A DLM/FD Method for Simulating Particle Motion in Viscoelastic Fluid

A Method for Simulating Particle Motion in Viscoelastic Fluid Tsorng-Wha Pan and Roland Glowinski Department of Mathematics Universit of Houston Houston, Teas Ma 3, 6 International Workshop on Fluid-Structure Interaction Problems Institute for Mathematical Sciences, NUS 6 This work is supported b NSF. (UH) Ma 3, 6 / 58

Outline.. 3. (UH) Ma 3, 6 / 58

Fig. : (Left) Spheres falling in % polo in 98% water draft, kiss and chain. The chain since the forces in a viscoelastic fluid are aggregative. Chained spheres form a long bod which is stable when its long ais is vertical. (Middle) Spheres are falling with their long ais vertical and along the stream, stable and permanentl chained. (Right) Clinders falling in a viscoelastic fluid turn their long ais parallel to the flow b the elasticit, the eact opposite of a Newtonian fluid. (D.D. Jospeh. Interrogations of Direct Simulation of Solid-Liquid Flow. efluids, ). (UH) Ma 3, 6 3 / 58

Particle motion in viscoelastic fluid Ω Fig. : An eample of flow region with one ball in IR 3. Ω is a bounded region in IR d (d =, 3) containing a viscoelastic fluid of either Oldrod-B or FENE-CR tpe and a ball B(t). B(t) is the particle region with G(t) as the mass center. Γ = Ω; γ(t) = B(t). B (UH) Ma 3, 6 4 / 58

Fluid flow governing equations ( ) u ρ f t +(u )u =ρ f g p+µ D(u)+ σ p in Ω \ B(t), () u = in Ω \ B(t), () u(, ) = u (), Ω \ B(), (with u = ), (3) u = g on Γ, (with g n dγ = ), (4) Γ C t +(u )C ( u)c C( u)t = f (C) (C I) in Ω \ B(t), (5) λ C(, ) = C (), Ω \ B(), C = C L on Γ. (6) u is the flow velocit, p is the pressure, and g is the gravit. D(u) = u + ( u) t, (v )w = { d j= v j w i j } d i=. On Γ, Γ is the upstream part and n is the outer normal unit vector. (UH) Ma 3, 6 5 / 58

Fluid flow governing equations For the fluid, ρ f is the densit, η is the viscosit, µ = η λ /λ is the solvent viscosit, η = η µ is the elastic viscosit, λ is the relaation time, and λ is the retardation time. The polmeric stress tensor σ p in () is given b σ p = η λ f (C)(C I), where the conformation tensor C is smmetric and positive definite and I is the identit matri. Setting f equal to unit corresponds to the Oldrod-B model while L f (C) = L tr(c) corresponds to the FENE-CR model, where tr(c) is the trace of the conformation tensor C and L is the maimum etension of the immersed polmer coils and referred to as the etensibilit of the immersed polmer coils. The Oldrod-B model then is a special case associated with infinite etensibilit. (UH) Ma 3, 6 6 / 58

Particle motion The motion of particle B satisfies the Euler-Newton s equations: v(, t) = V(t) + ω(t) G, {, t} B(t) (, T ), (7) dg = V, (8) dt dv M p dt = M d(i p ω) p g + F H, = T H, (9) dt G() = G, V() = V, ω() = ω. () M p is the mass, I p is the inertia, G is the mass center. V is the velocit of the mass center, and ω is the particle angular velocit. The hdrodnamical forces and torque are F H = σn dγ, T H = G σn dγ. γ γ (UH) Ma 3, 6 7 / 58

formulation For simplicit, we consider one solid ball B(t) in the fluid. To obtain a fictitious domain formulation formulation for above problem () (), we define first the following functional spaces V g (t) = { v v (H (Ω)) d, v = g (t) on Γ}, L (Ω) = { q q L (Ω), q d = }, V CL (t) = { C C (H (Ω)) d d, C = C L (t) on Γ }, V C = { C C (H (Ω)) d d, C = on Γ }, Λ(t) = H (B(t)) d. For <, > B(t), it can be defined as follows: <µ, v> B(t) = (µ v+ µ: v)d, µ H (B(t)) d, v H (Ω) d. B(t) Ω (UH) Ma 3, 6 8 / 58

formulation A distributed Lagrange multiplier/fictitious domain () formulation of the equations ()-() reads as follows: For a.e. t >, find u(t) V g (t), p(t) L (Ω), C(t) V C L (t), V(t) IR d, G(t) IR d, ω(t) IR d, λ(t) Λ(t) such that ρ f { u + (u )u} v d + µ D(u) : D(v) d Ω t Ω p v d v ( σ p ) d Ω Ω +( ρ dv f ρ s ){M p dt Y + I dω p dt θ} () =<λ, v Y θ G> B(t) +ρ f g vd+( ρ f )M p g Y, Ω ρ s {v, Y, θ} (H (Ω)) d IR d IR d, a.e. t (, T ), D particulate flow of Oldrod-B fluids: Singh, Joseph, Hesla, Glowinski, Pan. J. Non-Newtonian Fluid Mech. 9 () & Hao, Pan, Glowinski, Joseph. J. Non-Newtonian Fluid Mech. 56 (9). (UH) Ma 3, 6 9 / 58

