In this paper, we present a new direct simulation techniquefor non-Brownian hard particle suspensions formulated witha viscoelastic (Oldroyd-B) fluid under simple shear flow in awell-defined bi-periodic domain. Through several exampleproblems, we discuss bulk rheological properties as well asmicro-structural developments due to complicated hydrody-namic interactions in such a system.
This work is an extension of our previous study[1] forsuspensions in a Newtonian fluid, which used a fictitiousdomain method, similar to the distributed Lagrangian mul-tipliers (DLM) method of Glowinski et al.[2], in that afixed regular mesh is used for the entire computation andthat hydrodynamic interaction is treated implicitly via acombined weak formulation. (The combined weak formu-
lation was first implemented by Hu[3] using the arbitraryLagrangian–Eulerian technique.) However, we combined thefictitious domain method with a well-defined bi-periodic do-main concept, the Lees–Edwards boundary condition (LEbc)for the simple shear flow[4], which could transform a sus-pension consisting of a large number of particles into a par-ticulate flow problem in a unit cell, diminishing finite sizeeffect of the computational domain. Our previous work hastwo distinct features:
(1) Sliding bi-periodic constraints:The sliding bi-periodicdomain concept of the LEbc for discrete particleshas been extended to continuous fields and com-bined with the velocity–pressure formulation of thefictitious-domain/finite-element method by introducingthe constraint equations;
(2) A rigid-ring description of particles:Inertialess parti-cles are described by their boundary only, eliminatingdomain discretization of particles, which allows easytreatments of boundary-crossing particles.
16 W.R. Hwang et al. / J. Non-Newtonian Fluid Mech. 121 (2004) 15–33
This scheme has been verified to give accurate solu-tions (particle velocity, fluid velocity, pressure and evenvelocity-gradient distributions) with suitable choice of thespatial discretization. In addition, a general expression ofthe bulk stress was derived, which involves only bound-ary integrals of the Lagrangian multipliers on the domainboundary and on the boundary of particles crossing thedomain boundary.
In the present paper, we consider the suspensions formu-lated with a viscoelastic (Oldroyd-B) fluid. To combine theviscoelasticity with our existing scheme, we employ a mixedfinite-element formulation developed by Guénette and Fortin[5], the Discrete Elastic-Viscous Stress Splitting (DEVSS)method, which appears to provide one of the most robust for-mulations currently available. The discontinuous Galerkin(DG) formulation of Fortin and Fortin[6] will be used for thediscretization of the constitutive equation. The use of DG isparticularly suited in this work not only because of its min-imal coupling between elements, avoiding a large numberof coupled equations, but also from the inherent discontinu-ous nature of the fictitious domain method. The viscoelasticstress is discontinuous across the particle boundary.
Concentrating on the simulations of circular disk par-ticles in 2-D, we discuss the bulk rheological propertiesand micro-structural developments in flowing suspensionsformulated with an Oldroyd-B fluid, through numericalexamples of the single-, two- and many-particle prob-lems. Because of the bi-periodicity in the computationaldomain, a system possessing a few particles in a singledomain represents a problem with a large number of par-ticles in an unbounded domain. Using the single-particleproblem, we report the steady bulk viscosity and the firstnormal stress difference coefficient as a function of theWeissenberg number and the solid area fraction. The re-sults show shear-thickening behaviors for both properties.Interestingly, our direct simulations in 2-D reproduce thecommon experimental observation that the first normalstress difference in a filled viscoelastic fluid is a power-lawfunction of the imposed shear stress such thatN1 ≈ τn witha power-law exponentn which is determined by the unfilledviscoelastic fluid[8–10]. (For the Oldroyd-B fluid, the ex-ponentn becomes 2.) From the two-particle problems, wereport clustering of particles, when they are closely located.It results in kissing-tumbling-tumbling phenomena of thetwo particles: they keep rotating around each other. Thistendency becomes stronger with increasing fluid elasticity.Through the many-particle problems, we discuss devel-opments of strong elongational flows between separatingparticles and the resultant non-uniformity and anisotropy inthe micro-structures will be highlighted.
The paper is organized as follows. First, we state thegoverning sets of equations for the fluid, the particles andthe hydrodynamic interactions in the strong form. Next, wepresent the combined weak formulation of the whole systemand discuss details of the numerical implementation meth-ods. Subsequently, we use three sets of example problems
– the single, two and many particles in a sliding bi-periodicdomain – to demonstrate the feasibility of our scheme and,through these examples, we discuss the rheological prop-erties and microstructural developments in suspensions for-mulated with the viscoelastic fluid.
2. Modeling
2.1. Problem definition
In this paper, we consider suspensions of freely suspendedrigid particles in an Oldroyd-B fluid under simple shear flow.We restrict ourselves to 2-D systems composed of circulardisk-like particles, neglecting inertia for both the fluid andthe particles. By combining the bi-periodic domain conceptof Lees and Edwards[4], such a suspension problem can betreated as a particulate flow problem in a unit cell or frame.In this way, we can diminish finite size effects of the compu-tational domain, eliminating complex wall interactions, andsolve the problem at a reasonable computational cost.
Fig. 1 shows the sliding bi-periodic frames with a pos-sible particle configuration in a single frame. As time goeson, each frame translates at its own average velocity of theflow inside the frame and thereby rows of the frames sliderelatively to one another by the amount∆, which is deter-mined by the given shear rateγ, the elapsed timet, and theheight of the frameH :
∆ = γHt. (1)
The sliding velocity of the frame is determined by the givenshear rate and a representative vertical position of the framebased on an arbitrary global reference. Since it is an inertialframe of reference, only the relative velocity inside the frameis important. In addition, the sliding frame is bi-periodic:the left and right boundaries satisfy the usual periodic con-dition and the upper and lower boundaries are subject tothe time-dependent sliding periodicity described byEq. (1).Therefore the motion of the rigid particles as well as of thefluid particles is subject to the time-dependent coupling be-tween the upper and lower boundaries, in addition to theusual periodic condition in the horizontal direction. For de-tails of the sliding bi-periodic frame, refer to Hwang et al.[1].
A sliding bi-periodic frame, denoted byΩ, is the com-putational domain of this work (Fig. 1). The four bound-aries of the domain are denoted byΓi(i = 1,2,3,4) and thesymbolΓ will be used for
⋃4i=1Γi. The Cartesianx andy
coordinates are selected as parallel and normal to the shearflow direction, respectively. Particles are denoted byPi(t)(i = 1, . . . , N) andN is the number of particles in a singleframe. We use a symbolP(t) for
⋃Ni=1Pi(t), a collective re-
gion occupied by particles at a certain timet. For a particlePi, Xi = (Xi, Yi), Ui = (Ui, Vi), ωi = ωik andΘi = Θik
are used for the coordinates of the particle center, the trans-lational velocity, the angular velocity and the angular rota-
W.R. Hwang et al. / J. Non-Newtonian Fluid Mech. 121 (2004) 15–33 17
Fig. 1. Sliding bi-periodic frames in a simple shear flow (left). A sliding bi-periodic frame is the computational domain and a possible particle configurationinside the domain is indicated (right).
tion, respectively; andk is the unit vector in the directionnormal to the plane.
2.2. Governing equations
We present the governing equation sets in the strong formfor suspensions of 2-D disk-like circular particles in anOldroyd-B fluid in a sliding bi-periodic frame, neglectinginertia for both the fluid and the particles.
