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© 2016 The Korean Society of Rheology and Springer 255
Korea-Australia Rheology Journal, 28(4), 255-265 (November 2016)DOI: 10.1007/s13367-016-0027-2
www.springer.com/13367
pISSN 1226-119X eISSN 2093-7660
Creeping flow of Herschel-Bulkley fluids in collapsible channels: A numerical study
Ali Amini, Amir Saman Eghtesad and Kayvan Sadeghy*
Center of Excellence for the Design and Optimization of Energy Systems (CEDOES), School of Mechanical Engineering, College of Engineering, University of Tehran, Tehran 11155-4563, Tehran, Iran
(Received March 23, 2016; final revision received July 16, 2016; accepted July 25, 2016)
In this paper, the steady flow of a viscoplastic fluid is modeled in a planar channel equipped with a deform-able segment in the middle of an otherwise rigid plate. The fluid is assumed to obey the Herschel-Bulkleymodel which accounts for both the yield stress and the shear-thinning behavior of physiological fluids suchas blood. To accommodate the large deformations of the flexible segment, it is assumed to obey the two-parameter Mooney-Rivlin hyperelastic model. The so-called fluid-structure interaction problem is thensolved numerically, under creeping-flow conditions, using the finite element package, COMSOL. It is foundthat the yield stress leads to a larger wall deformation and a higher pressure drop as compared with New-tonian fluids. This behavior is predicted to intensify if the fluid is shear-thinning. That is, for a given yieldstress, the pressure drop and the wall deformation both increase with an increase in the degree of the fluid'sshear-thinning behavior.
Keywords: collapsible channel, Herschel-Bulkley fluid, Mooney-Rivlin solid, COMSOL software
1. Introduction
Flow through collapsible tubes and channels constitutes
an important class of fluid-solid-interaction (FSI) prob-
lems with many applications in physiological and engi-
neering systems (Grotberg and Jensen, 2004). One can
mention, for example, blood flow through veins and arter-
ies, air flow through lungs, fluid flow through the urethra,
and food movement in the intestines. Simulating this type
of flow is not an easy task simply because the shape of the
deformable wall is not known a priori. To this should be
added the fact physiological fluids such as blood are
known to exhibit significant non-Newtonian behavior
under certain conditions. The nonlinearity of the consti-
tutive behavior of fluids such as blood together with the
large deformations which are involved in real situations
contributes to make studying such flows a difficult task. In
one of the very first attempts to address such problems,
Katz et al. (1969) proposed a lumped parameter model in
order to simulate the flow of a Newtonian fluid in a
deformable tube. Later, Morgan and Parker (1989) devel-
oped a one-dimensional model that was numerically
solved for a Newtonian fluid and a deformable wall which
obeyed the general tube law. The main feature of their
model was its capability in predicting the wave pressure
transferred by the vessel walls. Concerned with the math-
ematical aspect of the FSI problem and its hyperbolic gov-
erning equations, Elad et al. (1991) established a kind of
analogy between the incompressible flow through a
deformable channel and the compressible flow through
rigid boundaries. They showed that when the speed index,
as defined by Shapiro (1977), is higher than unity a dis-
continuous change in flow velocity occurs similar to that
observed for a shock wave in supersonic flows.
Tang et al. (1999) developed a 3D model with different
levels of stenosis using the commercial software, ADINA,
in which the wall was modeled as a hyperelastic solid
obeying the Ogden model with the fluid being Newtonian.
Bertram and Tscherry (2006) showed that the flexibility of
airways and blood vessels may result in a number of phe-
nomena such as flow limitation and self-excited vibration
of the walls (which causes wheezing in the airways). It
was demonstrated that when the cross-section of the tube
loses its circular shape, it becomes much more vulnerable
to further deformation and collapse. This motivated a
number of ensuing works dealing with the instability of
collapsible tubes (Jensen, 1990; Bertram et al., 1990;
Pihler-Puzovi¢ and Pedley, 2013; Kudenatti et al., 2012;
Pourjafar et al., 2015). There were also some efforts for
developing more reliable models to determine the stress
field in the deformed wall utilizing thick-wall hyperelastic
models instead of the simpler thin-wall assumptions
(Bertram and Elliott, 2003; Rohan et al., 2013; Kozlovsky
et al., 2014).
As to the role played by a fluid’s rheology, Janela et al.
