Content uploaded by Omer Atasi
Author content
All content in this area was uploaded by Omer Atasi on Sep 28, 2017
Content may be subject to copyright.
PHYSICAL REVIEW FLUIDS 2, 094304 (2017)
Effect of buoyancy on the motion of long bubbles in horizontal tubes
Omer Atasi,1,2,*Sepideh Khodaparast,2,†Benoit Scheid,1,‡and Howard A. Stone2,§
1Transfers, Interfaces and Processes, Universite Libre de Bruxelles, Brussels 1050, Belgium
2Department of Mechanical and Aerospace Engineering, Princeton University,
Princeton, New Jersey 08544, USA
(Received 21 November 2016; published 26 September 2017)
As a conﬁned long bubble translates along a horizontal liquidﬁlled tube, a thin ﬁlm of
liquid is formed on the tube wall. For negligible inertial and buoyancy effects, respectively,
small Reynolds (Re) and Bond (Bo) numbers, the thickness of the liquid ﬁlm depends only
on the ﬂow capillary number (Ca). However, buoyancy effects are no longer negligible as
the diameter of the tube reaches millimeter length scales, which corresponds to ﬁnite values
of Bo. We perform experiments and theoretical analysis for a long bubble in a horizontal
tube to investigate the effect of Bond number (0.05 <Bo <0.5) on the thickness of the
liquid ﬁlm and the bubble orientation at different capillary numbers 10−3<Ca <10−1.
We investigate several features of the lubricating ﬁlm around the bubble. (i) Due to the
gravitational effects, the ﬁlm deposited on the upper wall of the channel is thinner than the
ﬁlm at the bottom wall. We extend the available theory for the ﬁlm thickness at the front of
the bubble in a twodimensional geometry at low capillary numbers Ca <10−3and ﬁnite
Bo to account for the effect of larger Ca. The resulting model shows very good agreement
with the present experimental measurements. (ii) Due to the asymmetry in the liquid ﬁlm
thickness and the consequent drainage of the liquid from the top to the bottom of the tube,
the bubble is inclined relative to the channel centerline and our sideview visualizations
allow direct quantiﬁcation of the inclination angle, which increases with both Bo and Ca.
While the inclination angle at the top is smaller than that at the bottom of the tube, the
average of these two values follows the predictions of a mass balance analysis in the central
region of the bubble. (iii) The inclination of the bubble causes the thickness of the thin ﬁlm
at the back of the bubble to depend on the length of the bubble, whereas the thickness at
the front of the bubble does not depend on the bubble length.
DOI: 10.1103/PhysRevFluids.2.094304
I. INTRODUCTION
The motion of elongated bubbles in small liquidﬁlled conﬁned geometries is a hydrodynamic
problem with a large variety of industrial and medical applications in lubrication, heat and mass
transfer, oil extraction, and treatment of pulmonary disorders. The problem involves translation
of a lowviscosity ﬂuid at average speed Uinto a higherviscosity liquid with density, viscosity,
and surface tension of ρ,μ, and γ, respectively; the density of the bubble is negligible relative to
that of the continuous liquid phase (Fig. 1). Characterization of the motion and the shape of the
bubble requires consideration of the Reynolds number Re =ρUa
μ, where ais the radius of the tube,
in order to characterize the relative magnitudes of inertial and viscous effects, the Bond number
Bo =ρga2
γ, where gdenotes the gravitational acceleration, in order to characterize buoyancy effects
relative to interfacial tension effects, and the capillary number Ca =μU
γ, in order to assess the
relative magnitudes of viscous and interfacial tension effects. In microscale geometries, the effects
*oatasi@ulb.ac.be
†sepidehk@princeton.edu
‡bscheid@ulb.ac.be
§hastone@princeton.edu
2469990X/2017/2(9)/094304(17) 0943041 ©2017 American Physical Society
ATASI, KHODAPARAST, SCHEID, AND STONE
+
hfront
hfront
hback
hback
U
L
g
,
a
++


FIG. 1. Schematic of a typical elongated bubble translating at capillary number Ca ≈10−2in a horizontal
tube of radius a. The bubble is immersed in a viscous liquid of density ρand viscosity μ. Here Uand Lare the
velocity and the length of the bubble, respectively. A spherical cap is present at the front of the bubble for low
to moderate capillary numbers, followed by a transition region and a lubrication ﬁlm. Because gravity gacts
perpendicular to the tube axis, the bubble is inclined and the thin ﬁlm has different thicknesses at the front (h+
front
and h−
front) compared to the back (h+
back and h−
back). At the back of the bubble, a capillary wave is present [1].
of buoyancy and inertia are often negligible (Bo,Re 1) and the motion of elongated bubbles can
be described by a single dimensionless number, namely, the capillary number Ca. However, as the
dimensions of the geometry increase to millimeter lengths, gravitational effects have signiﬁcant
impacts on the shape and dynamics of the bubbles. In this paper we aim to quantify such effects for
long bubbles in millimeter diameter tubes at moderate capillary numbers and in the absence of inertia.
It was ﬁrst observed by Fairbrother and Stubbs [2] that in a circular tube ﬁlled with liquid a
bubble moves slightly faster than the bulk liquid due to the formation of a thin liquid ﬁlm between
the bubble and the boundary. In other words, the deposition of a thin liquid ﬁlm on the wall does not
allow the entire volume of the more viscous liquid to be pushed out of a closed geometry using a
less viscous ﬂuid. The amount of liquid left on the tube wall is observed to increase with the velocity
of the bubble. Accurate determination of the volume of liquid left on the bounding walls of the
geometry was pioneered by the theoretical work of Bretherton [1] and experimental measurements
of Taylor [3]. Bretherton’s theory predicts the thickness of the liquid ﬁlm hto increase with Ca2/3
for Ca <10−3in the viscocapillary regime
¯
hBr =h
a=1.34 Ca2/3.(1)
Later, Aussillous and Quéré provided a scaling analysis to extend Bretherton’s theory to a wider
range of Ca by considering the contribution of the liquid ﬁlm thickness in deﬁning the curvature at
the front of the bubble [4]:
¯
h=h
a=1.34 Ca2/3
1+3.35 Ca2/3.(2)
The empirical coefﬁcient in Eq. (2), 3.35, was found by ﬁtting the result of their scaling analysis to
the experimental measurements of Taylor for capillary numbers up to Ca =2[3].
