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Effect of buoyancy on the motion of long bubbles in horizontal tubes

September 2017
Physical Review Fluids 2(9)

DOI:10.1103/PhysRevFluids.2.094304

Authors:

Omer Atasi

Université Libre de Bruxelles

Sepideh Khodaparast

University of Leeds

Benoit Scheid

Université Libre de Bruxelles

As a confined long bubble translates along a horizontal liquid-filled 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 two-dimensional geometry at low capillary numbers Ca<10−3 and 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 side-view 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.

Schematic of a typical elongated bubble translating at capillary number Ca ≈ 10 −2 in a horizontal tube of radius a. The bubble is immersed in a viscous liquid of density ρ and viscosity μ. Here U and L are 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 g acts perpendicular to the tube axis, the bubble is inclined and the thin film has different thicknesses at the front (h +

…

Schematic of the experimental setup showing a capillary placed in a box filled with refractive index matching liquid to eliminate optical distortions.

…

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 bright-field microscopy. The bubble translates from the left to the right. Time-strip analysis is performed along a sample line such as section AA. The field of view in a typical image covers less than a 2-mm length of the tube. (b) Time-strip analysis result showing evolution of section AA in 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.

…

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.

…

Side-view 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 off-center 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 m r is 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.

…

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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 liquid-ﬁlled 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 two-dimensional 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 side-view 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 liquid-ﬁlled 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 low-viscosity ﬂuid at average speed Uinto a higher-viscosity 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

ATASI, KHODAPARAST, SCHEID, AND STONE

hfront

hback

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 semi-inﬁnite low-viscosity 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 cross-sectional plane, buoyancy causes asymmetry in the distribution of the liquid

ﬁlm around the bubble. In two-dimensional (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

094304-2

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 Saffman-Taylor problem

refers to the motion of a low-viscosity ﬂuid displacing a higher-viscosity ﬂuid in a Hele-Shaw 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] Saffman-Taylor ﬁnger, 2D

(Hele-Shaw)

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] Saffman-Taylor ﬁnger, 2D

(Hele-Shaw)

Theory, experiments Effect of buoyancy on the shape of the

front meniscus in tilted channels.

[5] Liquid slug bounded

between two air bubbles,

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,

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 cross-sectional 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.

094304-3

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 bright-ﬁeld side-view 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 two-phase 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

094304-4

EFFECT OF BUOYANCY ON THE MOTION OF LONG . . .

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 air-ﬁlled 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 home-made

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). Bright-ﬁeld 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 bright-ﬁeld

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 time-strip method to the sequence of shadowgraphy images [15].

094304-5

ATASI, KHODAPARAST, SCHEID, AND STONE

(a) (b) (c)A

A’ L

1 mm 2 s1 mm

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 bright-ﬁeld microscopy. The bubble translates from the left to the right.

Time-strip analysis is performed along a sample line such as section AA. The ﬁeld of view in a typical image

covers less than a 2-mm length of the tube. (b) Time-strip analysis result showing evolution of section AAin

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 air-glycerol 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 time-strip 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 time-strip 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 time-strip 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

side-view visualizations at ﬁnite Bo numbers with the classic predictions of bubble shape with

negligible buoyancy effects. (i) Due to non-negligible 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

094304-6

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 two-dimensional analysis proposed by

Jensen et al. [11] at low Ca and ﬁnite Bo in Hele-Shaw 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 self-sufﬁciency 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;

094304-7

ATASI, KHODAPARAST, SCHEID, AND STONE

I II III

Rf+

Rf-

y = +

y = -Rf-

y = Rf+

y = 0

y = r-(x)

y = r+(x)

FIG. 5. Side-view 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

off-center 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 two-dimensional 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 two-dimensional conﬁguration, in a Hele-Shaw 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

094304-8

EFFECT OF BUOYANCY ON THE MOTION OF LONG . . .

region δpCS ≈−

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≈−

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 two-dimensional 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 two-dimensional interface in a Hele-Shaw 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¯

