Deposition of shear‐thinning viscoelastic fluids by an elongated bubble in a circular channel regarding the weakly elastic regime

Abstract

Thin‐film deposition of fluids is ubiquitous in a wide range of engineering and biological applications, such as surface coating, polymer processing, and biomedical device fabrication. While the thin viscous film deposition in Newtonian fluids has been extensively investigated, the deposition dynamics in frequently encountered non‐Newtonian complex fluids remain elusive, with respect to predictive scaling laws for the film thickness. Here, we investigate the deposition of thin films of shear‐thinning viscoelastic fluids by the motion of a long bubble translating in a circular capillary tube. Considering the weakly elastic regime with a shear‐thinning viscosity, we provide a quantitative measurement of the film thickness with systematic experiments. We further harness the recently developed hydrodynamic lubrication theory to quantitatively rationalize our experimental observations considering the effective capillary number CaeCaeCa_\mathrm{e} and the effective Weissenberg number WieWieWi_\mathrm{e}, which describe the shear‐thinning and the viscoelastic effects on the film formation, respectively. The obtained scaling law agrees reasonably well with the experimentally measured film thickness for all test fluids. Our work may potentially advance the fundamental understanding of the thin‐film deposition in a confined geometry and provide valuable engineering guidance for processes that incorporate thin‐film flows and non‐Newtonian fluids.

Full text

Received: 8 December 2023 Revised: 29 February 2024 Accepted: 3 March 2024
DOI: 10.1002/dro2.121
RESEARCH ARTICLE
Deposition of shear-thinning viscoelastic fluids by an elongated
bubble in a circular channel regarding the weakly elastic regime
SungGyu Chun1Zhengyu Yang1Jie Feng1,2
1Department of Mechanical Science and
Engineering, University of Illinois
Urbana-Champaign, Urbana, Illinois, USA
2Materials Research Laboratory, University of
Illinois Urbana-Champaign, Urbana, Illinois,
USA
Correspondence
Jie Feng, Department of Mechanical Science
and Engineering, University of Illinois
Urbana-Champaign, Urbana, IL 61801, USA.
Email: jiefeng@illinois.edu
Funding information
American Chemical Society Petroleum
Research Fund, Grant/Award Number:
61574-DNI9
Abstract
Thin-film deposition of fluids is ubiquitous in a wide range of engineering and biological
applications, such as surface coating, polymer processing, and biomedical device fabri-
cation. While the thin viscous film deposition in Newtonian fluids has been extensively
investigated, the deposition dynamics in frequently encountered non-Newtonian com-
plex fluids remain elusive, with respect to predictive scaling laws for the film thickness.
Here, we investigate the deposition of thin films of shear-thinning viscoelastic fluids
by the motion of a long bubble translating in a circular capillary tube. Considering
the weakly elastic regime with a shear-thinning viscosity, we provide a quantitative
measurement of the film thickness with systematic experiments. We further harness
the recently developed hydrodynamic lubrication theory to quantitatively rationalize
our experimental observations considering the effective capillary number Caeand the
effective Weissenberg number Wie, which describe the shear-thinning and the vis-
coelastic effects on the film formation, respectively. The obtained scaling law agrees
reasonably well with the experimentally measured film thickness for all test fluids. Our
work may potentially advance the fundamental understanding of the thin-film deposi-
tion in a confined geometry and provide valuable engineering guidance for processes
that incorporate thin-film flows and non-Newtonian fluids.
INTRODUCTION
The deposition of a continuous thin liquid film through the displace-
ment of the liquid phase by air in a confined geometry is a phenomenon
that has been a crucial objective in many engineering and biological-
relevant settings, such as in enhanced oil recovery,1,2 drug delivery,3,4
gas-assisted injection molding,5coating processes,6,7 and biomedi-
cal engineering.8–11 Flow configurations employed to investigate the
aforementioned applications typically involve the formation of a thin
liquid film upon a long bubble advancing into a tube prefilled with
fluids.12,13 Correspondingly, the study of flow dynamics involving bub-
bles in a confinement has been extensively investigated across various
This is an open access article under the terms of the Creative Commons Attribution License, which permits use, distribution and reproduction in any medium, provided
the original work is properly cited.