2.2.1. Fluid domainThe set of equations for the fluid domain is given by
∇ · σ = 0, in Ω\P(t), (2)
∇ · u = 0, in Ω\P(t), (3)
σ = −pI + 2ηsD + τp, in Ω\P(t), (4)
λ∇τp + τp − 2ηpD = 0, in Ω\P(t), (5)
u = Ui + ωi × (x − Xi) on ∂Pi, i = 1, . . . , N. (6)
Eqs. (2)–(6)are equations for the momentum balance, thecontinuity, the constitutive relation of the Oldroyd-B fluidand the rigid-body condition on the particle boundary, re-spectively.σ, u, p, D, I, τp, ηs, ηp and λ are the stress,the velocity, the pressure, the rate of deformation tensor, theidentity tensor, the polymeric contribution to the extra-stresstensor, the viscosity of a Newtonian solvent, the polymerviscosity and the relaxation time, respectively. The symbol∇ denotes the upper-convected time derivative, defined as
∇τp ≡ ∂τp
∂t+ u · ∇τp − (∇u)T · τp − τp · ∇u.
The zero shear viscosity is defined asη0 = ηs + ηp.We need an initial condition forτp,
τp|t=0 = τp0, in Ω\P(t). (7)
We use the stress-free state,τp0 = 0, as the initial condi-tion for τp over the whole domain throughout the study. Onthe other hand, the initial conditions for the velocity of the
fluid and for the particle are not necessary in the absence ofinertia.
In Newtonian flow simulations, two conditions need tobe satisfied across the domain boundaryΓ : the continuityof the velocity field and the force balance. These conditionshave been discussed in details in our previous work[1]. Inviscoelastic flows, however, one needs an additional bound-ary condition, the inflow condition, forτp, since the poly-mer stress is convected with the fluid. Since material parti-cles which cross a boundary of the domain should re-appearon the corresponding periodic boundary, the inflow condi-tion in this bi-periodic problem implies the continuity of thepolymer stress between the periodic boundaries.
The three conditions along the horizontal direction be-tweenΓ2 andΓ4 can be summarized as follows:
u(0, y) = u(L, y), y ∈ [0, H ], (8)
t(0, y) = −t(L, y), y ∈ [0, H ], (9)
τp(0, y) = τp(L, y), y ∈ [0, H ], (10)
with the vectort denoting the traction force on the boundary.The conditions for the sliding periodicity in the vertical di-rection are more complicated because of the time-dependentcoupling betweenΓ1 andΓ3 as described inEq. (1). Beloware the kinematic condition for the continuity, the force bal-ance and the continuity of the polymer stress, respectively.
u(x,H; t) = u(x− γHt∗,0; t)+ f , x ∈ [0, L), (11)
t(x,H; t) = −t(x− γHt∗,0; t), x ∈ [0, L), (12)
τp(x,H; t) = τp(x− γHt∗,0; t), x ∈ [0, L), (13)
wheref = (γH,0) and·∗ denotes the modular function ofL. (For example,1.7L∗ = 0.7L and−1.7L∗ = 0.3L.)
Eqs. (8)–(13)completes the governing equations withEqs. (2)–(7)for the fluid domain in the strong form.
In the weak formulation, the kinematic constraints(Eqs. (8) and (11)) are usually combined with Lagrangianmultipliers and then the associated force balances (Eqs. (9)and (12)) are satisfied implicitly through the multipli-ers. In this regard, we will use the kinematic equations
18 W.R. Hwang et al. / J. Non-Newtonian Fluid Mech. 121 (2004) 15–33
(Eqs. (8) and (11)) in derivation of the weak form and wecall the two equationsthe sliding bi-periodic constraints.The continuity conditions forτp (Eqs. (10) and (13)) willbe treated separately through the jump convection term inthe discontinuous Galerkin (DG) formulation, which willbe discussed inSection 3.1.
2.2.2. Particle domainAs we did in [1] for particles in a Newtonian fluid, we
again use the rigid-ring description for the particle domain inthe viscoelastic fluid, which holds whenever inertia is negli-gible. With this description, a particle is considered as a rigidring, which is filled with the same fluid as in the fluid do-main, and the rigid-body motion is imposed only on the par-ticle boundary. The idea is similar to the original immersedboundary method of Peskin[12] in which the equations forthe fluid velocity is solved for both inside and outside of themoving boundary of zero mass. This description needs dis-cretization only along the particle boundary so that it givessignificant reduction in memory and it is easier to imple-ment, especially for boundary-crossing particles, comparedwith methods using domain discretization[2,13–15]. In ad-dition, the traction force on the particle boundary can beobtained as a part of the solution, when the constraint equa-tion of the rigid-body motion is combined with Lagrangianmultipliers [1].
The particlePi (i = 1, . . . , N) in an Oldroyd-B fluid isdescribed by the following set of equations:
∇ · σ = 0, in Pi(t), (14)
∇ · u = 0, in Pi(t), (15)
σ = −pI + 2ηsD + τp, in Pi(t), (16)
λ∇τp + τp − 2ηpD = 0, in Pi(t), (17)
u = Ui + ωi × (x − Xi) on ∂Pi(t). (18)
Eqs. (14)–(18)are equations for the momentum balance,the continuity, the constitutive relation (Oldroyd-B) and theboundary condition, respectively, which are exactly the sameas the fluid domain equations as inEqs. (2)–(6). The initialcondition for the polymer stress is again the stress-free state,as it should be inside the rigid-ring, i.e.
τp|t=0 = 0, in Pi(t). (19)
We do not need an inflow condition for the polymer stressτp, since there is no net convection of material particlesacross the particle boundary. The solution of the rigid-ringproblem (Eqs. (14)–(19)) inside the particle appears to bethe rigid-body motion imposed on the particle boundary:
u = Ui + ωi × (x − Xi) in Pi(t). (20)
Due to the rigid-body motion, the polymer stress inside therigid-ring remains zero. Finally, the motion of the particlecenter is given by the following advection equations:
dXi
dt= Ui, Xi|t=0 = Xi,0, (21)
dΘidt
= ωi, Θi|t=0 = Θi,0. (22)
Note thatEq. (22)is completely decoupled from the otherequations.
2.2.3. Hydrodynamic interactionIn order to determine the unknown rigid body motions
(Ui,ωi) of the particles, one needs balance equations fordrag forces and torques on particle boundaries. In the ab-sence of inertia and external forces or torques, particles areforce-free and torque-free:
F i =∫∂Pi(t)
σ · n ds = 0, (23)
T i =∫∂Pi(t)
(x − Xi)× (σ · n)ds = 0, (24)
whereT i = Tik andn is a normal vector on∂Pi pointingout of the particle.
We did not use an artificial particle-particle collisionscheme[2,13–15], because the particle overlap could beavoided for the multiple-particle problems we studied inthis paper by taking a relatively small time-step and asufficiently refined particle boundary discretization.
2.3. Boundary-crossing particles
When a particle crosses the domain boundary, the parti-cle or parts of the particle, which are present outside thecomputational domain, need to be relocated into the domain.The relocation possibly involves the change of the transla-tional velocity of the particle, since the upper (lower) slidingframe moves faster (slower) than the frame of computationby the amount of the velocity differenceγH . We discussedthe treatment of such a situation in details with the slid-ing bi-periodic domain in Hwang et al.[1]. Here we brieflysummarize the method for completeness of the paper.
The relocation can be made in two consecutive steps:relocation of the particle center and of the particle boundary.Both of them can be expressed in a single equation. Considertwo sets of the coordinates: the unprimed set for the originalcoordinate (before relocation) and the primed set for therelocated coordinate. As shown inFig. 2, a positionx =(x, y), which resides outside the domain, belongs to one ofthe four regions around the domain. For the given(γ, H, t),the relocated positionx′ = (x′, y′) is determined by theregion whichx belongs to:
upper zone :(x′, y′) = (x− γHt∗, y −H),
lower zone :(x′, y′) = (x+ γHt∗, y +H),
right or left zone :(x′, y′) = (x∗, y),otherwise :(x′, y′) = (x, y).
