Non-Newtonian fluidstructure interaction Flow of a viscoelastic Oldroyd-B fluid in a deformable channel

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Non-Newtonian fluid–structure interaction: Flow of a viscoelastic
Oldroyd-B fluid in a deformable channel
Evgeniy Boykoa,b,∗,1, Ivan C. Christova
aSchool of Mechanical Engineering, Purdue University, West Lafayette, Indiana 47907, USA
bDavidson School of Chemical Engineering, Purdue University, West Lafayette, Indiana 47907, USA
ARTICLE INFO
Keywords:
Fluid–structure interaction
Viscoelasticity
Oldroyd-B fluid
Lubrication theory
Pressure drop
Microfluidics
ABSTRACT
We analyze the steady non-Newtonian fluid–structure interaction between the flow of an Oldroyd-B
fluid and a deformable channel. Specifically, we provide a theoretical framework for calculating the
leading-order effect of the fluid’s viscoelasticity on the flow rate–pressure drop relation and on the
deformation of the channel’s elastic wall. We first identify the characteristic scales and dimension-
less parameters governing the fluid–structure interaction in slender and shallow channels. Applying
the lubrication approximation for the flow and employing a perturbation expansion in powers of the
Deborah number , we derive a closed-form expression for the pressure as a function of the non-
uniform shape of the channel in the weakly viscoelastic limit up to O(). Coupling the hydrodynamic
pressure to the elastic deformation, we provide the leading-order effect of the interplay between the
viscoelasticity of the fluid and the compliance of the channel on the pressure and deformation fields, as
well as on the flow rate–pressure drop relation. For the flow-rate-controlled regime and in the weakly
viscoelastic limit, we show analytically that both the compliance of the deforming top wall and the
viscoelasticity of the fluid decrease the pressure drop. Furthermore, we reveal a trade-off between the
influence of compliance of the channel and the fluid’s viscoelasticity on the deformation. While the
channel’s compliance increases the deformation, the fluid’s viscoelasticity decreases it.
1. Introduction
In recent years, the fluid–structure interaction (FSI) be-
tween viscous fluids and the soft deformable configurations
they flow through has received considerable attention in the
scientific community due to its relevance to microfluidic,
lab-on-a-chip, and soft robotics applications [1,2,3,4]. FSI
is not limited to Newtonian fluids, and it arises in various
microfluidic applications involving complex fluids, such as
ones containing proteins, colloidal dispersions, nucleic acids,
or polymeric solutions [5,6,7,8,9]. In these cases, the in-
terplay between the compliance of the confining boundaries
and the complex rheological behavior of the non-Newtonian
fluids involved affects the FSI in new ways that have not been
fully understood. Specifically, the rheological behavior of
the fluid is featured in the deformation of the soft fluidic con-
duit, as well as in the relationship between the pressure drop
Δand the volumetric flow rate [4].
Table 1lists a chronological selection of previous work
on the steady fluid-structure interaction between complex
non-Newtonian fluids and deformable configurations. From
this table, we conclude that the main focus of the previ-
ous theoretical studies to date has been on shear-dependent
power-law fluids. However, beyond shear thinning, complex
fluids are characterized by other rheological features such as
viscoelasticity, and thus, it is of fundamental and practical
∗Corresponding author
eboyko@purdue.edu ( Evgeniy Boyko); christov@purdue.edu (
Ivan C. Christov)
ORCID(s): 0000-0002-9202-5154 ( Evgeniy Boyko);
0000-0001-8531-0531 ( Ivan C. Christov)
1Present address: Faculty of Mechanical Engineering, Tech-
nion – Israel Institute of Technology, Haifa 3200003, Israel; evg-
boyko@technion.ac.il
importance to understand how these features affect the FSI.