(2010a; 2010b) considered flow of four different shear-
thinning fluids in a collapsible tube. In a similar study
(Hundertmark-Zaušková and Lukáčová-Medvid’ová, 2010),
use was made of the Carreau and Yeleswarapu models for
representing the fluid. Also, a 3D model of a blood bypass
was created in CFX software (Kabinejadian and Ghista,
2012) where the blood was modeled as a shear-thinning
fluid. These works have shown that shear-thinning plays a
key role in decreasing the wall shear stress and the pres-*Corresponding author; E-mail: sadeghy@ut.ac.ir
Ali Amini, Amir Saman Eghtesad and Kayvan Sadeghy
256 Korea-Australia Rheology J., 28(4), 2016
sure loss along the channel. Chen et al. (2015) relied on
the constant-viscosity Oldroyd-B model to demonstrate
that a fluid’s elasticity when combined with the wall’s
elasticity may give rise to free oscillations. In a more
recent work, Chakraborty et al. (2010), and Chakraborty
and Prakash (2015) relied on the variable-viscosity FENE-
P and Owens viscoelastic fluid models to show that the
shear-thinning characteristic of the blood is more import-
ant than its viscoelastic behavior in its flow through col-
lapsible channels. They showed that the flow pattern and
wall deformation for constant-viscosity elastic fluids were
almost identical to those of a Newtonian fluid of the same
viscosity. In contrast, they found dramatic change in the
flow characteristic when shear-thinning fluids are involved.
Surprisingly, there appears to be no published work address-
ing the effect of a fluid’s yield stress on its flow through
collapsible channels. In this paper, we try to address the
effect of a fluid’s yield stress on the deformation of col-
lapsible channels, to the best of our knowledge, for the
first time. We also investigate the effect of shear-thinning
and also external loading (pressure) on the deformation of
the complaint section of the channel.
The work is developed as follows: In the next section,
we present the mathematical formulations for the fluid and
the solid sides together with the required boundary con-
ditions. To that end, the deformable section of the channel
is modeled as a finite-thickness hyperelastic solid which
obeys the two-mode Mooney-Rivlin model with the fluid
obeying the Herschel-Bulkley model. We then proceed
with briefly describing the numerical method of solution,
i.e., the finite-element-method. Numerical results are pre-
sented next together with discussing their physical signif-
icance. The work is concluded by highlighting its major
findings.
2. Mathematical Formulation
Fig. 1 shows a schematic of the flow configuration
adopted for study in the present work. The channel is two-
dimensional with its lower wall made of a rigid material.
The upper wall is made of the same rigid material except-
ing its middle section which is made of a compliant mate-
rial. We have relied on the same dimensions as used in
Luo et al. (2007) in our simulations such that comparison
can be made with published data for Newtonian fluids at
a later stage of the work. To that end, we set: Lu = 7H, L =
5H, Ld = 7H, h = 0.1H where H is the channel's height and
t is the thickness of the deformable segment (see Fig. 1).
We present the equations governing the fluid and the solid
sides separately. This will be followed by presenting the
boundary conditions for the fluid and the solid, separately.
2.1. The fluid sideAs to the equations governing the fluid flowing through
the channel, we start with the Cauchy equations of motion
together with the continuity equation, i.e.,
, (1)
(2)
where the fluid has been assumed to be incompressible. In
Eq. (1), D/Dt is the material derivative, v is the velocity
vector, p is the isotropic pressure, and τ is the deviatoric
stress tensor. The stress tensor must be related to the
velocity field through an appropriate constitutive equation.
In the present work, it is assumed that the fluid of interest
obeys the Herschel-Bulkley model; that is (Macosko, 1994),
(3)
where τy is the yield stress, m is the consistency index, n
is the power-law index, and is the rate-of-
deformation tensor with the superscript T denoting the
transpose of the velocity-gradient tensor, . It is worth-
mentioning that the above model reduces to the power-law
model by simply setting . For non-zero τy the model
reduces to the well-known Bingham model for n = 1 with
m now serving as the plastic viscosity (Macosko, 1994).