© 2024 The Author(s). Droplet published by Jilin University and John Wiley & Sons Australia, Ltd.
geometries and fluid characteristics. The pioneering works were car-
ried out by Bretherton12 and Taylor.14 They considered the configura-
tion involving a long bubble moving within confined geometries of small
dimensions where gravitational and inertial effects have negligible
impact. The film dynamics are governed only by the interplay between
viscosity and surface tension, captured by the capillary number, Ca =
𝜂U∕𝜎,where𝜂,U,and𝜎are the fluid viscosity, the speed of the bub-
ble, and the surface tension of the fluid, respectively.7,15 When Ca ≪1,
Bretherton12 discovered that the liquid film thickness hfollows the
scaling relationship as h∕R∼Ca2∕3,whereRis the radius of the cap-
illary tube. This relation was later extended to cover a wider range
of Ca <2 in the scaling analysis proposed by Aussillous and Quéré,15
Droplet. 2024;3:e121. wileyonlinelibrary.com/journal/dro2 1of8
https://doi.org/10.1002/dro2.121

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TAB LE 1 Chronological selection of previous experimental, numerical, and theoretical studies on the film coating of viscoelastic fluids with
various geometries.
Reference Focus
Fluid model/Channel
geometry Scaling law for film thickness validated by experiments
Ro and Homsy16 Theoretical Oldroyd-B/Hele–Shaw h
H=1.337Ca2∕3−0.12(1 −S)WiCa1∕3+(𝛿2Ca2∕3)
Huzyak and Koelling21 Experimental Giesekus/cylindrical Not Applicable (N/A)
De Ryck and Quéré19 Experimental/Numerical N/A/dip-coating 1
b+h=(0.646 −0.76 ln n)k(3U)n
hn+1∕2𝜎2∕3
+2n
2n+1I(n)N
𝜎
3U
h
2n
Gauri and Koelling22 Experimental Giesekus/cylindrical m
mNewt.
=0.1217 ln De +0.8461
Kami¸sli and Ryan23 Theoretical/Experimental Power-law/cylindrical
and rectangular
N/A
Lee et al.24 Numerical Oldroyd-B, FENE-CR,
and FENE-P/
Hele–Shaw and slot
N/A
Yamamoto et al.25 Experimental N/A/cylindrical N/A
Kamisli26 Theoretical CEF/circular,
rectangular
N/A
Ashmore et al.20 Theoretical/Experimental CEF/roller h≈kUn
𝜎2∕(2n+1)
𝜅−3∕(2n+1)
mfor Wi ≪1
h≈NU2n
𝜎1∕2n
𝜅−1∕2n
mfor Wi ≫1
Boehm et al.27 Experimental Giesekus/square N/A
Chang et al.28 Experimental Carreau–Yasuda/
cylindrical
N/A
Datt et al.29 Theoretical/Numerical Second-order
fluid/dip-coating
h
HCa2∕3=0.946 −0.138WiCa−1∕3+⋅⋅⋅
Present work Experimental Shear-thinning
viscoelastic fluids/
cylindrical
h
RCa2∕3
e
=1.34 −0.175WieCa−1∕3
e+⋅⋅⋅
Note: The nondimensional groups appearing above are defined as follows: the capillary number Ca =𝜂U∕𝜎, the Weissenberg number Wi =Ψ
1U∕2𝜂H, and the
Deborah number De =𝜏
𝛾. Here, 𝜂is the fluid viscosity, Uis the bubble velocity, 𝜎is the surface tension of the fluid, Ψ1is the first normal stress difference
coefficient of the fluid, His the characteristic length scale of the flow geometry, 𝜏is the relaxation time of the fluid, and 
𝛾is the shear rate. In Ro and Homsy,16
Sis the ratio of the solvent viscosity to the sum of the polymer and solvent viscosity and 𝛿is the ratio between the relaxation time of the fluid and the
characteristic residence time in the gap. In De Ryck and Quéré,19 bis the radius of the wire, nis the power-law index of the fluid, and kis the empirical
constant obtained from experiments. I(n)=(∫∞
1(2x−3)(x−1)2n−1x−4n−1dx) is the integral obtained from the numerical solution. In Ashmore et al.,20 Nis the
material constant related to the first normal stress difference coefficient, nis the power-law index, and 𝜅mis the curvature of the static meniscus.
