A Hybrid Immersed Boundary-Lattice Boltzmann Method for Simulation of Viscoelastic Fluid Flows Interaction with Complex Boundaries

Full text

Commun. Comput. Phys.
doi: 10.4208/cicp.OA-2019-0158
Vol. 29, No. 5, pp. 1411-1445
May 2021
A Hybrid Immersed Boundary-Lattice Boltzmann
Method for Simulation of Viscoelastic Fluid Flows
Interaction with Complex Boundaries
M. H. Sedaghat1,∗, A. A. H. Bagheri2, M. M. Shahmardan2,
M. Norouzi2, B. C. Khoo3and P. G. Jayathilake4
1School of Mechanical Engineering, Shiraz University, Shiraz, Iran.
2Department of Mechanical Engineering, Shahrood University of Technology, Shahrood,
Iran.
3Department of Mechanical Engineering, National University of Singapore,
Singapore 117575.
4Department of Oncology, University of Oxford, Oxford, OX3 7DQ, UK.
Received 8 September 2019; Accepted (in revised version) 21 February 2020
Abstract. In this study, a numerical technique based on the Lattice Boltzmann method
is presented to model viscoelastic fluid interaction with complex boundaries which are
commonly seen in biological systems and industrial practices. In order to accomplish
numerical simulation of viscoelastic fluid flows, the Newtonian part of the momentum
equations is solved by the Lattice Boltzmann Method (LBM) and the divergence of the
elastic tensor, which is solved by the finite difference method, is added as a force term
to the governing equations. The fluid-structure interaction forces are implemented
through the Immersed Boundary Method (IBM). The numerical approach is validated
for Newtonian and viscoelastic fluid flows in a straight channel, a four-roll mill geom-
etry as well as flow over a stationary and rotating circular cylinder. Then, a numerical
simulation of Oldroyd-B fluid flow around a confined elliptical cylinder with different
aspect ratios is carried out for the first time. Finally, the present numerical approach
is used to simulate a biological problem which is the muco-ciliary transport process of
human respiratory system. The present numerical results are compared with appro-
priate analytical, numerical and experimental results obtained from the literature.
AMS subject classifications: 76-XX, 65-XX
Key words: Lattice Boltzmann method, immersed boundary method, viscoelastic fluid, complex
boundaries, muco-ciliary transport.
∗Corresponding author. Email addresses: h.sedaghat@shirazu.ac.ir (M. H. Sedaghat),
anaraki.mech@gmail.com (A. A. H. Bagheri), mmshahmardan@shahroodut.ac.ir (M. M. Shahmardan),
mnorouzi@shahroodut.ac.ir (M. Norouzi), mpekbc@nus.edu.sg (B. C. Khoo),
jayathilake.pahalagedara@oncology.ox.ac.uk (P. G. Jayathilake)
http://www.global-sci.com/cicp 1411 c
2021 Global-Science Press

1412 M. H. Sedaghat et al. / Commun. Comput. Phys., 29 (2021), pp. 1411-1445
1 Introduction
Fixed mesh methods have been commonly employed for simulations of complex bound-
ary problems to relieve computational cost associated with moving mesh methods. One
of the robust fixed mesh methods is the Immersed Boundary Method (IBM) which was
first proposed by Peskin [1] to model blood flow in human heart. In this method, La-
grangian points are located on the boundary of the object and Eulerian points are used to
represent the fluid phase around the object. The idea of the IBM is the discretization of
the momentum equations on the Eulerian points and the immersed boundary condition
is depicted by adding a body force term into the momentum equations. Thereby, in this
method the governing equations are decoupled from the computational mesh and as a
result it can easily simulate not only flow past stationary complex geometries, but also
flow interaction with moving objects. Early studies of the IBM also include simulating
two-phase immiscible flows [2, 3].
In previous studies, viscoelastic fluid flow in simple geometries have been solved
using finite difference [4–6], finite element [7, 8], spectral finite element [9] and finite vol-
ume [10–16] methods. De et al. [17] implemented finite volume and immersed boundary
methods for simulating viscoelastic fluid flows around stationary cylinders. Krishnan
et al. [18] used a similar method and presented a fully resolved simulation of particles
moving in a viscoelastic fluid. Recently, the Lattice Boltzmann Method (LBM) has been
introduced as a powerful method to simulate two phase flow [19], Newtonian as well as
non-Newtonian fluid flows. The LBM is based on the mesoscopic kinetic equations for
fluids which solve the discrete Boltzmann equation for the particle density distribution
function [20]. This method has also been implemented for analysis of viscoelastic fluid
flows and researchers have verified their approaches for benchmark problems including
lid-driven cavity and Poiseuille flows [21–25]. Malaspinas et al. [26] proposed a new ap-
proach based on the LBM in order to simulate linear and non-linear viscoelastic fluids. In
particular those described by the Oldroyd-B and FENE-P constitutive equations at low
Reynolds numbers. Their model has some redundant terms which lack a clear physical
meaning for the recovered stress tensor. Su et al. [27] modified the model of Malaspinas
et al. [26] and proposed a novel numerical scheme for the simulation of viscoelastic fluid
flows based on the LBM over a large range of Weissenberg numbers at low Reynolds
numbers. Their results showed that viscoelastic fluid flow in two-dimensional channel
flow is found to be in full accord with the analytical solution. Their discretization scheme
for incorporating viscoelastic stress into the LBM is dependent on constitutive models,
and is not general either. Zou et al. [28] using two open source Computational Fluid Dy-
namics (CFD) toolkits, namely OpenFOAM and OpenLB, proposed an integrated scheme
based on the Lattice Boltzmann method and Finite Volume Method (FVM) for modeling
incompressible polymer viscoelastic fluid flows. Their model has been critically vali-
dated using the Oldroyd-B model and linear PTT model under Poiseuille flow, Taylor-
Green vortex flow and 4:1 abrupt planar contraction flow. In a similar investigation,
Zou et al. [29] has modified their previous model by introducing an integrated Lattice

