time-dependent viscoelastic properties of oldroyd-b fluid ... · promising numerical technique in...

© 2017 The Korean Society of Rheology and Springer 137

Korea-Australia Rheology Journal, 29(2), 137-146 (May 2017)DOI: 10.1007/s13367-017-0015-1

www.springer.com/13367

pISSN 1226-119X eISSN 2093-7660

Time-dependent viscoelastic properties of Oldroyd-B fluid studied by

advection-diffusion lattice Boltzmann method

Young Ki Lee, Kyung Hyun Ahn* and Seung Jong Lee

Institute of Chemical Processes, School of Chemical and Biological Engineering, Seoul National University, Seoul 08826, Republic of Korea

(Received February 18, 2017; final revision received April 2, 2017; accepted April 9, 2017)

Time-dependent viscoelastic properties of Oldroyd-B fluid were investigated by lattice Boltzmann method(LBM) coupled with advection-diffusion model. To investigate the viscoelastic properties of Oldroyd-Bfluid, realistic rheometries including step shear and oscillatory shear tests were implemented in wide rangesof Weissenberg number (Wi) and Deborah number (De). First, transient behavior of Oldroyd-B fluid wasstudied in both start up shear and cessation of shear. Stress relaxation was correctly captured, and calculatedshear and normal stresses agreed well with analytical solutions. Second, the oscillatory shear test was imple-mented. Dynamic moduli were obtained for various De regime, and they showed a good agreement withanalytical solutions. Complex viscosity derived from dynamic moduli showed two plateau regions at bothlow and high De limits, and it was confirmed that the polymer contribution becomes weakened as Deincreases. Finally, the viscoelastic properties related to the first normal stress difference were carefullyinvestigated, and their validity was confirmed by comparison with the analytical solutions. From this study,we conclude that the LBM with advection-diffusion model can accurately predict time-dependent visco-elastic properties of Oldroyd-B fluid.

Keywords: rheology, viscoelastic fluid, Oldroyd-B, lattice Boltzmann method, advection-diffusion model

1. Introduction

Since its introduction two decades ago, the lattice Boltz-

mann method (LBM) has been extensively used as a

promising numerical technique in computational fluid

dynamics (CFD) (Aidun and Clausen, 2010; Succi, 2001).

Unlike the conventional CFD methods based on contin-

uum mechanics, the LBM models the mesoscopic dynam-

ics of fluids. Due to its kinetic nature, it has more

adaptability in modeling complex systems, where the clas-

sical CFD could be rarely applied; for example, the flow

in porous media (Ginzburg et al., 2015; Molaeimanesh

and Akbari, 2015), modeling of complex fluids such as

particulate suspensions (Gross et al., 2014; Kromkamp et

al., 2006; Kulkarni and Morris, 2008; Ladd, 1994; Lee et

al., 2015), liquid crystals (Denniston et al., 2001; Maren-

duzzo et al., 2007), and amphiphilic fluids (Love et al.,

2003; Saksena and Coveney, 2009).

Recently, polymeric liquids have also been extensively

studied in LBM framework. There have been several

efforts to correctly capture the effect of fluid elasticity by

modifying the equilibrium distribution (Qian and Deng,

1997) or by adding a Maxwell-like force to the system

(Ispolatov and Grant, 2002). The Jeffreys model was also

applied in LBM framework (Giraud et al., 1998; Lalle-

mand et al., 2003). However, all of them considered par-

tial elastic effect only and did not reproduce the viscoelastic

constitutive equation such as the Oldroyd-B or FENE-P

models which are widely used in polymer rheology (Bird

et al., 1987).

More recently, two novel LBM methods which accu-

rately describe the constitutive equations for polymeric

liquids were newly introduced. Onishi et al. (2005) pro-

posed an LBM model, in which the viscoelastic stress was

evaluated as a net effect of the motion of polymer chains,

and their dynamics was modeled at the mesoscopic level

based on Fokker-Planck equation. Even though their study

was limited to simple shear flow, the viscoelastic behavior

of Oldroyd-B fluid was correctly captured. A different

type of model, which was developed on the basis of mod-

ified advection-diffusion LBM was also introduced by

Malaspinas et al. (2010). The main idea of this scheme is

to compute each component of the conformation tensor by

its own distribution function set. By this approach, any

constitutive equation of the same form (including Olr-

doyd-B and FENE-P models) can easily be considered in

LBM framework.