More recently, several attempts have been made to investigate the gravitational effects on different
features of the shape and the dynamics of conﬁned elongated bubbles [5–10]. Such effects were gen
erally neglected in the small horizontal geometries discussed above. A summary of the most relevant
works, including studies of the motion of semiinﬁnite lowviscosity air ﬁngers progressing in liquid
ﬂows, is presented in Table I. In general, two main features have been reported in the literature.
(i) In the crosssectional plane, buoyancy causes asymmetry in the distribution of the liquid
ﬁlm around the bubble. In twodimensional (2D) geometries, this effect simply leads to a thinner
liquid ﬁlm at the top compared to the bottom of the tube. For example, Jensen et al. showed that
for Ca <10−3the liquid ﬁlm at the front of a long ﬁnger of gas translating in viscous liquid can
0943042
EFFECT OF BUOYANCY ON THE MOTION OF LONG . . .
TABLE I. Previous research on the motion of bubbles in horizontal and inclined channels with signiﬁcant gravitational effects. The SaffmanTaylor problem
refers to the motion of a lowviscosity ﬂuid displacing a higherviscosity ﬂuid in a HeleShaw geometry. Here θis deﬁned as the angle between the orientation of
the channel and the vertical direction, ARis the aspect ratio of the cross section of the channel deﬁned as the ratio of the width to the height, and Lis the length of
the bubble divided by the channel radius.
Reference Problem and geometry Approach Flow parameters Focus of study
[11] SaffmanTaylor ﬁnger, 2D
(HeleShaw)
Theory, experiments Ca <10−3,0<Bo <3.3 Effect of Ca and Bo on the shape of the
front meniscus, pressure jump across
the interface, and ﬁlm thickness at
the front.
[12] SaffmanTaylor ﬁnger, 2D
(HeleShaw)
Theory, experiments Effect of buoyancy on the shape of the
front meniscus in tilted channels.
[5] Liquid slug bounded
between two air bubbles,
2D
Theory 0 <Ca <0.05, 0 <Bo <2, 0 <θ <180◦Effect of buoyancy on the shape of the
front meniscus in tilted channels.
[6] Liquid slug bounded
between two air bubbles,
2D
Numerics 0.03 <Ca <0.4, 0 <Bo <0.6, 0 <θ <180◦Effect of buoyancy on the ﬂow pattern
in the liquid plug region.
[7] Inﬁnite ﬁnger, 3D
(rectangular cross section)
Numerics 0.002 <Ca <20, Bo =0,1,2.5, 1 <A
R<8 Effect of channel aspect ratio AR,Ca,
and Bo on the crosssectional shape
of the bubble and pressure jump
across the front meniscus.
[8] Slug ﬂow, 3D (circular cross
section)
Experiments 10−4<Ca <0.4 Effect of Ca, Bo, and Re on the
distribution of the liquid ﬁlm
thickness around the bubble.
[10] Slug ﬂow, 3D (circular cross
section)
Theory, experiments 0.03 <Ca <0.2, Bo =0.07,0.17,0.25 (i) Effect of buoyancy on the
distribution of the liquid ﬁlm
thickness around the bubble. (ii)
Velocity measurement in the liquid
plug.
Present
study
Isolated elongated bubble,
3D (circular cross section)
Theory, experiments 10−3<Ca <0.15, 0.05 <Bo <0.42, 10 <L<35 (i) Effect of Ca and Bo on the bubble
inclination angle and the ﬁlm
thickness. (ii) Effect of bubble length
on the ﬁlm thickness.
0943043
ATASI, KHODAPARAST, SCHEID, AND STONE
be predicted analytically by ¯
h±=¯
hBr
1±Bo , where the +and −superscripts correspond to the top
and the bottom of the channel, respectively [11]. However, this model has not been tested against
experimental measurements performed at ﬁnite Bo. Thus, here we modify the theory proposed by
Jensen et al. to account for ﬁnite values of Ca and test its validity against experimental measurements
for long bubbles in a tube of circular cross section.
(ii) In the direction of the ﬂow, a bubble is inclined with respect to the horizontal axis of symmetry
[see the schematic in Fig. 1(a)]. As a result, a uniform ﬁlm thickness region between the front and the
back menisci, as predicted by Bretherton, no longer exists when gravitational effects are signiﬁcant.
Leung et al. showed that the inclination of a bubble is caused by the drainage of the liquid in the
thin ﬁlm region from the top to the bottom of the tube. Their experimental quantiﬁcation of the
drainage volume followed well the predictions of a mass balance analysis proposed in their study
for Bo 0.25 [10]. The inclination angle was observed to be fairly constant along the length of
the bubble [8,10]; however, direct experimental quantiﬁcation of the angle and its correlation with
the ﬂow parameters has not been reported. In this study we quantify the angle of the inclination for
ﬁnite values of Bo and a range of Ca. Due to the inclination, the length of the bubble is expected
to be an important parameter in determining the thickness of the ﬁlm along the bubble, which we
demonstrate and quantify below.
In a continuous train of bubbles separated by slugs of liquid on the orders of the tube radius and
when buoyancy effects are signiﬁcant, consecutive bubbles are observed to inﬂuence one another
due to hydrodynamic interactions [6]. This effect modiﬁes the ﬂow pattern in front of a translating
bubble and therefore alters the shape of the front meniscus and the ﬁlm thickness with respect to
that of a single translating isolated bubble. To avoid such hydrodynamic interactions, here we study
isolated independent bubbles in horizontal tubes of circular cross section.
We provide a theoretical analysis by combining the available models in the literature for different
geometrical and ﬂow parameters, which predicts the evolution of the liquid ﬁlm thickness around
an elongated air bubble in a horizontal tube of circular cross section. Also, we report experimental
data to quantify the buoyancy effects on the motion of conﬁned elongated bubbles. In particular, we
perform brightﬁeld sideview microscopy to directly observe the asymmetries in the ﬂow caused
by buoyancy. This approach allows direct quantiﬁcation of the inclination angle and the distribution
of the ﬁlm thickness along the length of the bubble. Different Ca and Bo numbers are tested by
systematically varying the velocity of the liquid phase and tube diameter. We also identify the
inﬂuence of bubble length.
II. EXPERIMENTAL SETUP
A. Refractive index matching
Experiments were performed in glass capillaries of radii a=0.51, 0.75, 1.15, and 1.51 mm and
length L=10 cm, which were submerged in a refractive index matching box ﬁlled with glycerol.