Ω formulation q u(t) d =, q L (Ω), a.e. t (, T ), () <µ, u(, t) V(t) ω G> B(t) =, µ Λ(t), a.e. t (,T ), (3) ( ) C +(u )C ( u)c C( u)t :sd (4) Ω t f (C) = (C I):sd, s V C, and C = I in B(t), Ω λ dg dt = V, (5) C(, ) = C (), Ω, (6) G() = G, V() = V, ω() = ω, B() = B, (7) { u (), Ω \ B, u(, ) = V + ω G (8), B. (UH) Ma 3, 6 / 58

formulation In relation () we can replace D(u) : D(v)d b Ω u: vd due to that u is divergence free and satisfies the Ω Dirichlet boundar condition on Γ. For d =, ω(t) = {,, ω(t)} and θ = {,, θ} in (), (3) and (8). The gravit g in () can be absorbed into the pressure term. The Lagrange multiplier λ defined over B in () can be viewed as an etra bod force to maintain the rigid bod motion inside B. The conformation tensor C inside the rigid particle is etended as the identit tensor I as in (4) since the polmeric stress tensor σ p is zero inside the rigid particle. (UH) Ma 3, 6 / 58

Lie s scheme Consider the following initial value problem: dφ dt + A(φ) = on (, T ), φ() = φ (9) with < T < +. We suppose that operator A has a decomposition such as A = J j= A j with J. (UH) Ma 3, 6 / 58

Lie s scheme Let τ(> ) be a time-discretization step, we denote nτ b t n. With φ n denoting an approimation of φ(t n ), the Lie s scheme reads as follows: For n, assuming that φ n is known (with φ = φ ), compute φ n+ via dφ dt + A j(φ) = on (t n, t n+ ), φ(t n ) = φ n+(j )/J ; φ n+j/j = φ(t n+ ), () for j =,..., J. The Lie s scheme is first order accurate, but its low order of accurac is compensated b its simplicit, making it (relativel) eas to implement, and b its robustness. Chorin, Hughes, McCracken, Marsden. Comm. Pure Appl. Math. 3 (978) (UH) Ma 3, 6 3 / 58

Lie s scheme The Lie s operator scheme allows us to decouple the following difficulties: The incompressibilit condition, and the related unknown pressure. The advection terms. The rigid-bod motion in B h (t), and the related DLM λ h. (UH) Ma 3, 6 4 / 58

Lie s scheme The constitutive equation satisfied b the conformation tensor C is split with J = 3 for now to show how the factorization approach works out. Suppose that C n and u are known, we compute dc dt + (u )C = on (tn, t n+ ), () C(t n ) = C n ; C n+/3 = C(t n+ ), dc dt ( u)c C( u)t + f (Cn+/3 ) C = on (t n, t n+ ), λ () C(t n ) = C n+/3 ; C n+/3 = C(t n+ ), dc dt = f (Cn+/3 ) I on (t n, t n+ ), λ (3) C(t n ) = C n+/3 ; C n+ = C(t n+ ). (UH) Ma 3, 6 5 / 58

Lie s scheme We have derived the following two equivalent equations based on the factorization approach with s = f (C n+/3 ) for equations () and (): Lemma For a matri A and C = AA t, given the velocit u, λ (> ) and a constant s, (a). if A satisfies the equation da + (u )A =, then C satisfies dt the equation dc + (u )C = ; dt (b). if A satisfies the equation da dt + s A ( u)a =, then C λ satisfies the equation dc dt + s λ C ( u)c C( u) t =. Lozinski & Owens. J. Non-Newtonian Fluid Mech. (3). (UH) Ma 3, 6 6 / 58

FE approimation Fig. 3: Eamples of uniform finite element mesh for the flow region in IR (left) and IR 3 (right). Ω IR d (d = or 3). h is a space discretization step. T h is a uniform finite element mesh of Ω (See Fig. 3). T h is a uniform finite element mesh twice coarser than T h. (UH) Ma 3, 6 7 / 58