The model reduces to Newtonian fluid model by simply
setting , n = 1 with m now serving as the dynamic
viscosity. It should also be stressed that the material as
represented by Eq. (3) does not flow for τ < τy. Thus the
state of stress is unknown in the un-yielded (or, plug)
region for this popular rheological model. This is compu-
tationally awkward in the pressure/velocity finite volume
or finite element schemes where one normally solves first
for the velocity field and then for the deformation field. To
circumvent this problem, we have decided to rely on the
regularized version of the Herschel-Bulkley model which
reads as (Macosko, 1994):
(4)
where for sufficiently large a the true Herschel-Bulkley
model can be closely recovered. It is to be noted that, the
term in this equation is the second invariant of the
rate-of-deformation tensor which in Cartesian coordinate
ρDv
Dt------- = ∇– p + ∇ τ⋅
∇ v⋅ = 0
τ = m II2D
n 1–
2----------
+ τy
II2D1/2
--------------- 2D( )
2D = ∇vT + ∇v
∇v
τy = 0
τy = 0
τ = m II2D
n 1–
2----------
+ τy 1 exp– a II2D
1/2–( )[ ]
II2D1/2
----------------------------------------------------⎝ ⎠⎜ ⎟⎛ ⎞
2D( )
II2D
Fig. 1. (Color online) Schematic of the flow configuration used
in this study.
Creeping flow of Herschel-Bulkley fluids in collapsible channels: a numerical study
Korea-Australia Rheology J., 28(4), 2016 257
system can be written as:
(5)
where the flow has been assumed to be two-dimensional
in the Eulerian frame of reference, x = (x, y).
2.2. The solid sideIn Lagrangian frame of reference, X = (X, Y), the equa-
tions governing the solid can be written as (Lai et al.,
2009),
, (6)
(7)
where u(X) is the deformation field, ρs is the solid’s den-
sity, F is the deformation gradient tensor, and P is the
Piola-Kirchhof stress tensor defined by (Lai et al., 2009),
(8)
where σ = −pδ + τ is the total stress tensor with τ being its
deviatoric part. In the present work, we assume that the
solid is a hyperelastic material obeying the two-mode
Mooney-Rivlin model. The Mooney-Rivlin model reason-
ably well fits the rheological data for silicone-based rub-
ber-like materials and the blood vessels (Lai et al., 2009).
This is mainly because (unlike the neo-Hookean) it has an
extra parameter which enables it to account for the non-
zero second normal stress difference in elastic materials.
For this solid model, the strain energy function, W, can be
written as (Lai et al., 2009) :
(9)
where IB = trB and are the two
invariants of the Cauchy-Green strain tensor. Also, we
have: B = FFT where is the deformation gra-
dient tensor, with xi being the coordinates at time t of a
material point whose coordinates are Xj in the un-deformed
reference configuration (i, j = 1, 2, 3). For isochoric defor-
mations (i.e., where detF = 1 so that we have IIB = trB−1)
the stress tensor for the two-mode Mooney-Rivlin solid is
written as (Macosko, 1994; Lai et al., 2009):
(10)
where C1 > 0 and C2 > 0 are the material properties related
to the first and second normal stress differences, respec-
tively. It is to be noted that this solid model reduces to the
neo-Hookean model by simply setting C2 = 0 (Macosko,
1994). In dimensionless form the stress tensor for the
Mooney-Rivlin solid can be written as:
(11)
where is called the elasticity parameter. In this
equation, the tilde symbol (~) refers to dimensionless
quantities, but, for convenience, it will be dropped from
the rest of the work. It needs to be mentioned that we have
relied on C1 to make all stress terms dimensionless, and on
W for scaling the length. In dimensionless form the
momentum equation for the solid side becomes:
(12)
where can be referred to as the solid’s
Reynolds number even though the solid model adopted in
the present work represents non-viscous materials only. In
the present work, however, the left-hand-side of this equa-
tion is zero because we are interested in steady situation
only so that we have: .
2.3. Boundary conditionsThe above system of equations requires appropriate
boundary conditions to become amenable to a numerical
solution. To that end, the velocity is assumed to be uni-
form at the inlet of the channel. At the outlet, the gage
pressure is set equal to zero. For the rigid lower wall and
the rigid sections of the upper wall, we invoke the no-slip
and no-penetration boundary conditions. For the deform-
able section of the upper wall, only the no-slip condition
is used. (Since the solid is deforming, the no-slip condi-
tion means that at any point on the deformable section, the
fluid velocity is the same as that of the solid wall.) In addi-
tion, at the interface between the fluid and the solid, the
shear and normal stresses should be equal to each other.