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±

a, where

R±

fis the distance between the interface and the cell centerline (+) top and (−) bottom. Substituting

these expressions in Eq. (5), we obtain

±¯

R±

a=Bo

2R±

a2

∓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 

a1, 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,

prR+

a+R−

a=−2+Bo

2R+

a2

−R−

a2.(7)

094304-9

ATASI, KHODAPARAST, SCHEID, AND STONE

For Bo 1Eq.(7) can be approximated as

pr≈− 2a

R+

f+R−

.(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¯

r±=∓ 2a

R+

f+R−

−BoR±

a.(9)

Note that the choice of coordinate system gives that ∂¯

x¯

r+<0 and ∂¯

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

∂¯

∂˜

x±=Ca ∂2¯

∂˜

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 no-shear-stress condition at the interface, i.e., ∂¯

∂˜

y±=0aty=h±, where h±=1∓r±.The

pressure gradient along ˜

x±is given by the curvature gradient ∂¯

∂˜

x±=−∂˜

x±˜

h±. Integrating the

velocity ﬁeld along ˜

y±yields a third-order 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±

∞

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−

−BoR±

a,(14)

where a numerical solution of Eq. (12) shows that limX→∞ ∂XXH=1.337 [1], which ﬁnally leads

h±

∞

a=1.337 Ca2/3

R+

f+R−

±BoR±

.(15)

094304-10

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±

∞

=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.

094304-11

ATASI, KHODAPARAST, SCHEID, AND STONE

B. Inclination angle in the thin-ﬁlm 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

23

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 cross-sectional

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 non-negligible 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

094304-12

EFFECT OF BUOYANCY ON THE MOTION OF LONG . . .

top

bottom

Eq. (20)

Eq. (15) & Eq. (19)

hfront

0.05

0.10

0.15

0.20

0 0.03 0.06 0.09 0.12

top

bottom

Eq. (20)

Eq. (15) & Eq. (19)

hfront

0.05

0.10

0.15

0.20

0 0.03 0.06 0.09 0.12

top

bottom

Eq. (20)

Eq. (15) & Eq. (19)

hfront

0.05

0.10

0.15

0.20

0 0.03 0.06 0.09 0.12

top

bottom

Eq. (20)

Eq. (15) & Eq. (19)

hfront

0.05

0.10

0.15

0.20

0 0.03 0.06 0.09 0.12

(a) (b)

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 non-negligible 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.

094304-13

ATASI, KHODAPARAST, SCHEID, AND STONE

Bo = 0.056

Bo = 0.11

Bo = 0.25

Bo = 0.42

+ [10-3 rad]

0 0.03 0.06 0.09 0.12

- [10-3 rad]

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 non-negligible 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).

094304-14

EFFECT OF BUOYANCY ON THE MOTION OF LONG . . .

Bo = 0.056

Bo = 0.11

Bo = 0.25

Bo = 0.42

αave [10-3 rad]

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

0.5

1.0

1.5

2.0

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.

094304-15

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 two-dimensional 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 cross-sectional plane

as compared to a circular shape. If the cross-sectional 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 cross-sectional 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.S-FNRS for ﬁnancial support as well as the BELSPO agency under

Grant No. IAP-7/38 MicroMAST. S.K. appreciates the early mobility funding from Swiss National

Science Foundation (Grant No. P2ELP2-158896). We are grateful to the referees for their insightful

comments.

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Non-negligible buoyancy effect on bubbles travelling in horizontal microchannels of comparable size at small Bond numbers

Article

Full-text available

Oct 2024
INT J MULTIPHAS FLOW

Releasing long bubbles trapped in thin capillaries via tube centrifugation and inclination