In spite of a few studies on polymeric liquids, the time-

dependent viscoelastic properties have rarely been explored

by LBM. A simulation study of the viscoelastic properties

of polymeric liquid was carried out by Onishi et al.

(2005), but it was only verified for step shear test. For

oscillatory shear test, they reported only a phase shift

observed in shear stress signal without any quantitative

analysis. With Malaspinas et al. (2010)’s LBM model,

much less study was done for the time-dependent visco-*Corresponding author; E-mail: [email protected]

Young Ki Lee, Kyung Hyun Ahn and Seung Jong Lee

138 Korea-Australia Rheology J., 29(2), 2017

elastic properties of polymeric liquids. Even though it was

validated by several benchmark problems such as Taylor-

Green vortex decay, the four roll mill, and the Poiseuille

flow, the tests were limited only to the system in which

non-transient extra force is imposed; for example, con-

stant shear force (four rolls mill) and constant gravita-

tional acceleration (Poiseuille flow). In other words, the

reliability of this algorithm has never been checked with

extra forces (or flows) which are time dependent.

The goal of the present study is to assess advection-dif-

fusion LBM as a tool to investigate time-dependent vis-

coelastic properties of Oldroyd-B fluid, which is known to

obey the characteristics of Boger fluid (Boger, 1977; James,

2009). The advection-diffusion LBM has never been

applied to the study of viscoelastic properties of polymer

solutions under realistic rheometry frameworks such as

step shear and oscillatory shear flows. In addition, this

algorithm was not verified yet for the systems with time-

dependent extra forces as mention above. In this sense,

oscillatory shear flow can be a good benchmark problem

to show more potential of this algorithm. As far as we are

concerned, the rigorous study by LBM has never been

performed for time-dependent viscoelastic properties

(dynamic moduli, complex viscosity, and normal stress

differences) of polymeric liquids. Therefore, this study has

a significance as the first report, which strictly analyzes

the rheological properties of polymeric liquids in LBM

framework.

This paper is organized as follows. Details of back-

ground theories and numerical methods are introduced in

Section 2, and the simulation results are provided in Sec-

tion 3. In Section 3.1, the rheological properties of Old-

royd-B fluid are introduced in the step shear flow. Time

evolution of polymer stress and its shear rate dependency

are discussed. In Section 3.2, we focus on the dynamic

viscoelastic properties under the oscillatory shear flow.

Dynamic moduli and viscoelastic properties related to the

first normal stress difference are carefully investigated,

and their utility is explained from the theoretical basis.

Finally, conclusions are drawn in Section 4.

2. Theory and Numerical Methods

2.1. Oldroyd-B modelIn the present work, we are interested in the rheology of

incompressible viscoelastic fluids, which is often charac-

terized by the Oldroyd-B model (Bird et al., 1987). This

constitutive equation considers the contribution of poly-

mer chains in Newtonian solvent, and has been widely

applied to study the viscoelastic flow behavior of polymer

solutions. In the Oldroyd-B model, the polymer molecules

are modeled by two beads connected by a linear spring,

and the interaction between polymer molecules and sol-

vent is considered by the drag force exerted on the beads.

The deformation of polymer molecules is determined by

two competing processes, namely the stretching by the

velocity gradient and the relaxation by the elasticity of the

polymer chains. The flow is governed by the continuity

and momentum balance equations.

, (1)

. (2)

The symbols ρ, u, p, and ηs are the density, the velocity,

the pressure, and the solvent viscosity. The rate of defor-

mation tensor is given by , where the

superscript “T” denotes the transpose. Here, σp is the

polymer stress obtained from the constitutive equation.

The Oldroyd-B model is written in terms of the confor-

mation tensor C, which is a statistical indicator of the ori-

entation of polymer chains, and the relation between the

polymer stress and the conformation tensor can be

described by the following equations.

, (3)

. (4)

Here, ηp is the dynamic viscosity of the polymer and λ1

is its relaxation time.

2.2. Lattice Boltzmann methodIn the present study, the lattice Boltzmann method (LBM)

(Aidun and Clausen, 2010; Succi, 2001) was adopted as a

solver for viscoelastic fluids. The Navier-Stokes equations

were considered by a classical LBM scheme (He and Luo,

1997), and it was coupled with viscoelastic stress tensor

obtained from the Oldroyd-B model. To solve this con-

stitutive equation, a modified advection-diffusion LBM

scheme (Malaspinas et al., 2010) was adopted.