The diameters of the tubes were measured optically using precalibrated microscope objectives. The
region of interest (ROI) was observed through the straight sidewall of the box (Fig. 2). This approach,
together with the identical refractive indices of the working ﬂuid, tube material, and surrounding
glycerol pool (n=1.47), ensured no optical distortions of the ROI due to light refraction at the
outer and inner curved walls of the capillary [13]. Moreover, the refractive index matching design
prevented total internal reﬂections and thus allowed us to resolve the thin liquid ﬁlm formed between
the twophase interface and the inner tube wall.
B. Experiments
All experiments were performed at T=25 ◦C using glycerol as the continuous liquid phase
(viscosity μ=0.84 Pa s, density ρ=1186 kg/m3, and surface tension γ=0.063 N/m) and air
as the dispersed gas phase. The glycerol used in the experiments was kept in sealed containers to
minimize the effect of humidiﬁcation and the viscosity was measured before and after the tests to
0943044
EFFECT OF BUOYANCY ON THE MOTION OF LONG . . .
P)
a
FIG. 2. Schematic of the experimental setup showing a capillary placed in a box ﬁlled with refractive index
matching liquid to eliminate optical distortions.
ensure no water absorption. The glass capillary, which was ﬁxed to the sides of the refractive index
matching box, was connected to a syringe pump using a long ﬂexible tube at the entrance, while at
the exit another ﬂexible tube emptied the outlet ﬂuid at atmospheric pressure (Fig. 2). The entire
volume of all connection tubes was ﬁlled with glycerol. At the beginning of each experiment, a long
air ﬁnger was created at the inlet of the ﬂexible tube by slightly elevating the tube inlet relative to the
outlet. The airﬁlled inlet was then connected to the syringe pump needle that was ﬁlled with liquid
to create a conﬁned bubble of length L∼20a. The bubble was carried into the ROI as glycerol was
pumped at a constant ﬂow rate. This procedure ensured no acceleration or deformation of the bubble
caused by the injection geometry.
C. Visualization and image processing
Experimental measurements were obtained by processing the digital shadowgraphy images. The
refractive index matching box was placed between a green collimated light source and a homemade
tube microscope equipped with long working distance objectives of 5×and 10×magniﬁcation (with
spatial resolutions of, respectively, 2.44 and 1.22 μm per pixel). Brightﬁeld microscopy allowed
visualization and consequently detection of a sharp interface between the gas and the liquid phases
using a Canny edge detection algorithm [14]. Images of the ROI were recorded by a Nikon D5100
camera at 30 frames per second. Film thickness measurements were performed using the brightﬁeld
images captured at the front and the back of the bubble. The maximum relative error in the ﬁlm
thickness measurements reported for the smallest Ca is 14%, while the rest of the measurements
have a maximum experimental error of 5%.
Obtaining high spatial resolution in the images requires high magniﬁcation in the optical setup,
which consequently leads to a relatively small ﬁeld of view (FOV). For example, using the 5×
objective in our experiments, we could visualize only about 2 mm of the tube length in the images,
which is obviously not adequate to capture the axial evolution of the shape of the long bubbles
[Fig. 3(a)]. In order to construct an image of the full length of the bubble that does not ﬁt in the FOV,
we applied the postprocessing timestrip method to the sequence of shadowgraphy images [15].
0943045
ATASI, KHODAPARAST, SCHEID, AND STONE
(a) (b) (c)A
A’ L
1 mm 2 s1 mm
2a
FIG. 3. Sample visualization results for an elongated bubble of length L=22 mm in a horizontal tube of
radius a=1.51 mm at Ca =0.076 and Bo =0.42. The dashed horizontal line shows the centerline of the tube.
(a) Sample image of the grayscale brightﬁeld microscopy. The bubble translates from the left to the right.
Timestrip analysis is performed along a sample line such as section AA. The ﬁeld of view in a typical image
covers less than a 2mm length of the tube. (b) Timestrip analysis result showing evolution of section AAin
time. Note that the horizontal axis represents time. (c) The horizontal time axis in (b) is converted to the length
axis using the bubble velocity. The inclination angle is measured by ﬁtting a straight line to the bubble interface
as shown by the red lines. Note that images similar to the one presented in (c) are only used to measure the
length and the inclination angle of the bubble.
In this approach, we followed the evolution of a speciﬁed vertical line of pixels along the tube
diameter [Fig. 3(a)] for a long period of time starting before the arrival of the bubble front until after
departure of the back of the bubble [Fig. 3(b)]. Knowing the bubble velocity, time is then converted
to the displacement of the bubble and the shape of the bubble is constructed [Fig. 3(c)]. The ﬁnal
image was then processed to obtain the length and the inclination angle of the bubble. The difference
between the thickness of the liquid ﬁlm at the top and at the bottom and the inclination of the bubbles
are clearly noticeable in Figs. 3(b) and 3(c). In order to obtain the inclination angle, a straight line
was ﬁtted to the detected airglycerol interface. The maximum experimental error in identifying
the inclination angle at the lowest Bo and Ca is 30%, while the rest of the measurements contain
a maximum experimental error of 10%. The capillary number Ca =μU
γwas calculated based on
the bubble velocity measured by tracking the nose of the bubble. The high optical resolution used
in the present study allowed determination of Ca down to values on the order of Ca =10−6.The
spatial resolution of the timestrip image perpendicular to the ﬂow is ﬁxed by the physical pixel
size and the objective magniﬁcation. On the other hand, the axial resolution of the timestrip image
along the ﬂow depends on the image acquisition frequency and on the speed of the bubble. The
minimum and maximum axial resolutions of the timestrip images in the experiments reported here
were, respectively, 256 and 1.3 μm. Note that the low optical resolution observed at the nose and
the back of the bubble in Fig. 3(c) is due to the large velocity of the bubble at larger Ca. This effect,
however, does not inﬂuence measurement of the bubble length or the inclination angle.
The camera is leveled horizontally prior to the start of the experiments using a digital level with an
accuracy of 10−4rad. Furthermore, in order to ensure that the tube is horizontal, the refractive index
matching box is mounted on a stage that is translated in the axial xdirection. Two images are captured
as the box is moved between two points that are axially 5 cm apart. Comparison between these two
images ensures no inclination in the ROI within the accuracy of our measurements (10−4rad).
D. Experimental observation
Examples of the shape of the nose, the central region, and the back of the bubbles at different
Ca and Bo are shown in Fig. 4. Two major differences are observed when comparing the present
sideview visualizations at ﬁnite Bo numbers with the classic predictions of bubble shape with
negligible buoyancy effects. (i) Due to nonnegligible buoyancy effects, the liquid ﬁlm at the front
of the bubble is not uniformly distributed but instead is thinner at the top than at the bottom of the
tube. (ii) Furthermore, the back of the bubble is elevated relative to the nose. This feature was shown
0943046
EFFECT OF BUOYANCY ON THE MOTION OF LONG . . .