FE approimation The finite dimensional spaces for approimating V g (t), (H (Ω))d, L (Ω), L (Ω), V C L (t), and V C, respectivel, are defined b V gh (t) ={v h v h (C (Ω)) d, v h E (P ) d, E T h, v h Γ = g h (t)}, V h ={v h v h (C (Ω)) d, v h E (P ) d, E T h, v h Γ = }, L h ={q h q h C (Ω), q h E P, E T h }, L h ={q h q h L h, q h d = }, Ω V CLh (t) ={s h s h (C (Ω)) d d, s h E (P ) d d, E T h, s h Γ =C Lh (t)}, h V Ch ={s h s h (C (Ω)) d d, s h E (P ) d d, E T h, s h Γ = }, h where P is the space of the polnomials in d variables of degree, g h (t) is an approimation of g satisfing Γ g h(t) ndγ =, and Γ h = { Γ, g h(, t) n() < }. (UH) Ma 3, 6 8 / 58

Discrete Lagrange multiplier Choose { i } K i= as a set of points that covers B(t) evenl, and then define the discrete Lagrange multiplier space b K Λ h (t)={µ µ = µ j δ( j ), µ j IR d, j =,..., K}, (4) j= where δ( j ) is the Dirac measure at j. Fig. 4: An eample of collocation points on the particle surface. (UH) Ma 3, 6 9 / 58

Discrete Lagrange multiplier Then instead of the scalar product of (H (B h (t)) d, we shall use < µ, v > Bh (t) defined b <µ, v> Bh (t) = K µ j v( j ), µ Λ h (t), v V gh (t) or V h. (5) j= Using the above scalar product implies that the rigid bod motion of B(t) is forced via a collocation. (UH) Ma 3, 6 / 58

Appling the Lie s scheme to the discrete analogue of the problem ()-(8) with the C = AA t (and then equations for A) and the backward Euler s to some subproblems, we obtain u = u h, C = C h, G = G, V = V, ω = ω given, (6) for n, u n, C n, G n, V n, ω n being known, we compute u n+ 5, and p n+ 5 via the solution of u n+ 5 u n ρ f vd p n+ 5 vd=, v Vh, Ω t Ω (7) q u n+ 5 d=, q L h ; u n+ 5 V n+ g h, p n+ 5 L h. Ω (UH) Ma 3, 6 / 58

Net, we compute u n+ 5 and A n+ 5 via the solution of du(t) ρ f vd+ (u n+ 5 )u(t) vd=, v V n+, Ω dt h ; Ω u(t n ) = u n+ 5, u(t) V h, u(t) = g h (t n+ ) on Γ n+, [t n, t n+ ]; da(t) :sd+ (u n+ 5 )A(t):sd=, s VAh ; Ω dt Ω A(t n ) = A n, where A n (A n ) t = C n A(t) V n+ A Lh, t [t n, t n+ ]; and set u n+ 5 = u(t n+ ) and A n+ 5 = A(t n+ ), where Γ n+, = { Γ, g h (t n+ )() n() < }, V h = {v h v h (C (Ω)) d, v h E (P ) d, E T h, }, and V n+, h = {v V h, v = on Γ n+, }. (8) (9) (UH) Ma 3, 6 / 58

Then, compute u n+ 3 5 and A n+ 3 5 via the solution of and set u n+ 3 5 u n+ 5 ρ f vd+αµ u n+ 3 5 : vd=, Ω t Ω v V h ; u n+ 3 5 V n+ Ω ( An+ 3 5 A n+ 5 t g h, ( u n+ 3 5 )A n+ 3 5 + f (An+ 5 (A n+ 5 ) t ) A n+ 3 5 ) : sd =, λ s V Ah ; A n+ 3 5 V n+ A Lh, (3) (3) C n+ 3 5 = A n+ 3 5 (A n+ 3 5 ) t + t f (An+ 3 5 (A n+ 3 5 ) t ) λ I. (3) (UH) Ma 3, 6 3 / 58

Then, predict the position and the translation velocit of the center of mass as follows: Take V n+ 3 5, = V n and G n+ 3 5, = G n ; then predict the new position and translation velocit via the following sub-ccling and predicting-correcting technique For k =,,..., N, compute ˆV n+ 3 5,k = V n+ 3 5,k + ( ρ f /ρ s ) M p F r (G n+ 3 5,k ) t/n, (33) Ĝ n+ 3 5,k = G n+ 3 5,k + ( t/4n)(ˆv n+ 3 5,k + V n+ 3 5,k ), (34) V n+ 3 5,k = V n+ 3 5,k + ( ρ f /ρ s ) M p (F r (Ĝn+ 3 5,k ) (35) +F r (G n+ 3 5,k )) t/4n, G n+ 3 5,k = G n+ 3 5,k + ( t/4n)(v n+ 3 5,k + V n+ 3 5,k ), (36) end do; let V n+ 3 5 = V n+ 3 5,N, G n+ 3 5 = G n+ 3 5,N. (UH) Ma 3, 6 4 / 58