Article

Full-text available

Nov 2024

In confined systems, the entrapment of a gas volume with an equivalent spherical diameter greater than the dimension of the channel can form extended bubbles that obstruct fluid circuits and compromise performance. Notably, in sealed vertical tubes, buoyant long bubbles cannot rise if the inner tube radius is below a critical value near the capillary length. This critical threshold for steady ascent is determined by geometric constraints related to the matching of the upper cap shape with the lubricating film surrounding the elongated part of the bubble. Developing strategies to overcome this threshold and release stuck bubbles is essential for applications involving narrow liquid channels. Effective strategies involve modifying the matching conditions with an external force field to facilitate bubble ascent. However, it is unclear how changes in acceleration conditions affect the motion onset of buoyancy-driven long bubbles. This study investigates the mobility of elongated bubbles in sealed tubes with an inner radius near the critical value inhibiting bubble motion in a vertical setting. Two strategies are explored to tune bubble motion, leveraging variations in axial and transversal accelerations: tube rotation around its axis and tube inclination relative to gravity. By revising the geometrical constraints of the simple vertical setting, the study predicts new thresholds based on rotational speed and tilt angle, respectively, providing forecasts for the bubble rising velocity under modified apparent gravity. Experimental measurements of motion threshold and rising velocity compare well with theoretical developments, thus suggesting practical approaches to control and tune bubble motion in confined environments.

Liquid plugs in narrow tubes: application to airway occlusion

Article

Nov 2024

Georg F. Dietze

We study liquid plugs in the pulmonary airways based on the two-phase axisymmetric weighted residual integral boundary-layer model of Dietze et al. ( J. Fluid Mech. , vol. 894, 2020, A17), which was originally developed to study liquid films coating the inner surface of a cylindrical tube in interaction with a core gas flow. The augmented form of this model, which was never applied beyond a proof of concept, allows for the representation of liquid pseudo-plugs. Here, we demonstrate its predictive power vs experiments and direct numerical simulations, in terms of the dynamics of plug formation and the characteristics of developed liquid plugs, such as their shape, flow field, speed and length, as well as the associated wall stresses and their spatial derivatives. In particular, we show that the augmented model allows us to establish a direct continuation path from travelling-wave solutions (TWS) to travelling-plug solutions (TPS). We then apply the model to predict mucus plugs in the conducting zone of the tracheobronchial tree, based on the lung architecture model of Weibel. We proceed by numerical continuation of travelling-state solutions in terms of the airway generation, whereby we impose the wavelength of the linearly most-amplified convective instability (CI) mode or that of the absolute instability (AI) mode. We identify the critical airway generation for liquid-plug formation (TWS/TPS transition), maximum potential for wall-stress-induced epithelial cell damage and CI/AI transition, and investigate how these phenomena are affected by the main control parameters, i.e. airway orientation vs gravity, air flow rate, mucus properties and airway size.

Immiscible non-Newtonian displacement flows in stationary and axially rotating pipes

Article

Full-text available

Oct 2024
PHYS FLUIDS

We examine immiscible displacement flows in stationary and rotating pipes, at a fixed inclination angle in a density-unstable configuration, using a viscoplastic fluid to displace a less viscous Newtonian fluid. We employ non-intrusive experimental methods, such as camera imaging, planar laser-induced fluorescence (PLIF), and ultrasound Doppler velocimetry (UDV). We analyze the impact of key dimensionless numbers, including the imposed Reynolds numbers (Re, Re*), rotational Reynolds number ( Rer), capillary number (Ca), and viscosity ratio (M), on flow patterns, regime classifications, regime transition boundaries, interfacial instabilities, and displacement efficiency. Our experiments demonstrate distinct immiscible displacement flow patterns in stationary and rotating pipes. In stationary pipes, heavier fluids slump underneath lighter ones, resulting in lift-head and wavy interface stratified flows, driven by gravity. Decreasing M slows the interface evolution and reduces its front velocity, while increasing Re* shortens the thin layer of the interface tail. In rotating pipes, the interplay between viscous, rotational, and capillary forces generates swirling slug flows with stable, elongated, and chaotic sub-regimes. Progressively, decreasing M leads to swirling dispersed droplet flow, swirling fragmented flow, and, eventually, swirling bulk flow. The interface dynamics, such as wave formations and velocity profiles, is influenced by rotational forces and inertial effects, with Fourier analysis showing the dependence of the interfacial front velocity's dominant frequency on Re and Rer. Finally, UDV measurements reveal the existence/absence of countercurrent flows in stationary/rotating pipes, while PLIF results provide further insight into droplet formation and concentration field behavior at the pipe center plane.