(a)
Ca = 0.076Ca = 0.006
Ca = 0.012 Ca = 0.063
(b)
(c)
Ca = 0.006 Ca = 0.064
a = 0.75 mma = 1.15 mma = 1.51 mm
FIG. 4. Examples of bubble shapes at different Ca and Bo: (a) Bo =0.11, (b) Bo =0.25, and (c) Bo =0.42.
Bubbles move from the left to the right. Images contain experimental visualization of the nose, the central part,
and the back of the bubble. The central region demonstrates a typical zone between the bubble nose and back,
where the liquid ﬁlm at the top (bottom) acquires a linearly decreasing (increasing) thickness towards the back.
The back of the bubble exhibits a capillary wave, meaning a region of negative curvature [16]. The red dashed
lines indicate the centerline of the tube. Note that the optical magniﬁcations used in visualization of ﬂows at
different Bond numbers are not identical.
by Leung et al. [10] to be caused by the drainage of the liquid in the ﬁlm from the top to the bottom
of the tube.
III. THEORETICAL PREDICTIONS
In this section we provide theoretical considerations to predict quantitatively both the ﬁlm
thickness and the inclination angle as a function of Ca and Bo. The model presented here for the ﬁlm
thickness at the front of the bubble is mainly based on the twodimensional analysis proposed by
Jensen et al. [11] at low Ca and ﬁnite Bo in HeleShaw geometries. This model is extended here to
account for the effect of ﬁnite Ca using the correction that was originally suggested by Aussillous and
Quéré [4], and a physical argument is provided to show that the lateral curvature of the tube does not
affect the thin ﬁlm thickness. The basic steps of Jensen et al. [11] are included in this section, wherever
necessary, to ensure the selfsufﬁciency of the arguments. Moreover, we utilize the mass balance
analysis proposed by Leung et al. [10] to obtain an estimate for the average inclination angle of the
bubble. In the next section, these theoretical models are compared to our experimental measurements
of the ﬁlm thickness at the front and the inclination angle of the bubble, for a range of Ca and Bo (<1).
A. Thickness of the liquid ﬁlm at the front of the bubble
Following the classical approach of Bretherton, we divide the front of the bubble into three
distinct regions (Fig. 5): region I, the capillary static (CS) region, where the pressure is hydrostatic;
region II, the transition region, where the surface tension force is balanced by the viscous forces;
0943047
ATASI, KHODAPARAST, SCHEID, AND STONE
I II III
R
Rf+
Rf
a
g
x
y
y = +
y = Rf
y = Rf+
y = 0
s+
s
h+
h
h+
h
y = r(x)
y = r+(x)
FIG. 5. Sideview schematic of the front of a bubble in a horizontal tube of radius a. The bubble moves
from left to right and is inclined with respect to the centerline of the tube. The +and −superscripts stand for
the quantities at the top and the bottom, respectively. The bubble is divided in three regions. Region I indicates
the capillary static region, where the tip of the bubble forms a sphere of radius R. The tip of the bubble is
offcenter with respect to the tube and is located at (x,y)=(0,). Region II indicates the transition region in
which the viscous forces are balanced by the capillary forces due to the variation of the curvature along the
interface. Between I and II, a matching region mris present where the interface shapes of I and II match. The
position of the interface in the matching region with respect to the tube centerline is at y=±R±
f. Region III
indicates the thin ﬁlm region, where the axial ﬂux (in the reference frame of the bubble) is balanced by the
lateral drainage ﬂux of the liquid in the ﬁlm.
and region III, the thin ﬁlm region, where the axial ﬂux (in the reference frame translating with the
bubble) is balanced by the lateral drainage.
1. Capillary static region
In this region, our approach follows closely that of Jensen et al. [11]. However, we extend the
original formulation by adding the condition that at higher Ca the thickness of the ﬁlm is not
negligible compared to the radius of the front nose of the bubble [4]. Moreover, we consider the top
to bottom asymmetry in the interface height with respect to the tube centerline (Fig. 5).
The dimensionless ﬁlm thickness at the front of the bubble ¯
h±
∞in a twodimensional geometry
and at very low capillary numbers Ca →0 was determined by Jensen et al. [11]tobe
¯
h±
∞=hBr
1±Bo ,(3)
where hBr =1.337 Ca2/3is the classic prediction of the ﬁlm thickness by Bretherton [1]atverylow
Ca and negligible buoyancy effects and ±indicates the top (+)orthebottom(−) relative to the
axis of symmetry. Both of these analyses were conducted by neglecting the azimuthal curvature of
the bubble.
For a twodimensional conﬁguration, in a HeleShaw cell, the pressure difference caused by
capillary forces between the liquid and the gas in the thin ﬁlm region is δpf≈0, while in the CS
0943048
EFFECT OF BUOYANCY ON THE MOTION OF LONG . . .
region δpCS ≈−
γ
a, leading to a pressure difference between the thin ﬁlm and the CS region of
δpf−δpCS ≈γ
a. In a tube of circular cross section and for thin liquid ﬁlms, the pressure difference
caused by capillary forces between the liquid and the gas in the thin ﬁlm region is δpf≈−
γ
adue to
the azimuthal curvature and in the CS region it is δpCS ≈−
2γ
a, which leads to a pressure difference
between the thin ﬁlm and the CS region of δpf−δp CS ≈γ
a. Thus, the azimuthal curvature of the
interface does not change the pressure difference between the thin ﬁlm region and the CS region in
a tube when compared to the twodimensional case. We will therefore conduct this ﬁrst part of our
analysis by not considering the azimuthal curvature.
In the CS region, where the pressure is hydrostatic, the coordinate system (x,y) has its yorigin at
the center of the channel and its xorigin at the tip of the bubble (Fig. 5). Due to buoyancy, the tip of
the bubble is elevated with respect to the center plane of the geometry and is located at y=a.
In the liquid phase, the pressure gradient is given by ∂p
∂y =−ρg, where ρis the density and gthe
gravitational acceleration. Integrating the pressure gradient with the Laplace pressure jump condition
at the interface, i.e., at y=r±(x), pg−p=∓γκ±, where pgis the pressure inside the bubble, γ
the surface tension, and κthe curvature of the interface, yields p+ρgy −pg=ρgr±±γκ±.The
curvature of a twodimensional interface in a HeleShaw conﬁguration is given by κ±=∂xxr±
[1+(∂xr±)2]3/2.