Recently, Ramos-Arzola & Bautista [17] studied theo-
retically the fluid–structure interaction between a simplified
Phan-Thien–Tanner (PTT) fluid [18,19] flow and a slen-
der and shallow deformable microchannel. Using lubrica-
tion theory and linear elasticity and neglecting the solvent
contribution, Ramos-Arzola & Bautista [17] derived an im-
plicit nonlinear first-order ordinary differential equation for
the flow rate–pressure relation, which depends on the com-
pliance parameter and the product PTT 2, where PTT is
the extensibility parameter of the PTT model, and   is the
Weissenberg number defined in Sec. 2.1. For a fixed flow
rate, their results predicted a decrease in the pressure drop
with increasing PTT 2. However, such a reduction in the
pressure drop arises due to shear-thinning effects of the PTT
model, which are manifested when PTT 2increases, and
is consistent with results of previous theoretical studies em-
ploying the simpler shear-thinning power-law model [14].
Moreover, for PTT = 0, when the PTT model corresponds
to the Oldroyd-B model, the solution of Ramos-Arzola &
Bautista [17] for the − Δrelation reduces to the Newto-
nian relations derived by Christov et al. [20] and Shidhore
& Christov [21], which are independent of the fluid’s vis-
coelasticity. Previous investigations of an Oldroyd-B fluid
in a rigid but non-uniformly shaped channel [22] showed
that the Oldroyd-B model’s flow differs from a Newtonian
one, which introduces viscoelastic corrections to the pres-
sure drop. Thus, one should anticipate that the viscoelastic-
ity of the complex fluid affects the fluid–structure interac-
tion, even under the Oldroyd-B model.
To the best of the authors’ knowledge, the fluid–structure
interaction between a constant-shear-viscosity viscoelastic
(Boger) fluid and a three-dimensional deformable channel
E. Boyko and I. C. Christov: Preprint submitted to Elsevier Page 1 of 12
arXiv:2210.13268v2 [physics.flu-dyn] 23 Jan 2023
Flow of a viscoelastic Oldroyd-B fluid in a deformable channel
Table 1
Chronological selection of previous experimental, numerical, and theoretical studies on the steady fluid–structure interaction of
complex, non-Newtonian fluids flowing in deformable configurations.
Reference Focus Geometry/Solid model Fluid/model
Chakraborty et al. [10,11] Numer. Two-dimensional channel with a short collapsible
(elastic) segment
Oldroyd-B, FENE-P, and
Owens
Yushutin [12] Theor. Axisymmetric tube as Winkler foundation Power law, moderate
Reynolds number
Tanner et al. [5] Exptl./
Numer.
Thin tube with non-axisymmetric large
deformation; simulations in ‘static’ geometry
Carboxymethyl-cellulose
aqueous solution/Carreau
Raj & Sen [6] Exptl./
Theor.
Slender and shallow rectangular channel with
elastic plate-like top wall
Polyethylene oxide solution/
Newtonian model only
Del Giudice et al. [7] Exptl. Square-cross-section channel extruded from large
block of PDMS
Polyethylene oxide solution
Raj et al. [8] Exptl. Axisymmetric tube extruded from large block of
PDMS
Xanthan gum solution
Poroshina & Vedeneev [13] Theor. Slender and thin axisymmetric shell with axial
tension
Power law, high Reynolds
number
Nahar et al. [9] Exptl. Slender and thin tube, non-axisymmetric
deformations
Carboxymethyl-cellulose,
polyacrylamide aqueous
solutions
Anand et al. [14] Theor./
Numer.
Slender and shallow rectangular channel with
elastic plate-like top wall
Power law
Anand & Christov [15] Theor./
Numer.
Slender and thin axisymmetric Donnell shell with
axial bending
Power law
Vedeneev [16] Theor. Geometrically nonlinear axisymmetric shell without
bending; hyperelastic material
Power law, high Reynolds
number
Ramos-Arzola & Bautista [17] Theor. Slender and shallow rectangular channel with
elastic plate-like top wall
Simplified PTT
Present work Theor. Slender and shallow rectangular channel with
elastic plate-like top wall
Oldroyd-B
has not been analyzed in the literature, even for “simple”
models such as Oldroyd-B and FENE-CR in the weakly vis-
coelastic limit, which motivates this study.