In LBM, macroscopic dynamics is described by the lat-

tice Boltzmann equation (LBE) which is an approximated

and discretized form of the Boltzmann equation. The

LBM introduces virtual particles which are the packets of

mesoscopic particles and the evolution of these particles

(so called probability distribution function) is described by

streaming and collision steps. In the streaming step, the

probability distribution function is propagated with the lat-

tice velocity vector ci to the next neighbor lattice node.

This process is described by the left-hand side of Eq. (5),

where x is the position of the lattice node at time t, ci is

the discrete lattice velocity, and fi is the distribution func-

tion for i direction.

∇ u⋅ = 0

∂tu + u ∇⋅( )u = 1

ρ---∇ pI– 2ηsD σp+ +( )⋅

D = ∇u ∇u( )T+( )/2

σp = ηp

λ1

----- C I–( )

dC

dt------- =

1

λ1

-----– C I–( ) + C ∇u⋅ ∇u( )T+ C⋅

Time-dependent viscoelastic properties of Oldroyd-B fluid studied by advection-diffusion lattice Boltzmann method

Korea-Australia Rheology J., 29(2), 2017 139

. (5)

After the streaming step, the probability distribution func-

tion is determined at each lattice node by the collision pro-

cess, which is described by the right-hand side of Eq. (5).

In this study, Bhatnagar-Gross-Krook (BGK) collision

operator (Qian et al., 1992) was used, in which the

momentum conservation is constrained by the equilibrium

distribution, , which is given by Eq. (6). The equilib-

rium distribution can be derived by the truncated form of

the Maxwell distribution, which is known as a good

approximation for small Mach numbers (He and Luo,

1997).

. (6)

In this study, the D2Q9 lattice model is used which is

designed to consider nine direction velocities in two-

dimensional (2D) space (Succi, 2001). The speed of sound

can be described by , where and

are the lattice spacing and time step, respectively. t is

the dimensionless relaxation time of the solvent, and it

is directly connected to the kinematic viscosity by

(Qian et al., 1992). Direction dependent

weight coefficients wi and the lattice velocity ci are given

in Table 1.

To impose the external force F in the system, Guo's

forcing scheme (Guo et al., 2002) was adopted, and it can

be incorporated to Eq. (7).

. (7)

The macroscopic properties of the solvent such as the

density ρ and the velocity u are derived by the zeroth and

first velocity moments of the distribution function fi, and

they are given by Eqs. (8) and (9).

, (8)

. (9)

In addition, the solvent stress can be calculated by Eq.

(10), where is pressure (Krüger et al., 2009).

. (10)

To solve the Oldroyd-B constitutive equation, the advec-

tion-diffusion LBM (D2Q5) is adopted (Malaspinas et al.,

2010). The main idea of this scheme is to compute each

component of the conformation tensor Cαβ by its own dis-

tribution function set hαβ. The governing equation can be

derived as follows.

(11)

where ϕ is the relaxation parameter of the advection-dif-

fusion scheme and Gαβ is the source term (for αβ com-

ponents), which depends on the constitutive equation.

Here, the velocity u is obtained by classic LBM part,

given in Eq. (9). It has been reported that the accuracy of

the advection-diffusion LBM strongly depends on ϕ. In

the present work, this value was set to ϕ = 0.51, and reli-

able results could be obtained for all simulation condi-

tions. Detailed validation is provided in Appendix.

The equilibrium distribution function, is given by

Eq. (12).

(12)

where cl is a scaling factor ( ), and wi,2 are

the weight coefficients. The weight coefficients, wi,2 and

the lattice velocity ξi are presented in Table 2.

The conformation tensor Cαβ is computed by Eq. (13).