Since the gas pressure is constant, we can set pg=0 without loss of generality. Furthermore, we
can express the pressure in terms of the reduced pressure pr=p+ρgy such that pris independent
of yand use the scales afor rand γ
afor prto ﬁnally obtain
¯
pr=Bo¯
r±±∂¯
x¯
x¯
r±
[1 +(∂¯
x¯
r±)2]3/2,(4)
where Bo =ρga 2
γis the Bond number and an overbar denotes dimensionless variables. Note that
Eq. (4) is the same as the intermediate result of Jensen et al. [11] and is obtained in the same manner.
Next we multiply Eq. (4)by∂¯
x¯
r±and integrate assuming ¯
prconstant to ﬁnd
¯
pr¯
r±=Bo
2¯
r±2∓1
[1 +(∂¯
x¯
r±)2]1/2+c±,(5)
where c±is the integration constant. The hypothesis of constant reduced pressure in the CS region
will be veriﬁed subsequently. We evaluate this equation in the matching region, where we assume
that viscous effects become signiﬁcant. In the matching region ∂¯
x¯
r±1 and ¯
r±=±
R±
f
a, where
R±
fis the distance between the interface and the cell centerline (+) top and (−) bottom. Substituting
these expressions in Eq. (5), we obtain
±¯
pr
R±
f
a=Bo
2R±
f
a2
∓1+c±.(6)
Setting R±
f=ain Eq. (6), we recover the formulation proposed by Jensen et al. [11]forprin
the limit of vanishing Ca and R±
f≈a. We can evaluate Eq. (5) at the tip of the bubble, where
∂¯
x¯
r±→∓∞and ¯
r±=
a, in which
a1, and subtract the equation corresponding to the bottom
from the top to ﬁnd c+=c−as in Ref. [11]. We note that is an intermediate variable, which is
introduced to make the discussion more clear, but its speciﬁc value does not enter in the derivation.
We then evaluate Eq. (6) at the top and at the bottom and do the subtraction top minus bottom to
obtain pr,
¯
prR+
f
a+R−
f
a=−2+Bo
2R+
f
a2
−R−
f
a2.(7)
0943049
ATASI, KHODAPARAST, SCHEID, AND STONE
For Bo 1Eq.(7) can be approximated as
¯
pr≈− 2a
R+
f+R−
f
.(8)
Note that ¯
prdoes not depend on the axial coordinate, ¯
x, as we assumed before integrating Eq. (4).
Having the expression of ¯
pr, we substitute it in Eq. (4) and evaluate this equation in the matching
region, i.e., for ∂¯
x¯
r±1, to ﬁnd
∂¯
x¯
x¯
r±=∓ 2a
R+
f+R−
f
−BoR±
f
a.(9)
Note that the choice of coordinate system gives that ∂¯
x¯
x¯
r+<0 and ∂¯
x¯
x¯
r−>0 (Fig. 5).
2. Transition region
In the transition region, we start by considering the lubrication approximation and the balance
between the pressure gradient and the viscous forces (per volume) along the xdirection. The steps
are standard, as ﬁrst developed by Bretherton [1]. In dimensionless variables, we have
∂¯
p
∂˜
x±=Ca ∂2¯
u
∂˜
y±2,(10)
where ¯
u=u
Ub, with Ubthe bubble velocity; ˜
y±=∓
¯
y+1; and ˜
x±=¯
x+¯
s±, with sbeing a shift in
the axial coordinate, as deﬁned in Fig. 5, and ¯
s=s
a.WeintegrateEq.(10) twice with ¯
u=−1at ˜
y±=0
and the noshearstress condition at the interface, i.e., ∂¯
u
∂˜
y±=0aty=h±, where h±=1∓r±.The
pressure gradient along ˜
x±is given by the curvature gradient ∂¯
p
∂˜
x±=−∂˜
x±˜
x±˜
x±˜
h±. Integrating the
velocity ﬁeld along ˜
y±yields a thirdorder differential equation, where the variables have been
rescaled as ¯
¯
h±=˜
h±
Ca2/3and ¯
¯
x±=˜
x±
Ca1/3,
¯
¯
h±3d3¯
¯
h±
d¯
¯
x±3=3( ¯
¯
h±−¯
¯
h±
∞),(11)
where ¯
¯
h±
∞=h±
∞
aCa2/3. We obtain the classical form by deﬁning H=¯
¯
h±
¯
¯
h±
∞
and X=¯
¯
x±
¯
¯
h±
∞
,
H3d3H
dX3=3(H−1).(12)
We now express the second derivative ∂xxr±, with the variables from the transition region:
∂xxr±=∓ 1
a¯
¯
h±
∞
∂XXH. (13)
The curvature expressed with the variables from the transition region and from the CS region should
match in a common overlap region, in which one can rewrite Eq. (9)intheform
∓1
¯
¯
h±
∞
lim
X→∞ ∂XXH=∓ 2a
R+
f+R−
f
−BoR±
f
a,(14)
where a numerical solution of Eq. (12) shows that limX→∞ ∂XXH=1.337 [1], which ﬁnally leads
to
h±
∞
a=1.337 Ca2/3
2a
R+
f+R−
f
±BoR±
f
a
.(15)
09430410
EFFECT OF BUOYANCY ON THE MOTION OF LONG . . .
This result generalizes the development of Jensen et al. [11] to the case of moderate values of Ca
and the development of Klaseboer et al. [17] to the case of ﬁnite Bo. From Eq. (15)weareableto
recover the three limiting cases presented previously in the literature.
(i) At very low capillary numbers Ca →0 and consequently very thin lubricating liquid ﬁlms,
the nose of the bubble approximately ﬁts the entire gap between the top and bottom planes (or the
tube diameter), thus R±
f≈a. In this limit and for Bo >0, we recover
h±
∞
a=1.337 Ca2/3
1±Bo ,(16)
which was originally obtained by Jensen et al. [11].