Abbreviations: CEF,Criminale–Ericksen–Filbey; N/A, not applicable.
where the radius of curvature of the static meniscus is accounted
as R−hrather than R, yielding a semiempirical equation as h∕R=
1.34Ca2∕3∕1+3.35Ca2∕3. Here, the empirical coefficient is deter-
mined from the correlation in Taylor’s experimental measurements.14
However, the deposition of non-Newtonian viscoelastic fluids by an
elongated bubble in a confined geometry has been much less under-
stood, although a lot of working fluids, such as polymer solutions,
colloidal suspensions, and biologically relevant fluids, show viscoelas-
tic behaviors in many practical applications.11,13,16–18 Tab l e 1lists a
summary of the experimental, numerical, and theoretical studies about
the liquid film deposition dynamics in non-Newtonian fluids. The early
studyledbyRoandHomsy
16 presented a theoretical analysis with per-
turbation expansion, taking into account the effects of normal stress
and shear-thinning viscosity, to assess the film thickness by bubble
motion in a Hele–Shaw cell. Using the Oldroyd-B constitutive model,
they found that the film thickness is slightly reduced due to resis-
tance to stream-wise strain as the liquid becomes more elastic. Next,
De Ryck and Quéré19 found good agreement between the experimen-
tal results for concentrated polymer solutions coating on a fiber and
the prediction of a viscoelastic constitutive model applicable to steady
viscometric flows. Huzyak and Koelling21 performed an experimental
investigation for a long gas bubble penetrating through a tube filled
with viscoelastic Boger fluids. They only reported the strong thicken-
ing of the liquid film deposited on the wall compared to Newtonian
results as the liquid becomes more viscoelastic because of the exten-
sionally thickening viscosity at the extension-dominated flow near the
bubble tip. This hypothesis was later confirmed by Gauri and Koelling22
with particle-tracking velocimetry. Kamisli and Ryan23 performed the
experiments on the liquid displacements using shear-thinning and vis-
coelastic fluids in both circular tubes and noncircular channels, and

CHUN ET AL.3of8
they also only reported that the liquid film thickness of viscoelastic flu-
ids with a constant viscosity deposited on the tube wall is thicker than
that of comparable Newtonian fluids. Lee et al.24 used the Oldroyd-
B, finitely extensible nonlinear elastic-Chilcott & Rallison (FENE-CR),
and finitely extensible nonlinear elastic with Peterlin approximation
(FENE-P) constitutive models for the viscoelastic fluid and performed
the numerical simulation for the motion of a bubble in the Hele–Shaw
and slot-coating configurations. Their numerical results have shown
that the liquid film thickness decreases at low values of Ca and the
Weissenberg number Wi =Ψ
1U∕2𝜂H,whereΨ1and Hare the first
normal stress difference coefficient and the characteristic length scale
oftheflowgeometry,andincreasesathighvaluesofCa and Wi,but
there is no experimental validation. Here, Wi represents the ratio of
first normal stress differences to the viscous shear stresses and reflects
the importance of elasticity in the flow. Yamamoto et al.25 and Boehm
et al.27 conducted an experimental study of gas penetration through
viscoelastic fluids in a cylindrical tube and a square channel, respec-
tively, and both studies have shown that the film thickens with strong
elastic effects. Ashmore et al.20 presented a theoretical and experi-
mental investigation of the film thickness produced by a single roller
in a non-Newtonian fluid bath based on the Criminale–Ericksen–Filbey
constitutive model, in the regimes where the fluid rheology is either
weakly elastic or dominated by shear thinning, or strongly elastic
and dominated by elastic stresses. They experimentally observed the
film thickness increased in the strongly elastic limit. Unlike the afore-
mentioned studies with a straight channel, Chang et al.28 performed
experiments with long bubbles penetrating in a suddenly contracting
and expanding cylindrical tube. The authors observed that the cou-
pling of the geometrical changes and fluid viscoelasticity of the fluids
affects the liquid mass fraction coverage. A recent theoretical work
from Datt et al.29 investigated how the viscoelasticity of the fluid
influences the thickness of the deposited film considering a second-
order model for Boger fluids, and they have shown that the weak
viscoelasticity decreases the thickness of the deposited film compared
to its Newtonian value with a scaling law incorporating Ca and Wi,
but no experimental validation is performed. Aforementioned studies
are mostly limited to numerical or theoretical perspectives, and there
are few available scaling laws to capture the film thickness for vis-
coelastic fluids. In particular, to the best of our knowledge, a systematic
experimental characterization to obtain a scaling law for the deposited
film thickness of shear-thinning viscoelastic fluids in the weakly elastic
regime is still lacking.