Net compute {u n+ 4 5, λ n+ 4 5, V n+ 4 5, ω n+ 4 5 } via the solution of u n+ 4 5 u n+ 3 5 ρ f vd + βµ u n+ 4 5 : vd Ω t [ Ω ] +( ρ f V n+ 4 5 V n+ 3 5 ω n+ 4 5 ω n ) M p Y + I p θ ρ s t t =< λ n+ 4 5, v Y θ G n+ 3 5 >B n+ 3 +( ρ 5 f /ρ s )M p g Y, h v V h, Y IR d, θ IR d, < µ, u n+ 4 5 V n+ 4 5 ω n+ 4 5 G n+ 3 5 > n+ 3 =, µ Λ n+ 3 5 B 5 h ; h u n+ 4 5 Vg n+ h, λ n+ 4 n+ 5 Λ 3 5 h, and set C n+ 4 5 = C n+ 3 5, and then let C n+ 4 5 = I in B n+ 3 5 h. (37) (UH) Ma 3, 6 5 / 58

Then take V n+, = V n+ 4 5 and G n+, = G n+ 3 5 ; and predict the final position and translation velocit as follows: For k =,,..., N, compute ˆV n+,k = V n+,k + ( ρ f /ρ s ) M F r (G n+,k ) t/n, (38) Ĝ n+,k = G n+,k + ( t/4n)(ˆv n+,k + V n+,k ), (39) V n+,k = V n+,k + ( ρ f /ρ s ) M (F r (Ĝn+,k ) (4) +F r (G n+,k )) t/4n, G n+,k = G n+,k + ( t/4n)(v n+,k + V n+,k ), (4) end do; let V n+ = V n+,n, G n+ = G n+,n. (UH) Ma 3, 6 6 / 58

Finall, compute u n+ via the solution of u n+ u n+ 4 5 ρ f vd + γµ u n+ : vd, Ω t Ω = η v ( f (C n+ 4 5 )(C n+ 4 5 I))d, v Vh ; u n+ V n+ λ Ω g h. (4) We complete the final step b setting C n+ = C n+ 4 5, and ω n+ = ω n+ 4 5. In the above, Vg n+ h = V gh (t n+ ), Λ n+s h = Λ h (t n+s ), V n+ A Lh = V ALh (t n+ ), B n+s h = B h (t n+s ) where the spaces V ALh (t) and V Ah for A are defined similar to those C Lh (t) and C h, and α + β + γ =, where α, β, γ. (UH) Ma 3, 6 7 / 58

Solution strategies In the algorithm (6)-(4), we have obtained a sequence of simpler subproblems via Lie s scheme: A degenerated quasi-stokes problem (7) is solved b an Uzawa/ preconditioned conjugate gradient algorithm operating in the space L h as in Glowinski: Handbook of Analsis (3), Glowinski, Pan & Periau: Comput. Appl. Mech. Engrg. (998). The advection problems (8) and (9) are solved b a wave-like equation (Dean & Glowinski: C.R. Acad. Sci. Paris, Série (997)) which is an eplicit and does not introduce numerical dissipation. Since the advection problem is decoupled from the other ones, we can choose a proper sub-time step so that the CFL condition is satisfied. Problem (37), concerning the rigid bod motion enforcement, is a saddle point problem and is solved b a conjugate gradient developed in Glowinski, Pan, Hesla & Joseph: Int. J. Multiphase Flow (999). (UH) Ma 3, 6 8 / 58

Solution strategies Problems (3) and (4) are classical elliptic problems which can be solved b a matri-free fast solver from FISHPAK b Adams et al.: NCAR (98). Here we have chosen the use of uniform mesh so that a matri-free fast solver can be used to solve classical elliptic problems arising in problems (3), (37) and (4), which is especiall important for three dimensional flow problems we have proposed here since the memor cost can be reduced a lot. In (33)-(36) and (38)-(4), we have used a predicting-correcting scheme to obtain the position of the mass center and the translation velocit of the particle. In order to let the short range repulsive force to be activated effectivel, we have, tpicall, chosen N = sub-time steps to move the particle in one time step in the simulation. (UH) Ma 3, 6 9 / 58