(ii) For negligible buoyancy effects Bo →0, no asymmetry is present at the nose of the bubble,
thus R±
f=Rf. In this limit, if we consider moderate Ca the radius of the spherical cap of the bubble
no longer ﬁts the radius of the tube, i.e., a= Rf, and Eq. (15) is reduced to the form presented by
Aussillous and Quéré [4] and discussed further by Klaseboer et al. [17]:
h±
∞
Rf
=1.337 Ca2/3.(17)
(iii) Finally, for Ca →0, Bo →0, and R±
f≈a, we recover the expression of Bretherton [1]:
h±
∞
a=1.337 Ca2/3.(18)
The distance between the cell or tube centerline and the interface in the matching region R±
fand
the liquid thickness CNh±
∞should ﬁt the radius of the tube. This gives an additional equation as
proposed by [17]
a=CNh±
∞+R±
f,(19)
where CNis a constant and h±
∞is given by Eq. (15). Notice that combining Eqs. (15) and (19)inthe
limiting case when Bo =0 and R+
f=R−
f, we recover an equation similar to that in Refs. [4,17],
h±
∞
a=1.337 Ca2/3
1+1.337CNCa2/3,(20)
where CN=2.5 by ﬁtting Eq. (20) to the experimental measurements of Taylor [3,4] and CN=2.79
following a theoretical analysis proposed by Klaseboer et al. [17]. However, Klaseboer et al. [17]
pointed out that the exact value of CNdepends on where Rf±is evaluated in the matching region and
can take a value up to 2.9 if Rf±is evaluated for H>106, i.e., X>10 000, and 2.34 if it is evaluated
for H=6×102, i.e., X=50. We choose CN=2.5 in this work since it has been shown to ﬁt the
experimental measurements. It should be noted that as the ﬁlm gets thicker with increasing Ca, for a
given value of the tube radius a,CNcould decrease since the liquid plus the bubble should ﬁt in the
tube radius. However, experimental results of Taylor [3,4] show no such effect for Ca up to 2. On
the other hand, the inﬂuence of Bo on CNneeds further investigation. Equations (15) and (19)form
a system of four equations with four unknowns R+
f,R−
f,h+
∞, and h−
∞, which is solved numerically
using the NSolve function of Wolfram Mathematica. Note that much above a critical Bond number,
i.e., Bo Boc, the pressure in the bottom ﬁlm becomes hydrostatic, i.e., ρgRf−Uμl
(h−
∞)2, where lis
the length of the transition region and scales with l∼Ca1/3, while h−
∞∼Ca2/3, thus Boc∼1
1−Ca2/3.
The critical value for the Bond number is Boc=1 for vanishing Ca, i.e., Rf±≈a, however, for
moderate Ca, i.e., Rf±= a,thevalueofBo
cincreases with Ca. Equation (15)isvalidforBo<Boc.
09430411
ATASI, KHODAPARAST, SCHEID, AND STONE
B. Inclination angle in the thinﬁlm region
Leung et al. [10] directly visualized the drainage ﬂow from the top to the bottom of the channel in
the thin ﬁlm region. Using a mass balance analysis in the central region of the bubble and assuming
the thin ﬁlm limit, they demonstrated that the drainage was causing the thin ﬁlm to be inclined at an
angle. Their main point was that, in the reference frame translating with the bubble velocity, liquid
ﬂows in the ﬁlm. The volume of liquid in the ﬁlm that is drained from the top to the bottom of the
channel increases as one proceeds towards the back of the bubble, leading to less liquid ﬂowing in
the ﬁlm as the back of the bubble is reached at the top of the channel; this description rationalizes
the thinning of the ﬁlm at the top. From their mass balance [Eq. (14) from Ref. [10]] we can obtain
tan α±=∓
h∞θ=π
23
3Ca Bo,(21)
where h∞(θ=π
2) is the ﬁlm thickness at the center plane of the channel and is approximately given
by h∞(θ=π
2)≈h+
∞+h−
∞
2. It should be noted that possible deformation of bubble in the crosssectional
plane is assumed to be negligible in order to obtain Eq. (21).
IV. EXPERIMENTAL RESULTS AND DISCUSSION
In this section we present results of our experimental measurements as a function of capillary Ca
and Bond Bo numbers for the range of 5×10−4<Ca <0.12 and 0.05 <Bo <0.42. The largest
value of the Reynolds number in our experiments was Re =0.023, therefore, inertial effects are
negligible. We distinguish features of the thin ﬁlm at the top and bottom of the front of the bubble
as well as quantifying the inclination of the bubble due to buoyancy effects. Moreover, we present
measurements of the ﬁlm thickness at the back of the bubbles, which are different from those
obtained at the front due to the inclination of the bubble. Our main goal in this section is to report
our experimental results, organized systematically using Ca and Bo, and to compare with the theory
presented in the preceding section.
A. Thickness of the liquid ﬁlm at the front of the bubble
Experimental measurements of the ﬁlm thickness at the front of the bubble are performed at
both the top and bottom of the tube. Results of these measurements are presented in Fig. 6for four
different values of the Bond number. Note that these experimental data points may correspond to
bubbles of different lengths. Nevertheless, the dispersion of the experimental points are within the
experimental error and the ﬁlm thicknesses at the front of the bubble h±
front do not depend signiﬁcantly
on the length of the bubble. Consistent with the theoretical prediction, the ﬁlm thickness at the front
of the bubble is thicker at the bottom than the top, while the difference between the values of the top
and bottom ﬁlm thicknesses at the front become larger as the Bo increases.
The data in Fig. 6are compared with two models presented in Sec. III, one being the extended
Bretherton without buoyancy effect as presented in Eq. (20)[4] and a second model derived here,
which was inspired by Jensen et al. accounting for buoyancy effects. The extended Bretherton
formulation (solid line in Fig. 6) always lies in between the experimental measurements at the top
and bottom of the tube. The theoretical prediction obtained by solving the system of equations (15)
and (19) (dotted lines in Fig. 6) follows the experimental measurements well. However, we note that
an underprediction of the ﬁlm thickness at the bottom and an overprediction of that at the top are
present for all cases, which can be due to nonnegligible lateral drainage from the top to the bottom
of the channel in the transition region.
A scaling argument supporting this idea is the following. The time taken for liquid to be convected
in the transition region is tc≈l
Ub, where lis the length of the transition region. The drainage time
of the liquid from top to bottom is td≈πa
Ud, where Udis the drainage velocity. One can estimate the
09430412
EFFECT OF BUOYANCY ON THE MOTION OF LONG . . .