In this work, we report an experimental investigation of the film
deposition dynamics when an elongated bubble is translating in a cir-
cular capillary tube filled with shear-thinning viscoelastic fluids, that
is, the polyacrylamide (PAA) and polyvinyl alcohol (PVA) solutions.
Our experiments involve the flow regimes in the weakly elastic limit
with a shear-thinning viscosity. Rheological experiments were also
performed to characterize the test fluids. Instead of describing the
film thickness as a function of the zero-shear-rate capillary number,
Ca0=𝜂
0U∕𝜎,where𝜂0is the zero-shear-rate viscosity of the fluid,
we consider the combination of two different dimensionless numbers,
the effective Weissenberg number, Wie=Ψ
1U∕2𝜂eRand the effec-
tive capillary number,Cae=𝜂
eU∕𝜎, to describe the thin-film deposition
of shear-thinning viscoelastic fluids. Here, 𝜂eis the effective viscos-
ity corresponding to the representative shear rate in the film as U∕h.
We further consider the hydrodynamic lubrication theory that yields
a thin-film equation to obtain the scaling law for the film thickness
based on Caeand Wiein the weakly elastic limit. Finally, we show that
the scaling law predicts the experimentally measured film thickness
reasonably well for a wide range of Caeand Wie.
RESULTS AND DISCUSSION
The rheological properties of the PAA and PVA solutions are shown in
Figure 1.InFigure1a, we note that the PAA solutions display relatively
high shear-thinning behaviors in the range of shear rates over seven
orders of magnitude (10−3s−1≪
𝛾≪104s−1) under steady shear
conditions. Here, we use the Carreau model to fit the viscosity as an
explicit function of the shear rate30,31 with full consideration of the
viscosity plateaus at the very low or high shear rates and the shear-
thinning behavior at the intermediate shear rates. The constitutive
equation of the Carreau model is 𝜂=𝜂
∞+(𝜂0−𝜂
∞)(1 +(𝜆
𝛾)2)(n−1)∕2,
where 𝜂0,𝜂∞,andnare the zero-shear-rate viscosity, infinite-shear-
rate viscosity, and the power-law index, respectively. Here, 𝜆is the
FIGURE 1 Rheogram of all the working fluids. (a) Rheological data for shear viscosity 𝜂as a function of shear rate 
𝛾for the PAA and PVA
solutions with different mass fractions. The dashed lines represent the fitting curves obtained with the Carreau model. (b) Rheological data for the
first normal stress difference N1as a function of 
𝛾for the test fluids. N1is fitted by the function N1=𝜏
11 −𝜏
22 =m1
𝛾2n
Ras shown by the dashed
lines.20 (c) Rheological data for storage and loss moduli G′and G′′ as a function of shear rate for the test fluids. The parameters of the rheological
models obtained from the fitting are summarized in Table 2. Error bars are smaller than the symbols. PAA, polyacrylamide; PVA, polyvinyl alcohol.

4of8 DROPLET
TAB LE 2 Rheological properties of the PAA and PVA solutions
with different mass fractions.
Fluids 𝝈(mN/m) 𝝀(s) 𝜼0(Pa s) 𝜼∞(Pa s) nm
1
PAA 2 wt% 69.1±0.7 0.04 0.26 1.3×10−30.57 0.2
PAA 3 wt% 63.8±1.60.24 2.9 2.5×10−30.43 1.4
PAA 4 wt% 49.1±1.9 9.83 150.7 1.8×10−30.33 40
PVA 10 wt% 63.5±1.244.0 3.2 1.0×10−30.87 0.04
Abbreviations: PAA, polyacrylamide; PVA, polyvinyl alcohol.
inverse of a characteristic shear rate at which shear-thinning becomes
apparent. The PVA solution, however, shows a relatively constant
viscosity at a large range of shear rate32 as listed in Table 2.Inthe
following, we will consider the values of the rheological parameters as
in Table 2unless otherwise specified.