M. H. Sedaghat et al. / Commun. Comput. Phys., 29 (2021), pp. 1411-1445 1413
Boltzmann and finite volume schemes to improve the grid mapping. These latter stud-
ies [28, 29] have two important shortcomings. The First one is that, OpenFOAM and
OpenLB use staggered and non-staggered grids, respectively, and therefore certain spa-
tial interpolation schemes should be introduced to map physical parameters from one
scheme to another at each time step and this leads to increase in computational time.
The second shortcoming is that these studies have reported some benchmark validation
results only at low Reynolds numbers.
In the last decade or so, the LBM, which is also called lattice Boltzmann equation or
Boltzmann cellular automata in the literature, has emerged as a new and effective nu-
merical approach of computational fluid dynamics and achieved considerable success
in simulating fluid flows and associated transport phenomena. The lattice Boltzmann
model is based on microscopic models and mesoscopic kinetic equations. The LBM con-
structs simplified kinetic models that incorporate the essential physics of microscopic
or mesoscopic processes so that the macroscopic averaged properties obey the desired
macroscopic equations. In the LBM, a simplified version of the kinetic equation is used
rather than solving complicated kinetic equations such as the full Boltzmann equation. It
has an advantage when implementing fully parallel algorithms [30].
Prominent features of Lattice Boltzmann Method have persuaded scientists to im-
plement the Lattice Boltzmann Method coupled with the Immersed Boundary Method
(IB-LBM) for simulating fluid flows around complex geometries. The IB-LBM was orig-
inally introduced by Feng and Michaelides [31]. Niu et al. [32] developed a momentum
exchange based the IB-LBM to simulate flows around fixed and moving particles. Wu et
al. [33] used the IB-LBM to model a deformable and moving solid bodies suspended in
Newtonian fluid. Wu and Shu [34] used an implicit form of the IBM and developed a
boundary condition enforced IB-LBM. In another investigation, Wu and Shu [35] intro-
duced a novel IB-LBM to simulate migration of a neutrally buoyant particle in Newto-
nian flow. An inspection of the literatures reveals that even in these settings, Newtonian
flow behavior has been studied much more extensively than the viscoelastic flow char-
acteristics. The major contribution of the present paper is on simulation of viscoelastic
fluid flow interacting with complex geometries. For this purpose, the Lattice Boltzmann-
Finite Difference (LB-FD) method is presented which is coupled with an implicit form of
Immersed Boundary Method presented by Wu and Shu [35]. In the present work, this
numerical approach is validated for flow past a stationary/rotating cylinder. Then, the
method is applied to simulate muco-ciliary transport process of the human respiratory
system.
The rest of the paper is organized as follows: In Section 2, the governing and con-
stitutive equations of the viscoelastic fluid flow are described. The numerical method is
described in detail in Section 3. In Section 4, the present numerical method is validated
for some benchmark problems and then the muco-ciliary transport process is simulated
with viscoelastic properties of the mucus layer.

1414 M. H. Sedaghat et al. / Commun. Comput. Phys., 29 (2021), pp. 1411-1445
2 Model formulation
2.1 Governing equations
The governing equations of incompressible viscoelastic fluid flow for the IBM formula-
tion can be written as follows [35]:
~
∇·~u=0, (2.1a)
∂~u
∂t+~u.~
∇~u+1
ρ
~
∇p=1
ρ
~
∇.σ+~
f, (2.1b)
~
f(~x,t) = ZΓ
~
F(s,t)δ(~x−~
X(s,t))ds, (2.1c)
∂~
X(s,t)
∂t=~
U(~
X(s,t),t) = ZΩ
~u(~x,t)δ(~x−~
X(s,t))d~x, (2.1d)
where ~xis the Eulerian coordinates, ~uis the fluid velocity vector, ρis the fluid density, pis
the static pressure, tis time, σis the stress tensor and ~
fis the boundary force acting on the
fluid field due to the immersed object. Also, ~
Xis the Lagrangian coordinates representing
the boundary of the object, ~
Uis the velocity vector of Lagrangian nodes, sis the arc length
of Lagrangian nodes and ~
Fis the boundary force density. In addition δ(~x−~
X(s,t)) is a
Dirac delta function. Eqs. (2.1c) and (2.1d) describe the interaction between the immersed
boundary and the fluid by distributing the boundary forces at the Lagrangian points (Γ)
to the nearby Eulerian grid points (Ω) and interpolating the velocity at the Eulerian points
to the Lagrangian points.
2.2 Constitutive equation
One of the advantages of the present numerical method is that all kinds of constitutive
equations of viscoelastic fluid which stress tensor can be decomposed into elastic and
viscous parts can be employed. Here, we use the Oldroyd-B model as the constitutive
equation of the fluid. The Oldroyd-B model is a well-known viscoelastic constitutive
equation which is useful for describing rheological behaviors of dilute polymeric solu-
tions at high rates of deformation and Boger liquids. In this equation, the stress tensor σ
can be decomposed into two parts,
σ=σN+σE. (2.2)
The first term is related to the Newtonian solvent and the second is related to the poly-
meric additives. The contribution of the Newtonian solvent is:
σN=2ηND. (2.3)
The polymeric contribution is derived based on the upper convected Maxwell (UCM)
model:
σE+λ∇
σE=2ηED. (2.4)

M. H. Sedaghat et al. / Commun. Comput. Phys., 29 (2021), pp. 1411-1445 1415
In these expressions, ηNis the viscosity of the Newtonian solvent and ηEis the viscosity
of the elastic additives. The viscosity of the polymeric solution is:
η0=ηN+ηE. (2.5)
In Eq. (2.4), λis the relaxation time, and the upper convected derivative of σEis defined
as: ∇
σE=∂σE
∂t+~u.∇σE−σE.∇~u−∇~uT.σE. (2.6)
In Eqs. (2.3) and (2.4), Dis the rate of deformation which is defined as:
D=1
2∇~u+∇~uT. (2.7)
The Oldroyd-B constitutive equation can be derived from a molecular model in which
the polymer molecules are idealized as infinitely extensible Hookean springs connecting
two Brownian beads [36]. This model specially has been used for predictions of simple
shear flows that are in qualitative agreement with measurements for some Boger fluids
[37]. If the solvent term in the Oldroyd-B equation is set to zero (ηN=0), the equation
is reduced to the upper-convected Maxwell (UCM) equation. The Oldroyd-B equation is
also suitable to model shear and pressure-driven flows of dilute polymeric solutions.
By defining the non-dimensional velocity and length as ~u∗=~uURe f and ~x∗=~xLRe f
respectively, the non-dimensional form of above governing equations give the following
non-dimensional numbers:
σ∗
E=σELRe f
η0URe f
, Re =ρUR e f LRe f
η0
,We =λURe f
LRe f
,β=ηE
η0
. (2.8)
In Eq. (2.8), URe f and LRe f are the reference velocity and length, respectively, Re is the
Reynolds number, We is the Weissenberg number and βis the viscosity ratio.
3 Numerical method
In this section, the hybrid Immersed Boundary-Lattice Boltzmann-Finite Difference
method is introduced. In this approach, the Newtonian part of the momentum equa-
tions is solved by Lattice Boltzmann method. The elastic part of the stress tensor (σE) is
solved by the Finite Difference method and the effect of the immersed objects are added
to the LBM equations as force terms.
Although in pure LBM, physical domain and fluid properties must be imported into
LB equations based on non-dimensional parameters, by introducing transformation coef-
ficients the real value of physical domain as well as fluid properties can be imported into
the momentum equations in the current method. In addition by choosing proper values
for grid and time spacing, more accurate values for velocity and viscosity transformation
coefficients have been obtained. Therefore, without considering the limitation of Mach