3. Numerical Method
At present, for solving the equations governing the
fluid-solid interaction for viscoelastic fluids one has no
choice other than developing his/her code (Chen et al.,
2015; Chakraborty et al., 2010; Chakraborty and Prakash,
2015). The situation becomes totally different if the non-
Newtonian fluid is inelastic. For such non-Newtonian flu-
ids (which are called generalized Newtonian fluids or
GNFs) one can rely on commercial software such as Flu-
ent or CFX for solving the fluid equations and on com-
mercial software such as Abaqus and/or Ansys for solving
the coupled solid equations (Vierendeels et al., 2007;
Kabinejadiana and Ghista, 2012). In practice, this is often
realized to be quite inconvenient. Fortunately, there are
three finite element fluid packages which are self-con-
tained in this regard, i.e., they do not depend on another
solid software in the course of their simulations. These
software are: Adina, Fidap, and COMSOL. We have
already mentioned the performance of Adina in simulating
FSI problems (Tang et al., 1999). Similarly, Fidap has
II2D=1
2---– tr 2D( )( )2 tr 2D( )2( )–[ ]= 4–
∂vx
∂x-------
∂vy
∂y-------
⎝ ⎠⎛ ⎞+
∂vx
∂y-------
∂vy
∂x-------
⎝ ⎠⎛ ⎞
2
det F( ) = 1
ρs
∂2u
∂t2
--------
X
= ∇X P⋅
P = σ F1–( )
T
2W = C1 IB 3–( ) + C2 IIB 3–( )
IIB = 12--- trB( )2 tr B
2( )–[ ]
F = ∂xi/∂Xj
σ = ps
– δ +2∂W
∂IB--------B
∂W
∂IIB---------B
1––⎝ ⎠
⎛ ⎞ = ps
– δ +2C1B 2– C2B1–
σ̃ = ps
– δ +B̃ − ζ B̃1–
ζ = C2/C1
Res
∂2u
∂t2
--------
X
= ∇X P⋅
Res = ρs C1R2/η
2( )
0 = ∇X P⋅
Ali Amini, Amir Saman Eghtesad and Kayvan Sadeghy
258 Korea-Australia Rheology J., 28(4), 2016
been used with great success for simulating flow of New-
tonian fluids in collapsible tubes (Luo and Pedley, 1995).
For GNF fluids, however, it requires developing a user-
defined function (UDF) for taking care of the fluid’s shear-
dependent viscosity. In contrast, COMSOL does not
require such an extra and tedious step simply because it
has built-in functions for a variety of GNF fluids. For
Newtonian fluids, this software has shown its efficiency in
simulating flow through flexible micro-channels (Ozsun et
al., 2013). It has also proven very successful for simulat-
ing the flow of blood through flexible vein and venous
valve (Wijeratne and Hoo, 2008). In the latter work, blood
was assumed to obey the Quemada model which belongs
to the class of GNF fluids. The Bingham model also
belongs to this class of non-Newtonian fluids. Therefore,
it appears that COMSOL should face with no major dif-
ficulty in simulating the flow of a Bingham fluid in our
challenging fluid-solid interaction problem, as depicted in
Fig. 1. For this reason, we have decided to rely on the
solver of this versatile software for solving the coupled
governing equations developed in the previous section for
the fluid and the solid. A segregated iterative method
(SIM) embedded in this software is used for this purpose
(Janela et al., 2010a).
In the SIM, the velocity and the pressure fields for the
fluid are computed at each interaction step based on the
deformation experienced by the compliant wall in the pre-
vious step. Once the flow field is known, the fluid stress
tensor at the wall is calculated in a semi-implicit way from
the following equation (Janela et al., 2010b; Hundertmark-
Zaušková and Lukáčová-Medvid’ová, 2010),
(13)
where it is assumed that the shear-dependent viscosity has
already been found knowing the local shear rates for the
previous interaction. In this equation the superscripts i and
i−1 denote the parameter values at the present and previ-
ous iteration steps, respectively. The stress tensor from
this equation is then translated into the force vector and is
applied on the wall boundaries. That is, these forces are
the boundary conditions for the solid domain so that wall
deformation for the current interaction step can be com-
puted (Hundertmark-Zaušková and Lukáčová-Medvid’ová,
2010). For the newly-obtained geometry, the mesh grid is
updated, the flow field is re-computed for this grid, and
this procedure is repeated. At the end of each interaction
step, the convergence criteria are checked, and the numer-
ical procedure is continued until the convergence criterion
is met for each material. The convergence criterion adopted
in this work was set at 0.01% of the relative error for each
of the variables in the solution vector. This vector is
formed by the horizontal and vertical components of the
velocity and the pressure of the fluid nodes, and also the
horizontal and vertical displacements of the solid nodes.