Figure 1b shows the first normal stress difference N1as a function of
shear rates, which links to the degree of viscoelasticity of the test flu-
ids. The data are fitted using a power-law function N1=m1
𝛾2n,where
the values of m1and nare listed in Table 2.20 For a shear strain ampli-
tude of 1%, the oscillation frequencies are varied across five decades,
between 0.01 and 1000 rad/s ( 
𝛾=5.73 to 5.73 ×104s−1), and the
resulting storage and loss moduli (G′and G′′) are shown in Figure 1c.
We note that the bubble motion experiments were conducted in the
range of representative shear rate in the film 
𝛾h=U∕h=30–550 s−1,
which corresponds to the range of angular frequency (0.18–9.7rad/s).
Therefore, in our experiments, viscoelastic effects are nonnegligible in
all test fluids as shown in Figure 1.
We conducted bubble motion experiments in a circular glass capil-
lary tube with a length of 300 mm and an internal radius R=0.47 mm.
The glass capillary was positioned in a vertical orientation, and the
central section was placed inside a transparent rectangular container
filled with glycerin to minimize optical distortions arising from the
curvature of the tube wall.6,33 We connected one end of the capillary
tube to a syringe filled with the test fluid using a flexible connection
tube. The other end of the capillary was open to the atmosphere.
The test fluid was first injected into the capillary and then pushed
with air at a constant flow rate using a syringe pump (11 Pico Plus
Elite, Harvard Apparatus). The flow rate was set to a very low value
of ≈20 μL∕min to smoothly transfer a liquid plug from the flexible
connecting tube into the inlet of the glass capillary. Once the front
of the air plug reached the inlet, the syringe pump was adjusted to
the desired flow rate accordingly. Optical images of the regime of
interest were recorded at a rate of 60 frames per second using a digital
camera (20.9 Megapixel, D7500, Nikon), as illustrated in Figure 2a.
The average velocity of the bubble was determined by tracking the
gas–liquid interfaces at the liquid and bubble front tips using the
images captured with the resolution of ≈9μm per pixel. In the current
experiment, Bo =𝜌gR2∕𝜎 < 0.04 and Re =𝜌UR∕𝜂e≪1, where 𝜌and
gare the fluid density and gravity acceleration, respectively, so gravity
and inertia effects are negligible34,35 (see Table 3for ranges of the
nondimensional numbers in current experiments). We also confirm
that the experimental results for the film thickness did not change if
the direction of the flow was from top to bottom in Figure 2.
The schematic of a bubble translating in a circular capillary as
well as the representative images of the bubble front are shown in
Figure 2b and c, respectively. We measure the film thickness using the
following mass balance analysis as shown in the inset of Figure 2a.The
film thickness is determined by monitoring the changes in the length
of the liquid plug, Lp. We note that we controlled Lp≈7cm ≫R.Asthe
liquid plug advances inside the tube, Lpdiminishes due to the film depo-
sition onto the tube wall. By examining the positions of both the front
menisci of the liquid plug and the bubble, we compute the velocities at
these two points, denoted as Upf and U, respectively. Thus, assuming
a homogeneous deposition of the liquid film inside the channel and a
cylindrical shape of the liquid plug, a mass balance on the moving plug
of length Lpyields the following equation36,37:
h
R=1−1+1
U
dLp
dt =1−Upf
U,(1)
which will be used to determine the film thickness in all experiments.
We show the film thickness measurements in the experiments for
different test fluids as a function of the zero-shear-rate capillary num-
ber Ca0in Figure 3a. Similar to the Newtonian fluids, we observe the
increase in the film thickness with Ca0because of the increase in the
viscous effects. However,compared to the deposited film thickness of a
Newtonian fluid predicted by Aussillous and Quéré,15 the bubble forms
a thinner liquid film in both the PAA and PVA solutions because of the
shear-thinning effect. The thinner liquid film formed in the PAA solu-
tions compared to that in the PVA solutions at the same Ca0results
from the stronger shear-thinning effect in the PAA solutions than that
in the PVA solutions, as indicated by the power-law indices of PAA
solutions compared to that of PVA solutions in Table 2.