In this work, we provide a theoretical framework for cal-
culating the leading-order effect of the fluid’s viscoelastic-
ity on the flow rate–pressure drop relation and on the defor-
mation of the channel’s elastic wall. Our framework cap-
tures the effect of the viscoelasticity of the fluid using the
Oldroyd-B model. In Sec. 2, we present the problem formu-
lation and the dimensional governing equations. We further
identify the characteristic scales and dimensionless parame-
ters governing the fluid–structure interaction and provide the
reduced lubrication equations for an Oldroyd-B fluid flow
in a slender and shallow channel in dimensionless form. In
Sec. 3, we present a low-Deborah-number lubrication anal-
ysis and derive a closed-form expression for the pressure as
a function of the non-uniform shape of the channel up to
O(). Coupling the resulting expression for the pressure to
the elastic deformation, in Sec. 4we provide analytical solu-
tions for the pressure distribution and pressure drop account-
ing for the leading-order effect of the fluid’s viscoelasticity
and the channel wall’s compliance. As a concrete example of
our theoretical approach, in Sec. 5, we present results for an
Oldroyd-B fluid in a deformable channel with a compliant
top wall modeled using the Kirchhoff–Love plate-bending
theory. We conclude with a discussion in Sec. 6.
2. Problem formulation and governing
equations
We study the steady fluid–structure interaction between
a non-Newtonian viscoelastic dilute polymer solution and a
slender, shallow and deformable channel of length 𝓁, width
, and (deformed) height , where     𝓁, as shown
in Fig. 1. The fluid flow has a velocity field 𝒗= (, , )
and pressure distribution , which are induced by the im-
posed flow rate . We seek to determine the resulting axial
pressure drop Δ≜(= 0) − (=𝓁)for a given . The
channel’s top wall is soft, while its sidewalls are assumed to
be rigid. As the fluid flows through the channel, the fluid
stresses deform the fluid–solid interface along the channel’s
top wall. We denote by (, )the vertical displacement
of the fluid–solid interface, so that its position is given by
=(, ) = 0+(, ), where 0is the undeformed
E. Boyko and I. C. Christov: Preprint submitted to Elsevier Page 2 of 12
Flow of a viscoelastic Oldroyd-B fluid in a deformable channel
w
y
x
z
b
E
l
a
s
t
i
c
s
o
l
i
d
Vi
s
c
oe
l
asti
c
p
o
l
ymer
s
o
l
u
t
i
o
n
∆p
uy(x,z)
h(x,z)h0
q
Figure 1: Schematic illustration of the problem geometry: a
three-dimensional deformable channel of length 𝓁with an ini-
tially rectangular cross-section of width and height 0. A
viscoelastic dilute polymer solution flows steadily within the
channel, driven by an imposed flow rate . The fluid flow
stresses cause a deformation (, )of the fluid–solid inter-
face, which affects the pressure drop Δover the axial distance
𝓁. The channel’s sidewalls at = ±∕2, as well as the bottom
surface at = 0, are assumed to be rigid. The elastic top wall
has a thickness .
height of the channel (i.e., in the absence of the flow). We
further assume that the top wall of the channel has a constant
thickness and constant material properties (i.e., a Young’s
modulus and a Poisson’s ratio ).
We consider low-Reynolds-number flows, so that the fluid
inertia is negligible compared to viscous (Newtonian and
viscoelastic) stresses. In this limit, the continuity and mo-
mentum equations governing the fluid flow take the form
𝛁⋅𝒗= 0,𝛁⋅𝝈=𝟎,(1)
where 𝝈is the stress tensor given by
𝝈= −𝑰+ 2𝑬
Solvent
+𝝉
Polymer
.(2)
The first term on the right-hand side of Eq. (2) is the pressure
contribution, the second term is the viscous stress contribu-
tion of the Newtonian solvent with a constant viscosity ,
where 𝑬= (𝛁𝒗+ (𝛁𝒗)T)∕2 is the rate-of-strain tensor, and
the last term, 𝝉, is the polymer contribution to the stress ten-
sor, for which a separate constitutive equation needs to be
specified [23].