(25)
The positionx can be either a particle centerX or apoint on the particle boundary. The last line inEq. (25) is
W.R. Hwang et al. / J. Non-Newtonian Fluid Mech. 121 (2004) 15–33 19
Fig. 2. A description of a particle which crosses the domain boundary.The part of a particle which is present outside the domain is relocatedaccording toEq. (25). If a particle crosses the upper and thereby lowerboundaries, thex directional translational velocities for different parts ofthe particle are different by the amount ofγH .
introduced to attain the consistency in notation. Since theupper (lower) frame translates faster (slower) than the frameof computation by the amount ofγH (−γH), the relocationinvolves changes in thex directional velocity componentU,if a particle crosses upper or lower boundaries. Again thechange in the translational velocity can be expressed in asingle equation for both a particle center and a point on theparticle boundary and is determined by the region where theoriginal (unprimed) position is located:
upper zone :U ′ = U − γH,
lower zone :U ′ = U + γH,
otherwise :U ′ = U.
(26)
Again, the last line is added for consistency in notation.
3. Numerical methods
3.1. Combined weak formulation with DEVSS/DG
Following the combined weak formulation of Glowinskiet al. [2] in which the hydrodynamic force and torque act-ing on the particle boundary are canceled exactly, we de-rived the weak form for the Newtonian system together withthe rigid-ring description of the particle and with the slidingbi-periodic domain constraints in our previous work[1]. Inthis work, we extend our previous formulation to incorporatewith the viscoelastic fluid. We employ the DEVSS method,a mixed finite-element formulation developed by Guénetteand Fortin[5], which appears to provide one of the most ro-bust formulations currently available. The DG formulationof Fortin and Fortin[6] is used for the discretization of theconstitutive equation. The combination of the DEVSS for-mulation with DG has been verified to produce a remarkablystable solution, in particular, for flows with a geometricalsingularity[7].
For the DEVSS formulation, we introduce an extra vari-ablee, the viscous polymer stress.
e = 2ηpD. (27)
Then one can rewrite the momentum equations for the fluiddomain (Eq. (2)) and for the particle domain (Eq. (14))with the viscous polymer stress, which gives extra stabilityin the discretized equations compared with the formulationwithout e.
As we did in the previous work, we introduce three dif-ferent Lagrangian multipliersλh, λv and λp,i, which areassociated with the kinematic constraint equation for theperiodicity in the horizontal direction (Eq. (8)), the con-straint equation for the sliding periodicity in the verticaldirection (Eq. (11)) and the rigid-ring constraint along theith particle (Eqs. (6) and (18)):
λh = (λhx, λ
hy) ∈ L2(Γ4)
2,
λv = (λvx, λ
vy) ∈ L2(Γ3)
2,
λp,i = (λp,ix , λ
p,iy ) ∈ L2(∂Pi(t))
2.
The choice ofΓ4 rather thanΓ2 (or, Γ3 rather thanΓ1) isarbitrary, but the sign of the multiplier will change accord-ing to the choice. In our previous work, we showed that themultipliers λh andλv are the traction force on the domainboundaryΓ and that the multiplierλp,i is the traction forceacting on the particle boundary∂Pi(t).
Introducing the separate functional spacesU, P, S andE for u, p, τp ande, respectively, over the whole domain,including the interior of the particle, the combined weakformulation for the whole domain can be constructed, alongwith the DG formulation for the constitutive equation. Thefinal weak form can be stated as follows:
For t > 0, find u ∈ U, p ∈ P, τp ∈ S, e ∈ E, λh ∈L2(Γ4)
2, λv ∈ L2(Γ3)2, λp,i ∈ L2(∂Pi(t))
2, Ui ∈ R2 andωi ∈ R (i = 1, . . . , N) such that
−∫Ω
p∇ · v dA+∫Ω
2η0D[u] : D[v] dA−∫Ω
e : D[v] dA
+ (λh, v(0, y)− v(L, y))Γ4 + (λv, v(x,H; t)
− v(x− γHt∗,0; t))Γ3 +N∑i=1
(λp,i, v − V i + χi
× (x − Xi))∂Pi =∫Ω
τp : D[v] dA, (28)
∫Ω
q∇ · u dA = 0, (29)
−∫Ω
es : D[u] dA+ 1
2ηp
∫Ω
es : e dA = 0, (30)
∫Ω
S : (λ∇τp + τp − 2ηpD[u])dA
− λ∑e
∫Γ ine
S : (τp − τextp )(u · n)ds = 0, (31)
(µh,u(0, y)− u(L, y))Γ4 = 0, (32)
20 W.R. Hwang et al. / J. Non-Newtonian Fluid Mech. 121 (2004) 15–33
for all v ∈ U, q ∈ P, S ∈ S, es ∈ E, µh ∈ L2(Γ4)2, µv ∈
L2(Γ3)2, µp,i ∈ L2(∂Pi(t))
2, V i ∈ R2 andξi ∈ R.In Eq. (31), n is the unit outward normal vector on the
boundary of elemente, Γ ine is the part of the boundary of
elemente whereu · n < 0, andτextp is the polymer stress in
the neighboring upwind element. The inner product(·, ·)Γjis the standard inner product inL2(Γj):
(µ, v)Γj =∫Γj
µ · v ds.
We have several remarks on the weak form inEqs. (28)–(35).
(1) The rigid-body motion on the particle boundary istreated as a constraint equation,the rigid-ring con-straint, which is satisfied in a weak sense with themultiplier λp (Eqs. (28) and (34)); the LEbc is satisfiedagain weakly through the sliding bi-periodic constraintswith the multipliersλh andλv (Eqs. (28), (32) and (33)).
(2) The inflow continuity condition of the polymer stress(Eqs. (10) and (13)) on the domain boundary has beentreated with the DG formulation (Eq. (31)), by takingthe external stressτext
p of the element, attached on thedomain boundaryΓ , from the stress at the coupled po-sition of the corresponding periodic boundary.
(3) The DG method, which uses discontinuous interpolationof the polymer stress, is particularly suited in this sim-ulation not only because of minimal coupling betweenelements, avoiding a large number of coupled equations,but also from the inherent discontinuous nature of thefictitious domain method, since the viscoelastic stress isdiscontinuous across the particle boundary.
(4) Since the sliding bi-periodic constraints (Eqs. (8) and(11)) impose only the relative relation in the velocity,one needs to specify the reference velocity at a singlepoint in the domain. To obtain the simple shear flow inthex direction with the upper sliding boundary velocityof (1/2)γH and that of the lower one of−(1/2)γH , weassign a zero value for the velocity in both directions atthe center ofΓ4.
(5) The pressure level of the fluid domain is determinedby specifying one of the normal component of the La-grangian multipliers onΓ (λh
x or λvy), since the multi-
plier can be identified as the traction force.(6) In the rigid-ring description, the pressure inside the par-
ticle is an undetermined constant. Our numerical methodwith the fictitious domain method chooses a value forthe pressure. However, the value of the pressure in-side the rigid ring does not affect other results such as
the rheological material functions that are presented inSection 4.
(7) When a particle crosses the domain boundary, a partof the particle needs to be relocated as described inEq. (25), which is possibly followed by the change inthe rigid-body motion as mentioned withEq. (26). Insuch a situation,Eqs. (28) and (34)need to be modifiedas follows:
wheref ′ = (−γH,0), when the originalx belongs tothe upper zone, and it becomes(γH,0) for x in the lowerzone (Fig. 2). (Refer to Hwang et al.[1] for details.)