Next, we plot the film thickness h∕hCa0,wherehCa0∕R=
1.34Ca2∕3
0∕(1 +3.35Ca2∕3
0), as a function of the zero-shear-rate
Weissenberg number, Wi0=Ψ
1U∕2𝜂0R, as shown in Figure 3b,
respectively. Here, we obtain the values of the first normal stress
difference coefficient Ψ1from the experimental data in Figure 1b using
the relation Ψ1=N1∕
𝛾2
R=m1∕
𝛾2n−2
Rwith 
𝛾R=U∕Ras the charac-
teristic shear rate at the bubble front.38 For the current range of Wi0
(Table 3), the film-thinning behavior is observed, which we believe may
originate from the coupling of the weakly elastic and shear-thinning
effects. First, in the weakly elastic regime, such a thinning behavior
in the range of Wi0is in good qualitative agreement with both the
theoretical results from Ro and Homsy16 and the numerical simulation
of Lee et al.24 In essential, previous studies found two major competing
forces due to elasticity in the capillary-transition regime of the bubble
front: the shear stress gradients in the film direction (y-axis) and the
normal stress gradients in the flow direction (x-axis) (see Figure 2b).
The shear stress gradient represents the viscous stresses acting to
drag a fluid element past the bubble front to form the fluid film, while
the normal stress gradients are caused by the restoring force in the
polymer molecules to oppose the stretch in the flow direction. The
normal stress gradients are found to be negative and larger in magni-
tude than the gradients in shear stress with the weakly elastic effect;
therefore, the normal stress gradients act to resist the flow along the

CHUN ET AL.5of8
FIGURE 2 Experimental setup and bubble motion configuration. (a) Schematic of the experimental configuration. A cylindrical glass tube is
filled with a solution of the test fluid. The central part of the circular glass capillary is submerged in a bath of glycerin to decrease the optical
distortion from the curvature of the tube wall. Inset: schematic of the liquid plug pushed by an air bubble confined in a circular capillary.
(b) Schematic of a translating air bubble confined in a circular tube. (c) Representative experimental images of the bubble front as it translates in a
circular capillary filled with the PAA solutions of different mass fractions at 2 wt% (Ca0=0.14), 3 wt% (Ca0=0.92), and 4 wt% (Ca0=1.83), and
the PVA solution of a mass fraction of 10 wt% (Ca0=0.25). The scale bar is 0.5 mm. PAA, polyacrylamide; PVA, polyvinyl alcohol.
TAB LE 3 Ranges of the dimensionless number (Re for Reynolds, Bo for Bond, Ca0for zero-shear-rate capillary, Caefor effective capillary, Wi0
for zero-shear-rate Weissenberg, and Wiefor effective Weissenberg) used to characterize the flow.
Re =𝝆UR
𝜼e
Bo =𝝆gR2
𝝈Ca0=𝜼0U
𝝈Cae=𝜼eU
𝝈Wi0=𝚿1U
2𝜼0RWie=𝚿1U
2𝜼eR
Range 9.0×10−5to 9.2×10−23.0×10−2to 4.4×10−25.0×10−3to 8.21.2×10−3to 1.51.6×10−1to 12.46.7×10−1to 67.7
wall, triggering film thinning.16,24 Second, the shear-thinning rheology
may also contribute to the film-thinning trend shown in Figure 3b,
since the weakly shear-thinning PVA solution shows a less significant
film thinning compared with the strongly shear-thinning PAA solution.
In real experiments, it is difficult to separate the coupling of the
shear thinning and elastic effects from the fluid rheology. Therefore,
analyses of the results with one dimensionless number, Ca0or Wi0
alone, cannot fully capture the viscoelastic effects on the film depo-
sition dynamics. In the following, we will revisit briefly the recently
developed thin-film model considering the theoretical work from Datt
et al.29 and our previous work37 to enable the consideration of a scaling
law to collapse the experimental results.
Following the thin-film equation derived in Datt et al.29 for the vis-
coelastic Landau–Levich problem where a film is entrained by pulling
a solid plate from a quiescent liquid bath, we write the thin-film
equation for the schematic of Figure 2b, with the frame of reference
situated at the bubble tip, as
𝜎d3h
dx3−3𝜂U
h3(h−h∗)−3Ψ1U2
2h3
dh
dx Fh
h∗=0,
with F(x)=1−6x−1+6x−2,(2)
where h∗is an integration constant that controls the flux in the
liquid film, which is the film thickness as h(−∞)→h∗.29 Here, we

6of8 DROPLET
FIGURE 3 Measurements for nondimensional liquid film thickness. (a) Nondimensional liquid film thickness h∕Ras a function of Ca0. The black
line represents the prediction from hCa0∕R=1.34Ca2∕3
0∕(1 +3.35Ca2∕3
0).15 (b) Nondimensional film thickness compared with the Newtonian case
(h∕hCa0) as a function of Wi0. The experimental measurements are shown as open symbols, and error bars are smaller than the symbols. PAA,
polyacrylamide; PVA, polyvinyl alcohol.