1416 M. H. Sedaghat et al. / Commun. Comput. Phys., 29 (2021), pp. 1411-1445
number (Ma) and single relaxation time in LBM, the physical values of viscosity and ve-
locity of the fluid can be imported. The transformation coefficients which transform the
physical domain to the Lattice Boltzmann domain are introduced as follows:
Chx=∆xphy
∆xlb ,Chy=∆yphy
∆ylb ,Ct=∆tphy
∆tlb ,Cρ=ρphy
ρlb
0
, (3.1)
where ∆xphy and ∆yphy are gird spacing in xand ydirection, ∆tphy is time spacing and
∆ρphy is density for physical domain. Additionally, ∆xlb and ∆yl b are grid spacing, ∆tlb is
time spacing and ∆ρlb
0is reference density in LBM domain. In addition, Chx,Chy,Ctand Cρ
are related to space in xand ydirections, time and density transformations, respectively.
In this study, ”lb” superscript is related to LBM variables and for simplicity, physical
variables are shown without any superscript. In Eq. (3.1), by considering ∆x=∆yand
∆xlb =∆ylb, we can write Chx=Chy=Ch. The velocity transformation coefficients can be
obtained as follows:
Cu=~u
~ul b =Ch
Ct
. (3.2)
Although, all kinds of LB equations can be used in the present method, the pressure base
LB equations proposed by He and Luo [38] are used as the base of our numerical method.
These pressure base LB equations are used to simulate incompressible fluid flows, spe-
cially the pressure driven flows. The pressure based Lattice Boltzmann equation with
external forces can be written as:
pα~xl b +~eαδt,tl b +δt−pα~xlb ,tlb=−1
τpα~xlb,tlb −pαe q ~xlb ,tlb +~
Fαδt, (3.3a)
Fα=ωαPlb
01−1
2τ~eα−~ulb
c2
s
+~eα.~ulb
c4
s
~eα·~
flb , (3.3b)
where pαis the pressure distribution function, pαeq is its corresponding equilibrium dis-
tribution function for the discrete velocity ~eα,Fαis the discrete force term, ~
flb is the force
density, ωαis weighting coefficient which depends on the selected lattice velocity model,
τ=νlb /c2
s∆tlb +0.5 is the single relaxation time, νlb is the kinematic viscosity in the LBM
domain, cs=c/√3 is the sound speed and c=∆xlb /∆tl b is the lattice speed where ∆xlb and
∆tlb are the lattice constant and the time step size, respectively. The equilibrium pressure
distribution function pαeq is defined as:
pαeq =ωαPlb +Pl b
01+3(~eα·~ulb )
c2+9(~eα·~ulb )2
2c4−3~ul b
2
2c2#), (3.4)
where Plb =c2
sρlb . Here, the D2Q9lattice velocity model is used, which discrete velocity

M. H. Sedaghat et al. / Commun. Comput. Phys., 29 (2021), pp. 1411-1445 1417
Figure 1: Lattice arrangements for 2-D problems, D2Q9.
vectors are as follows:
~eα=










[0,0],α=0,
chcos(α−1)π
2, sin(α−1)π
2i,α=1−4,
√2chcos(α−5)π
2+π
4, sin(α−1)π
2+π
4i,α=5−8.
(3.5)
The weighting coefficients ωiare defined as:
~eα=












4
9,α=0,
1
9,α=1−4,
1
36,α=5−8.
(3.6)
Fig. 1 shows the lattice arrangements for 2-D problems (D2Q9). In addition the veloc-
ity and pressure in the LBM can be obtained as [39]:
Plb =∑
α
pα, (3.7a)
Plb
0~ul b =∑
α
eαpα+∆tlb
2c2
s~
flb . (3.7b)
In Eqs. (3.3b) and (3.7b), ~
flb consists of various different forces: ~
flb
IB M which is related to
the force that the immersed body imposes on the fluid and ~
flb
Ewhich is related to the
elastic part of the stress tensor in viscoelastic fluids and other external force fields (~
flb
O)
such as gravity, magnetic field etc. These forces are explained in the following sections in
details.

1418 M. H. Sedaghat et al. / Commun. Comput. Phys., 29 (2021), pp. 1411-1445
3.1 Immersed boundary forces
In order to satisfy the no-slip boundary condition between the fluid and the immersed
object, the IB-LBM is employed [35]. In this method, the interpolated velocity of the fluid
at the boundary point is enforced to be equal to the velocity of the boundary by a set of
velocity corrections. Indeed, the velocity of the fluid field is defined as follows:
~ul b =~ulb,∗+∆~ulb . (3.8)
In Eq. (3.8), ~ulb,∗is the intermediate fluid velocity and ∆~ulb is the velocity correction. The
intermediate fluid velocity ~ulb,∗is given as:
~ul b,∗=1
P0∑
α
~eαpα. (3.9)
The velocity correction, ∆~ulb , can be calculated as:
∆~ulb =1
2
~
flb ∆tl b . (3.10)
Furthermore, by interpolating the value of the boundary velocity correction ∆~
Ulb , the
fluid velocity correction, ∆~ulb , can be determined as follows:
∆~ulb (~xlb ,tlb) = ZΓ
∆~
Ulb (~
X,tlb )δ(~xlb −~
Xlb (slb,tl b ))dsl b, (3.11)
where Dij(xij −Xl)is Delta function which is defined as:
δ(~xl b −~
Xlb (slb,tl b )) = Dij (xlb
ij −Xlb
l) = δ(xlb
ij −Xlb
l)δ(ylb
ij −Ylb
l), (3.12a)
δ(r) = 


1
h1+cosπr
2,|r|≤2,
0, |r|>2.
(3.12b)
In Eq. (3.12), his the mesh spacing of Eulerian mesh around the boundary. Eq. (3.11) can
be written as:
∆~ulb (~xlb ,tlb) = ∑
l
∆~
Ulb (~
Xlb
l,tlb )Dij(xlb
ij −Xlb
l)∆slb
l. (3.13)

M. H. Sedaghat et al. / Commun. Comput. Phys., 29 (2021), pp. 1411-1445 1419
The boundary velocity correction, ∆~
Ulb, can be then calculated as follows:
A~
X=~
B, (3.14a)
X=n∆~
Ulb
1,∆~
Ulb
2,···,∆~
Ulb
moT, (3.14b)
A=




δ11 δ12 ··· δ1n
δ21 δ22 ··· δ21
.
.
..
.
.....
.
.
δm1δm2··· δmn











δB
11 δB
12 ··· δB
1m
δB
21 δB
22 ··· δB
2m
.
.
..
.
.....
.
.
δB
n1δB
n2··· δB
nm