Because of the nonlinear nature of the problem at hand, an
under-relaxation factor of α = 0.4 is applied in order to
smoothen the obtained values at each interaction step.
Therefore, the updated vector of variables is then calcu-
lated as:
. (14)
In the next step, the wall is modeled using N elements
in the COMSOL package. The same first-order triangular
elements are also used for the fluid domain (Janela et al.,
2010b). Due to the simple nature of the N1 + N2 elements
utilized in this work (for solving the velocity and the pres-
sure fields in the fluid domain), it was necessary to apply
the “streamline diffusion method” in order to stabilize the
solution. Fig. 2 represents a magnified view of the grid
used for the simulations near the elastic boundary (which
extends from −250 to +250). It is to be noted that the
dimensions shown in this figure are in micrometers. As
can be seen in Fig. 2, versatile triangular elements, suit-
able for large deformations, have been chosen for discret-
izing the geometry.
In order to check the accuracy of the solution scheme, it
was used to reproduce numerical results reported by Luo
et al. (2007) for the flow of a Newtonian fluid in the chan-
nel depicted in Fig. 1 with the only difference being that
the compliant segment was made of a Hooken solid instead
of the Mooney-Rivlin solid. Fig. 3 shows a comparison
between the two sets of results. As can be seen in this fig-
ure, our numerical scheme is doing a nice job for New-
τi = p
i– δ +μ γ·( )
i 1–∇ufluid
i ∇ufluid
i( )T
+( )[ ]
Un = U
n 1– + α U
newU
n 1––( )
Fig. 2. The triangular elements used in COMSOL software for
the simulations.
Fig. 3. (Color online) A comparison between our numerical
results (red line) with published numerical results reported (blue
line) as reported in Ref. 23.
Creeping flow of Herschel-Bulkley fluids in collapsible channels: a numerical study
Korea-Australia Rheology J., 28(4), 2016 259
tonian/Hookean combination. (The deformable segment
stretches from x = 5 cm to x = 10 cm in this figure.)
The next step was to determine the appropriate grid size
to ensure mesh-independent results. Three different grids
were tried for this purpose labeled as: “fine”, “normal”,
and “coarse” corresponding to 32864, 20629, and 5904
elements, respectively. The regulating parameter in the
Herschel-Bulkley model, a, was set equal to 109, as rec-
ommended in Chen et al. (2014), for these simulations. It
was found that the normal grid can ensure grid-indepen-
dent results (Amini, 2015; Eghtesad, 2016).
4. Results and Discussions
In this section, we present our new numerical results for
the flow of a Herschel-Bulkley fluid through the collaps-
ible channel shown schematically in Fig. 1. We have been
able to obtain converged results for different set of param-
eters. We were primarily interested in investigating the
effect of yield stress and the power-law index on the flow
characteristic. For curiosity, the effect of external loading
as caused by muscular contraction (which drives blood
against gravity from our legs to our heart) is also studied.
Only typical results are presented here; see Amini (2015)
for more results. To that end, we set ζ = 2 and Re = 10−5
which correspond to an inlet velocity of 1 μm/s.
4.1. Effect of the yield stressIn this sub-section we address the effect of the yield
stress on the flow characteristics. To that end, the yield
stress is first made dimensionless by introducing the ratio:
where V is the average velocity (which is
the same as the uniform inlet velocity in our case). This
ratio will be referred to as the Bingham number in the fig-
ures to be presented shortly (Macosko, 1994). In all these
simulations, it is assumed that the deformation is caused
by a change in the outside pressure.
Fig. 4 shows the effect of the Bingham number on the
displacement of the deformable section of the channel
depicted in Fig. 1. As can be seen in Fig. 4, the fluid's yield
stress plays a pivotal role in determining the wall's defor-
mation. Obviously, by an increase in the fluid's yield stress
the wall displacement is increased.
Fig. 5 shows the effect of the Bingham number on the
velocity profiles at the middle of the channel. This figure
includes the case of Newtonian fluids for comparison pur-
poses. For Newtonian fluids, the velocity profile looks
more or less like a parabolic profile. However, by an
increase in the fluid's yield stress the centerline velocity is
deceased and the profile becomes progressively flatter
(say, for a given flow rate). This suggests that a plug
might have been formed in the central region of the chan-
nel. The effect of the yield stress on the centerline velocity
can be explained by noting that with an increase in the
fluid's yield stress, the wall bends outward more severley,
and so the flow passage widens. Since the flow rate is
constant, this gives rise to a drop in velocity, as can be
seen in Fig. 5.