We describe the viscoelastic behavior of the complex
fluid using the Oldroyd-B constitutive model [24], which is
a well-established continuum model for viscoelastic fluids
with constant shear viscosity (i.e., Boger fluids) [23]. Impor-
tantly, the Oldroyd-B constitutive model can be derived from
microscopic principles by modeling polymer molecules as
elastic dumbbells being advected and stretched by the flow
and having a linear restoring force [25]. In the Oldroyd-B
model, the deviatoric stress tensor is the sum of the New-
tonian solvent and polymer contributions, as shown in Eq.
(2). At steady state, the polymer contribution 𝝉to the fluid’s
stress tensor satisfies the upper-convected Maxwell consti-
tutive equation
𝝉+[𝒗⋅𝛁𝝉− (𝛁𝒗)T⋅𝝉−𝝉⋅(𝛁𝒗)] = 2𝑬,(3)
where is the polymer contribution to the shear viscosity
at zero shear rate, and is the longest relaxation time of the
polymers [23,25,26].
Using Eqs. (2) and (3), the stress tensor 𝝈can be also
expressed as
𝝈= −𝑰+ 20𝑬
Newtonian
−[𝒗⋅𝛁𝝉− (𝛁𝒗)T⋅𝝉−𝝉⋅(𝛁𝒗)]

Viscoelastic
,
(4)
where 0=+is the total zero-shear-rate viscosity of
the polymer solution.
Substituting Eq. (4) into the second relation in Eq. (1)
provides an alternative form of the momentum equation:
𝛁=0∇2𝒗−𝛁⋅[𝒗⋅𝛁𝝉− (𝛁𝒗)T⋅𝝉−𝝉⋅(𝛁𝒗)],(5)
which, as we show below, is convenient for assessing the
viscoelastic effects on the steady flow and pressure fields of
an Oldroyd-B fluid.
2.1. Scaling analysis and non-dimensionalization
In this work, we analyze the fluid–structure interaction
of a slender and shallow deformable channel in which  
  𝓁. We consider a flow-rate-controlled situation, in
which the characteristic axial velocity scale is set by the
flow rate as =∕(0).
We introduce dimensionless variables based on lubrica-
tion theory [27,28,29,30,31,22]:
=
, =
0
, =
𝓁,(6a)
=
, =
, =
,(6b)
=2𝓁
0
, =
0
, =
,(6c)
 =𝓁
0
, =𝓁
0
, =2𝓁
0
,(6d)
 =𝓁
0
, =𝓁
0
, =𝓁
0
,(6e)
where is the characteristic scale of deformation of the
top wall and we have introduced two dimensionless parame-
ters that quantify the slenderness and the shallowness of the
channel,
≜0
𝓁1and ≜0
1,(7)
which are assumed to be small; the viscosity ratios,
≜
+
=
0
and ≜1 −
=
0
,(8)
and the Deborah and Weissenberg numbers,
 ≜
𝓁=
0𝓁and   ≜
0
=
2
0
.(9)
E. Boyko and I. C. Christov: Preprint submitted to Elsevier Page 3 of 12
摘要：

Non-Newtonianuidstructureinteraction:FlowofaviscoelasticOldroyd-BuidinadeformablechannelEvgeniyBoykoa,b,<,1,IvanC.ChristovaaSchoolofMechanicalEngineering,PurdueUniversity,WestLafayette,Indiana47907,USAbDavidsonSchoolofChemicalEngineering,PurdueUniversity,WestLafayette,Indiana47907,USAARTICLEINFOK...

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