,
B=










~
Ulb
1
~
Ulb
2
.
.
.
~
Ulb
m











−




δ11 δ12 ··· δ1n
δ21 δ22 ··· δ21
.
.
..
.
.....
.
.
δm1δm2··· δmn
















~ul b,∗
1
~ul b,∗
2
.
.
.
~ul b,∗
n











. (3.14c)
Here, mis the number of Lagrangian points on the boundary and nis the number of
Eulerian points. Ulb
l(l=1,2,···,m)is the velocity vector of Lagrangian points and ∆~
Ulb
l
(l=1,2,···,m)is the unknown velocity correction vector at those points. In Eq. (3.14),
δij =Dij(xlb
ij −Xlb
l)∆xlb ∆ylb and δB
ij =Dij(xlb
ij −Xlb
l)∆slb
lwhere ∆slb
lis the arc length of a
boundary element. Therefore, according to the Eq. (3.10), the relationship between the
force density, ~
Flb , and the fluid velocity correction, ∆~
Ulb
l, can be written as:
~
Flb
IB M =2∆~
Ulb
l
∆tlb . (3.15)
By interpolating the boundary forces acting on the fluid nodes, the fluid forces which
must be added to the LB equations can be determined as follows:
~
flb
IB M =∑
l
~
Flb
IB M Di j (xlb
ij −Xlb
l∆slb
l). (3.16)
3.2 Elastic forces contributed by fluid
As mentioned in Section 2, the stress tensor of fluid, σ, is decomposed into the Newto-
nian and elastic parts. The Newtonian part of σis solved by the LBM as described in
the numerical method. Using the physical velocity obtained from LB equations, the elas-
tic part of the stress tensor (Eq. (2.4)) is solved by the finite difference method. These
equations are formulated explicitly, and the first-order forward finite difference in time
and the second-order central finite difference in space (FTCS) are used for numerical dis-
cretization [40]. The polymeric contribution σEis calculated and then the elastic force of
the stress tensor can be updated as:
~
fE=1
ρ
~
∇.σE. (3.17)

1420 M. H. Sedaghat et al. / Commun. Comput. Phys., 29 (2021), pp. 1411-1445
~
fEis defined as the physical variables, so it should be multiplied by the force transforma-
tion coefficient to be transformed into LB force as follows:
~
flb
E=fE
Ct2
Ch
. (3.18)
The value of the total force which should be added to LB equations as the extra force term
is given by:
~
flb =~
flb
IB M +~
flb
E+~
flb
O. (3.19)
In Eq. (3.19) ~
flb represents the total body force due to the presence of submerged objects
(~
flb
IB M ), the viscoelastic behavior of fluid (~
flb
E) and other external force fields (~
flb
O).
In summary, the implementation of the present numerical scheme can be outlined as
follows:
i. Set the initial values of the physical geometry ∆x=∆y,∆tand by setting ∆xlb =
∆ylb =∆tlb =1 and calculate the spatial, temporal and density transformation coef-
ficients by using Eq. (3.1).
ii. Set the Newtonian kinematic viscosity and velocity in all nodes and use velocity
transformation coefficients (Eq. (3.2)) and compute velocity in all nodes in the LBM
domain.
iii. Use Eq. (3.3a) to get the pressure distribution function at time level t=tn(initially
setting Fα=0) and compute density and velocity in LBE using Eqs. (3.7a) and (3.7b).
iv. Use velocity calculated in step iii and by knowing the velocity vector of Lagrangian
(boundary) points, calculate the immersed boundary force density ~
flb
IB M by using
Eqs. (3.14)-(3.16).
v. Use the velocity transformation coefficient and change the obtained velocity from
Step iii to physical velocity and use them to find the elastic stress tensor σEby
solving the constitutive equations of viscoelastic fluid (Eq. (2.4)).
vi. Compute the elastic force ~
flb
Eby using Eqs. (3.17) and (3.18).
vii. Calculate the total force term (~
flb ) by using Eq. (3.19)
viii. Compute Fαby using Eq. (3.3b) and also calculate the new velocity by using
Eq. (3.7b) at t=tn+1.
ix. Repeat Step ii to viii until convergence is achieved.

M. H. Sedaghat et al. / Commun. Comput. Phys., 29 (2021), pp. 1411-1445 1421
4 Results and discussion
In this section the present numerical results are compared with pertinent analytical and
numerical studies, and the advantages of the present approach over others are also dis-
cussed.
4.1 Viscoelastic fluid flow in a channel
In this part, the present approach is verified in the absence of immersed bodies and other
external forces by setting ~
flb
IB M =0 and ~
flb
o=0 in Eq. (3.19). For this purpose, the steady
viscoelastic plane Poiseuille flow as well as unsteady viscoelastic Womersley flow has
been studied here.
4.1.1 Steady plane Poiseuille flow
In this section numerical simulations for the viscoelastic plane Poiseuille flow driven by a
fixed velocity profile at the entrance of the channel is simulated and the obtained results
have been compared with the analytical solution. The channel has a length of 10H and
width of H (H=0.4 m). The uniform grid is 400×40 which gives the spacing of mesh as
∆x=∆y=0.01 m. At the entrance of the channel (upstream), a parabolic velocity profile
is applied and also the value of the elastic stress tensor is set to zero. Therefore, the
boundary condition at the inlet of the channel can be derived as follows:
ux=Umax 
1− 
y
H/2!2
,uy=0, σExx =0, σExy =0, σEyy =0. (4.1)
At the outlet, the Neumann boundary conditions are imposed for the flow components:
∂ux
∂x=0, ∂uy
∂x=0, ∂σExx
∂x=0, ∂σExy
∂x=0, ∂σEyy
∂x=0. (4.2)
At walls, the following boundary conditions are imposed:
∂ux
∂y=0, uy=0, ∂σExx
∂y=0, ∂σExy
∂y=0, ∂σEyy
∂y=0. (4.3)
Under the steady state conditions, there exists an exact solution for the Oldroyd-B fluid
flow which is given by [16]:
u∗
x=1−(|2y∗|)2, (4.4)
σ∗
Exy =β∂u∗
x
∂y∗, (4.5)
σ∗
Exx =2Weβ∂u∗
x
∂y∗2
, (4.6)
σ∗
Eyy =0. (4.7)

1422 M. H. Sedaghat et al. / Commun. Comput. Phys., 29 (2021), pp. 1411-1445
(a)
(b)
Figure 2: (a) Comparison of numerical and analytical solutions of the fully developed nondimensional velocity
profiles versus y∗at We =0.6,β=0.7 and Re=45; (b) Numerical and analytical solutions of the profiles of
non-dimensional elastic shear stress component for three cases of viscosity ratio (β=0.3, 0.5, 0.7) at W e =0.6
and Re=45.
Fig. 2(a) shows the comparison of numerical and analytical solutions of the non-
dimensional velocity profiles (u∗
x) versus y∗at We =0.6, β=0.7 and Re=45. Here, the
velocity field is identical to the Newtonian fluid flow through the channel. Fig. 2(b) shows
the analytical and numerical comparison of non-dimensional shear stress component of
the elastic stress tensor (σ∗
Exy ) at Re=45 and We =0.6 and also for three different cases
(β=0.3, 0.5 and 0.7). In addition, Fig. 3 shows the variation of non-dimensional normal
stress component of the elastic stress tensor (σ∗
Exx ) with respect to the y∗for Re=45 and
various values of We and for three different viscosity ratios (β=0.3, 0.5 and 0.7). Accord-
ing to Figs. 2 to 3, we can clearly see that the present numerical solution of channel flow
agrees very well with the analytical solutions (Eqs. (4.4)-(4.6)) even for large values of We
number.