Solution strategies Problem (3) gives a simple equation at each grid point which can be solved easil if we use the trapezoidal quadrature rule to compute the integrals as in Hao, Pan, Glowinski & Joseph: J. Non-Newtonian Fluid Mech (9). The discretized conformation tensor C n is clearl smmetric, and positive definite, b construction. We have split the constitutive equation for C b the Lie s scheme and the factorization to obtain (9), (3) and (3). In Lozinski and Owens: J. Non-Newtonian Fluid Mech. (3), the have split the equation with J =. Here we have decoupled the advection term so that we can appl our wave-like equation to solve the advection problem for A in (9). To us, (9) and (3) are simpler subproblems. (UH) Ma 3, 6 3 / 58

in an Oldrod-B fluid The computational domain is Ω = (,)(,6) initiall, then it moves verticall with the center of the disk. The disk diameter is d =.5 and its densit is ρ s=.. The initial position of the disk center is (.35,.5). The fluid densit is ρ f = and its viscosit is η =.. The relaation time and retardation time are λ = and λ = λ /4, respectivel. The particle Renolds number is Re = ρ f Ud, the Debra η number is De = λ U, the Mack number is d M = DeRe, and the elasticit number is E = De/Re where U is the averaged terminal speed. 6 5 4 3.5 Fig. 5. Computational domain Ω and a disk at its initial position. (UH) Ma 3, 6 3 / 58

in an Oldrod-B fluid V.5.4.3.....3 h=/96, dt=.4 h=/96, dt=. h=/44, dt=. 3 4 5 6 7 8 t V.3.....3.4 h=/96, dt=.4 h=/96, dt=. h=/44, dt=. 3 4 5 6 7 8 t Fig. 6. Histories of the -component (left) and -component (right) of the translational velocit: Re =.8, M =.3, De =.57, E = 3.. Hao, Pan, Glowinski, Joseph. J. Non-Newtonian Fluid Mech. 56 (9). (UH) Ma 3, 6 3 / 58

in viscoelastic fluids The computational domain is Ω = (,)(,6) initiall, then it moves verticall with the center of the disk. The disk diameters are d =.5 and the densit is ρ s=.. The initial position of the disk centers are (.35,.5) and (.65,.5). The fluid densit is ρ f = and its viscosit is η =.. The relaation time λ is either.5 and ; and the retardation time λ is λ /4. The maimal etension L of the immersed polmer coils is 5,, and 5. 6 5 4 3.5 Fig. 7. Computational domain Ω and two disks at the initial position. (UH) Ma 3, 6 33 / 58

Two disk in viscoelastic fluids 5-5 - -5 Oldrod-B FENE-CR: L=5 FENE-CR: L= FENE-CR: L=5.5 5-5 - -5 Oldrod-B FENE-CR: L=5 FENE-CR: L= FENE-CR: L=5.5 Fig. 8. Trajectories of two disks: (left) λ =.5 & E=.6 and (right) λ = & E=3.. (UH) Ma 3, 6 34 / 58

Two disk in viscoelastic fluids Horizontal velocit Horizontal velocit..5 -.5 Oldrod-B FENE-CR: L=5 FENE-CR: L= FENE-CR: L=5 -. 4 6 8 t..5 -.5 Oldrod-B FENE-CR: L=5 FENE-CR: L= FENE-CR: L=5 -. 4 6 8 t Fig. 9. Horizontal velocit of two disks: (top) λ =.5 & E=.6 and (bottom) λ = & E=3.. (UH) Ma 3, 6 35 / 58

Vertical velocit Vertical velocit -.5 -. -.5 -. Two disk in viscoelastic fluids Oldrod-B FENE-CR: L=5 FENE-CR: L= FENE-CR: L=5 -.5 4 6 8 t -.5 -. -.5 -. Oldrod-B FENE-CR: L=5 FENE-CR: L= FENE-CR: L=5 -.5 4 6 8 t Fig.. Vertical velocit of two disks: (top) λ =.5 & E=.6 (Oldrod-B: Re=., M=.55, De=.33; FENE-CR of L=5: Re=.97, M=.49, De=.36) and (bottom) λ = & E=3. (Oldrod-B: Re=.8, M=.37, De=.664; FENE-CR of L=5: Re=.95, M=.35, De=.65). (UH) Ma 3, 6 36 / 58