. (13)

The source term Gαβ is determined according to the con-

stitutive equation, and for Oldroyd-B model, it is given by

Eq. (14). Here, we note that no summation convention is

applied in Eqs. (11)-(14).

fi x ciΔt, t Δt+ +( ) − fi x, t( ) = 1

τ---– fi x, t( ) − f i

eqx, t( )[ ]

+ 11

2τ-------–⎝ ⎠

⎛ ⎞FiΔt

f ieq

f ieq

x( ) = wiρ 1ci u⋅

cs

2----------

ci u⋅( )2

2cs

4----------------

u u⋅

2cs

2---------–+ +

cs = 1/3Δx/Δt Δx Δt

ν = cs

2τ 1/2–( )Δt

Fi = wi

ci u–

cs

2-----------

ci u⋅( )

cs

4--------------+ ci F⋅

ρ = i∑ fi

ρu = i∑ fici +

F

2---Δt

p = cs

2ρ

σs,αβ = p– I − 1τ

2---–⎝ ⎠

⎛ ⎞ i∑ fi f i

eq–( )cαicβi

hiαβ x ξiΔt, t Δt+ +( ) −hiαβ x,t( ) = 1

ϕ---– hiαβ x,t( ) hiαβ

eq– Cαβ, u( )[ ]

+ 11

2ϕ-------–⎝ ⎠

⎛ ⎞Gαβ

Cαβ

---------hiαβ

eqCαβ, u( )

hiαβ

eq

hiαβ

eqCαβ, u( ) = wi 2, Cαβ 1

ξi u⋅

cl

2----------+⎝ ⎠

⎛ ⎞

cl = 1/ 3Δx/Δt

Cαβ = i∑ hiαβ +

Gαβ

2---------

Table 1. Weight coefficients, wi and the lattice velocity ci in D2Q9 lattice model.

i = 0 i = 1 i = 2 i = 3 i = 4 i = 5 i = 6 i = 7 i = 8

wi 4/9 1/9 1/9 1/9 1/9 1/36 1/36 1/36 1/36

ci (0, 0) (1, 0) (0, 1) (−1, 0) (0, −1) (1, 1) (−1, 1) (−1, −1) (1, −1)

Table 2. Weight coefficients, wi,2 and the lattice velocity xi in the

D2Q5 lattice model.

i = 0 i = 1 i = 2 i = 3 i = 4

wi,2 1/3 1/6 1/6 1/6 1/6

ξi (0, 0) (1, 0) (0, 1) (−1, 0) (0, −1)



. (14)

Finally, the polymer stress σp is calculated by Eq. (3)

after conformation tensor Cαβ is obtained, and it is again

used in Eqs. (5) and (7) as an external forcing term F

(= ) to solve the velocity field of the solution.

3. Results and Discussion

3.1. Step shear testFirst, step shear flow was applied. The stress develop-

ment and its relaxation were carefully investigated in two-

dimensional (2D) channel flow. The simulation parame-

ters were set as described below (all variables are denoted

in lattice units, LU). Dimensionless relaxation time of the

solvent was set as τ = 1.0 (kinematic viscosity of solvent,

νs = 1/6), and kinematic viscosity of polymer was νp = 2/

3. Eventually, total kinematic viscosity of fluid ν0 = νs + νp

was used, and the relative kinematic viscosity was defined

as β = νs/ν0 = 1/5. The density of fluid was ρ = 1, and the

total dynamic viscosity of the fluid corresponded to η0 =

ρν0 = ν0. Polymer relaxation time was set as λ1 (λ) = 105,

and retardation time was defined by λ2 = βλ1. These mate-

rial properties correspond to polymer solution which has

properties of ν0 = 5 × 10−6 m2/s, λ1 = 5/3 s, and λ2 = 5/6 s

(in lattice unit length and unit time scale; Δx = 10 μm and

Δt = 1/6 × 10−4 s, respectively), and these are very similar

to the properties of the Boger fluid used in experiments

(Nam et al., 2010). The size of the simulation domain was

100 × 102 (width × height, including wall nodes). The

channel height was 100, based on the parameter set given

above, which is very close to the usual gap size of 1 mm

in the rotational rheometer. Shear rate was set as = 10−7

and 10−6, which corresponds to the Weissenberg number

(Wi = ) 0.01 and 0.1 respectively. The moving wall

boundary condition was used in both upper and lower

walls to impose Couette flow, and the periodic boundary

condition was applied to the flow direction. For moving

wall boundary condition, the halfway bounce-back

method (Nguyen and Ladd, 2002) was used with addi-

tional treatment in advection-diffusion part (details can be

found in Malaspinas et al., 2010).

For Oldroyd-B model, the transient stress in simple

shear flow can be derived analytically. With definitions of

flow direction 1, and shear gradient direction 2, the poly-

mer contribution of stress tensor for shear (σp,12) and for

normal (σp,11) could be derived as follows.