M. H. Sedaghat et al. / Commun. Comput. Phys., 29 (2021), pp. 1411-1445 1423
Figure 3: Numerical and analytical solutions of the profiles of non-dimensional elastic normal stress component
for three cases of viscosity ratio (β=0.3, 0.5, 0.7) at Re=45 and various amount of We.
4.1.2 Unsteady 2D Womersley flow
The two-dimensional Womersley flow (pulsatile flow in two-dimensional channel) is em-
ployed as the second test case to validate the proposed numerical scheme for unsteady
flow. The geometric configuration of the Womersley flow is identical to that of the plane
Poiseuille flow, but the flow is driven by a periodic pressure gradient at the entrance of
the channel. In this case a finer grid is used and the grid spacing is ∆x=∆y=0.0025 m and
the time step of the computation is ∆t=10−5s. Here, the amplitude of the oscillations is
denoted by A, and the frequency by ω
∂p
∂x=−Acos(ωt). (4.8)

1424 M. H. Sedaghat et al. / Commun. Comput. Phys., 29 (2021), pp. 1411-1445
Figure 4: Comparison of numerical and analytical solutions of fully developed non-dimensional velocity profiles
of Womersley Newtonian flow at five different times after initial run with respect to the y∗at β=0.7 and
Re=30.
The maximum velocity of the flow, reached in the middle of the channel at time t=n2π/ω
for any integer value n, is found to be:
Umax =A
8νL2
y. (4.9)
By submitting this expression for the pressure gradient into Navier-Stokes equation, the
velocity profile of Newtonian pulsatile flow in two-dimensional channel can be calcu-
lated as [41]:
ux=ℜ


A
iωρ 
1−coshh1
√2(γ+iγ)2y
Lyi
coshh1
√2(γ+iγ)i
eiωt


, (4.10)
where ℜdenotes the real part of the solution and γis the Womersley parameter as follows
[41]:
γ=Ly
2rω
ν. (4.11)
The convergence criterion is that:
max
u(xi,t+T)−u(xi,t)
u(xi,t+T)<10−6. (4.12)
Fig. 4 shows the comparison of numerical and analytical solutions of fully developed
non-dimensional velocity profiles of Womersley Newtonian flow at five different times

M. H. Sedaghat et al. / Commun. Comput. Phys., 29 (2021), pp. 1411-1445 1425
Figure 5: Variation of fully developed non-dimensional velocity profiles of viscoelastic Womersley flow for Re=30,
We =0.6 and β=0.3, 0.5 and 0.7 at three different times t=T/4,t=T/2 and t=T.
after initial run with respect to the y∗for β=0.7 and Re=30. This figure shows reasonable
agreement between our numerical results and those of the analytical solutions.
To further validate present numerical method, a two dimensional viscoelastic Wom-
ersley flow problem is reported here. As Fig. 5 shows, the viscosity ratio (β) has great
influence on velocity profile. Variation of non-dimensional elastic tensor components at
Re=30 and t=T/4 as a function of We for β=0.3, 0.5 and 0.7 are illustrated in Fig. 6.
Fig. 6(a) shows that both We and βaffect σ∗
Exx, but the effects of βis more significant.
In contrast to steady viscoelastic Poiseuille flow, Fig. 6(b) illustrates that σ∗
Eyy in the vis-

1426 M. H. Sedaghat et al. / Commun. Comput. Phys., 29 (2021), pp. 1411-1445
(a) (b) (c)
Figure 6: Variation of (a) σ∗
Exx , (b) σ∗
Eyy and (c) σ∗
Exy of viscoelastic Womersley flow for three cases of viscosity
ratio (β=0.3, 0.5, 0.7), Re=30 and various amounts of We at t=T/4.
coelastic Womersley flow is not zero and its value is enhanced by increasing the value
of β. This figure also shows that the effects of We on the σ∗
Eyy is more considerable at
higher values of viscosity ratio (β=0.7). Fig. 6(c) shows that in the steady viscoelastic
Poiseuille flow, σ∗
Exy is only influenced by β(Eq. (4.5)) but in viscoelastic Womersley flow,
both effects of We and βare important.
4.2 Viscoelastic four-roll mill
For the next validation case, a 2D four-roll mill geometry which consists of four cylin-
ders that rotate in a way to create a stagnation point between the rollers is studied. The
schematic geometry of this problem is plotted in Fig. 7. In this geometry the rollers can be
replaced by a body force which has the same effect of producing four vortices at the loca-
tion of the rollers [42,43]. The geometry is considered as a square of size [0,2π]×[0,2π].
The periodic boundary condition is imposed on each side of the square. The force term
which should be added to the momentum equations to simulate the rollers is given
by [44]:
~
f∗=2(sin(x)cos(y),−cos(x)sin(y)). (4.13)

M. H. Sedaghat et al. / Commun. Comput. Phys., 29 (2021), pp. 1411-1445 1427
Figure 7: Schematic geometry of the 2D four-roll mill.
(a) (b)
(c) (d)
Figure 8: Compression of Newtonian four-roll velocity distribution between analytical solution and current
numerical simulation for horizontal (a, b) and vertical (c, d) velocity component.
In a Newtonian Stokes flow by considering LR e f =2πand URe f =LRe f the four-roll velocity
field is given by [44]:
~u∗=(sin(x)cos(y),−cos(x)sin(y)). (4.14)
Fig. 8 indicates a good agreement between the numerical and analytical solution of the
horizontal and vertical velocity distribution of the Newtonian Stokes flow through the

