Creeping flow of Herschel-Bulkley fluids in collapsible ...Creeping flow of Herschel-Bulkley fluids...

© 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: [email protected]

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⋅



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.



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).



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).



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.



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.



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.



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

References

Amini, A., 2015, Steady Flow of Herschel-Bulkley Fluids in Col-

lapsible Channels: a Numerical Study, MSc Thesis, University

of Tehran.

Bertram, C.D., C.J. Raymond, and T.J. Pedley, 1990, Mapping of

instabilities during flow through collapsed tubes of different

length, J. Fluids Struct. 4, 125-153.

Bertram, C.D. and N.S.J. Elliott, 2003, Flow-rate limitation in a

uniform thin-walled collapsible tube, with comparison to a uni-

form thick-walled tube and a tube of tapering thickness, J. Flu-

ids Struct. 17, 541-559.

Bertram, C.D. and J. Tscherry, 2006, The onset of flow-rate lim-

itation and flow-induced oscillations in collapsible tubes, J.

Fluids Struct. 22, 1029-1045.

Chakraborty, D. and J.R. Prakash, 2015, Viscoelastic fluid flow

in a 2D channel bounded above by a deformable finite-thick-

ness elastic wall, J. Non-Newton. Fluid Mech. 218, 83-98.

Chakraborty, D., M. Bajaj, L. Yeo, J. Friend, M. Pasquali, and

J.R. Prakash, 2010, Viscoelastic flow in a two-dimensional col-

lapsible channel, J. Non-Newton. Fluid Mech. 165, 1204-1218.

Chen, S.G., Q.C. Sun, F. Jin, and J.G. Liu, 2014, Simulations of

Bingham plastic flows with the multiple-relaxation-time lattice

Boltzmann model, Sci. China-Phys. Mech. Astron. 57, 532-540.

Chen, X., M. Schäfer, and D. Bothe, 2015, Numerical modeling

and investigation of viscoelastic fluid-structure interaction

applying an implicit partitioned coupling algorithm, J. Fluids

Struct. 54, 390-421.

Eghtesad, A.S., 2016, Pulsatile Flow of Herschel-Bulkley Fluids

in Collapsible Channels: A Numerical Study, MSc Thesis, Uni-

versity of Tehran.

∇U

γ·



Elad, D., D. Katz, E. Kimmel, and S. Einav, 1991, Numerical

schemes for unsteady fluid flow through collapsible tubes, J.

Biomed. Eng. 13, 10-18.

Hundertmark-Zaušková, A. and M. Lukáčová-Medvid’ová, 2010,

Numerical study of shear-dependent non-Newtonian fluids in

compliant vessels, Comput. Math. Appl. 60, 572-590.

Grotberg, J.B. and O.E. Jensen, 2004, Biofluid mechanics in flex-

ible tubes, Annu. Rev. Fluid Mech. 36, 121-147.

Janela, J., A. Moura, and A. Sequeira, 2010a, A 3D non-New-

tonian fluid-structure interaction model for blood flow in arter-

ies, J. Comput. Appl. Math. 234, 2783-2791.

Janela, J., A. Moura, and A. Sequeira, 2010b, Absorbing bound-

ary conditions for a 3D non-Newtonian fluid-structure interac-

tion model for blood flow in arteries, Int. J. Eng. Sci. 48, 1332-

1349.

Jensen, O.E., 1990, Instabilities of flow in a collapsed tube, J.

Fluid Mech. 220, 623-659.

Kabinejadian, F. and D.N. Ghista, 2012, Compliant model of a

coupled sequential coronary arterial bypass graft: Effects of

vessel wall elasticity and non-Newtonian rheology on blood

flow regime and hemodynamic parameters distribution, Med.

Eng. Phys. 34, 860-872.

Katz, A.I., Y. Chen, and A.H. Moreno, 1969, Flow through a col-

lapsible tube: Experimental analysis and mathematical model,

Biophys. J. 9, 1261-1279.

Kozlovsky, P., U. Zaretsky, A.J. Jaffa, and D. Elad, 2014, General

tube law for collapsible thin and thick-wall tubes, J. Biomech.

47, 2378-2384.

Kudenatti, R.B., N.M. Bujurke, and T.J. Pedley, 2012, Stability of

two-dimensional collapsible-channel flow at high Reynolds

number, J. Fluid Mech. 705, 371-386.

Lai, W.M., D.H. Rubin, and E. Krempl, 2009, Introduction to

Continuum Mechanics, 4th ed., Butterworth-Heinemann, Oxford.

Luo, X., B. Calderhead, H. Liu, and W. Li, 2007, On the initial

configurations of collapsible channel flow, Comput. Struct. 85,

977-987.

Luo, X.Y. and T.J. Pedley, 1995, A numerical simulation of

steady flow in a 2-D collapsible channel, J. Fluids Struct. 9,

149-174.

Macosko, C.W., 1994, Rheology: Principles, Measurements and

Applications, 1st ed., Wiley-VCH, Newyork.

Morgan, P. and K.H. Parker, 1989, A mathematical model of flow

through a collapsible tube-I. model and steady flow results, J.

Biomech. 22, 1263-1270.

Ozsun, O., V. Yakhot, and K.L. Ekinci, 2013, Non-invasive mea-

surement of the pressure distribution in a deformable micro-

channel, J. Fluid Mech. 734, R1.

Pihler-Puzovi¢, D. and T.J. Pedley, 2013, Stability of high-Reyn-

olds-number flow in a collapsible channel, J. Fluid Mech. 714,

536-561.

Pourjafar, M., H. Hamedi, and K. Sadeghy, 2015, Stability of

power-law fluids in creeping plane Poiseuille: The effect of

wall compliance, J. Non-Newton. Fluid Mech. 216, 22-30.

Rohan, C.P-Y., P. Badel, B. Lun, D. Rastel, and S. Avril, 2013,

Biomechanical response of varicose veins to elastic compres-

sion: A numerical study, J. Biomech. 46, 599-603.

Shapiro, A.H., 1977, Steady flow in collapsible tubes, J. Biomech.

Eng.-Trans. ASME 99, 126-147.

Tang, D., C. Yang, and D.N. Ku, 1999, A 3-D thin-wall model

with fluid-structure interactions for blood flow in carotid arter-

ies with symmetric and asymmetric stenoses, Comput. Struct.

72, 357-377.

Vierendeels, J., L. Lanoye, J. Degroote, and P. Verdonck, 2007,

Implicit coupling of partitioned fluid-structure interaction prob-

lems with reduced order models, Comput. Struct. 85, 970-976.

Wijeratne, N.S. and K.A. Hoo, 2008, Numerical studies on the

hemodynamics in the human vein and venous valve, American

Control Conference, Seattle, USA, 147-152.

Creeping flow of Herschel-Bulkley fluids in collapsible ...Creeping flow of Herschel-Bulkley fluids...

Documents

Transcript of Creeping flow of Herschel-Bulkley fluids in collapsible ...Creeping flow of Herschel-Bulkley fluids...