From the results depicted in Fig. 5 one can conclude that
the fluid having the largest yield stress is associated with
the largest frictional losses, despite having a relatively
lower maximum velocity. Fig. 6 shows that this is indeed
the case. That is, fluids having a larger yield stress are
associated with higher pressure drops along the channel.
For such fluids, although there exists an un-yielded region
in the domain, the flow has to reduce its velocity from
maximum at the centerline to zero at the wall in a much
shorter distance. In practice, this results in a higher shear
rate and a larger wall shear stress. Thus the pressure drop
Bn = Ty
m----- H/V( )n
Fig. 4. (Color online) Effect of the Bingham number on the wall
deformation (n = 1). In this figure “wall length” means the axial
distance along the deformable segment of the upper plate which
is stretched from x = 0 mm to x = 500 mm.
Fig. 5. (Color online) Effect of the Bingham number on the
velocity profile obtained at the middle of the channel (n = 1).
Ali Amini, Amir Saman Eghtesad and Kayvan Sadeghy
260 Korea-Australia Rheology J., 28(4), 2016
should increase by an increase in the yield stress, as can
be seen in Fig. 7. It should also be noted that the pressure
inside the channel plays a very important role in deform-
ing the flexible segment. Actually, it serves as a part of the
boundary conditions which must be met by the deform-
able wall.
In obtaining the above results it has tacitly been assumed
that the deformation is initiated by changing the external
pressure (which is uniform along the outer surface of the
deformable segment). To see what happens when the
external pressure is fixed (say, at 0.0775 Pa) and it is the
internal pressure which causes the deformation, we have
plotted Fig. 7. This figure again reveals that the fluid’s
yield stress plays an important role in dictating the wall
displacement. Interestingly, unlike the previous case, the
deformation of the segment is not symmetrical. This is not
surprising realizing the fact that the internal pressure var-
ies (nonlinearly) along the deformable segment (see Fig.
6). In fact, depending on the inlet pressure and the pres-
sure outside the segment, the variation of the inside pres-
sure by the action of the shear stress (which depends on
the Bn number) causes the segment to deflect inwardly or
outwardly in a rather nonlinear fashion.
4.2. Effect of shear-thinningThe power-law index, n, in the Herschel-Bulkley model
allows it to account for the shear-thinning behavior of
physiological fluids such as blood and industrial materials
such as polymeric liquids. To investigate the role played
by shear-thinning in the deformation of compliant section
of the channel depicted in Fig. 1, we fix the yield stress
(say, at 0.0006 Pa) and lower the power-law exponent, n,
starting from one. Fig. 8 shows the effect of shear-thinning
on the velocity profiles at the middle of the channel and
also the pressure variation along the channel's centerline.
As can be seen in this figure, the asymmetry of the veloc-
Fig. 6. (Color online) Effect of the Bingham number on the pres-
sure drop along the channel centerline (n = 1).
Fig. 7. (Color online) Effect of the Bingham number on the ver-
tical displacement of the deformable segment along its length
(n = 1) when the outside pressure is fixed.
Fig. 8. (Color online) Effect of the power-law index, n, on the velocity profiles at the middle of the channel (left plot) and pressure
variation along the centerline (right plot).
Creeping flow of Herschel-Bulkley fluids in collapsible channels: a numerical study
Korea-Australia Rheology J., 28(4), 2016 261
ity profiles becomes more evident when n is decreased.
Also, the profiles become flatter by a decrease in n (i.e.,
by an increase in the degree of shear-thinning). The asym-
metry of the velocity profiles is not surprising realizing
the fact that the lower wall is completely rigid. On the
other hand, the prediction that the velocity profiles are
flatter for shear-thinning fluids suggests that a larger por-
tion of the channel is now occupied by un-yielded fluid.
With the velocity profiles being affected by n, one would
expect to see a strong effect on the pressure variation, too.
Another look at Fig. 8 reveals that this is indeed the case.
Surprisingly, however, the pressure gradient is predicted to
be larger for shear thinning fluids as compared with the
Newtonian fluids. That is, for a fixed outlet (gauge) pres-
sure of zero, the solver evidently needs a larger inlet pres-
sure for a given flow rate.