Equilibrium shape of a bubble in a liquid-filled horizontal capillary

Article

Feb 2024
INT J MULTIPHAS FLOW

Experimental observation of a confined bubble moving in shear-thinning fluids

Article

Full-text available

Dec 2022

The motion of a long gas bubble in a confined capillary tube is ubiquitous in a wide range of engineering and biological applications. While the understanding of the deposited thin viscous film near the tube wall in Newtonian fluids is well developed, the deposition dynamics in commonly encountered non-Newtonian fluids remains much less studied. Here, we investigate the dynamics of a confined bubble moving in shear-thinning fluids with systematic experiments, varying the zero-shear-rate capillary number

Ca_0

in the range of O(10^{-3}\unicode{x2013}10^2) considering the zero-shear-rate viscosity. The thickness of the deposited liquid film, the bubble speed and the bubble front/rear menisci are measured, which are further rationalized with the recent theoretical studies based on appropriate rheological models. Compared with Newtonian fluids, the film thickness decreases for both the carboxymethyl cellulose and Carbopol solutions when the shear-thinning effect dominates. We show that the film thickness follows the scaling law from Aussillous & Quéré ( Phys. Fluids , vol. 12, no. 10, 2000, pp. 2367–2371) with an effective capillary number

Ca_e

, considering the characteristic shear rate in the film as proposed by Picchi et al. ( J. Fluid Mech. , vol. 918, no. A7, 2021, pp. 1–30).

Ca_e

is calculated by the Carreau number and the power-law index from the Carreau–Yasuda rheological model. The shear-thinning effect also influences the bubble speed and delays the transition to the parabolic region in the bubble front and rear menisci. In particular, a high degree of undulations on the bubble surface results in an intricate rear viscosity distribution for the rear meniscus and the deviation between the experiments and theory may require a further investigation to resolve the axial velocity field. Our study may advance the fundamental understandings and engineering guidelines for coating processes involving thin-film flows and non-Newtonian fluids.

Experimental observation of a confined bubble moving in shear-thinning fluids

Preprint

Full-text available

Nov 2022

Ca_0

in the range of

O(10^{-3}-10^2)

considering the zero-shear-rate viscosity. The thickness of the deposited liquid film, the bubble speed and the bubble front/rear menisci are measured, which are further rationalized with the recent theoretical studies based on appropriate rheological models. Compared with Newtonian fluids, the film thickness decreases for both the carboxymethyl cellulose and Carbopol solutions when the shear-thinning effect dominates. We show that the film thickness follows the scaling law from \citet{aussillous2000quick} with an effective capillary number

Ca_e

, considering the characteristic shear rate in the film as proposed by \citet{picchi2021motion}.

Ca_e

is calculated by the Carreau number and the power-law index from the Carreau-Yasuda rheological model. The shear-thinning effect also influences the bubble speed and delays the transition to the parabolic region in the bubble front and rear menisci. In particular, a high degree of undulations on the bubble surface results in intricate rear viscosity distribution for the rear meniscus and the deviation between the experiments and theory may require a further investigation to resolve the axial velocity field. Our study may advance the fundamental understandings and engineering guidelines for coating processes involving thin-film flows and non-Newtonian fluids.