, (15)

. (16)

Transient shear stresses σp,12 at two different shear rates

(Wi = 0.01 and Wi = 0.1) are plotted in Fig. 1a. Both

stresses increase, and they reach the steady state at t larger

than 3λ. Even though different levels are observed for two

shear rates, the results exactly match with the analytical

solution given by Eq. (15). The normal stresses σp,11 are

plotted in Fig. 1b. They also show similar transient behav-

ior with σp,12, and are in good agreement with the analyt-

ical solution given in Eq. (16).

Stress relaxation after cessation of shear flow was also

investigated. At first, shear flow was applied in the same

manner to the previous test. Shearing was stopped after

shear stress reached steady state (t = 10λ), and stress

relaxation was observed (Fig. 2). In this test, the relaxation

behavior was the same for both Wi = 0.01 and Wi = 0.1

when the stress and time are normalized. For Oldroyd-B

fluids, stress relaxation can be predicted by the analytical

solution (Eq. (17)), and the simulation agrees well with

the predictions regardless of Wi.

. (17)

Oldroyd-B fluid (or Boger fluid) does not show any

shear rate dependency in shear viscosity (Boger, 1977),

and this Newtonian behavior is also reflected in our sim-

ulation. In Fig. 3a, the polymer stress with respect to shear

G = 1

λ1

-----– C I–( ) + C ∇u⋅ ∇u( )T+ C⋅

∇ σp⋅

γ·

λ1γ·

σp 12, t( ) = σp 21, t( ) = ηpγ· 1 e

t/λ1

–

–( )

σp 11, t( ) = 2ηpλ1γ·2

1 et/λ

1–

–( ) + 2ηpλ1γ·2

tet/λ

1–

σp 12, t( ) = σp 12, t0( )et t

0–( )/λ

1–

Fig. 1. (Color online) Transient behavior of polymer stress at Wi

= 0.01 and 0.1. (a) Time evolution of σp,12. (b) Time evolution of

σp,11. Black dashed lines are the analytical solutions obtained by

Eqs. (15) and (16). The stress was normalized by the reference

value (analytical solution for Wi = 0.1 at t = 10λ).



rate (~ Wi) is plotted, and it can be observed that both

shear stress and normal stress show a power-law relation-

ship (with different slopes, 1 and 2, respectively) with

respect to shear rate. These stresses can be converted to

the forms of shear viscosity (polymer part. ηp = σp,12/ )

and the coefficient of the first normal stress difference (Ψ1

= (σ11 − σ22)/ ), respectively. In Fig. 3b, both material

functions show constant values independent of shear rate.

In particular, shear viscosity exactly corresponds to the

input material property νp = 2/3.

3.2. Oscillatory shear testOscillatory shear flow has been widely applied to char-

acterize the viscoelastic properties of complex fluids

(Ferry, 1980). In particular, small amplitude oscillatory

shear (SAOS) has been used as a common technique for

measuring linear viscoelasticity, in which the strain ampli-

tude is small and shear stress is linear to shear strain (Bird

et al., 1987). To confirm whether above viscoelastic func-

tions are reproduced in the present work, simulations were

carried out for oscillatory shear flow. Same material prop-

erties and simulation domain used in step shear test were

also applied, and the strain γ was imposed as a function of

time as in Eq. (18).

. (18)

Here, γ0 is the strain amplitude, ω is the angular fre-

quency (ω = 2πf, f : frequency), and t denotes the simu-

lation time. The calculations were performed at fixed γ0 =

0.1 in the range of angular frequencies ω = 6.28 × 10−7−

1.26 × 10−4 Δt−1 (lattice unit), which corresponds to the

angular frequency ω* ~ 0.24 rad/s − 47.5 rad/s. These sim-

ulation conditions also correspond to Deborah number

(λ1ω), De = 0.06 − 12.67, and the range covers most of the

experimental conditions (Nam et al., 2010). Figure 4

shows simulation results at De = 6.28. Time lag between

strain and stress is notable compared to the pure viscous

fluid, which has been widely reported for complex fluids

with large relaxation time. The stress signal can be for-

mulated as below form.

(19)

where σ0 is the stress amplitude, and δ1 is the phase angle

between strain and stress.