Two disk in viscoelastic fluids 3 - - -3-4 t= t=5 t=.5 t=5 t= t=5 t=3 t=35 t=4 t=45 t=5 t=55-5 -6-7 -8-9 - - - -3 t=6 t=65 t=7 t=75 t=8 t=85 t=9 t=95 t=.5 3 - - -3-4 t= t=5 t= t=5.5 t= t=5 t=3 t=35 t=4 t=45 t=5-5 -6-7 -8-9 - - - -3 t=55 t=6 t=65 t=7 t=75 t=8 t=85 t=9 t=95 t=.5 3 - - -3-4 t= t=5 t= t= t=5 t=5 t=3 t=35 t=4 t=45 t=5.5-5 t=55-6 -7-8 -9 - - - -3 t=6 t=65 t=7 t=75 t=8 t=85 t=9 t=95 t=.5 3 - - -3-4 t= t=5 t= t=5 t= t=5 t=3 t=35 t=4 t=45 t=5.5-5 -6-7 -8-9 - - - -3 t=55 t=6 t=65 t=7 t=75 t=8 t=85 t=9 t=95 t=.5 Fig.. Positions of two disks : (left two) FENE-CR of L =5, E =.6, (middle left two) Oldrod-B, E =.6, (middle right two) FENE-CR of L =5, E = 3., and (right two) Oldrod-B, E = 3.. (UH) Ma 3, 6 37 / 58

Particle in an Oldrod-B fluid Si disks form a long chain after drafting, kissing and chaining, which is different from the well known phenomenon called drafting, kissing and tumbling for disks settling in Newtonian fluid. The dimension of computational domain is Ω = (,)(,7) initiall, then it moves verticall with the center of the lowest disk. The disk diameters are d =.5 and the densit is ρ s=.. The fluid densit is ρ f = and its viscosit is η =.6. The relaation time and retardation time are λ =.3 and λ = λ /8, respectivel. The mesh sizes for the velocit field, etra stress tensor and pressure are h = /96, /96, and /48, respectivel; and the time step is.4. The averaged terminal speed is U =.535. The particle Renolds number is Re = ρ f Ud η =.476. The Debra number is De = λ U d =.798. The Mack number is M = DeRe =.343. The elasticit number is E = De/Re =5.48. (UH) Ma 3, 6 38 / 58

Particle in an Oldrod-B fluid Fig. : The snap shots of particle position and velocit field at t =,,, 4, 6, 8,, 4, 6, 8, and 3 (from left to right). (UH) Ma 3, 6 39 / 58

Particle in a FENE-CR fluid Si disks form a long chain after drafting, kissing and chaining, which is different from the well known phenomenon called drafting, kissing and tumbling for disks settling in Newtonian fluid. In the FENE-CR model, the polmer etension limit is set to be L = 5 in the function f (C) = L L/(L L tr(c)) where C is the conformation tensor. The viscoelastic fluid with a shorter polmer etension limit can not hold all s together. The dimension of computational domain is Ω = (,)(,7) initiall, then it moves verticall with the center of the lowest disk. The disk diameters are d =.5 and the densit is ρ s=.. The fluid densit is ρ f = and its viscosit is η =.6. The relaation time and retardation time are λ =.3 and λ = λ /8, respectivel. The mesh sizes for the velocit field, etra stress tensor and pressure are h = /96, /96, and /48, respectivel; and the time step is.4. The averaged terminal speed is U =.37. The particle Renolds number is Re = ρ f Ud η =.66. The Debra number is De = λ U d =.6847. The Mack number is M = DeRe =.944. The elasticit number is E = De/Re =5.48. (UH) Ma 3, 6 4 / 58

Particle in a FENE-CR fluid Fig. 3: The snap shots of particle position and velocit field at t =, 4, 6, 8,, 8,, 4, 6, and 3 (from left to right) in a FENE-CR fluid with L = 5. (UH) Ma 3, 6 4 / 58

Horizontal velocit Vertical velocit..5 -.5 Comparison: case Oldrod-B FENE-CR: L=5 FENE-CR: L= FENE-CR: L=5 -. 5 5 5 3 t.5 -.5 -. -.5 -. Oldrod-B FENE-CR: L=5 FENE-CR: L= FENE-CR: L=5 -.5 5 5 5 3 t 3 - - Oldrod-B FENE-CR: L=5 FENE-CR: L= FENE-CR: L=.5 Fig. 4: Histories of the particle velocit (left) and trajectories of si disks (right). (UH) Ma 3, 6 4 / 58

Comparison: case.5 t=..5 t=..5 t=.5 t=.5 t=4.5 t=4.5 t=6.5 t=6.5 t=8.5 t=8.5 t=.5 t=.5.5 -.5.5.5 -.5.5.5 -.5.5.5 -.5.5.5 -.5.5.5 -.5.5.5 -.5.5.5 -.5.5.5 -.5.5.5 -.5.5.5 -.5.5.5 -.5 - -.5 -.5 - -.5 -.5 - -.5 -.5 - -.5 -.5 - -.5 -.5 - -.5 -.5 - -.5 -.5 - -.5 -.5 - -.5 -.5 - -.5 -.5 - -.5 -.5 - -.5 -.5 OL FENE OL FENE OL FENE OL FENE OL FENE OL FENE Fig. 5: Comparison of numerical results in Oldrod-B fluid (OL) and FENE-CR fluid (FENE) for L = 5: t =,, 4, 6, 8, and (from left to right). (UH) Ma 3, 6 43 / 58