3.2. Bulk stress
The bulk stress is the average stress over the domain andit can be expressed, for a volumeV , as the sum of the fluidcontribution and the particle contribution[16]:
〈σ〉 = 1
V
∫V
σ dV = 1
V
∫Vf
σ dV + 1
V
∫Vp
σ dV,
where〈·〉 denotes the averaged quantity inV , Vf andVp arethe volume occupied by the fluid and the particle, respec-tively. For an Oldroyd-B fluid in the absence of inertia, onegets
〈σ〉 = −p′I + 2ηs〈D〉 + τ′p + 1
V
∫∂Vp
xt dA. (37)
In this equationp′ and τ′p are the averaged contributions
of the pressure and the polymer stress from the fluid do-main: i.e.
∫VfpdV/V and
∫Vf
τp dV/V , respectively. Onecould evaluate the bulk stress directly fromEq. (37). To doso, one needs the identification of the particle domain, inthe regular mesh problem like this, to evaluatep′ and thisprocedure inevitably involves an additional approximation.In this regard, we prefer to use an alternative expression,which involves only boundary integrals of the traction forcealong the domain boundary and on the boundary of particlescrossing the domain boundary, as we did in our previouswork for the Newtonian system[1]. Since the Lagrangianmultipliersλh, λp andλp,i, which are obtained as a part ofthe solution, can be related directly to the traction forces,the boundary integrals can easily be evaluated. By using thetensor identity∂(xiσkj)/∂xk = σij , in the momentum bal-ance and applying the divergence theorem over the volumeV , the bulk stress can be expressed in terms of the tractionforce on the domain boundary[1]:
〈σ〉 = 1
V
∫∂V
xt dS
(= 1
A
∫Γ
xt ds
). (38)
The expression in the parenthesis is the 2-D form for thesliding bi-periodic frame of areaA.
W.R. Hwang et al. / J. Non-Newtonian Fluid Mech. 121 (2004) 15–33 21
The identity between the traction force and the Lagrangianmultipliers were discussed for the Newtonian system in de-tail [1]. Such a relationship is still valid in the viscoelasticflow, since the traction force includes the contribution fromthe polymer stress,t = n · (−pI + 2ηsD + τp). In this re-gard, we present the final expression of the bulk stress with-out proof in the present paper. (Details of the derivation canbe found in[1].)
When all particles are completely immersed in the domain(i.e., no boundary-crossing particle), each component of thebulk stress can be obtained by the following equations:
〈σ11〉 = 1
A
∫ L
0(x− γHt∗ − x)λv
x(x)dx
+ 1
H
∫ H
0λhx(y)dy, (39a)
〈σ22〉 = − 1
L
∫ L
0λvy(x)dx, (39b)
〈σ12〉 = 1
A
∫ L
0(x− γHt∗ − x)λv
y(x)dx
+ 1
H
∫ H
0λhy(y)dy = − 1
L
∫ L
0λvx(x)dx. (39c)
If there areNc particles crossing the domain boundary, weneed a correction term by replacing the particle contributionof the relocated coordinatex′ by that of the original coordi-natex. The bulk stress in this case can be written as follows:
〈σ′〉 = 〈σ〉 + 1
A
Nc∑k=1
∫∂Pk
(x − x′)λp,k(x′)ds, (40)
where〈σ〉 is the result of the boundary integral fromEq. (39).It is worthwhile to notice that the Lagrangian multiplier
for the rigidity constraint over the particle domain was iden-tified as a body force in the original DLM method of Glowin-ski et al. [2] and in its modified version by Patankar et al.[14].
3.3. Implementation
3.3.1. Spatial discretizationFor the discretization of the weak form, we use reg-
ular quadrilateral elements with continuous bi-quadraticinterpolation (Q2) for the velocityu, discontinuous linearinterpolation (P1) for the pressurep, continuous bi-linearinterpolation (Q1) for the viscous polymer stresse anddiscontinuous bi-linear interpolation (Qd
1) for the polymerstressτp. This combination of the discretizations foru, e
andτp has been verified to give the most stable result withthe DEVSS/DG formulation[7]. The use of the discontin-uous interpolation of the pressure appears to be mandatory,since an arbitrary location of a particle boundary inducesdiscontinuity in the pressure across the boundary[17].
As we did in Ref.[1], we use point collocation to dis-cretize the weak form of the rigid-ring constraint (Eqs. (28)and (34)or (36)). For example, the integral inEq. (36)hasbeen approximated as follows:
(µp,i(x′),u(x′)− Ui + ωi × (x − Xi))∂Pi
≈Mi∑k=1
µp,ik · u(x′
k)− (Ui + ωi × (xk − Xi)), (41)
whereMi, xk, x′k andµ
p,ik are the number of the colloca-
tion points on∂Pi, the original (before relocation) of thekthcollocation point, the relocated coordinate of thekth col-location point and the collocated Lagrangian multiplier atx′k, respectively. We define equally distributed collocation
points on the particle boundary based on the original coor-dinate and the number of the collocation pointsM is takento be proportional to the radius of the particle. In our previ-ous work, we analyzed effects of the number of collocationpoints and found that approximately one collocation pointin an element appears to give the most accurate solution[1].A small change inM from the optimal number does not af-fect solutions much, but it gives additional control to avoidparticle collision, although it hardly occurs in our simula-tion with a reasonable time step size. (For details, refer toHwang et al.[1].) In addition, as demonstrated inEq. (41),the use of point collocation circumvents tedious boundaryintegrals over the splitted particle boundary, when the par-ticle crosses the computational domain.
To discretize the boundary integrals for the horizontal pe-riodicity in Eqs. (28) and (32), we use the nodal collocationmethod, point collocation at all nodes, since the facing el-ements betweenΓ2 andΓ4 are always conforming in oursimulation with the regular discretization of the computa-tional domain. On the other hand, for the boundary integralsassociated with the vertical sliding periodicity (Eqs. (28)and (33)), care should be exercised in choosing a properdiscretization of the multiplierλv. The reason is that con-nection of the facing elements betweenΓ1 andΓ3 are ingeneral non-conforming and time-dependent. The situationis analogous to the mortar/finite element contact descrip-tion in frictional contact surface problems in solid mechan-ics, where the traction and kinematic compatibility shouldbe approximated across non-conforming interfaces[18]. Inthose problems, it has been known that the optimal conver-gence rate is only obtained when integral representation ofthe contact constraint based on mortar element methods areutilized [19]. We tested several different interpolations forthe multiplier space ofλv in our previous work and we ver-ified that the integral method with continuous linear inter-polation of the multiplier space gives the best solutions[1].We use the same technique in this work.
3.3.2. Time integrationInitially ( t = 0), the viscoelastic polymer stress is set to
be zero over the entire computational domain (Eq. (35)). We
22 W.R. Hwang et al. / J. Non-Newtonian Fluid Mech. 121 (2004) 15–33
solve the momentum equation with the constraint equations(Eqs. (28)–(30), (32)–(34)) implicitly to get the rigid-bodyvelocity of the particle and the velocity distribution at theinitial time step. (See Step 3 withEq. (44)) below.) Then, atevery time step, the following procedures are conducted.
Step 1. Get the particle configurationXn+1i (i = 1, . . . , N)
by integrating the kinematic equation inEq. (21) (andEq. (22)) using the explicit second-order Adams–Bashforthmethod (AB2):
Xn+1i = Xn
i ++t(32Un
i − 12Un−1
i ). (42)
Step 2. Solve the viscoelastic polymer stressτn+1p by in-
tegrating the evolution equation of the polymer stressτp(Eq. (31)) using the explicit second-order Adams–Bashforthmethod (DG/AB2):
[M]
(τn+1
p − τnp
+t
)= 3
2g(un, τnp)−
1
2g(un−1, τn−1
p ), (43)
where [M] is the mass matrix and the vectorg is the forc-ing term which appears in the evolution equation ofτp(Eq. (31)).