Normal stress is plotted in Fig. 4b. In the Oldroyd-B

model, the first normal stress difference N1 is oscillating

with a frequency 2ω about nonzero mean value N1,0

because the first normal stress difference depends only on

the strain amplitude, not on its direction (Ferry, 1980;

Nam et al., 2008). The phase angle δ2 can be defined in a

similar way (Bird et al., 1987), and the dependence of nor-

mal stress on time can be as follows.

(20)

where N1,2 is the magnitude of the oscillating normal

stress.

From the shear stress signal, the dynamic moduli can be

derived according to the following equations.

, (21)

(22)

γ·

γ·2

γ = γ0 sin ωt( )

σ12 = σ0 sin ωt δ1+( )

N1 = N1 0, + N1 2, sin 2ωt δ2+( )

G′ = σ0

γ0

----- cosδ1

G″ = σ0

γ0

----- sinδ1

Fig. 3. (Color online) (a) Polymer stress with respect to Weis-

senberg numbers Wi. (b) Kinematic shear viscosity (polymer

part) νp and the coefficient of first normal stress difference Ψ1.

Fig. 2. (Color online) The time evolution of shear stress with

respect to the dimensionless time (t/λ) during the inception of a

step shear and subsequent cessation. The shear starts at t/λ = 0

and the stress reaches to steady state at t/λ = 3. Shear flow is

switched off after t/λ = 10, and the stress begins to relax.



where G' and G'' are the storage modulus and the loss

modulus.

In Fig. 5a, the dynamic moduli are plotted as a function

of Deborah number (De). At low Deborah number (De <

1), both storage modulus and loss modulus increase with

slopes 2 and 1, respectively, and they coincide at De ~ 1.

At higher Deborah number (De > 1), the storage modulus

shows a plateau, while the loss modulus keeps on increas-

ing with De. This means that the solvent contribution

becomes more dominant for De > 1. In small strain ampli-

tude region, the dynamic moduli of Oldroyd-B fluids can

be derived as follows, and our simulation corresponds

well with the analytical solution.

, (23)

. (24)

The complex viscosity can also be obtained as in Eq.

(25).

. (25)

As shown in Fig. 5b, the complex viscosity shows two

plateau regions. At low De limit (De < 0.1), the complex

viscosity shows the first plateau, which corresponds to η0.

With the increase in De, it shows shear thinning behavior,

and eventually reaches a secondary plateau (De ~ 10),

which corresponds to ηs. This result means that the poly-

mer contribution becomes relatively weak as De increases,

and it corresponds to the well-known rheological charac-

teristics of Boger fluids (Nam et al., 2010).

A popular way to show the response of a viscoelastic

fluid upon the oscillatory shear is to plot a closed loop of

stress versus strain or stress versus rate of strain, namely

Lissajous plot. In the Lissajous plot, an elliptical curve is

observed in the linear viscoelastic regime, while non-ellip-

tical shape is observed in the nonlinear viscoelastic regime

G′ ω( ) = η0ωλ1 λ2–( )ω

1 λ1

2ω

2+

-----------------------

G″ ω( ) = η0ω1 λ+ 1λ2ω

2

1 λ1

2ω

2+

------------------------

η*ω( ) =

G′2 ω( ) G″2ω( )+

ω-------------------------------------------

Fig. 5. (Color online) (a) Dynamic moduli and (b) complex vis-

cosity obtained by LBM simulation (symbols) and theoretical

prediction (lines). They are plotted as a function of Deborah

number (λ1ω). The parameter sets are η0 = 5/6, β = 1/5, λ1= 105,

and λ2 = βλ1. Cross-over point of dynamic moduli corresponds to

De = 1, and the plateau modulus (GN) is observed at high De. η0

and ηs correspond to the complex viscosity at low De limit and

at high De limit, respectively.

Fig. 4. (Color online) (a) Sinusoidal input shear strain of ampli-

tude γ0 = 0.1 at angular frequency ω = 6.28 × 10−5 Δt−1

(De = 6.28) produces sinusoidal total shear stress. The total

shear stress (black solid line) is plotted together with polymer

stress (red dot). Time lag between stress and strain is clearly

observed, which is defined as the phase angle δ1. (b) The first

normal stress difference oscillates with an angular frequency 2ω

with a nonzero mean value (N1,0).