1428 M. H. Sedaghat et al. / Commun. Comput. Phys., 29 (2021), pp. 1411-1445
four rolls. Next, Oldroyd-B fluid is sent through the four-roll geometry. The initial values
of the elastic stress tensor (σ∗
E) is set to zero. The periodic boundary conditions are used
for the stress tensor and velocity in both horizontal and vertical directions. The total
viscosity of the fluid is considered as η0=1.5 Pa.s and the viscosity ratio is set to β=1/3
[43].
Local elongation rate is calculated as:
˙
ε=∂ui
∂xi
. (4.15)
By using the local elongation rate at the centre of the box (x=y=π), the effective Weis-
senberg number which is an actual Weissenberg number by the local rate of strain at the
hyperbolic point can be defined as [42]:
Wie f f =˙
εWi. (4.16)
Fig. 9 shows the variation of elongation and effective Weissenberg number versus We
at the centre of an Oldroyd-B four-roll mill geometry for present numerical simulation
and study of Thomases and Shelley [43] and Malaspinas et al. [42] in the steady state
condition. This figure also shows a reasonable agreement between the current numerical
simulation and previous studies.
4.3 Newtonian fluid flow around a confined circular cylinder
Newtonian fluid flow around complex geometries can be simulated easily by omitting
the elastic effect of fluid (~
flb
E=0 in Eq. (3.19)). In this section simulations of Newtonian
fluid flows around stationary and rotating circular cylinders, which are confined at the
center, are carried out and the results are compared with previous numerical results. The
schematic geometry of this problem is shown in Fig. 10. All the boundary conditions for
velocity are the same as in Section 4.1.
4.3.1 Flow around a stationary confined circular cylinder
By setting the angular velocity of the cylinder to zero, (ΩC=0 in Fig. 10) the simulations
of Newtonian fluid around a stationary confined circular cylinder is carried out in this
section. The results of the present numerical simulation are compared with the results of
Bharti et al. [45]. As shown in Fig. 10 the channel has a blockage ratio ε=H/D=4 and
the lengths of the upstream and the downstream of the cylinder are chosen as Lu/D=10
and Ld/D=40. Fig. 11 compares the pressure coefficient on the circular cylinder with the
numerical results of Bharti et al. [45] for Re=10, 20, 40. The local pressure coefficient in
this section is computed from the following relation:
CP=Static Pressure
Dynamic Pressure =P(θ)−P∞
1
2ρU2
max
, (4.17)

M. H. Sedaghat et al. / Commun. Comput. Phys., 29 (2021), pp. 1411-1445 1429
(a)
(b)
Figure 9: Compression of (a) elongation rate and (b) effective Weissenberg number versus We of an Oldroyd-
B four-roll mill geometry between current numerical simulation and study of Thomases and Shelley [43] and
Malaspinas et al. [42].
Figure 10: Schematic geometry of the flow around confined cylinder in current research.
where P∞is related to the outlet pressure of the channel. As shown in Fig. 11, both re-
sults are in a good agreement. Additionally, a reasonable agreement can be observed by
comparison of total drag coefficient for a range of Reynolds numbers as given in Table 1.

1430 M. H. Sedaghat et al. / Commun. Comput. Phys., 29 (2021), pp. 1411-1445
Figure 11: Comparison of pressure coefficient (Cp) over the surface of the cylinder for the range of Reynolds
number (Re) between present study and Bharti et al. [45].
Fig. 12 shows the evolution of the streamline pattern behind the cylinder as Reynolds
number increases. As the value of the Reynolds number increases, the wake region is
characterized by a pair of counter-rotating vortices which remain attached. As Fig. 12
(a-d) shows that by increasing the value of Reynolds number from Re=10 to Re=40 the
wake length and also the width of the wake increases and the point of separation on the

M. H. Sedaghat et al. / Commun. Comput. Phys., 29 (2021), pp. 1411-1445 1431
Table 1: Comparison of total drag coefficient of Newtonian fluid flow around a confined circular cylinder.
Re Bharti et al. [45] Present Study
5 6.42 6.12
10 3.75 3.64
20 2.47 2.41
30 2.01 1.96
40 1.77 1.72
Figure 12: Streamlines of Newtonian fluid around a stationary circular cylinder at various Reynolds numbers.

1432 M. H. Sedaghat et al. / Commun. Comput. Phys., 29 (2021), pp. 1411-1445
(a) Re=75
(b) Re=100
Figure 13: Variation of drag coefficient on the cylinder versus time for Newtonian fluid flow around circular
cylinder at (a) Re=75 and (b) Re=100.
surface of the cylinder moves forward as commonly seen in well-established literatures.
Fig. 12 (e-f) also indicates that by increasing the value of Reynolds number, the viscous
forces are no longer sufficient to reverse these disturbances and the wake becomes unsta-
ble. This instability intensifies with increasing Reynolds number. To better understand
this unsteady problem, the variation of total drag coefficient around cylinder at Re=75
and 100 versus time has been plotted in Fig. 13 and it shows that a fully time-periodic
flow regime exists behind the cylinder. To test the grid independency of these simula-
tions, the model is run with different gird resolution. Table 2 lists the average total drag
coefficient on different meshes for Re=75 and 100. As this table represents the grid-
independent solution is obtained when using a mesh size larger than 1863×150. The
non-dimensional Strouhal number is used to describe the time period of oscillations or
the frequency of vortex shedding of this flow as follows:
St =fCD
Umax
, (4.18)

M. H. Sedaghat et al. / Commun. Comput. Phys., 29 (2021), pp. 1411-1445 1433
Table 2: Values of average total drag coefficient of Newtonian fluid on a confined circular cylinder obtained on
different meshes for Re=75 and 100.
PPPPPPP
P
Re
Nx×Ny1239×100 1489×120 1863 ×150 2738×220
75 1.48 1.456 1.443 1.443
100 1.404 1.383 1.347 1.35
where fcis the frequency of oscillations of drag coefficient with time. Based on Eq. (4.14)
the value of Strouhal number for the mentioned confined flow around the circular cylin-
der at Re=75 and 100 are 0.37 and 0.386, respectively.
4.3.2 Flow around a rotating confined circular cylinder
A cursory inspection of the aforementioned as well as other authoritative surveys clearly
show that the bulk of the literature for flow over cylinders relates to the case of stationary
cylinders. In this section, simulation of Newtonian fluid flow around a rotating confined
circular cylinder is carried out and the results are compared with the previous investiga-
tions. In this case as shown in Fig. 10 circular cylinder is placed at the middle height of
the channel and has a constant counterclockwise angular velocity Ωc. To show the grid-
independency of the code developed here, Table 3 shows the variation of average drag
coefficient on the cylinder for Re=20 and 30, and Ωc=10 rad/s. Fig. 14 compares the
pressure coefficient (Cp) on the lower channel wall for ε=H/D=10/7 with the results of
Figure 14: Comparison of pressure coefficient (Cp) of Newtonian fluid flow around rotating circular cylinder on
the lower channel wall for ε=10/7 between present study and Champmartin et al. [46].

