become exceedingly small in certain situations. In numerical simulations, terefore, numerical truncation errors may allow particles to actually touc, or even overlap. 3.. Sedimentation We next discuss te numerical results obtained using te above algoritm for te particleparticle interactions. Te particles are suspended in Newtonian and Oldroyd-B fluids. Te parameter c in te Oldroyd-B model will be assumed to be 7, i.e., η p = 7 η s. Te parameter in te particle-particle and particle-wall force models k and k w are equal to 0 4 wic was picked to ensure tat te particles do not overlap more tan a one undredt of te velocity element size. We will also assume tat all dimensional quantities are in te CGS units Newtonian Fluid We begin by investigating te case of two circular particles sedimenting in a cannel filled wit a Newtonian fluid. One of te objectives of tis study is to sow tat te particle trajectories before collision are independent of te mes resolution and te time step. We ave used two regular triangular meses to sow tat te results converge wit mes refinement. In a triangular element tere are six velocity and tree pressure nodes, and tus te size of te velocity elements is one alf of tat of te pressure elements. Te particle domain is also discretized using a triangular mes similar to te one used in []. Te size of velocity elements for te first mes is /96, and for te second mes is /44. Te size of particle elements for te first mes is /70, and for te second mes is /5. Te number of velocity nodes and elements in te first mes are 36 and 5596, respectively. In te second mes, tere are velocity nodes and 446 elements. Te time step for tese simulations is 0.00 or For tese calculations η = 0.08 and ρ L =.0, and te particle diameter and density are 0. and., respectively. Te initial velocity distribution in te fluid, and te particle velocities are: u 0 = 0, U = U 0, 0 0 = ω = ω = Te cannel widt is and eigt is 6. Te simulations are started at t=0 by dropping two particles at te center of te cannel at y= 5.0 and 5.3. arcive/ddj/00/papers/moddlm-collisions/manuscript.doc 9

10 It is well known tat wen two particles are dropped close to eac oter in a Newtonian fluid tey interact by undergoing repeated drafting, kissing and tumbling [9,0]. From Figure 3 and 4 we note tat initially bot particles accelerate downwards until tey reac te terminal velocity at t 0.4. From tese figures we also note tat te particle in te wake falls more rapidly tan te particle in front, as te drag acting on it is smaller. Tus, te gap between tem decreases wit time and tey touc or kiss eac oter at t=~0.4 (see Figure 5). After toucing eac oter, te particles fall togeter wit an approximately constant velocity of ~ 4.0. Te Reynolds number based on tis velocity is ~0. Te maximum overlap between te particle is less tan , wic is approximately one undredt of te velocity element size. Also notice tat wen te upper particle touces te lower particle te y-components of teir velocities become approximately equal until tey tumble and subsequently separate from eac oter. Te tumbling of particles takes place because te configuration were tey are aligned parallel to te flow direction is unstable (see Figure 5c). Te velocity distribution around sedimenting particles is sown in Figure 5a-c. In Figure 5a te two particles are aligned approximately parallel to te sedimentation direction. Te particles are toucing eac oter. Figure 5b sows te velocity distribution at t=0.65 wen te particles are beginning to tumble. Te last figure in te sequence sows te velocity distribution wen te particles ave separated from eac oter. Tese figures sow tat te velocity at te nodes tat are closer to te upper particle is constrained by te upper particle, and tat for te nodes closer to te lower particle is constrained by te lower particle. Te metod terefore allows te particles to touc as well as sligtly overlap eac oter. In Figures 3 and 4 we ave sown te x and y-components of positions and velocities as a function of time for te two meses described above using te time steps of 0.00 and From Figure 3 we note tat wen te number of nodes used is approximately doubled te x- and y- components of velocities and positions for bot particles (marked and ) before collision, i.e., for t<0.4, remain approximately te same. Te results obtained for te finer mes are denoted by ' and '. Similarly, a comparison of te curves marked ( and ') and ( and ') in Figure 4 sows tat wen te time step is reduced by a factor of.5 te temporal evolutions of te particle velocities and trajectories do not cange significantly wic sows tat te results for t<0.4 are also independent of te time step. Te motion of particles for t>0.4 is influenced by te particle-particle collision and te instability tat makes te alignment of particles parallel to te flow direction unstable. Te trajectories for t>0.4 are terefore sensitive to te mes size and te time step. We may terefore conclude tat for t<0.4 te time duration for wic te particle-particle interactions do not significantly influence te motion of te particles te results are independent of te mes resolution. arcive/ddj/00/papers/moddlm-collisions/manuscript.doc 0