To explain the rather unexpected prediction that Her-
schel-Bulkley fluid needs a larger pressure gradient for a
fixed flow rate, we resort to Fig. 9 which shows the flow-
curve for a Herschel-Bulkley fluid as a function of n in
simple shear. This figure shows that for Herschel-Bulkley
fluids, there exists a critical shear rate (equal to 1 s−1) below
which lower n corresponds to a larger viscosity. With this
in mind, we have plotted the shear rate distribution along
the upper plate (Fig. 10) and realized that indeed the shear
rates on this plate are lower than 1 s−1. This means a larger
shear stress (as can be seen in Fig. 10c) which reflects
itself by a larger pressure gradient (Amini, 2015).
Fig. 11 shows the effect of power-law index on the
deformable segment’s vertical displacement. The strong
effect of n on the deformation of the segment is evident in
this figure. This is not surprising realizing the fact that, at
any given section, a lower n corresponds to a higher pres-
sure) simply because the shear rates everywhere are lower
than the critical rate. A higher internal pressure results in
a larger outward wall displacement when n is decreased,
as can be seen in Fig. 11.
4.3. Effect of the external loadingExternal loading is an important mechanism for the fluid
transport in collapsible tubes. Normally, one would expect
to see a suppression of the wall displacement by an
Fig. 9. (Color online) Effect of the power-law index, n, on the
dynamic viscosity profiles for a Herschel-Bulkley fluid in simple
shear.
Fig. 10. (Color online) Effect of the power-law index, n, on: (a)
the shear rate distribution, (b) viscosity variation, and (c) shear
stress distribution along the deformable section of the collaps-
ible channel.
Ali Amini, Amir Saman Eghtesad and Kayvan Sadeghy
262 Korea-Australia Rheology J., 28(4), 2016
increase in the outside pressure. To check if this is indeed
true in our case, we have carried out simulations for the
case of m = 0.01 Pa·s, and τy = 0.001 Pa, and n = 0.5. For
this set of simulations, the pressure inside the channel was
initially set equal to zero. As such, negative values of the
external loading correspond to a vacuum whereas positive
values mean a compression. Fig. 12 shows the wall dis-
Fig. 11. (Color online) Effect of the power-law exponent, n, on
the displacement of the deformable segment of the channel.
Fig. 12. (Color online) Effect of external loading, Pext, on the
shape of the deformable section of the channel.
Fig. 13. (Color online) Velocity magnitude and wall Von Mises stress contours when relative vacuum (upper plot) or relative pressure
(lower plot) is applied on the deformable section of the channel.
Creeping flow of Herschel-Bulkley fluids in collapsible channels: a numerical study
Korea-Australia Rheology J., 28(4), 2016 263
placement for both negative and positive values of the
ambient (gauge) pressure exerted on the deformable sec-
tion of the upper wall. As can be seen in this figure, the
external pressure dramatically affects the wall deforma-
tion. As stated earlier, there is a strong coupling between
the flow kinematics and the wall displacement. An out-
wardly-distorted wall enlarges the channel and reduces the
velocity gradients so that the pressure loss due to the shear
stress decreases. On the other hand, when the wall is forced
to deflect inwardly (say, by applying a positive pressure),
the channel's height is reduced and the resistance against
the flow is intensified. It is interesting to note that even
when the external pressure is equal to the internal pressure
(i.e., Pex = 0) the wall still deflects outwardly.
Fig. 13 shows how for a Herschel-Bulkley fluid the
velocity field and also the wall's Von Mises stress are con-
trolled by the external loading. This figure clearly shows
the crucial role played by the external loading on the flow
characteristics near the deformable section of the channel.
The upper plot in Fig. 13 shows the fluid velocity profiles
at different sections of the channel for the case in which
the ambient pressure is negative. On the other hand, the
plot at the bottom of this figure corresponds to a positive
outside pressure. The dark red color in the latter plot (say,
in the middle of the channel) shows that the fluid is
moving with a slightly larger velocity as compared with
the upper plot. This highly accelerated flow, which is
related to the constant inlet flow condition and the fluid
incompressibility, leads to a higher shear stress and con-
sequently to a larger pressure loss. This can best be seen
in Fig. 14 which shows how the pressure varies along the
channel's centerline for different values of the external
pressure.
