Time-dependent viscoelastic properties of Oldroyd-B fluid ... · promising numerical technique in computational fluid dynamics (CFD) (Aidun and Clausen, 2010; Succi, ... and the interaction
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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