11 3... Oldroyd-B Fluid Wen two particles sediment in a cannel filled wit a viscoelastic liquid te stable configuration is te one were te particles are aligned parallel to te flow direction. For tese simulations te cannel widt is and eigt is 6. Te velocity element size is /96, and te size of te particle elements is /70. Te fluid density is.0, and te particle density is.07. Te fluid viscosity is 0.4 and relaxation time is 0.3, and te particle diameter is 0.. Te time step used for all results reported tis subsection is In Figures 6a and b, te positions of te two particles and tra distributions for t=.0 and 3.0 are sown. Te particles were dropped at a distance of 0.5D from eac oter at t=0. Te Reynolds number of te particles is 0.48 and De=.75. Also note tat since te velocity of te particle on te top is larger, te distance between tem decreases wit time. Te distance decreases to approximately 0.000D at t=.0. From Figure 6a-b, were we ave sown te isovalues of tra, we note tat tra in front of te leading particle and around te particles is relatively large. Tis figure also sows tat it takes some time for te fluid to relax back to te state of equilibrium. Tis gives rise to te caracteristic streak lines tat are indicative of te yperbolic nature of te constitutive equation for te stress. For tis case since te Mac number is 0.9 and te elasticity number is 3.59, tere is a strong tendency for te particles to align parallel to te falling direction. Te velocity distribution around particles is sown in Figure 6c. Wen more tan two particles sediment in a viscoelastic liquid te stable configuration is te one were tey are aligned parallel to te sedimentation direction. To simulate tis, we consider sedimentation of ten particles in a cannel of widt and eigt 8. Te fluid density is.0, and te particle density is.07. Te fluid viscosity is 0.4 and relaxation time is 0.3, and te particle diameter is 0.. Te velocity element size is /96, and te size of te particle elements is /70. Te simulations are started at t=0 by placing te particles in a vertical arrangement along te cannel center. Te distance between te particles is.5d and te center of te first particle is at 5.0. Te particles are dropped wit zero linear and angular velocities, and te fluid is in te quiescent relaxed state. Te average Debora number is.73, te average Reynolds number is 0.5, te Mac number is 0.93 and te elasticity number is Te distribution of tra and te particle positions at t=0.88 is sown in Figure 7. We note tat te particles are aligned approximately parallel to te sedimentation direction and te distance between te particles of te cain is very small. In fact, some of te particles are sligtly overlapping eac oter. Te repulsive body force is needed to ensure tat te overlap is sufficiently small. 3.. Time Evolution of Gap Widt in Newtonian Fluids arcive/ddj/00/papers/moddlm-collisions/manuscript.doc