(Hyun et al., 2011). This trend in the Lissajous curve is

well reproduced through our simulation. In Fig. 6, stress

(shear and normal stress difference) versus strain is plot-

ted. At low De (De = 0.06), both total shear stress σ12 and

polymer shear stress σp,12 maintain ellipsoidal shape, but

the curves change to non-elliptical shape as De increases.

The area inside of the Lissajous plot provides information

on the relative contribution of polymer phase in the fluid.

The smaller area is observed for polymer stress than total

shear stress, and the more reduced area is clearly observed

as De increases. This can be interpreted as a reduced con-

tribution of polymers at large strain amplitude, which also

corresponds to the trend of the complex viscosity in Fig.

5b. Contrast to the shear stress σ12, the first normal stress

difference N1 does not show elliptic curve at all, and their

phase angle δ2 also shows quite different behavior from δ1.

In our simulation, δ2 decreases with De, and it shows

minus (−) values after De ~ 1. On the other hand, δ1 is

always positive. In addition, N1 vs strain curve changes its

shape after De ~ 1 (convex to concave), which corresponds

to experiments with Boger fluid (Nam et al., 2010).

In the small amplitude oscillatory shear (SAOS) limit,

the first normal stress difference N1 can be reformulated as

follows. n1,0 can be defined in terms of zero mean value of

N1 (N1,0) and strain amplitude γ0, which corresponds to the

storage modulus G' of Oldroyd-B fluid (Bird et al., 1987;

Ferry, 1980).

. (26)

The other quantities ( and ) can also be related

to the normal stress difference, and they are co-related

with phase angle (Nam et al., 2008;

Nam et al., 2010).

, (27)

. (28)

The characteristics of and are clearly reflected

in our simulation. In Fig. 7a, G' obtained from shear stress

and n1,0 derived from normal stress are plotted together.

Even though they are obtained from different material

functions, both results are well coincided with each other,

and moreover, they correspond well with analytical solu-

tions.

Finally, the quantities and are plotted in Fig.

7b. With respect to De, the slope of is 3 at low De,

while it becomes −1 at high De. The slope of shows

a sudden decrease after increasing with the slope of 2. At

higher De, a sudden increase is captured with an increase

of De, and eventually, it reaches a plateau. These obser-

vations not only follow theoretical predictions (Eqs. (27)

and (28)), but also coincide to the reported rheological

behavior of Boger fluids (Nam et al., 2010).

4. Conclusions

Time-dependent viscoelastic properties of Oldroyd-B

fluids were investigated by the lattice Boltzmann method

n1 0, = N1,0/γ 0

2 = G′ ω( )

n1 2, ′ n1 2, ″

δ2 = tan1–

n1 2, ′/n1 2, ″( )

n1,2′ = N1 2, cosδ2/γ 0

2 = G″ ω( )−

1

2---G″ 2ω( )

n1,2″ = N1 2, cosδ2/γ 0

2 = G– ′ ω( )+

1

2---G′ 2ω( )

n1,2′ n1,2″

n1,2′ n1,2″n1,2′

n1,2″

Fig. 6. (Color online) Lissajous plots: stress versus strain. The shear stress and the first normal stress difference are represented for

various Deborah numbers (De = λ1ω), and the curves are normalized by their amplitudes. The simulation parameters are η0 = 1/6, β

= 1/5, λ1 = 105, and λ2 = βλ1.



(LBM) coupled with advection-diffusion model. To quan-

tify the viscoelastic properties of polymer solutions, real-

istic rheometry was adopted in LBM framework. Firstly, a

transient behavior was investigated by step shear test.