Step 3. Get the remaining solutions(u, p, e,Ui, ωi,λp,λh,
λv) for the (n + 1)th step by solvingEqs. (28)–(30)and(32)–(34)implicitly using Xn+1
i andτn+1p (i = 1, . . . , N):
−∫Ω
pn+1∇ · v dA+∫Ω
2η0D[un+1] : D[v] dA
−∫Ω
en+1 : D[v] dA+ (λh,n+1, v(0, y)− v(L, y))Γ4
+ (λv,n+1, v(x,H)− v(x− γHtn+1∗,0))Γ3
+N∑i=1
(λp,i,n+1, v − V i + χi × (x − Xn+1i ))∂Pi
=∫Ω
τn+1p : D[v] dA, (44a)
∫Ω
q∇ · un+1 dA = 0, (44b)
−∫Ω
es : D[un+1] dA+ 1
2ηp
∫Ω
es : en+1 dA = 0, (44c)
(µh,un+1(0, y)− un+1(L, y))Γ4 = 0, (44d)
(µv,un+1(x,H)− un+1(x− γHtn+1∗,0))Γ3 =(µv,f )Γ3,
(44e)
(µp,i,un+1 − Un+1i + ωn+1
i × (x − Xn+1i ))∂Pi = 0.
(44f)
We have some remarks for each step:
• In the first step, the method needs the present particle po-sition, the previous particle velocity and the present parti-cle velocity to predict the next particle position. However,
when the present particle center has come from outside ofthe upper or lower boundaries via relocation (Eq. (25)),one has to modify the previous velocity according tochanges in thex directional translational velocity of theparticle (Eq. (26)), since the next particle position shouldbe evaluated based on the present relocated particle po-sition and the present particle velocity. The modificationof the AB2 method applies only on thex direction and itcan be written, for theith particle, whose center comesacross the upper or lower boundary, as follows:
Xn+1i = X′n
i ++t(32U
ni − 1
2U′(n−1)i ), (45)
whereX′ni , Xn+1
i , Uni , andU ′(n−1)i are the present (relo-
cated if necessary) particle position, the next particle po-sition, the present velocity, and the modified previous ve-locity, respectively. The modified previous velocityU ′n−1
i
is determined byEq. (26)in Section 2.3.• In the Step 2,Eq. (43)can be solved at the element level,
leading to a minimal coupling between elements. In ad-dition, the method provides easy treatments for the exter-nal inflow stressτext
p across the upper and lower slidingboundaries inEq. (31), which involves the time-dependentnon-conforming connection between facing elements. Thestability and the second-order accuracy in the solutionof the evolution equation ofτp with the combination ofthe DG method in space and the AB2 method in time(DG/AB2) have been verified by Hulsen et al.[20].
• In the last step,Eq. (44) leads to the equation with asparse symmetric matrix with many zeroes on the diag-onal, which has been solved by a direct method basedon a sparse multifrontal variant of Gaussian elimination(HSL/MA41) [21–23].
4. Results
In this section, we present numerical results from the threesets of example problems: a single particle, two particlesand six particles in the sliding bi-periodic frame. Because ofthe bi-periodicity of the computational domain, a few parti-cle system in a single domain represents a large number ofparticles in an unbounded domain of the same configurationin simple shear flow. There are only two parameters in theOldroyd-B fluid: the Weissenberg numberWeand the ratioof the solvent viscosity to the polymer viscosityηs/ηp. TheWeissenberg number is defined as
We= γλ,
and we useηs = ηp in all example problems.Hereafter, for the purpose of the consistency, we will dis-
cuss the bulk suspension behavior by presenting the bulkshear stress〈σ12〉 and the bulk first normal stress〈N1〉, nor-malized byη0γ and 2ηpλγ
2, respectively. In fact, the nor-malized quantities can be identified by the relative shear vis-
W.R. Hwang et al. / J. Non-Newtonian Fluid Mech. 121 (2004) 15–33 23
Fig. 3. The periodic configuration of the single-particle problem.
cosity ηr and by the relative first normal stress differencecoefficientΨ1r, which can be defined as follows:
ηr = 〈σ12〉η0γ
and Ψ1r = 〈N1〉/γ2
2ηpλ= 〈N1〉
2ηpλγ2. (46)
4.1. Single particle in a sliding bi-periodic frame
The first test problem is a single particle of radiusr sus-pended freely at the center of the sliding bi-periodic do-main of size 1× 1 (i.e. L = H = 1). As mentionedearlier, the reference velocity has been specified by zeroat the center of the left domain boundary so that the up-per boundary translates at the velocity(1/2)γH and thelower one translates at−(1/2)γH . As a result, the parti-cle does not translate relatively to the domain but rotates atthe fluctuating angular velocityω(t). As shown inFig. 3,the problem represents a regular configuration of a largenumber of such a system and the initial configuration ofthe particle is reproduced periodically with the time periodT = L/γH .
4.1.1. Convergence testWe begin by checking the convergence of our code using
the single-particle problem withr = 0.2 and We = 0.5.We used three different meshes: 25-by-25, 50-by-50 and100-by-100, denoted byh = 1/25, 1/50 and 1/100, respec-tively. The number of the collocation points on the particleboundary are in turns 32, 64, and 128, which have beenverified optimal in our previous work forr = 0.2 [1]. Fig. 4shows the evolution of the angular velocityω (normalizedby −γ), the relative shear viscosityηr and the relative firstnormal stress differenceΨ1r.
The results show good convergence for all the quantitieswith mesh refinement. In addition, a smaller time step+tthan those used inFig. 4gives the same results, although notpresented here. Theh = 1/50 mesh appears to be a propercompromise between the accuracy and the computationalcosts (in memory and CPU time).
4.1.2. Time-dependent bulk suspension behaviorSince there is no relative change in the particle configu-
ration inside a frame, the single-particle problem appears agood example in investigating the sole effects of the fluidelasticity(We) and the solid fraction(φ) on the bulk suspen-
sion behavior. At the same time, we want to emphasize thatthe present results from the single-particle problem whichuses the artificial regular particle configuration are differ-ent from the suspensions with many particles, especially for
Fig. 4. The convergence result of the single-particle problem withr = 0.2andWe= 0.5. (a) The normalized angular velocity; (b) the relative shearviscosity; (c) the relative first normal stress difference coefficient.
24 W.R. Hwang et al. / J. Non-Newtonian Fluid Mech. 121 (2004) 15–33
Fig. 5. The effect of elasticity in the single-particle problem withr = 0.2.(a) The normalized angular velocity; (b) the relative shear viscosity; (c)the relative first normal stress difference coefficient.
highly concentrated systems, where the microstructural in-formation affects the bulk properties significantly.
We begin with discussions about the time-dependent be-havior of the suspension. Plotted inFig. 5a–care the angularvelocity of the particle, the relative shear viscosity and therelative first normal stress coefficient of the single-particleproblem with r = 0.2 for various Weissenberg numbers.All data show temporal fluctuations and converge to steadyoscillation for larget. The fluctuation originates from thetime-periodic change in the relative configuration of thesliding frame as shown inFig. 3. For example, the mag-nitude of the angular velocity becomes (local) maximumwhen the distance between particles (of neighboring upperor lower frames) is minimized, like the initial configura-
tion in Fig. 3, whereas it gives a local minimum value forthe configuration of the largest inter-particle distance (theintermediate configuration inFig. 3.) The relative shearviscosity appears the opposite behaviour to the normalizedangular velocity: i.e., the viscosity reaches the maximumvalue when the inter-particle distance (of the upper andlower frames) is the largest, and vice versa. The first normalstress difference reaches the maximum (minimum) valuein between the minimum and the maximum inter-particledistances, while the distance decreases (increases).