12 In situations in wic te lubrication force dominates all oter contributions to te total ydrodynamic force, te equations of motion of te particles can be solved explicitly, yielding explicit formulas for te time evolution of te gap widt (t), as sown by Hesla [3]. Tese formulas apply to te "collision" between a free particle and a (stationary) rigid bounding wall, as well as to te "collision" between two free particles, te wall acting as a large, funny-saped particle constrained to remain motionless. In tis section we compare Hesla s formula for (t) wit te DLM results for collisions between a particle and a wall, as well as for collisions between two particles. Te domain used in tese calculations is square saped wit sides 0 R. Te velocity element size is /40, and te size of te particle elements is /30. Te fluid viscosity is varied. Te fluid density is.0. Te particle density is varied and its radius is 0.. Te initial fluid velocity in te domain is assumed to be zero and te time step is Gravity-Driven "Collision" For a gravity-driven "collision," te gap widt (t) must continue to decrease indefinitely. For example, wen a negatively buoyant free circular particle sediments toward a stationary flat wall in two dimensions, Hesla sows tat (wen te lubrication force is dominant) for large t, (M m)gt = + C + o() (8) / (t) C were C is a constant, M is te mass of te particle, m is te mass of te displaced fluid, g is te acceleration of gravity, and C is a constant tat depends on te initial conditions. Te constant C depends on te fluid viscosity and te particle radius. Its exact value must be determined using te metod of asymptotic analysis. In Figure 8a we ave plotted as a function of time for two different values of te / (t) particle density for te case were particle sediments under gravity over a fixed wall, and in Figure 8b similar results are sown for te case of two particles. In te two particle case te lower particle is assumed to be fixed and te upper particles sediments under gravity. Te fluid properties are kept fixed and te initial particle velocity is assumed to be zero. From tese two figures we note tat increases wit time but te curvature decreases / (t) wit time. Also note tat for te curves marked for t>~0.7 in Figure 8a and for t>~. in Figure 8b arcive/ddj/00/papers/moddlm-collisions/manuscript.doc

13 / (t) increases linearly wit time, as suggested by expression (8). Te curves marked also sow a similar linear beavior. Tese results sow tat simulations correctly capture te asymptotic beavior of a particle sedimenting onto a orizontal wall or anoter fixed particle. In Figure 8a were te particle approaces a wall curve becomes approximately orizontal wen te distance / (t) between te particle and te wall is approximately equal to te size of one velocity element, indicating tat in simulations te particle decelerates faster tan tat is given by asymptotic analysis. Tis sudden decrease in te particle velocity is due to te discrete incompressibility constraint. In Figure 8b owever te particle deceleration is relatively slower and te particle distance from te wall becomes muc smaller tan te size of one velocity element Non-Driven "Collision" For cases were tere is no net buoyancy force difference driving te "collision", Hesla sows tat te gap widt (t) between te two surfaces will never become smaller tan a certain (positive) minimum value, wic depends on te initial conditions of te problem. For example, wen a neutrally buoyant free circular particle is "trown" toward a stationary flat wall in two dimensions, Hesla sows tat (wen te lubrication force is dominant) MU + C 0 / / 0 were 0 =(0) and U 0 is te magnitude of te initial velocity, and tat for large t, (30) log C = M 3/ were C depends on te initial conditions. t + C + o() For a fixed value of 0, in Figure 9a we ave plotted / (3) as a function of U 0 for te case were a particle approaces a wall and in Figure 9b for te case were a particle approaces a fixed particle of te same size. Te value 0 for all cases in Figure 9a is 4R and in Figure 9b is R. From tese figures we note tat, as expected, / increases wen initial particle velocity is increased and / arcive/ddj/00/papers/moddlm-collisions/manuscript.doc 3

14 varies linearly wit U 0, as suggested by Hesla s analysis. In bot figures for te last point te value of is less tan te size of te velocity element, and tus te truncation errors are of te same order as. In Figure 0a we ave plotted log( ) as a function of t for te case were a particle approaces a wall, for four different values of U 0, and in Figure 0b for te case were a particle approaces a fixed particles of te same size ead on. In tese calculations te initial fluid velocity is assumed to be zero and only te initial particle velocity is varied. In Figure 0a for all four cases and in Figure 0b for te tree cases, log( ) varies linearly wit time as, as suggested by expression (3). From (3) we note tat te slope in te linear region is α. Our simulations, terefore, correctly capture te asymptotic particle motion as well as te role of lubrication forces as te particle approaces te wall. 4. Conclusions Te modified DLM metod developed in tis paper allows particles to come arbitrarily close to eac oter, including sligtly overlap eac oter. Wen conflicting rigid body motion constraints from two different particles are applicable on a velocity node, te constraint from te particle tat is closer to te node is used and te oter constraint is dropped. A repulsive force is applied wen te particles overlap eac oter to limit te overlap. In our simulations, te particles are allowed to overlap as muc as /00 of te velocity element size. Te modified distributed Lagrange multiplier/fictitious domain metod could be used to study te particle-particle interactions as well as te collisions between particles suspended in te Newtonian and Oldroyd-B fluids. In order to validate our code we ave simulated te time dependent interactions between two particles sedimenting in Newtonian and Oldroyd-B fluids. We ave verified tat te results are independent of te mes resolution as well as te size of time step. Our simulations sow tat wen particles are dropped in a cannel, and te viscoelastic Mac number (M) is less tan one and te elasticity number (E) is greater tan one, te particles form a cain parallel to te flow direction and te particles in te cain touc eac oter. Te particles dropped in a Newtonian liquid undergo caracteristic drafting, kissing and tumbling. Te particles touc eac oter during te kissing pase. Our results for te time dependent motion of a particle moving towards a wall are in agreement wit te asymptotic analytical results for a particle trown wit a fixed velocity towards te wall and also for te case of a particle sedimenting on a orizontal surface. arcive/ddj/00/papers/moddlm-collisions/manuscript.doc 4