1434 M. H. Sedaghat et al. / Commun. Comput. Phys., 29 (2021), pp. 1411-1445
Table 3: Values of average total drag coefficient of Newtonian fluid on a rotating circular cylinder obtained on
different meshes for Re=20 and 30 and ΩC=10 rad/s.
PPPPPPP
P
Re
Nx×Ny989×80 1114×90 1239×100 1489 ×120
20 2.5157 2.484 2.476 2.477
30 2.034 2.021 2.009 2.006
Champmartin et al. [46]. The pressure coefficient is defined as follows:
CP=P(x)−P∞
ηΩC
, (4.19)
where P∞is related to the outlet pressure of the channel and ηis the viscosity of the fluid.
These results indicate a good agreement between the present and previous simulations
indicating the validity of the present numerical approach.
Streamline and velocity vector around the cylinder at Re=20 and 30 for various
amounts of cylinder angular velocity has been plotted in Fig. 15. The results show
that when the angular velocity is increased the fluid tends to rotate around the cylin-
der. Streamlines also indicate that when the values of Ωcis increased the wake becomes
unstable and the viscous forces are no longer sufficient to overcome these disturbances.
This instability grows further as Reynolds number increases.
4.4 Viscoelastic fluid flow around a confined cylinder
By considering the effects of elastic as well as immersed boundary forces in the present
numerical method, the simulation of viscoelastic fluid flow around submerged bodies
can be carried out. In this section, as a benchmark case, the results of two-dimensional
Oldroyd-B fluid flow around a circular cylinder confined between two parallel plates
have been compared with the previous numerical studies. The geometry and the corre-
sponding boundary conditions used in this section is the same as the study of Dou and
Phan-Thien [47] which is shown in Fig. 10 with the following specifications:
The cylinder radius is R=3.188×10−3m, blockage ratio is ε=H/D=2, channel length
is 30Rand the lengths of the upstream and the downstream of the cylinder are chosen
to be 6Dand 8D, respectively. The fluid parameters are from the work of McKinley,
Armstrong and Brown [48], which is for a 0.31 wt.% PIB Boger fluid solution. The fluid
density is ρ=8.7534×10−3kg/m3, the total viscosity is η0=13.76 Pa.s, the viscosity ratio is
β=0.41 and the relaxation time is λ=0.794 s. It should be noted that the reference length
is LRe f =D/2 and the reference velocity is URe f =Uave based on Fig. 10 and Eq. (2.8). In
order to compare the results with the study of Dou and Phan-Thien [47], the Reynolds
number is kept less than 10−2.
Fig. 16 shows the comparison of the non-dimensional axial and lateral velocity pro-
file between the present numerical solution and the results of Dou and Phan-Thien [47].

M. H. Sedaghat et al. / Commun. Comput. Phys., 29 (2021), pp. 1411-1445 1435
Figure 15: Streamlines and velocity vector of Newtonian fluid flow around a rotating cylinder for (a) Re=20
and (b) Re=30.
These results show reasonable agreements between the present work and the work of
Dou and Phan-Thien [47]. Additionally, Table 4 shows very similar results for the to-
tal drag coefficient of Oldroyd-B fluid for the confined circular cylinder obtained from

1436 M. H. Sedaghat et al. / Commun. Comput. Phys., 29 (2021), pp. 1411-1445
Figure 16: Comparison of (a) non-dimensional axial velocity (b) non-dimensional lateral velocity profile of
Oldroyd-B fluid around confined circular cylinder between the present numerical solution and study of Dou and
Phan-Thien [47] for We =0.4 (maximum value is denoted by a circle, and minimum value by a rectangle).
Table 4: Comparison of total drag coefficient of Oldroyd-B fluid around a confined circular cylinder.
We Dou and Phan-Thien [47] Present Study
0.1 10.3230 10.3039
0.2 10.0591 10.1871
0.3 9.8290 9.6814
0.4 9.6736 9.4768
0.5 9.5952 9.0584
1 10.0972 9.8841
1.4 11.2959 11.8380
the present numerical simulations and the study of Dou and Phan-Thien [47] at various
values of Weissenberg numbers.
Next, use of the present numerical method to study viscoelastic fluid flow around
a confined elliptical cylinder is showcased for the first time to the best of the authors’

M. H. Sedaghat et al. / Commun. Comput. Phys., 29 (2021), pp. 1411-1445 1437
Figure 17: Non-dimensional (a) axial and (b) lateral velocity profile of Oldroyd-B fluid around confined circular
and elliptical cylinders for We =0.4.
knowledge. Here, the geometry and the fluid specification is the same as the study of
Dou and Phan-Thien [47] except that the circular cylinder is replaced by a horizontal
elliptical cylinder. The minor axis of the elliptical cylinder is D/2 and the aspect ratio is
αdefined as the ratio of the major axis of cylinder to the minor axis of it. Fig. 17 shows
the non-dimensional axial and lateral velocity distributions for the circular (α=1) and

1438 M. H. Sedaghat et al. / Commun. Comput. Phys., 29 (2021), pp. 1411-1445
Table 5: Values of total drag coefficient of Oldroyd-B fluid around confined circular and elliptical cylinders.
We α=1α=1.5 α=2
0.1 10.3039 12.6146 15.1885
0.2 10.1871 12.3127 14.8435
0.3 9.6814 12.0714 14.5762
0.4 9.4768 11.9975 14.5454
0.5 9.0584 11.8645 14.4078
elliptical cylinders with two different aspect ratios (α=1.5 and 2). The values of the
total drag coefficient around a circular and two different elliptical cylinders are given in
Table 5. The results indicate that at a given value of Weissenberg number, the total drag
coefficient increases as the aspect ratio increases because the frictional drag increases with
the aspect ratio due to increased surface area.
4.5 Muco-ciliary clearance
Upon validation of the numerical approach in previous sections, the present numerical
method is now examined for a biological problem which involves unsteady viscoelastic
flow, muco-ciliary transport of lungs.
Muco-ciliary transport is one of the main protection mechanisms of the upper air-
ways. The inner walls of the airways are covered by the airway surface liquid (ASL)
and ASL consists of two layers, the inner layer is called the periciliary liquid layer (PCL),
which is a watery lubricating fluid, and the outer layer is called the mucus layer, which
consists of highly viscous and viscoelastic fluid. An array of cilia is immersed in the PCL.
Each cilium performs a repetitive beat cycle consisting of recovery and effective strokes.
During the effective stroke each cilium makes contact with the PCL layer and propels it
forward which leads to transport of the mucus layer, with entrapped particles, from the
airways.
The cilia motion is prescribed for the model and the cilia beat pattern in our study is
the one reported in Fulford and Blake [49]. The cilia beat pattern and a schematic geom-
etry of this problem with appropriate boundary conditions are shown in Fig. 18. More
details about the application of the present numerical method for muco-ciliary transport
can be found at [50–52], but without proper validation of the numerical approach. For
the present simulation, the ciliary length is LCil ia =6µm [53], the spacing between any
two neighboring cilia along the epithelium wall is chosen as d=3µm [54,55], the depth of
the PCL is LPCL =6µm [56], the depth of the mucus layer is LM=4µm [53], the viscosity
of PCL is ηPCL =0.001 Pa.s [53] and the viscosity of mucus layer is ηM=0.0482 Pa.s [53].
As mentioned before ηMis decomposed into viscous and elastic parts. In this study
we assume that the standard viscosity of Newtonian part of mucus is as the same as
the viscosity of PCL i.e. ηM,N=0.001 Pa.s and the elastic part of the mucus viscosity is
ηM,E=0.0472 Pa.s. For these parameters the viscosity ratio is β=ηM,E/ηM=0.98. The