Dispersion of a passive scalar around a Taylor bubble

Article

Full-text available

Nov 2022

The motion of Taylor bubbles in capillaries is typical of many engineering and biological systems, ranging from subsurface flows to small-scale reactors. Although the hydrodynamics of elongated bubbles has been the object of several studies, the case where a solute is transported in the surrounding liquid and surface mass-transfer mechanisms act on the solid wall or the bubble–fluid interface is much less understood. To fill this gap, we investigate the transport problem around a confined Taylor bubble to access the competition between advection, diffusion and surface mass transfer in the different regions of the bubble. With this aim, we derive a one-dimensional advection–diffusion–mass-transfer equation where the transport mechanisms are described through an effective velocity, an effective diffusion coefficient and an effective Sherwood number. Our model generalises the Aris–Taylor dispersion to the case of a Taylor bubble and clarifies the impact of surface mass transfer in the advection and diffusion dominated regimes for both the front and rear menisci. The model recovers the typical Pèclet square relationship of the effective diffusion coefficient, which also depends on the film thickness. Also, when the Pèclet number balances with the Sherwood number, there exist conditions that lead to the formation of hot spots of concentration. We show that the typical shape oscillations of the bubble rear locally enhance superficial mass transfer. Finally, we study the transport problem in the uniform film region where the concentration field can be found analytically.

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

Article

Full-text available

Apr 2024

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 Cae

Ca_\mathrm{e}

and the effective Weissenberg number Wie

Wi_\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.

Motion of a confined bubble in a shear-thinning liquid

Article

Full-text available

May 2021

The motion of a gaseous Taylor bubble in a capillary tube is typical of many biological and engineering systems, such as small-scale reactors and microfluidic devices. Although the dynamics of a bubble in a Newtonian liquid has been the subject of several studies since the seminal works of Taylor (J. case where the fluid exhibits a shear-thinning behaviour is much less understood. To fill this gap, we study the dynamics of a bubble that moves in a shear-thinning fluid whose viscosity is described by the Ellis viscosity model. With this aim, we derive a lubrication model in the film region to identify the scaling laws for the bubble speed, the film thickness and the pressure drop as a function of the Ellis number and the degree of shear thinning. Our model generalizes Bretherton's results to shear-thinning fluids by identification of a universal scaling law for the effective viscosity that accounts for the interplay of the zero-shear-rate and shear-thinning effects. The film thickness follows a 2/3 scaling law with respect to the capillary number based on the proposed effective viscosity. The ratio between the bubble speed and the average velocity of the fluid ahead of the bubble is a function of the effective capillary number only. We show that some portions of the bubble are dominated by the zero-shear-rate effect discussing the extent to which the use of the power-law viscosity model can be legitimized. Finally, we study the location of the recirculating vortices ahead of the bubble.

Droplets in Microchannels: Dynamical Properties of the Lubrication Film

Article

Full-text available

Aug 2015
PHYS REV LETT

We study the motion of droplets in a confined, micrometric geometry, by focusing on the lubrication film between a droplet and a wall. When capillary forces dominate, the lubrication film thickness evolves nonlinearly with the capillary number due to the viscous dissipation between the meniscus and the wall. However, this film may become thin enough (tens of nanometers) that intermolecular forces come into play and affect classical scalings. Our experiments yield highly resolved topographies of the shape of the interface and allow us to bring new insights into droplet dynamics in microfluidics. We report the novel characterization of two dynamical regimes as the capillary number increases: (i) at low capillary numbers, the film thickness is constant and set by the disjoining pressure, while (ii) above a critical capillary number, the interface behavior is well described by a viscous scenario. At a high surfactant concentration, structural effects lead to the formation of patterns on the interface, which can be used to trace the interface velocity, that yield direct confirmation of the boundary condition in the viscous regime.

Quick deposition of a fluid on the wall of a tube

Article

Full-text available

Oct 2000

We are interested in the amount of fluid left behind a drop moved inside a capillary tube. Long ago, Taylor showed that for very viscous liquids moved at small velocities, the film thickness is a monotonic increasing function of the capillary number. New data obtained with liquids of low viscosity are reported here and compared with Taylor's law. Two successive effects are observed: above a threshold in capillary number, the film is thicker than a Taylor film; at a very high speed, the deposition law becomes a decreasing function of the drop velocity. Both behaviors are analyzed thanks to scaling arguments and shown to be consequences of inertia.