top
bottom
Eq. (20)
Eq. (15) & Eq. (19)
hfront
0
0.05
0.10
0.15
0.20
Ca
0 0.03 0.06 0.09 0.12
top
bottom
Eq. (20)
Eq. (15) & Eq. (19)
hfront
0
0.05
0.10
0.15
0.20
Ca
0 0.03 0.06 0.09 0.12
top
bottom
Eq. (20)
Eq. (15) & Eq. (19)
hfront
0
0.05
0.10
0.15
0.20
Ca
0 0.03 0.06 0.09 0.12
top
bottom
Eq. (20)
Eq. (15) & Eq. (19)
hfront
0
0.05
0.10
0.15
0.20
Ca
0 0.03 0.06 0.09 0.12
(a) (b)
(c) (d)
FIG. 6. Top (open symbols) and bottom (closed symbols) ﬁlm thickness measurements at the front of the
bubble for different Bond numbers: (a) Bo =0.056, (b) Bo =0.11, (c) Bo =0.25, and (d) Bo =0.42. The thin
black solid line is the prediction given by Eq. (20) with the empirical constant CN=2.5. The dotted lines are the
predictions obtained by solving the system of equations (15)and(19) numerically. The results computed with
Eq. (15) are in good agreement with the experiments. As Bo increases the agreement becomes less satisfactory
due to the nonnegligible drainage of liquid in the transition region. For the smallest Bo, the dashed and dotted
lines converge towards the solid line.
drainage velocity by balancing gravity and viscous shear along the vertical direction Udμ
h∞(θ≈π
2)2≈ρg,
i.e., Ud≈ρgh∞(θ=π
2)2
μ. Using the scales l∼Ca1/3and h∞(θ=π
2)∼Ca2/3, we obtain the ratio
between the two time scales td
tc∼1
Bo Ca2/3. For sufﬁciently small values of Bo and Ca, the ratio of
time scales is large enough to ensure no inﬂuence of the lateral drainage, but for the upper bounds
of these parameters in our experiments the ratio decreases towards 3, meaning that drainage in the
transition region begins to be signiﬁcant. It should be noted that the larger difference between the
experiments and the proposed model for the ﬁlm thickness at bottom compared to that at the top,
especially at higher Bond numbers (Bo =0.25 and 0.42), may also be a consequence of enhanced
drainage for thicker ﬁlms.
09430413
ATASI, KHODAPARAST, SCHEID, AND STONE
Bo = 0.056
Bo = 0.11
Bo = 0.25
Bo = 0.42
+ [103 rad]
0
2
4
6
8
Ca
0 0.03 0.06 0.09 0.12
 [103 rad]
0
2
4
6
8
Ca
0 0.03 0.06 0.09 0.12
(a) (b)
FIG. 7. Inclination angle of the bubble at the (a) top and (b) bottom as a function of capillary number Ca
for different Bond numbers Bo. The error bars (visible when larger than the symbol size) correspond to the
standard deviation of the measurements obtained in three independent experimental tests.
B. Inclination of the bubble
Measurements of the inclination angle αof the bubble at the top α+and bottom α−of the tube are
presented in Fig. 7. The inclination angle increases at higher Bo due to the intensiﬁed drainage from
the top to the bottom of the tube. We observe that the inclination angle at the top of the bubble is
smaller than that at bottom of the tube. This feature is believed to be due to the slight deformation of
the circular cross section of the bubble due to the buoyancy, which is not included in the theoretical
analysis provided here. Such deformations due to gravity have been reported by de Lozar et al. [7].
Since the scaling analysis discussed in Eq. (21) is performed considering a circular cross section all
along the bubble, we compare the theoretical prediction with the average inclination angle calculated
for the top and bottom αave =α++α−
2. The result of this comparison is presented in Fig. 8, where
excellent agreement is obtained between the experimental measurements and the prediction of the
mass balance analysis (21). Note that according to both theory and the present experiments, buoyancy
effects are observable even at the lowest Bond number Bo =0.056. Moreover, the dependence of
the inclination angle on the capillary number diminishes at higher Ca. Figure 8shows that the mass
balance argument provided by Leung et al. [10] can provide a reliable prediction method for the
average inclination angle of the bubble in the presence of nonnegligible buoyancy.
C. Liquid ﬁlm at the back of the bubble
Unlike the ﬁlm thickness at the front of the bubble, the ﬁlm thickness varies with the length of
the bubble at the back due to the drainage of the liquid ﬁlm and the consequent inclination of the
bubble. This effect is observed in our experimental measurements of the ﬁlm thickness at the back
of the bubble h±
back. Figure 9presents the ratio of the ﬁlm thickness at the back of the bubble to
that at the front ¯
h±
back/¯
h±
front versus the dimensionless length of the bubble ¯
L=L/a obtained in the
experiments. Three main features are clearly observed in Fig. 9. (i) The ratio ¯
h±
back/¯
h±
front is linearly
correlated to the dimensionless length of the bubble ¯
L=L/a. (ii) The magnitude of the slopes of
the best linear ﬁts to experimental data are slightly larger at the bottom compared to the top of the
tube. This ﬁnding once again shows that the cross section of the bubble loses its circular shape
towards the back of the bubble. (iii) For the limit of ¯
L→0 all the linear trends at both the top and
the bottom of the tube reach unity (Fig. 9).
09430414
EFFECT OF BUOYANCY ON THE MOTION OF LONG . . .
Bo = 0.056
Bo = 0.11
Bo = 0.25
Bo = 0.42
αave [103 rad]
0
2
4
6
Ca
0 0.03 0.06 0.09 0.12
FIG. 8. Mean inclination angle of bubbles computed by averaging the measurements at the top and the
bottom αave =(α++α−)/2. Solid lines present the prediction of Eq. (21).
At low Ca and ﬁnite Bo, the ﬁlm thickness at the top of the tube and at the nose of the bubble
is less than 10 μm. As this thickness decreases towards the back of the bubble, it can eventually
reach values smaller than 1 μm for long bubbles of L∼10 mm. Thin ﬁlms of partially wetting
liquids with thicknesses below 1 μm are known to be metastable and thus may dewet the wall of the
channel [18–20]. This effect causes dewetting on the top wall of the channel, which was observed
in our experiments at low capillary numbers Ca <10−3. Therefore, not only the mean liquid ﬁlm
Bo = 0.056
Bo = 0.11
Bo = 0.25
Bo = 0.42
h±
back / h±
front
0
0.5
1.0
1.5
2.0
L
0 10203040
FIG. 9. Ratio of the back to front ﬁlm thicknesses ¯
h±
back/¯
h±
front obtained in the experiments versus the
dimensionless length of the bubble ¯
L=L/a. Open and closed symbols correspond to the measurements at the
top and bottom of the channel, respectively. Solid lines represent the best linear ﬁt to the experimental data.
Here ¯
h±
back/¯
h±
front shows a linear trend with ¯
Land approaches unity in the limit ¯
L→0.
09430415
ATASI, KHODAPARAST, SCHEID, AND STONE
thickness but also the minimum ﬁlm thickness caused by the effect of buoyancy at the top rear of the
bubble, which depends on the bubble length, must be considered in studies where the ﬁlm thickness
has a critical effect on the ﬁnal results, e.g., critical heat transfer determination for a heated pipe.
