An extended Bretherton model for long Taylor bubbles at moderate capillary numbersEvert Klaseboer, Raghvendra Gupta, and Rogerio Manica

Citation: Physics of Fluids (1994-present) 26, 032107 (2014); doi: 10.1063/1.4868257 View online: http://dx.doi.org/10.1063/1.4868257 View Table of Contents: http://scitation.aip.org/content/aip/journal/pof2/26/3?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Film drainage of viscous liquid on top of bare bubble: Influence of the Bond number Phys. Fluids 25, 022105 (2013); 10.1063/1.4792310 The effect of viscoelasticity on the stability of a pulmonary airway liquid layer Phys. Fluids 22, 011901 (2010); 10.1063/1.3294573 Capillary origami in nature Phys. Fluids 21, 091110 (2009); 10.1063/1.3205918 Liquid film dynamics in horizontal and tilted tubes: Dry spots and sliding drops Phys. Fluids 19, 042102 (2007); 10.1063/1.2714569 Upward-propagating capillary waves on the surface of short Taylor bubbles Phys. Fluids 18, 048103 (2006); 10.1063/1.2192781

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PHYSICS OF FLUIDS 26, 032107 (2014)

An extended Bretherton model for long Taylor bubblesat moderate capillary numbers

Evert Klaseboer,1 Raghvendra Gupta,1,2 and Rogerio Manica1

1Institute of High Performance Computing, 1 Fusionopolis Way, Singapore 1386322Birla Institute of Technology and Science, Pilani, K K Birla Goa Campus, Zuaringar, 403726 Goa, India

(Received 11 October 2013; accepted 27 February 2014; published online 20 March 2014)

When (long) bubbles are transported in tubes containing a fluid, the presence of a thinfilm of fluid along the tube walls causes the velocity of the bubble to be different fromthe average fluid velocity. Bretherton [“The motion of long bubbles in tubes,” J. FluidMech. 10, 166 (1961)] derived a model to describe this phenomenon for pressuredriven flows based on a lubrication approach coupled with surface deformation ofthe bubble. Bretherton found that the parameter governing the physics involved isthe capillary number (Ca) which expresses the relationship between speed of thebubble, surface tension, and viscosity of the liquid. The results of Bretherton arehere re-derived and analyzed in a slightly more perspicuous manner. Incorporatingthe condition that the bubble-film combination should fit inside the tube results inan expression very similar to the one found empirically by Aussillous and Quere[“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 isdone in terms of the pressure buildup which originates from the interplay betweensurface tension and lubrication due to the thin layer of fluid near the tube wall.C© 2014 AIP Publishing LLC. [http://dx.doi.org/10.1063/1.4868257]

I. INTRODUCTION

Reynolds1 introduced lubrication theory to study thin liquid films between rigid surfaces.Later deformable surfaces were analyzed as well by other authors. Landau and Levich2 noted that:“Numerous attempts at evaluating the thickness of dragged layer of fluid found in literature containsome incorrect assumptions in the very basis of the method of computation, thus leading to erroneousformulae for the value of this thickness.” Even though tremendous progress has been made in ourunderstanding of the physics involved for such problems, this does not exclude that substantialimprovements of our theoretical knowledge are still possible.

Thin films are often difficult to measure, yet, if care is taken to use the right physics at the rightplaces, deformable thin films can be modeled with surprising accuracy. For example, air lubricationbetween a deformable tap and a recording head3 or interactions involving drops4 and bubbles,5, 6

provide excellent agreement with experimental data without any fitting parameter.Thin films are also important in the study of movement of long bubbles through small tubes,

often referred to as Taylor bubbles.7 Fairbrother and Stubbs8 were probably the first to realize thatthe speed of such bubbles is not the same as the average speed of the fluid in the tube, due to theliquid that “has been left behind by the liquid moving in front of the bubble” and “becomes part ofthe liquid behind the rear meniscus.”

Bretherton9 derived a model to describe a Taylor bubble based on a lubrication approach coupledwith surface deformation of the bubble. The effect of inertia10, 11 was deemed to be negligible inhis analysis (low Reynolds number). Furthermore, the flow considered was pressure driven – notbuoyancy driven due to gravity.

1070-6631/2014/26(3)/032107/8/$30.00 C©2014 AIP Publishing LLC26, 032107-1

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032107-2 Klaseboer, Gupta, and Manica Phys. Fluids 26, 032107 (2014)

FIG. 1. Bubble moving with a velocity U through a channel of radius r. The constant film thickness at the “flat” section (orfilm region) is “h = b.” At the front of the bubble there is a transition region in which the pressure builds up. The pressurecurve is also shown.

In Figure 1, we illustrate a typical Taylor bubble with front radius RF. The pressure p, is definedto be zero in front of the bubble. For long bubbles, the middle section (film region) of the bubble hasa constant film thickness, b, which was calculated by Bretherton9 to be

b

r= 0.643 (3Ca)2/3 (1)

in which r is the tube radius and Ca is the capillary number defined as Ca = μU/σ , with U thevelocity of the bubble (which is different from the average liquid velocity!), μ the viscosity of thefluid medium, and σ its surface tension.