The effect of gravity on liquid plug propagation in a two-dimensional channel

Article

Full-text available

Mar 2005

The effect of plug propagation speed and gravity on the quasisteady motion of a liquid plug in a two-dimensional liquid-lined channel oriented at an angle α with respect to gravity is studied. The problem is motivated by the transport of liquid plugs instilled into pulmonary airways in medical treatments such as surfactant replacement therapy, drug delivery, and liquid ventilation. The capillary number Ca is assumed to be small, while the Bond number Bo is arbitrary. Using matched asymptotic expansions and lubrication theory, expressions are obtained for the thickness of the trailing films left behind by the plug and the pressure drop across it as functions of Ca, Bo, α and the thickness of the precursor films. When the Bond number is small it is found that the trailing film thickness and the flow contribution to the pressure drop scale as Ca2/3 at leading order with coefficients that depend on Bo and α. The first correction to the film thickness is found to occur at O(Ca) compared to O(Ca4/3) in the Bo=0 case. Asymmetry in the liquid distribution is quantified by calculating the ratio of liquid volumes above and below the centerline of the channel, VR;. VR=1 at Bo=0, indicating a symmetric distribution, and decreases with Bo and Ca, but increases with the plug length Lp. The decrease of VR with Ca suggests that higher propagation speeds in small airways may result in less homogenous liquid distribution, which is in contrast to the expected effect in large airways. For given values of the other parameters, a maximum capillary number Cac is identified above which the plug will eventually rupture. When the Bond number becomes equal to an orientation-dependent critical value Boc, it is found that the scaling of the film thickness and pressure drop change to Ca1/2 and Ca1/6, respectively. It is shown that this scaling is valid for small increments of the Bond number over its critical value, Bo=Boc+BCa1/6, but for higher Bond numbers the asymptotic approach breaks down.

Studies in electro-endosmosis - VI. The "bubble tube" method of measurement

Article

An extended Bretherton model for long Taylor bubbles at moderate capillary numbers

Article

Feb 2014

When (long) bubbles are transported in tubes containing a fluid, the presence of a thin film of fluid along the tube walls causes the velocity of the bubble to be different from the average fluid velocity. Bretherton ["The motion of long bubbles in tubes," J. Fluid Mech. 10, 166 (1961)] derived a model to describe this phenomenon for pressure driven flows based on a lubrication approach coupled with surface deformation of the bubble. Bretherton found that the parameter governing the physics involved is the capillary number (Ca) which expresses the relationship between speed of the bubble, surface tension, and viscosity of the liquid. The results of Bretherton are here re-derived and analyzed in a slightly more perspicuous manner. Incorporating the condition that the bubble-film combination should fit inside the tube results in an expression very similar to the one found empirically by Aussillous and Quéré ["Quick deposition of a fluid on the wall of a tube," Phys. Fluids 12, 2367 (2000)] of the Taylor ["Deposition of a viscous fluid on the wall of a tube," J. Fluid Mech. 10, 161 (1961)] experimental data. Our expression is valid for Ca values up to Ca = 2.0, but approaches Bretherton's result for low values of Ca. The analysis is done in terms of the pressure buildup which originates from the interplay between surface tension and lubrication due to the thin layer of fluid near the tube wall.

Application of micro particle shadow velocimetry μPSV to two-phase flows in microchannels

Article

Jun 2014
INT J MULTIPHAS FLOW

Micro particle shadow velocimetry (μμPSV) is performed in the present study for simultaneous velocity measurement and interface tracking in both liquid-liquid and gas-liquid two-phase flows through circular microchannels of 500 μμm diameter. The back-lit illumination using a non-coherent LED light source, combined with full refractive index matching of the liquid phases, the tube wall material and the channel exterior medium, allowed velocimetry to be done both within and around the liquid droplets, and even close to the interfaces and boundaries. Moreover, post-processing methods are proposed and implemented in order to resolve motion of isolated gas bubbles and immiscible liquid droplets in laminar flows quantitatively. In particular, simultaneous measurements of local instantaneous phase velocities and flow rates, liquid film dynamics and its thickness, shape and volume of the dispersed phase, and development length in front and at the back of the bubbles are obtained using one single sequence of grey scale shadowgraphy images. Such results are valuable for validation of corresponding numerical simulation codes. It is believed that this approach significantly reduces the size and cost of the experimental setup while increasing the accuracy and reliability of the measurements.