V. CONCLUSION
In this study we investigated the effect of buoyancy on the thickness of the liquid ﬁlm formed
around elongated conﬁned bubbles, which translate in a horizontal tube of circular cross section.
We performed systematic experiments, in which the dimensionless parameters, namely, capillary Ca
and Bond Bo numbers, were studied. In general, buoyancy effects were observed even at very low
Bond numbers Bo =0.056 and were ampliﬁed at higher Bo.
At ﬁnite Bo, the ﬁlm thickness at the front of the bubble is thinner at the top than at the bottom.
We extended the theory originally proposed by Jensen et al. in a twodimensional conﬁguration
for low capillary numbers Ca <10−3and ﬁnite Bo [11], to account for the ﬁlm thickening effect
observed at moderate capillary numbers [4,17]. The resulting theoretical correlation quantitatively
predicts the present experimental measurements with less than 12% relative error. In general, the
theory underpredicts the effect of buoyancy on the ﬁlm thickness especially at higher Ca and Bo,
which may be a consequence of the lateral drainage in the transition region.
Due to buoyancy, a lateral drainage of liquid exists from the top to the bottom, which leads to
the thinning of the liquid ﬁlm towards the back of the bubble at the top. Therefore, the body of
the bubble is inclined relative the centerline of the tube so that the back of the bubble is elevated.
We quantiﬁed this inclination angle at the top and the bottom of the bubble. While the angle of
inclination remains constant along the bubble, it is slightly larger at the bottom of the bubble. We
believe that this effect is a consequence of the deformation of the bubble in the crosssectional plane
as compared to a circular shape. If the crosssectional deformation of the bubble is neglected, a mass
balance analysis for the liquid around the bubble in the central region by Leung et al. provides a
theoretical prediction for the inclination angle [10]. We showed that this prediction can well predict
the mean value of our experimental measurements of the inclination angles at the top and bottom
of the bubble. Additionally, our ﬁndings suggest that the role of the drainage of the liquid in the
transition region at the nose of the bubble and deformation of the bubble in the crosssectional plane
is an avenue for future theoretical analysis. Furthermore, future theoretical studies should investigate
how the nondimensional liquid thickness in the matching region CNdepends on Bo.
ACKNOWLEDGMENTS
O.A. and B.S. thank the F.R.SFNRS for ﬁnancial support as well as the BELSPO agency under
Grant No. IAP7/38 MicroMAST. S.K. appreciates the early mobility funding from Swiss National
Science Foundation (Grant No. P2ELP2158896). We are grateful to the referees for their insightful
comments.
[1] F. P. Bretherton, The motion of long bubbles in tubes, J. Fluid Mech. 10,166 (1961).
[2] F. Fairbrother and A. E. Stubbs, Studies in electroendosmosis. Part VI. The “bubbletube” method of
measurement, J. Chem. Soc. 527 (1935).
[3] G. I. Taylor, Deposition of a viscous ﬂuid on the wall of a tube, J. Fluid Mech. 10,161 (1961).
[4] P. Aussillous and D. Quéré, Quick deposition of a ﬂuid on the wall of a tube, Phys. Fluids 12,2367 (2000).
[5] V. Suresh and J. B. Grotberg, The effect of gravity on liquid plug propagation in a twodimensional
channel, Phys. Fluids 17,031507 (2005).
[6] Y. Zheng, H. Fujioka, and J. B. Grotberg, Effects of gravity, inertia, and surfactant on steady plug
propagation in a twodimensional channel, Phys. Fluids 19,082107 (2007).
[7] A. de Lózar, A. Juel, and A. L. Hazel, The steady propagation of an air ﬁnger into a rectangular tube,
J. Fluid Mech. 614,173 (2008).
09430416
EFFECT OF BUOYANCY ON THE MOTION OF LONG . . .
[8] Y. Han and N. Shikazono, Measurement of the liquid ﬁlm thickness in micro tube slug ﬂow, Int. J. Heat
Fluid Flow 30,842 (2009).
[9] R. Gupta, H. Hagnefelt, D. F. Fletcher, and B. S. Haynes, Proceedings of the Seventh International
Conference on Multiphase Flow, Tampa, University of Florida, Gainesville, FL, 2010.
[10] S. Y. Leung, R. Gupta, D. F. Fletcher, and B. S. Haynes, Gravitational effect on Taylor ﬂow in horizontal
microchannels, Chem. Eng. Sci. 69,553 (2012).
[11] M. H. Jensen, A. Libchaber, P. Pelcé, and G. Zocchi, Effect of gravity on the SaffmanTaylor meniscus:
Theory and experiment, Phys. Rev. A 35,2221 (1987).
[12] E. Brener, M. Rabaud, and H. Thomé, Effect of gravity on stable SaffmanTaylor ﬁngers, Phys.Rev.E48,
1066 (1993).
[13] R. Budwig, Refractive index matching methods for liquid ﬂow investigations, Exp. Fluids 17,350 (1994).
[14] S. Khodaparast, N. Borhani, and J. R. Thome, Application of micro particle shadow velocimetry μPSV
to twophase ﬂows in microchannels, Int. J. Multiphase Flow 62,123 (2014).
[15] N. Borhani, B. Agostini, and J. R. Thome, A novel time strip ﬂow visualisation technique for investigation
of intermittent dewetting and dryout in elongated bubble ﬂow in a microchannel evaporator, Int. J.
Multiphase Flow 53,4809 (2010).
[16] M. D. Giavedoni and F. A. Saita, The rear meniscus of a long bubble steadily displacing a Newtonian
liquid in a capillary tube, Phys. Fluids 11,786 (1999).
[17] E. Klaseboer, R. Gupta, and R. Manica, An extended Bretherton model for long Taylor bubbles at moderate
capillary numbers, Phys. Fluids 26,032107 (2014).
[18] A. Sharma and E. Ruckenstein, Dewetting of solids by the formation of holes in macroscopic liquid ﬁlms,
J. Colloid Interface Sci. 133,358 (1989).
[19] G. Callegari, A. Calvo, and J. P. Hulin, Dewetting processes in a cylindrical geometry, Eur. Phys. J. E 16,
283 (2005).
[20] A. Huerre, O. Theodoly, A. M. Leshansky, M.P. Valignat, I. Cantat, and M.C. Jullien, Droplets in
Microchannels: Dynamical Properties of the Lubrication Film, Phys.Rev.Lett.115,064501 (2015).
09430417