For small Weissenberg number cases, the angular veloc-ity and the bulk shear viscosity converge to the result ofthe same single-particle problem in the Newtonian fluid.(The Newtonian results were taken from our previous study
Fig. 6. The effect of particle size in the single-particle problem withWe = 1. (a) The normalized angular velocity; (b) the relative shearviscosity; (c) the relative first normal stress difference coefficient.
W.R. Hwang et al. / J. Non-Newtonian Fluid Mech. 121 (2004) 15–33 25
[1].) However, for non-small Weissenberg number cases,we found a remarkable decrease in the absolute value ofthe angular velocity, about 40% reduction in magnitude ofthe angular velocity withWe = 2.5. On the other hand,the shear viscosity and the first normal stress coefficientincrease with the Weissenberg number. That is, both prop-erties are shear-thickening. (This issue will be discussed inthe next section more details with the time-averaged steadyproperties.)
In Fig. 6a–cwe present those three data with a fixedWeissenberg number (We = 1) for various particle radii.Comparison with the pure fluid case has been made for thebulk shear viscosity and the first normal stress coefficient.For smallr, both properties converge to the pure Oldroyd-Bfluid results without the particle (r = 0). All data shows pe-riodic fluctuation, as mentioned before, and the fluctuationamplitude increases with the particle radius. Again, one canfind decrease in the absolute value of the angular velocityand increase of the bulk shear viscosity and of the first nor-mal stress coefficient with the increasing particle radius. Inthe Newtonian system, it is well known that the bulk shearviscosity increases with increasing size of the particle size.The same is true for the viscoelastic system and for the firstnormal stress coefficient. In fact there is a simple scalingbetween the first normal stress and the bulk shear stress insuspensions of the Oldroyd-B fluid, which will be discussedin the next section.
4.1.3. Steady bulk suspension propertiesOne can obtain the steady bulk suspension properties by
taking the time-average of the steady oscillation parts in
Fig. 7. The steady time-averaged bulk suspension properties as a function of the solid area fractionφ for different Weissenberg numbers (φmax = π/4).(a) The relative shear viscosity and (b) the relative first normal stress difference coefficient.
the time-dependent behavior of the single-particle results,presented in the previous section. Plotted inFig. 7a and bare the steady bulk shear viscosity and the first normal stresscoefficient evaluated in such a way for several combinationsof the Weissenberg numbers and the solid area fractionsφ.
As shown inFig. 7a, the bulk shear viscosity increaseswith the Weissenberg number as well as the solid area frac-tion. That is, the shear viscosity of the bulk suspensionshows shear-thickening behavior, when formulated with anOldroyd-B fluid. For a small Weissenberg number (We =0.05), the shear viscosity converges to that of the Newto-nian system. (The Newtonian results has been adopted fromHwang et al.[1].) In addition, for small value ofφ, the shearviscosity of the viscoelastic system converges to Einstein’sanalytic result for a dilute Newtonian system, i.e.ηr = 1+2φ[24].
The steady time-averaged first normal stress coefficient(Fig. 7b) shows the analogous behavior: it increases withthe Weissenberg number and with the solid fractionφ.In fact, the relative first normal stress coefficient appearsshear-thickening and converges to the pure Oldroyd-B fluidresult of the value ‘1’ for smallφ. In contrast to the vis-coelastic results presented here, the time-averaged value ofthe first normal stress was found zero in the single-particleproblem in the Newtonian system in the sliding bi-periodicframe[1].
Patankar and Hu[26] state that for a 2-D dilute suspensionin a second-order fluid (SOF), the relative increase of theviscosity and the first normal stress coefficient due to thepresence of the particles is the same and identical to therelative increase of the effective viscosity in the Newtonian
26 W.R. Hwang et al. / J. Non-Newtonian Fluid Mech. 121 (2004) 15–33
Fig. 8. The ratio of the relative first normal stress difference coefficientto the relative shear viscosity as a function of the solid area fraction fordifferent Weissenberg numbers.
system. That is,ηr = 1+ 2φ andΨ1r = 1+ 2φ. In order tocheck the SOF limit, we plotted the ratio of the two,Ψ1r/ηr,in Fig. 8. (See also the SOF limit compared inFig. 7a andb.) Nearφ = 0, the deviation from 1 seems to be of secondorder in φ. For the dilute Oldroyd-B fluid, the SOF limitseems to be fulfilled also up to a moderate Weissenbergnumber. However, for the non-dilute case,Ψ1r/ηr > 1, i.e.the normal stress increases faster than the shear stress withthe solid fraction, (but not as fast as the square of the shearstress,Ψ1r/η
2r < 1, which is discussed below.)
Regarding bulk suspension behavior, one common exper-imental observation is that the first normal stress differencein a filled viscoelastic fluid is a power-law function of the
Fig. 9. The scaling between the steady time-averaged first normal stress difference and the shear stress in suspensions formulated with an Oldroyd-Bfluid for various solid area fractionφ (ηp = ηs = 1).
imposed shear stress such thatN1 ≈ τn with an exponentn that appears to depend on the specific matrix fluid usedin preparing the suspensions[8–10]. With this argument,the n value in the Oldroyd-B fluid becomes 2, since it isindependent of the volume fraction. In order to check theappearance of such a relationship in our simulation results,we plotted the steady time-averaged first normal stress dif-ference as a function of the steady bulk shear stress usingthe log–log scale inFig. 9. Interestingly, our single-particlesimulations in a 2-D sliding frame reveal a set of parallelstraight lines which gradually shift to the shear stress axiswith increasing solid area fractions, for the wide range ofφ, from the low particle concentration (3.14%) to the ex-tremely high concentration (50.3%). The implication in thisplot is that the first normal stress difference relatively de-creases with increasing solid area fraction when comparedat a constant values of the shear stress. In fact, the firstnormal stress difference increases with the solid area frac-tion as shown inFig. 7b, even faster than the shear stress(see Fig. 8), but not as fast as the square of the shearstress does.
The data inFig. 9suggest that the normal stress differencecan be conveniently expressed by using a shift factorβ(φ),which only depends on the solid fractionφ, analogous tothe approach of Mall-Gleissle et al.[10]:
〈N1〉(φ, 〈σ12〉) = 2ηpλ
η20
β(φ)〈σ12〉2. (47)
With the definition ofEq. (47), β(φ = 0) = 1. Substitutingthe definition ofηr andΨ1r (Eq. (46)) into Eq. (47), we get
β(φ) = Ψ1r
η2r. (48)
W.R. Hwang et al. / J. Non-Newtonian Fluid Mech. 121 (2004) 15–33 27
Fig. 10. The ratioΨ1r/η2r as a function of the solid area fractionφ,
using the steady time-averaged values of the relative first normal stressdifferenceΨ1r and the relative viscosityηr . The SOF limit is given by1/(1 + 2φ).
In order to show that the right-hand-side ofEq. (48) isindeed independent from the Weissenberg numberWe,we have plotted inFig. 10 the values ofΨ1r/η
2r , using
the steady time-averaged values ofΨ1r and ηr, as a func-tion of φ. The values decrease monotonically within therange of φ in our simulations. The dependence on theWeissenberg number is very minor and the approxima-tion by a single functionβ(φ) is very good for the valuesof We considered here. InFig. 10 we have also plottedthe SOF limitβ(φ) = 1/(1 + 2φ), and this simple rela-tion gives a surprisingly good fit for the shift factor up toφ = 0.5.