4.4. The ΔP−Q relationship
Finding the relationship between the flow rate passing
through any arbitrary passage and the corresponding pres-
sure drop is of outmost importance in fluid mechanics. For
Newtonian fluids in Poiseuille flow, under laminar condi-
tions, there is a linear relationship between the pressure
drop and the flow rate. The question then arises as to what
it would be if the fluid is viscoplastic and/or the wall is
deformable. To address this issue, four different cases are
studied here, and for each of them the relationship between
pressure and flow rate was obtained. Fig. 15 shows the
ΔP−Q curve for each case. As can be seen in this figure,
the pressure drop for Herschel-Bulkley fluids is less than
that for Newtonian fluids. And, for each fluid, wall flex-
ibility lowers the pressure drop. To explain these results it
should be noted that, for a deformable wall (in the absence
of external loading) the fluid pressure enlarges the channel
by pushing the upper wall outwards. A wider channel
means that the average and the maximum velocities are
reduced in the mid-section of the channel so that the
velocity and thereby the shear rates are reduced. This low-
ers the shear stresses and consequently the pressure loss.
The extent to which the pressure drop is reduced depends
directly on the wall displacement. A larger displacement
means that a wider space is available for the fluid to pass
through, and this causes a lower pressure drop along the
channel. This is why the “deformable segment” curves in
this figure deviate more from the rigid ones when the flow
rate is increased. This argument is also valid for shear-
thinning fluids.
5. Concluding Remarks
In this paper, we have tried to investigate the effects of
Fig. 14. (Color online) Pressure variations along the channel
centerline for different outside pressures.
Fig. 15. (Color online) Pressure drop as a function of the flow
rate for rigid and flexible channels.
Ali Amini, Amir Saman Eghtesad and Kayvan Sadeghy
264 Korea-Australia Rheology J., 28(4), 2016
a fluid's rheology (say, its yield stress and its shear-thin-
ning behavior) in a planar channel equipped with a deform-
able segment in the middle of its upper (otherwise rigid)
wall. To that end, the fluid was assumed to obey the Her-
schel-Bulkley model. To accommodate large deformations
of the flexible segment, use was made of the two-param-
eter Mooney-Rivlin model. The so-called fluid-structure
interaction problem was then solved numerically under
creeping conditions using the finite element package,
COMSOL. It is found that the yield stress leads to a larger
wall deformation and a higher pressure drop as compared
with its Newtonian counterpart. This behavior is intensi-
fied when the fluid becomes more shear-thinning. That is,
for a given yield stress, the pressure drop and the wall
deformation both increase by an increase in the degree of
shear-thinning. It was also shown that the wall flexibility
contributes to a lower pressure drop in comparison with a
rigid boundary. In the former case, it is predicted that the
pressure inside the channel pushes the compliant wall out-
wardly and widens the channel. A wider channel (for a
fixed flow rate) means that the maximum velocity is
reduced, and consequently the shear stress is decreased.
Work is currently ongoing to investigate the effect of a
pulsatile driving pressure, typical of physiological sys-
tems, on the FSI problem.
Acknowledgements
The authors wish to express their sincerest thanks to Iran
National Science Foundation (INSF) for supporting this
work under contract number 95815139. Special thanks are
also due to the respectful reviewers for their constructive
comments.
List of Symbols
a : Regulating parameter
C1 : Mooney-Rivlin first elastic parameter
C2 : Mooney-Rivlin second elastic parameter
D : Rate of deformation tensor
E : Elastic modulus
f : Body force vector
F : Deformation gradient tensor
h : Thickness of the deformable section
H : Height of the channel
I : Identity tensor
J : Determinant of the deformation gradient tensor
Ldown : Length of the downstream rigid section of the
wall
Lup : Length of the upstream rigid section of the wall
Lw : Length of the compliant wall
m : Consistency index
n : Power-law index
N : Number of elements in COMSOL
p : Fluid pressure
P : Piola-Kirchhof stress tensor
t : Time
u : Fluid horizontal component velocity
Uin : Fluid inlet velocity
: Fluid velocity gradient
v : Fluid vertical component velocity
W : Mooney-Rivlin strain energy
x, y : Coordinate in Eulerian frame of reference
X, Y : Coordinate in Lagrangian frame of reference
Greek Symbols
I : First invariant of the finger tensor
II : Second invariant of the finger tensor
α : Under-relaxation factor
: Shear rate
ε : Strain tensor
η : Shear-dependent viscosity
κ : Bulk modulus
ρ : Fluid density
σ : Total stress tensor
τ : Fluid shear stress
τy : Fluid’s yield stress
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