The rear meniscus of a long bubble steadily displacing a Newtonian liquid in a capillary tube

Article

Apr 1999

In this work the interfacial shapes and the flow occurring at the trailing meniscus of a long bubble is numerically analyzed. The technique employed solves the complete set of governing equations simultaneously. The numerical results reported complete previous descriptions of the creeping flow regime; the influence of the inertia forces on the free surface shapes, interfacial undulations, and flow patterns is also analyzed.

Effects of gravity, inertia, and surfactant on steady plug propagation in a two-dimensional channel

Article

Aug 2007

Liquid plugs may form in pulmonary airways during the process of liquid instillation or removal in many clinical treatments. Studies have shown that the effectiveness of these treatments may depend on how liquids distribute in the lung. Better understanding of the fundamental fluid mechanics of liquid plug transport will facilitate treatment strategies. In this paper, we develop a numerical model of steady plug propagation driven by gravity and pressure in a two-dimensional liquid-lined channel oriented at an angle alpha with respect to gravity. We investigate the effects of gravity through the Bond number, Bo, and alpha the plug propagation speed through the capillary number, Ca, or the Reynolds number, Re; the plug length LP, and the surfactant concentration C0. Without gravity, i.e., Bo=0, the plug is symmetric, and there are two regimes for the flow: two wall layers and two trapped vortices in the core. There is no flow interaction between the upper and lower half plug domains. When Bo!=0 and alpha!=0, pi, fluid is found to flow from the upper precursor film, through the core and into the lower trailing film. Then the number of vortices can be zero, one, or two, depending on the flow parameters. The vortices have stagnation points on the interface when C0=0, however when the surfactant is present (C0>0), the vortices detach from the interface and create saddle points inside the core. The front meniscus develops a capillary surface wave extending into the precursor film. This is where the film is thinnest and thus the wall shear stress is highest, as high as ~100 dyn/cm2 in adult airways, which indicates a significant risk of pulmonary airway epithelial cell damage. Adding surfactant can decrease the peak magnitude of the shear stress, thus reducing the risk of cell damage. The prebifurcation asymmetry of the plug is quantified by the volume ratio, Vr, defined as the ratio of the liquid above to that below the center line of the channel. Vr is found to increase with LP, Ca, Re, and C0, while it decreases with Bo. The total mass left behind in the trailing films increases with Bo for any alpha at alpha>2pi/5, Ca and alpha for any value of Bo>0.

Studies in electroendosmosis. part vi. The bubble-tube methods of measurement

Article

Jan 1935
J Chem Soc

Gravitational Effect on Taylor Flow in Horizontal Microchannels

Article

Feb 2012
CHEM ENG SCI

The effect of the gravitational force has generally been ignored in studies of the hydrodynamics, mass and heat transfer characteristics of Taylor flow in microchannels. In this study, the effects of gravity on the bubble behavior and heat transfer rates for horizontal Taylor flow in millimeter-sized channels were studied experimentally. Significant bubble asymmetry was observed for nitrogen-ethylene glycol Taylor flow in three circular channels of different internal diameters (d=1.12, 1.69, 2.12 mm giving Bond numbers (Bo) of 0.287, 0.653 and 1.028). A parabolic drainage velocity profile inside the liquid film surrounding the bubble was noted through performing Micro-Particle Image Velocimetry (μPIV) measurements in that region. A simplified model similar to Nusselt's solution for laminar film condensation on a horizontal tube was derived to describe the drainage velocity profile in the liquid film at a cross-section away from the bubble ends and was able to explain the observed tilt of the bubble. Performing a non-dimensional analysis of the model, the ratio of the average drainage to bubble velocities was found to be a function of the Bond and Capillary (Ca) numbers. The gravitational effect on the bubble behavior vanishes for small value of Bo and/or Ca or as Ca→∞Ca→∞. Heat transfer experiments in a 2.00 mm tube (Bo=0.915) showed that gravitational effects had no significant effect on heat transfer in the Taylor flow regime for the conditions studied.