It would be worthwhile to mention the previous resultsfrom 2-D direct numerical simulations on bulk suspensionproperties by Patankar and Hu[25] for a Newtonian fluid and[26] for a viscoelastic fluid. They solved the single-particleproblem in a one-side periodic channel under simple shearflow with non-zero Reynolds number (less than 0.035) us-ing a relatively small fluid elasticity (We less than 0.2),for a very dilute system (φ less than 0.001). They claimedthat there is no effect of the fluid elasticity on the bulkviscosity and the first normal stress, as expected from ourresults inFig. 7a and bwithin such a low particle frac-tion. However, the effect of fluid elasticity on these prop-erties have appeared significant in our present results, asthe particle fraction increases and as the fluid elasticity in-creases.
Before closing the section, it would be interesting to seethe steady time-averaged angular velocity of the particle.As indicated inFig. 11, it decreases substantially with theWeissenberg number and with the solid fraction, and theeffect of the Weissenberg number is more pronounced thanthe solid fraction.
Fig. 11. The steady time-averaged angular velocity of the particle as afunction of the solid area fractionφ for different Weissenberg numbers.
4.2. Two particles in a sliding bi-periodic frame
The two-particle problem is constructed to investigate theeffect of the hydrodynamic interaction between two (almost)isolated particles. It is stated as follows: Two identical par-ticles of radiusr = 0.12 are suspended freely in the slidingbi-periodic domain of size 1×1. The initial positions of theparticles are chosen symmetrically as shown inFig. 12. Thedistance from the particle center to the horizontal centerlineof the domain is denoted byD. Again the reference velocityhas been specified by zero at the center of the left domainboundary (Γ4) so that the upper left particle is supposed tomove to the right direction and the lower right one to theleft direction. In fact, the identical problem has been solvedin our previous work for suspensions in a Newtonian fluid,where we showed that the bulk viscosity is a function ofD
and increases with theaveragedinter-particle distance: i.e.,minimum forD = 0 and maximum forD = 0.25. (Refer toHwang et al.[1] for details.)
However, in the viscoelastic fluid, we could not performsuch a systematic approach to investigate the effect of hy-drodynamic interaction, because there is no steady configu-
Fig. 12. The initial particle configuration of the two-particle problem.
28 W.R. Hwang et al. / J. Non-Newtonian Fluid Mech. 121 (2004) 15–33
Fig. 13. The comparison of the orbits of the left particle in the two-particleproblem in Newtonian and Oldroyd-B fluids (D = 0.025 andr = 0.12).
ration of the particles, that can be identified by a single pa-rameterD: the configuration of the two particle continuouslychanges. In fact, the motion of the two particles show strongtendency toward clustering. Plotted inFig. 13are the orbitsof the upper left particle in two Oldroyd-B fluids,We= 0.5and 1, in comparison with the orbit in a Newtonian fluid forD = 0.025. The motion of the other particle is completely
Fig. 14. The evolution of the particle configuration in the two-particle problem withD = 0.025, r = 0.12 and We = 0.5, which shows thekissing-tumbling-tumbling phenomena. The quantity plotted in gray scale is the trace of the polymer conformation tensorA and the particles are describedby their collocation points. (a)γt = 5.75 (t/λ = 11.5); (b) γ t = 8.125; (c) 9.375; (d) 10.25; (e) 14.5; (f) 15.5; (g) 16.875; (h) 19.375; (i) 24.875. Thesubfigure (g) is near the instant when the two particles are apart the largest (denoted by (×) in Fig. 13).
Fig. 15. The comparison of the particle orbits in a Newtonian and anOldroyd-B (We = 0.5) fluid for D = 0.05. The initial position of theparticle is marked with a×.
symmetric with respect to the point (0.5, 0.5), because of thesymmetry of the problem. It is well-known that the closelylocated two particles show kissing-tumbling-separation phe-nomena in simple shear flow in the Newtonian fluid, as isalso indicated inFig. 13. However, the figure reveals that thetwo particles in an Oldroyd-B fluid keep rotating around eachother, i.e. we find kissing-tumbling-tumbling-(· · · ) phenom-ena. In other words, the particles show the tendency for clus-tering, when they are closely located initially, unlikely to theNewtonian case. As shown inFig. 13, this tendency appearsto get more apparent with the larger Weissenberg number.
W.R. Hwang et al. / J. Non-Newtonian Fluid Mech. 121 (2004) 15–33 29
Fig. 16. The location(X, Y) and the rigid-body velocity(U, V, ω) ofthe left particle in the two-particle problem withD = 0.025, r = 0.12and We= 0.5. The translational and angular velocity components werenormalized by byγH and γ, respectively.
In order to have a closer look, we show several snapshotsin our result withWe= 0.5, while the kissing and the firsttumbling occurs, presenting evolutions of the trace of thepolymer conformation tensorA and of the particle configu-ration. The polymer conformation tensorA is defined as
A = I + λ
ηpτp.
The particles are described by their collocation points.1
Firstly, the two particles kiss (Fig. 14a and b) and tumble(Fig. 14c and d) and they start to separate (Fig. 14e andf) each other. However, when approaching the maximumseparation distance (Fig. 14g), they begin to recoil back(Fig. 14h) and tumble again (Fig. 14i). The whole process re-peats continually, reducing the maximum distance betweenthe two particle, as was presented inFig. 13.
Fig. 15shows another example for this phenomena for alargerD = 0.05 with We = 0.5.2 The two particles kiss,tumble and separate, when they are first encountered. How-ever, in the next turn, they keep rotating around each otheras forD = 0.025.
In essence, the two particles appear to behave as if theywere connected with a elastic spring. However, at present,we do not have any postulation yet to explain this phenomenaof clustering. It needs further investigation.
The evolution of the position and the velocity of the upperleft particle are plotted in time inFig. 16 for the case withD = 0.025 andWe= 0.5. The period involved in the rota-tion of the two clustered particles reduces as time goes on.When the two particles located vertically (as inFig. 14c), the
1 The movie files forWe = 0.5 and 1 of the two-particle problemare accompanied with the paper under the names Movie 1 and Movie 2,respectively.
2 The corresponding movie file is named Movie 3.
magnitude of the velocity componentU becomes the maxi-mum and the particle rotates at the highest velocity. On theother hand, if they are aligned in the horizontal direction (asin Fig. 14g), the translational velocity component becomesalmost zero (then changes its sign) and the particle rotatesat the slowest velocity.
The relative shear viscosity and the relative first normalstress difference coefficient in case ofD = 0.025 have beencompared for a Newtonian and Oldroyd-B fluids inFig. 17.The bulk shear stress appears to increase slightly with in-creasing elasticity, but there is hardly any apparent increasein the normal stress coefficient. The reason might be thesmall solid fraction, about 9%, of the problem, which is ex-pected considering the single-particle result inFig. 7.