15 Acknowledgements Tis work was partially supported by National Science Foundation KDI Grand Callenge grant (NSF/CTS ), te Department of Basic Energy Science at DOE and te University of Minnesota Supercomputing Institute. 5. References [] R. Glowinski, T.W. Pan, T.I Hesla and D.D. Josep, A distributed Lagrange multiplier/fictitious domain metod for particulate flows. Int. J. of Multipase Flows. 5 (998), [] P. Sing, D.D. Josep, T.I Hesla, R. Glowinski and T.W. Pan, Direct numerical simulation of viscoelastic particulate flows, J. of Non Newtonian Fluid Mecanics 9, (000). [3] T.I. Hesla, No collisions in viscous fluids!, in preparation (00). [4] P.Y. Huang, H.H. Hu and D.D. Josep, Direct simulation of te sedimentation of elliptic particles in Oldroyd-B fluids, J. Fluid Mec. 36 (998), [5] H.H. Hu Direct simulation of flows of solid-liquid mixtures. Int. J. Multipase Flow, (996), [6] P.Y. Huang, J. Feng, H.H. Hu and D.D. Josep, Direct simulation of te motion of solid particles in Couette and Poiseuille flows of viscoelastic fluids, J. Fluid Mec. 343 (997), [7] R. Glowinski, T.W. Pan and J. Periaux, A Lagrange multiplier/fictitious domain metod for te numerical simulation of incompressible viscous flow around moving rigid bodies, C.R. Acad. Sci. Paris 34 (997), [8] V. Girault, R. Glowinski and T.-W. Pan, A fictitious domain metod wit distributed multiplier for te Stokes problem, Applied nonlinear analysis, Kluwer Academic/Plenum Publiser, 999, [9] A. Fortes, D.D. Josep and T. Lundgren, Nonlinear mecanics of fluidization of beds of sperical particles, J. Fluid Mec. 77 (987), [0] D.D. Josep, A. Fortes, T. Lundgren and P. Sing, Nonlinear mecanics of fluidization of beds of speres, cylinders, and disks in water, Advances in Multipase Flow and Related Problems (Edited by Papanicolau, G.) 6 (987), arcive/ddj/00/papers/moddlm-collisions/manuscript.doc 5