Simulation was carried out for Wi = 0.01 and 0.1, and the

tests for both start up shear and cessation of shear were

implemented. Calculated shear and normal stresses were

well matched to analytical solutions, and the Newtonian

behavior was also correctly captured. To investigate more

complex viscoelastic properties of the fluid, simulation

was carried out in an oscillatory shear flow. The wide

range of Deborah number, De ~ 0.1 − 10, was considered,

and reliable results were obtained for all conditions. The

time lag between shear strain and shear stress was observed,

and frequency 2ω in the first normal stress was also cor-

rectly reproduced. Dynamic moduli calculated from the

shear stress showed good agreement with known analyt-

ical solutions, and complex viscosity derived from dynamic

moduli also correctly followed the general behavior of

Oldroyd-B fluid. Complex viscosity showed two plateau

regions at both low and high De limits, and they corre-

sponded to η0 and ηs, respectively. It was also confirmed

that the polymer contribution becomes reduced as De

increases, which can also be confirmed through Lissajous

plot. Finally, viscoelastic quantities which are related to

the first normal stress difference N1 were carefully ana-

lyzed. n1,0 was obtained from zero mean value of N1, and

it was compared to storage modulus derived from the

shear stress. Even though these properties were derived

from two different viscoelastic origins, they were exactly

matched with each other. The other normal stress quanti-

ties and , were also analyzed. Their complex

behavior with respect to De was captured in the simula-

tion, and they also showed a good agreement with ana-

lytical solutions. From this study, it can be confirmed that

time-dependent viscoelastic properties of Oldroyd-B fluid

can be accurately predicted by LBM coupled with advec-

tion-diffusion model, especially in the framework of real-

istic rheometry. Even though present study is limited to

Oldroyd-B model, expansion to more realistic models is

straightforward, and the study of more complex fluids will

be available with this simulation framework.

Acknowledgment

This work was supported by the National Research

Foundation of Korea (NRF) grant funded by the Korea

government (MSIP) (No. 2016R1E1A1A01942362).

Appendix

In the advection-diffusion scheme, the accuracy of the

solution is determined by the relaxation parameter ϕ

(Malaspinas et al., 2010), and the proper choice of ϕ is

essential. To study the effect of ϕ, oscillatory shear test

was carried out for various ϕ. Simulation condition was

set as follows. νs = 1/6 (τ = 1.0), νp = 2/3, β = νs/ν0 = 1/

5, λ1(λ) = 105, and λ2 = βλ1. Strain γ was imposed as a

function of time as in Eq. (18). Strain amplitude γ0 = 0.1

and angular frequency ω = 6.28 × 10−5 Δt−1 were used in

the test. This condition corresponds to Deborah number

(λ1ω), De = 6.28. The size of the simulation domain was

100 × 102. The moving wall boundary condition was

imposed in both upper and lower walls, and the periodic

boundary condition was applied to the flow direction.

Comparison between LBM results and analytical solu-

tions were implemented, and to quantify the difference of

two solutions, L2-error given in Eq. (A1) was used. Here,

ALBM is a solution obtained by LBM simulation, and

AAnalytical is the predicted value by analytical solutions. In

the present test, the shear stress σ12 and the first normal

stress difference N1 were analyzed. The analytical solu-

tions for each rheological property were obtained by solv-

ing Eqs. (19)-(28).

n1,2′ n1,2″

Fig. 7. (Color online) (a) Storage modulus (G') and the oscilla-

tory property of the first normal stress difference n1,0 defined by

Eq. (26). (b) The oscillatory properties ( and ) of the

first normal stress difference. is plotted as an absolute

value because it changes sign during an oscillation.

n1 2, ′ n1 2, ″n1 2, ″



. (A1)

In the test, L2-error was measured in the time range ωt

= 0~π/4, and it was discretized as the total number of bins

N = 250 in the analysis.

In Fig. A1, time evolution of the stress signal is plotted

for various ϕ (ϕ = 0.9, ϕ = 0.6, and ϕ = 0.51). At ϕ = 0.9,

inaccurate solutions are clearly captured in both σ12 and

N1. Discrepancies between LBM simulation and analytical

solutions are observed not only in the maximum peak but

also in phase angle, and with a decrease of ϕ, more accu-

rate solutions are obtained. To quantify solution depen-

dency for the parameter ϕ, L2-errors of σ12 and N1 are

analyzed (Fig. A2). In case of shear stress, at ϕ = 0.9,

error (σ12) = 9.45 × 10−2; at ϕ = 0.6, error (σ12) = 5.38 ×

10−2; at ϕ = 0.51, error (σ12) = 1.14 × 10−2. In case of the

first normal stress difference, at ϕ = 0.9, error (N1) = 2.78

× 10−1; at ϕ = 0.6, error (N1) = 1.12 × 10−1; at ϕ = 0.51,

error (N1) = 1.46 × 10−2. In the test limit, large error was

captured in N1 than σ12 (about 200%), and the error dif-

ference between two properties becomes reduced with the

decrease in ϕ (from 300% at ϕ = 0.9 to 130% at ϕ = 0.51).

In both cases, a more accurate solution was obtained as ϕ

decreased, and finally the most accurate solution was con-

firmed at ϕ = 0.51.

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