Finally, we present the mesh refinement results for theprevious two symmetrically located-particle problem (D =0.025,r = 0.12 andWe= 0.5), which has been investigatedextensively in our paper. In this case, two closely-locatedparticles interact with each other and the accuracy of thesolution within the interparticular region is much more in-volved. We tested three meshes: 25-by-25, 50-by-50 and100-by-100, denoted byh = 1/25, 1/50 and 1/100, re-spectively, and the comparison has been made for a largenumber of time steps (γ t ≥ 25), during which the firstkissing-tumbling-tumbling phenomena of the two particlesoccurs. The number of collocation points for each particle(and the time step used) are in turns 24(+t = 0.01), 48(0.005) and 96(0.0025) for theh = 1/25, 1/50 and 1/100mesh, respectively.Fig. 18shows the evolution of the rela-tive viscosityηr and the relative first normal stress diffenceΨ1r for the three mesh problems. Plotted inFig. 19 is theorbit of the left particle for each case. Obviously, the overallbehavior does not change significantly with the mesh size.The results indicate good convergence with the mesh refine-ment, before the instance when the two particles approachesthe maximum interparticle distance (nearγ t = 15). (SeeFig. 14for comparison.) Then there appears a small discrep-ancy in the relative viscosity and the first normal stress co-efficient. The reason is that the particles computed with thefiner mesh follow a slightly larger orbit and thereby the timefor the particle recoil increases with the mesh refinement.Such a small error is then accumulated for a large numberof time steps and causes the phase shift in the bulk stressresult later.
4.3. Many particles in a sliding bi-periodic frame
Now we proceed to more complex problems: six randomlydistributed particles in the sliding bi-periodic frame. Themain objective here is to see how complicated particle–fluidand particle–particle interactions affect the microstructuralbehavior in concentrated suspensions with a viscoelasticfluid. We consider two different initial configurations of sixparticles: particles with different radii and with equal ra-dius. The solid area fractions of the two cases are the same(φ = 0.296). Again the size of the sliding frame is 1× 1
30 W.R. Hwang et al. / J. Non-Newtonian Fluid Mech. 121 (2004) 15–33
Fig. 17. The relative shear viscosity (a) and the relative first normal stress difference coefficient (b) of the two-particle problems in Newtonian andOldroyd-B fluids (D = 0.025 andr = 0.12).
and the reference velocity is specified by zero at the centerof the left domain boundary (Γ4). The two initial configura-tions are plotted inFig. 20, where the particles are describedby their collocation points. The 50-by-50 mesh is used for
Fig. 18. The mesh refinement result of the two-interacting-particle problem(D = 0.025, r = 0.12 andWe= 0.5). (a) The relative shear viscosity;(b) the relative first normal stress difference coefficient.
Fig. 19. The mesh refinement result of the two-interacting-particle problem(D = 0.025, r = 0.12 andWe= 0.5). The orbit of the left particle.
Fig. 20. Two initial particle configurations of different size particles (left)and of the same size particles (right). The two examples have the samesolid area fractionφ = 0.296. Particles are described by their collocationpoints.
the computation and two Weissenberg number flows withWe= 0.5 and 1 are tested.
Fig. 21shows the instantaneous distributions of the traceof tensorA of the different-sized particle case and of theequal-size case forWe = 0.5.3 Firstly, one can observestrong elongational flows generated between separating par-
3 A movie file has been prepared for the different sized particle casewith We= 0.5 and named Movie 4.
W.R. Hwang et al. / J. Non-Newtonian Fluid Mech. 121 (2004) 15–33 31
Fig. 21. The distribution of the trace of the conformation tensor in thesix-particle problem withWe= 0.5. (a) The different-sized particle caseat γ t = 6 (t/λ = 12) and (b) the equal-sized particle case atγ t = 6.3.
ticles. Secondly, another high stretched region occurs whentwo particles approach (or kiss) each other closely. Thirdly,there is also weak elongational flow region between par-ticles aligned parallel to the flow direction. The value oftr(A) in such a region is sometimes lower than the value 2.5,which is the value of the pure Oldroyd-B fluid under thesame condition. Therefore, the resultant microstructure be-comes highly non-uniform and anisotropic. Especially, thereseems to be a typical orientation for the highly stretched re-gions which originate from separating particles. The angleis about 20 [deg.] measured from the flow direction. In ad-dition, such a non-uniform and anisotropic microstructurebecomes more pronounced in the higher Weissenberg num-ber flow, as shown inFig. 22.4
4 The corresponding movie file has been prepared and named Movie 5.
Fig. 22. The distribution of the trace of the conformation tensorin the six-particle problem with different radii withWe = 1 atγ t(= t/λ) = 6.384.
The presence of the oriented highly elongational flows isparticularly interesting, since it induces polymer moleculesto align in such a direction, which could affect themicro-rheological behavior of the suspension. For ex-ample, the stretched molecules enhance the nucleationand crystallization process in case of semi-crystallinepolymers during processing[27,28], which of courseaffects the mechanical properties of the final product.(Refer to Schrauwen et al.[11] for experimental obser-vations about flow effects on the impact toughness inthe injection-molded products of a hard particle-filledsemi-crystalline polymer.)
Finally the relative shear viscosity and the relative firstnormal stress difference coefficient in the six-particle prob-lems were presented inFig. 23 along with the comparisonwith the Newtonian system and with the correspondingsingle-particle problem having the same solid fraction. (TheNewtonian results have been adopted from our previouswork [1].) The bulk shear viscosity of the six-particle prob-lem is always larger than that of the corresponding singleparticle case for both Newtonian and viscoelastic systems,which is the effect of increased hydrodynamic interactions.In addition, there seems to be an increase in the relative vis-cosity with the Weissenberg number, which agrees with thesingle-particle result inFig. 7a. However, due to too muchfluctuations in our data, we postpone the conclusion on theother parameter dependencies: e.g., dependence of the firstnormal stress coefficient on the strength of hydrodynamicinteraction and on the Weissenberg number. Computationswith a large number of particles for a long time seemto be necessary to investigate definite relations on suchmatters.
32 W.R. Hwang et al. / J. Non-Newtonian Fluid Mech. 121 (2004) 15–33
Fig. 23. The comparison of the relative shear viscosity (a) and the relativefirst normal stress difference coefficient (b) for different particle configu-ration and the Weissenberg number along with the single-particle results.
5. Conclusions
In this work, a new finite-element formulation for di-rect numerical simulations of particle suspensions in anOldroyd-B fluid has been developed and implemented, byextending the authors’ previous scheme for suspensionsin the Newtonian fluid. The main features of our presentscheme can be summarized as follows:
• The sliding bi-periodic domain concept of Lees and Ed-wards for discrete particles has been extended to contin-uous field problems and combined with the DEVSS/DGfinite-element method for accurate and stable computationof viscoelastic flows.
• The freely suspended particles are described by therigid-ring problem, which eliminates the need for particledomain discretization and allows easy treatment of theboundary-crossing particles.
• A simple expression of the bulk stress has been estab-lished, which involves only boundary integrals of the La-
grangian multipliers along the domain boundary and alongthe boundary of particles crossing the domain boundary.
Concentrating on 2-D circular disk particles, we discussedthe bulk rheology of the suspensions and the micro-structuraldevelopments through the numerical examples of single-,two- and many-particle problems, which represents a largenumber of such systems in the unbounded domain. Beloware the summary of our observations from the example prob-lems:
• From the single-particle problems, we reported the bulkshear viscosity and the first normal stress difference co-efficient, from very dilute to highly concentrated suspen-sions. Both of them show shear-thickening behavior. Thecommon experimental scaling of the first normal stress tothe bulk shear stress has been reproduced.
• Clustering of the two particles has been reported, whichis quite distinct from the Newtonian system, and the ten-dency of such a phenomena has been found to increasewith the elasticity in the fluid.
• From the many-particle problems, we observed the pres-ence of strong elongational flows between separating par-ticles, which leads to highly oriented and non-uniformmicro-structures in suspensions of the viscoelastic flows.
Acknowledgements
This work was supported by the Dutch Polymer Institute.The authors thank Prof. Gareth H. McKinley from M.I.T.for fruitful suggestions on the comparison to experimentalresults.
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