assume the liquid film to be steady in the bubble frame and the film
merges into the stationary bubble with the speed of Uas shown in
Equation (2). By matching the film curvature to that of the bubble
front, the boundary condition d2h∕dx2(∞)→1∕Ris obtained.12 The
second-order fluid model is considered in the above equation.29 When
Ψ1=0, Equation (2) reduces to the thin-film equation for the classical
Bretherton problem with a Newtonian fluid.12 We modify the approach
of Datt et al.29 by introducing scales in the Bretherton problem so
that
H(x)=h(x)
RCa2∕3,X=x
RCa1∕3,(3)
which transforms Equation (2)to
d3H
dX3−3(H−H∗)
H3−3WiCa−1∕3dH
dX
F(H∕H∗)
H3=0,(4)
where H∗=h∗∕(RCa2∕3). In the limit of WiCa−1∕3≪1, the dimension-
less film thickness H∗is determined by numerically solving Equation (4)
with the boundary conditions of H(−∞)→H∗and d2H∕dX2(∞)→1by
regular perturbation expansion. Up to (WiCa−1∕3), we obtain
H∗=1.34 −0.138WiCa−1∕3.(5)
For the Newtonian case, the value of the dimensionless entrained
film thickness is known to be H∗=1.34 in the Bretherton problem,
but H∗becomes a function of the dimensionless parameter WiCa−1∕3
in the case of constant viscosity viscoelastic fluids.29,39 However, we
consider shear-thinning viscoelastic fluids in the current experiments.
In particular, we note that both Picchi et al.40 and our previous work37
have shown that the effect of the shear-thinning viscosity on the film
thickness can be described by Caetheoretically and experimentally,
respectively, where Caeis the effective capillary number considering
the characteristic shear rate in the film (Table 3). Therefore, we recast
Equation (5)withCaeconsidering the effective viscosity in the film as
h∗
RCa2∕3
e
=1.34 −0.138WiCa−1∕3
e.(6)
We further compare two definitions for Wi in Equation (6) by val-
idation of the experimentally measured film thickness. As shown by
Figure 4a, the experimental data of the film thickness do not collapse
into Equation (6)withWi0, which is based on the zero-shear-rate vis-
cosity and does not capture the shear-thinning effect. On the other
hand, as confirmed in Figure 4b, the theoretical prediction using
Equation (6)withWiebased on the effective viscosity using the char-
acteristic shear rate in the film agrees reasonably well for WieCa−1∕3
e
up to the value of 5. At small values of WieCa−1∕3
e≪1, the film thick-
ness is similar to the Newtonian values. As the values of WieCa−1∕3
e
increase, the film thickness starts to decrease due to the increase in
the viscoelasticity of the fluids. In addition, we obtain h∕(RCa2∕3
e)=
1.34 −0.175WieCa−1∕3
eby fitting. The discrepancy between the fitting
relation and Equation (6) may come from the fact that the second-
order fluid model underestimates the film-thinning effect from the
fluid viscoelasticity. Physically, the parameter WieCa−1∕3
eis equivalent
to the Deborah number, representing the ratio between the character-
istic viscoelastic timescale of the fluid Ψ1∕2𝜂eand the characteristic
residence time of a fluid element in the capillary-transition regime
RCa1∕3
e∕U.16 By systematic experiments, we show that the entrained
film thickness is a function of the parameter WieCa−1∕3
ein the weakly
elastic regime, and we obtain a scaling argument to predict the film
thickness applicable for the shear-thinning viscoelastic fluids up to
Wie≈5.8andCae≈0.7 for the first time.