16 6. Figures Figure. A typical rectangular domain used in our simulations; Γ is te upstream portion of Γ. Figure. Te imaginary particle used for computing te repulsive force acting between a particle and a wall. Figure 3. Te velocity components u and v and te positions x and y of te sedimenting circles are sown as a function of time. Te size of te velocity elements is /96, and te size of te particle elements is /70. Te time step for te curves marked and is 0.00 and for te curves marked and is Figure 4. Te velocity components u and v and positions x and y of te sedimenting circles are sown as a function of time. For curves marked and te size of te velocity elements is /96 and te size of te particle elements is /70 and for te curves marked and te size of te velocity elements is /44, and te size of te particle elements is /5. Te time step is Figure 5. Te size of te velocity elements is /96, and te size of te particle elements is /70. Te time step is Te velocity fields at t=0.5, 0.65 and are sown in (a), (b) and (c), respectively. Figure 6. Te size of te velocity elements is /96, and te size of te particle elements is /70. Te time step is Te two particles are dropped at t=0 at y=5.0 and y=5.3 along te cannel center. (a) Isovalues of tra at t=.0, (b) Isovalues of tra at t=3.0, (c) Te velocity field at t=.0. Figure 7. Isovalues of tra at t=. for ten particles sedimenting in a cannel filled wit Oldroyd-B fluid is sown. Te parameters are te same as in Figure 6. Figure 8. is plotted as a function of time. Te circle is released from a state of rest. (a) Circle / (t) sediments onto a fixed wall. Te particle density for te curve marked is.0 and for is.05. Te fluid viscosity is 0.08 and 0 =3R. (b) Circle sediments onto a fixed circle of te same diameter. Te particle density for te curve marked is.0 and for is.. Te fluid viscosity is.0 and 0 =0.7R. Figure 9. Te asymptotic distance / is plotted as a function of te initial particle velocity. (a) Circle is trown towards a fixed wall. Te particle density is 5.0 and te fluid viscosity is 0.. (b) Circle is trown towards anoter fixed circle of te same diameter. Te particle density is 5.0 and te fluid viscosity is 0.8. arcive/ddj/00/papers/moddlm-collisions/manuscript.doc 6

17 Figure 0. log( ) is plotted as a function of time for several different values of te initial velocity. Te parameters are te same as in Figure 9. (a) Circle is trown towards a fixed wall. (b) Circle is trown towards anoter fixed circle of te same diameter. arcive/ddj/00/papers/moddlm-collisions/manuscript.doc 7

18 Γ 4 Ω Γ P Γ Γ 3 Γ Γ U Figure. arcive/ddj/00/papers/moddlm-collisions/manuscript.doc 8

19 d i Γ j Figure arcive/ddj/00/papers/moddlm-collisions/manuscript.doc 9

20 .0 x.0 x x x' x' t u t -.0 u u u' u' Figure 3a-b arcive/ddj/00/papers/moddlm-collisions/manuscript.doc 0

21 y y y y' y' t 0.0 t v v v v' v' -5.0 Figure 3c-d arcive/ddj/00/papers/moddlm-collisions/manuscript.doc

22 .0 x.0 x x x' x' t.0 t u u u u' u' -.0 Figure 4a-b arcive/ddj/00/papers/moddlm-collisions/manuscript.doc

23 y y y y' y' t 0.0 t v v v v' v' -5.0 Figure 4c-d arcive/ddj/00/papers/moddlm-collisions/manuscript.doc 3

24 Figure 5a. arcive/ddj/00/papers/moddlm-collisions/manuscript.doc 4

25 Figure 5b. arcive/ddj/00/papers/moddlm-collisions/manuscript.doc 5

26 Figure 5c. arcive/ddj/00/papers/moddlm-collisions/manuscript.doc 6

27 Figure 6a. arcive/ddj/00/papers/moddlm-collisions/manuscript.doc 7

28 Figure 6b. arcive/ddj/00/papers/moddlm-collisions/manuscript.doc 8

29 Figure 6c. arcive/ddj/00/papers/moddlm-collisions/manuscript.doc 9

30 Figure 7. arcive/ddj/00/papers/moddlm-collisions/manuscript.doc 30

31 0 / (t) a t / (t) 0 8 b t Figure 8. (a) and (b). arcive/ddj/00/papers/moddlm-collisions/manuscript.doc 3

32 / U 0 60 / U 0 Figure 9 a and b. arcive/ddj/00/papers/moddlm-collisions/manuscript.doc 3

33 log t U0=5.0 U0=6.0-4 U0=6.5 U0=7.0-7 log 5 3 U0=0.8 U0=0.9 U0=.0 U0= t -3 Figure 0 a and b. arcive/ddj/00/papers/moddlm-collisions/manuscript.doc 33