M. H. Sedaghat et al. / Commun. Comput. Phys., 29 (2021), pp. 1411-1445 1439
(a)
(b)
Figure 18: (a) The beat cycle of each cilium [49]. (b) A schematic illustration for the muco-ciliary transport
problem.
standard value of the relaxation time of the mucus is about λ=0.034 s [53] and the den-
sity of the fluid is ρ=1000 kg/m3[53]. The computational mesh is chosen as 195×50
(∆x=∆y≈2×10−7), the number of control points along each cilium is 20. The time step
is ∆t=8×10−7s. The velocity field of muco-ciliary transport process based on our nu-
merical simulation has been represented in a video in the online supplement.
In our previous studies [50–52], free slip boundary condition in the LBM (which is re-
ported by Succi [57]) was used as the boundary condition on the upper layer. This bound-
ary condition does not have a good accuracy in estimating the velocity near the top layer.
Therefore, in this study, we improved this boundary condition and hence zero-gradient
boundary condition has been employed to the upper layer. Our simulation using this
boundary condition and the mentioned standard parameters in healthy state predicts a

1440 M. H. Sedaghat et al. / Commun. Comput. Phys., 29 (2021), pp. 1411-1445
time average mucus velocity of u0=85.7µm/s where u0is calculated as:
u0=1
TC(Lt−LPCL)ZTC
0ZLt
LPCL
uMdydt. (4.20)
In Eq. (4.9) Ltis the total depth of ASL, TCis the consecutive beat cycle and uMis the
mucus velocity.
Mean mucus velocity in healthy state is affected by many physiological factors such
as age, sex, posture, sleep and exercise. In addition, the method chosen to measure mucus
transport velocity may also affect the results, as is the case for the determination of CBF
[58]. Thus previous studies showed a wide range of muco-ciliary clearance values within
a group of healthy persons. Yeates et al. [59] reported mucus flow in the trachea of 42
healthy nonsmoking adults to average 60µm/s with a coefficient of variation of 75%.
Foster et al. [60] showed that the mucus velocity in the main bronchi was 40µm/s and
tracheal mucus velocity 92µm/s in a group of seven healthy subjects. Friedman et al.
[61] measured mean mucus velocity in tracheal in 12 normal nonsmoking subjects and
found values in the range of 111.6−190µm/s. Salathe et al. [62] reported a range of
values of between 67 and 333µm/s for mucus flow. Previous investigations also reported
a wide range of mean mucus velocity based on simulation of muco-ciliary transport.
Numerical simulation of Lee et al. [54] and Jayathilake et al. [55] by considering mucus
as a Newtonian fluid predicted a mean mucus velocity of 44.38µm/s using the standard
parameter set. Smith et al. [53] by considering mucus as a linear Maxwell viscoelastic
fluid showed a mean mucus velocity of 38.3µm/s. The present numerical simulation has
a reasonable prediction of mucus flow with the most of previous reported results.
5 Conclusions
In this study, a physically appealing numerical approach was presented to study vis-
coelastic fluid flow interaction with complex boundaries. In this hybrid method, the
Newtonian part of the momentum equations was solved by the Lattice Boltzmann
method and the effect of submerged objects as well as the elastic behavior of the fluid
on flow pattern was added to the Boltzmann equations as the force terms.
The main features of the current numerical method are summarized as follows:
•In this method, the main part of the momentum equations (which contains non-
linear terms) has been solved by the LBM, which has many advantages including
clear physical pictures, easy implementation, and availability of full-parallelization
algorithms. The finite difference method was employed for solving the constitu-
tive equation of viscoelastic fluid. Therefore, this hybrid method does not have
the common restrictions seen for traditional CFD methods such as use of staggered
grids, divergence of the numerical solution due to the existence of nonlinear ad-
vection term in the momentum equations and solving the Poisson equation for the
pressure.

M. H. Sedaghat et al. / Commun. Comput. Phys., 29 (2021), pp. 1411-1445 1441
•Use of the finite difference method to solve the constitutive equation of viscoelastic
fluid has the following advantages:
i. The appropriate simulation of the problem can be accomplished based on
the physical rheological properties of the fluid while in pure LBM only non-
dimensional parameters have been considered.
ii. In contrast with previous pure LBM for simulating of limited model of vis-
coelastic fluid, this method has the capability of solving the all the constitutive
equation of viscoelastic fluid in which stress tensor can be decomposed into
elastic and viscous parts.
•By introducing transformation coefficients in this method, some restriction in pure
LBM has been relieved. For instance, by choosing proper values for grid and time
spacing, more accurate values for velocity and viscosity transformation coefficients
have been obtained. Therefore, regardless of restriction of Mach number (Ma) and
single relaxation time in LBM, the physical values of viscosity and velocity of the
fluid can be imported in the current method.
•One of the advantageous of the current study is use of an implicit form of immersed
boundary method. Therefore, viscoelastic fluid flow around stationary and moving
complex bodies has been carried out easily without considering the complexities
of grid generation. However, this method has some disadvantageous specially for
unsteady fluid flows because after updating the velocity field the momentum equa-
tions is not solved based on these updated velocities. Some convergence criterion
has been considered in unsteady flow to overcome this issue.
•Due to the hybrid nature of this method, as remarked earlier, by removing the elas-
tic or immersed boundary forces from the momentum equations, different fluid
flows as well as some simple benchmarks e.g. like viscoelastic Poiseuille flow or
Newtonian fluid flow around cylinders can be implemented with this approach.
•One of the restrictions of this method is the different time step that may be chosen
for the LBM (for solving Newtonian part of the momentum equations) and finite
difference method (for solving elastic part of stress tensor). This increases the over-
all processing time, especially in unsteady problems.
The efficiency and capability of the present method was illustrated through several
benchmark problems. Oldroyd-B Womersley Flow as well as viscoelastic fluid flow
around a confined elliptical cylinder has been studied for the first time in this study.
Finally, muco-ciliary transport process was simulated using the present approach and
the results have a reasonable agreement with the previous results.

1442 M. H. Sedaghat et al. / Commun. Comput. Phys., 29 (2021), pp. 1411-1445
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