Even though our experimental results show good quantitative
agreement with the scaling argument, we note that the experimental
observation might only be limited to a weakly elastic limit. Since
the second-order fluid model used in the derivation for the scaling
law is expected to be valid for slow flows, we should realize that the

CHUN ET AL.7of8
FIGURE 4 Measurements for nondimensional liquid film thickness normalized by the shear-thinning scaling for all test fluids. Nondimensional
liquid film thickness normalized by the shear-thinning scaling h∕(RCae
2∕3) considering (a) Wi0Ca−1∕3
eand (b) WieCa−1∕3
ein Equation (6).
h∕(RCa2∕3
e)=1.34 −0.175WieCae
−1∕3is obtained by fitting. The experimental measurements are shown as open symbols, and error bars are
smaller than the symbols. PAA, polyacrylamide; PVA, polyvinyl alcohol.
scaling law might not be appropriate to apply at relatively large Wie.29
In addition, we note that the numerical simulation of Huzyak and
Koelling24 showed an initial decrease in the film thickness compared
to the Newtonian values at the weakly elastic limit, which is followed
by the film thickening at the strongly elastic limit. As the value of Wi
increases, their numerical results show that a steep stress boundary
layer is formed around the bubble front, accumulating normal stress
into the thin-film parallel flow regime and thus resulting in a positive
normal stress gradient in the capillary-transition regime. This gradient
reduces the strength of the recirculating regime near the bubble front
and moves the stagnation point toward the tip of the bubble by drag-
ging more fluid into the thin-film regime, increasing the film thickness.
Because of the limitation of the current experimental configuration,
we find that when Wie≫1, it is difficult to move the liquid plug in a
steady way with a low flow rate to maintain a small Re; hence, it is not
feasible for us to access the strongly elastic limit in the current study.
CONCLUDING REMARKS
In this work, we present an experimental investigation of the film
thickness generated by the motion of a long bubble translating in a
circular capillary tube filled with shear-thinning viscoelastic fluids. In
the weakly elastic limit, we show that the film thins compared with the
Newtonian limit. Considering the recently developed hydrodynamic
thin-film model, we rationalize the experimental measurements of
the film thickness by two nondimensional numbers, Caeand Wie,
considering both the shear-thinning and elastic effects. We show that
the dimensionless parameter WieCa−1∕3
ecan be applied to capture the
film thickness in current experiments, and we further obtain a scaling
law to predict the film thinning for shear-thinning viscoelastic fluid
in the weakly elastic limit. To fully characterize the film deposition
dynamics in viscoelastic fluids, our future work aims to experimentally
investigate the flow dynamics at the strongly elastic limit as well as the
transition regime from the weakly elastic to strongly elastic limits.
MATERIALS AND METHODS
We used PAA (MW ≈5000–6000 kg/mol; Sigma Aldrich) and PVA
(MW ≈85–124 kg/mol; Sigma Aldrich) solutions as the viscoelastic
test fluids. We prepared the PAA solutions with mass fractions of
2–4 wt%, and the PVA solution with a mass fraction of 10 wt%. Specif-
ically, the PAA solutions were prepared by slowly dissolving a known
weight of PAA powders in deionized water with continuous stirring
at room temperature. A similar preparation procedure was followed
for the PVA solution with a stirring temperature of 95◦C for 1 h. The
mixing was maintained for 24 h until clear, and homogeneous solutions
were obtained. We measured the rheology of the working fluids with a
controlled stress rheometer (DHR-3, TAInstrument) using a cone-plate
geometry (40 mm in diameter and 1◦in cone angle) under both steady
and oscillatory shear conditions. All rheological measurements were
made at 25◦C. The surface tensions of the test fluids were determined
by the pendant drop method.41,42
ACKNOWLEDGMENTS
S.C. and J.F. acknowledge the use of facilities and instrumentation at
the Materials Research Laboratory Central Research Facilities, Univer-
sity of Illinois, for the rheological experiments. S.C. and J.F. thank partial
support from the American Chemical Society Petroleum Research
Fund Grant No. 61574-DNI9.
CONFLICT OF INTEREST STATEMENT
The authors declare no conflicts of interest.
ORCID
Jie Feng https://orcid.org/0000-0002-4891-9214

8of8 DROPLET
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How to cite this article: Chun S, Yang Z, Feng J. Deposition of
shear-thinning viscoelastic fluids by an elongated bubble in a
circular channel regarding the weakly elastic regime. Droplet.
2024;3:e121. https://doi.org/10.1002/dro2.121