Comparison: case.5.5.5 -.5.5 t=.5.5 -.5 t=.5.5.5 -.5.5 t=.5.5 -.5.5 t=.5.5 -.5.5 t=4.5.5 -.5.5 t=4.5.5 -.5.5 t=6.5.5 -.5.5 t=6.5.5 -.5.5 t=8.5.5 -.5.5 t=8.5.5 -.5.5.5 t=.5 -.5 t= - -.5 -.5 - -.5 -.5 - -.5 -.5 - -.5 -.5 - -.5 -.5 - -.5 -.5 - -.5 -.5 - -.5 -.5 - -.5 -.5 - -.5 -.5 - -.5 -.5 - -.5 -.5 OL FENE OL FENE OL FENE OL FENE OL FENE OL FENE Fig. 6: Comparison of numerical results in Oldrod-B fluid (OL) and FENE-CR fluid (FENE) for L = 5: t =,, 4, 6, 8, and (from left to right). (UH) Ma 3, 6 44 / 58

Comparison.5.5.5 -.5.5.5 t=.5 -.5.5 t=.5.5 -.5.5.5 t=.5 -.5.5 t=.5.5 -.5.5.5 t=4.5 -.5.5.5 t=4.5 -.5.5.5.5 t=6 -.5.5.5 t=6.5 -.5.5.5.5 t=8 -.5.5.5 t=8.5 -.5.5.5.5 t=3 -.5 t=3 - -.5 -.5 - -.5 -.5 - -.5 -.5 - -.5 -.5 - -.5 -.5 - -.5 -.5 - -.5 -.5 - -.5 -.5 - -.5 -.5 - -.5 -.5 - -.5 -.5 - -.5 -.5 OL FENE OL FENE OL FENE OL FENE OL FENE OL FENE Fig. 7: Comparison of numerical results in Oldrod-B fluid (OL) and FENE-CR fluid (FENE) for L = 5: t =,, 4, 6, 8, and 3 (from left to right). (UH) Ma 3, 6 45 / 58

in viscoelastic fluids The computational domain is Ω = (,)(,5) initiall, then it moves verticall with the center of the disk. The disk diameters are d =.5 and the densit is ρ s=.5,.75, or.. The initial position of the disk centers are (.6, ), (.88, ) and (.6, ). The fluid densit is ρ f = and its viscosit is η =.6. The relaation time λ is either.75 or.5; and the retardation time λ is λ /8. The maimal etension L of the immersed polmer coils is 5,, and 5. 5 4.5 4 3.5 3.5.5.5 Fig. 8. Computational domain Ω and three disks at the initial position. (UH) Ma 3, 6 46 / 58

in viscoelastic fluids - - - - - - -3-3 -4-4 -5-5 -3-4 -5-6 -6-7 L=5 L= L=5 Oldrod-B -8-9 -6 L=5 L= L=5 Oldrod-B -7-8 L=5 L= L=5 Oldrod-B -7-8 Fig. 9: The disk trajectories: ρs =.5 (left),.75 (middle), and. (right) for E=3. (λ =.75). (UH) Ma 3, 6 47 / 58

in viscoelastic fluids -5-5 - - -5-5 - - -5 - -5 L=5 L= L=5 Oldrod-B -.5 L=5 L= L=5 Oldrod-B -5.5-3.5 L=5 L= L=5 Oldrod-B -5.5-3.5.5 Fig. : The disk trajectories: ρs =.5 (left),.75 (middle), and. (right) for E=6.4 (λ =.5). (UH) Ma 3, 6 48 / 58

in an Oldrod-B fluid in a vertical clinder The dimension of computational domain is Ω = (,D)(,D)(,4D) initiall, then it moves verticall with the center of the ball. The diameter and height of the clinder inside Ω are D and 4D, respectivel. The ball diameter is d = and its densit is ρ s=3.58. The fluid densit is ρ f =.868 and its viscosit is η =3. The relaation time and retardation time are λ =. and λ = λ /9, respectivel. The ratio of the ball diameter and the clinder diameter is α = d/d=.,., or.5. The particle Renolds number is Re = ρ f Ud, the η Debra number is De = λ U, the Mack number is d M = DeRe, and the elasticit number is E = De/Re where U is the averaged terminal speed. Fig.. Computational domain Ω and a ball for the case α =.. (UH) Ma 3, 6 49 / 58

in an Oldrod-B fluid in a Verical velocit -5 - -5 - α =. vertical clinder Bodart & Crochet [994] Hu, Patankar & Zhu [] h=/8 & dt=.5 h=/8 & dt=.95 h=/6 & dt=.5-5...3.4.5.6.7 t Fig.. The comparison of the vertical speed for the case α =.: U =5.54, Re =.8977, M =.8345, De =.7757, E =.864 for t =.5 and h = /6. Bodart & Crochet. J. non-newtonian Fluid Mech. 54 (994) Hu, Patankar & Zhu. J. Comput. Phs. 69 (). (UH) Ma 3, 6 5 / 58

in an Oldrod-B fluid in a Veritcal velocit -5 - -5 - α =. vertical clinder Bodart & Crochet [994] Hu, Patankar & Zhu [] h=/48 & dt=.45 h=/64 & dt=.5 h=/96 & dt=.9-5...3.4.5.6.7 t Fig. 3. The comparison of the vertical speed for the case α =.: U =.9, Re =.754, M =.6557, De =.695, E =.864 for t =.9 and h = /96. Bodart & Crochet. J. non-newtonian Fluid Mech. 54 (994) Hu, Patankar & Zhu. J. Comput. Phs. 69 (). (UH) Ma 3, 6 5 / 58

in an Oldrod-B fluid in a Vertical velocit -5 - α =.5 vertical clinder Bodart & Crochet [994] h=/3 & dt=.5 h=/48 & dt=. h=/64 & dt=.5-5...3.4.5.6.7 t Fig. 4. The comparison of the vertical speed for the case α =.5: U =3.9, Re =.64, M =.4, De =.956, E =.864 for t =.5 and h = /64. Bodart & Crochet. J. non-newtonian Fluid Mech. 54 (994) Hu, Patankar & Zhu. J. Comput. Phs. 69 (). (UH) Ma 3, 6 5 / 58

Particle in an Oldrod-B fluid Two balls form a vertical chain after drafting, kissing and chaining, which is different from the well known phenomenon called drafting, kissing and tumbling for disks settling in Newtonian fluid. The dimension of computational domain is Ω = (,)(,)(,4) initiall, then it moves verticall with the center of the lower ball. The ball diameters are all d =. and their densit is ρ s=.. The fluid densit is ρ f = and its viscosit is η =.6. The relaation time and retardation time are λ =.53846538 and λ = λ /8, respectivel. The mesh sizes for the velocit field, etra stress tensor and pressure are h = /64, /64, and /3, respectivel; and the time step is.5. The averaged terminal speed is U =.797. The particle Renolds number is Re = ρ f Ud η =.546. The Debra number is De = λ U d =.546. The Mack number is M = DeRe =.546. The elasticit number is E = De/Re =. (UH) Ma 3, 6 53 / 58

t=.5 t=3 t= t=33 t=35 t=6 Particle in an Oldrod-B fluid Fig. 5: (Left) The particle position at t =.5, 3,, 33, 35, and 6 (from left to right and top to bottom), (middle) eperimental results, and (right) the trajectories of two balls. 5-5 - -5 - -5-3 -35-4 -45-5 (UH) Ma 3, 6 54 / 58

in an Oldrod-B fluid A long bod like ellipsoid turns its broadside to the direction in Oldrod-B fluid. The dimension of computational domain is Ω = (,)(,)(,4) initiall, then it moves verticall with the center of the lower ball. The semi-ais of the ellipsoid are.5,.65, and.65 and its densit is ρ s=.. The fluid densit is ρ f = and its viscosit is η =.6. The relaation time and retardation time are λ = and λ = λ /8, respectivel. The mesh sizes for the velocit field, etra stress tensor and pressure are h = /64, /64, and /3, respectivel; and the time step is.. The averaged terminal speed is U =.7. The particle Renolds number is Re = ρ f Ud η =.9874. The Debra number is De = λ U d =.48. The Mack number is M = DeRe =.4. The elasticit number is E = De/Re =4.6. (UH) Ma 3, 6 55 / 58

t= t=5 t=5 t=75 t=5 t=3 in an Oldrod-B fluid Fig. 6: (Left) The particle positions at t =, 5, 5, 75, 5, and 3 (from left to right and top to bottom), (middle) eperimental results, and (right) the trajector of the ellipsoid mass center. 5-5 - -5 - -5-3 (UH) Ma 3, 6 56 / 58

Some future works Chain of man balls. The effect of coil etension L on the cluster of particles. The orientation of long bod in viscoelastic fluid. (UH) Ma 3, 6 57 / 58

Thank ou for our attention! (UH) Ma 3, 6 58 / 58