Equation (1) is only valid for very low Ca numbers (accurate within 10% for Ca < 5 × 10−3).An empirical “fix” was proposed by Aussillous and Quere,12 which was in good agreement with theexperiments of Taylor7 for capillary number values up to ∼2

b

r= 1.34 (Ca)2/3

1 + 2.5∗1.34 (Ca)2/3 . (2)

Aussillous and Quere12 state that: “A fit . . . can even be found (maybe coincidentally) to describequite precisely the data” using the form of Eq. (2) “where the coefficient 2.5 is empirical.” In thiswork, we will show that this “empirical model” can be derived theoretically based on Bretherton’s9

analysis with a few essential modifications (see Sec. II).Despite the progress made so far, in recent years there is renewed interest in understanding

the physics of Taylor bubbles (and droplets) in capillaries because of its numerous applications inmicrofluidics13 and chemical microprocessing.14–16 Gravitational effects on horizontal Taylor bub-bles flow were investigated by Leung et al.17 Numerical simulations were performed by Ratulowskiand Chang18 combined with matched asymptotic analysis in an axial symmetric configuration andby Halpern and Gaver,19 Suresh and Grotberg,20 and Zheng et al.21 in a 2D configuration. Pressuredrops, including inertial effects were investigated by Kreutzer et al.22 Reviews on the area were madeby Angeli and Gavriilidis23 and Gupta et al.24 and we refer to the references in those works for furtherexperimental and numerical studies on Taylor bubbles. Besides bubbles in circular tubes, bubbles inrectangular and square channels were studied by Abadie et al.25 and Cubaud and Ho,26 respectively.Taylor droplets were studied by Fischer et al.27 and Gupta et al.28 in view of their importance inheat transfer. For Taylor bubbles, the flow conditions and the types of fluids in these applicationscan vary over a wide range and so do the Capillary and Reynolds numbers, the non-dimensionalnumbers that characterize the flow. As the liquid film thickness is an important design parameter in

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032107-3 Klaseboer, Gupta, and Manica Phys. Fluids 26, 032107 (2014)

all of these applications, it is important to develop an expression for film thickness which is validover a wide range of capillary numbers with a firm theoretical basis.

For Taylor bubbles, the thickness of the film region was shown to be directly related to the speedof the bubble.29 The bubble speed is independent of the length (volume) of the bubble provided thatit is “long” enough. This also implies that the bubble shape at the front remains unaffected from thatat the back.30, 31 We will also investigate how long the bubble has to be for this condition to be validin Sec. III of this paper.

II. THE EXTENDED BRETHERTON MODEL

Both surface tension and lubrication are important in this problem and the solution shouldreflect the correct combination of the two effects. A theory combining the effects of deformation andlubrication can give very accurate results (when compared with experiments), for example, in thearea of film drainage and coalescence.32–34 The Young-Laplace equation describes the relationshipbetween the pressure p (in both the transition region and in the film region), the surface tension σ ,and the film thickness h

p = −σd2h

dx2− σ

r − b+ 2σ

RF. (3)

This equation states that there is a pressure jump across the bubble interface, equal to thecurvature of this interface times the surface tension. It is assumed that the pressure inside the bubble(the Laplace pressure) is constant and equal to 2σ /RF, where the Laplace radius, RF is unknown apriori. The first two terms on the right hand side in Eq. (3) are related to the curvature of the bubblesurface. The exact expression for the curvature is more complex in an axial symmetric context, butit can easily be shown that these two terms are the dominant terms for this particular problem. Wenow need to find an expression for the pressure buildup.

In the case of thin films (such as is the case here), we can use the lubrication approach1 to modelthe film that forms between the bubble and the tube. The approach followed here is essentially thesame as that in Bretherton.9 We will make a distinction between the flat film region (see Figure 1)and the transition region (or pressure build-up region, Figure 1). In order for the lubrication approachto be valid, the film must be relatively flat, such that the pressure in the direction normal to the wall(y) is constant. Thus, the pressure is only a function of the x-coordinate. The velocity in the film, u,can be a function of both x and y. The Navier-Stokes equations then simplify to

1

μ

dp

dx= d2u

dy2. (4)

In the flat film region, the film thickness is constant and equal to h = b.From Eq. (4) and applying this equation in the LABORATORY frame of reference, we get the

following boundary conditions: u = 0 at y = 0 (velocity at the tube wall is zero) and dudy = 0 at y

= b (no shear stress at the bubble surface). Since the pressure in the flat film is constant and equalto p = pF, this necessarily means that the velocity is everywhere zero, u = 0, in the film! This alsomeans that there is no energy dissipation in the film region.

In the BUBBLE frame of reference, the lubrication equation can be deduced as follows. Thebubble surface is not moving, while the tube is moving backwards with a speed −U. IntegratingEq. (4) twice, we obtain

u = 1

2μ

dp

dxy2 + cy + c2. (5)

With boundary conditions u = −U at y = 0 and dudy = 0 at y = h (no shear stress at the bubble

surface). Thus,

u = 1

μ

dp

dx

[y2

2− yh

]− U. (6)

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032107-4 Klaseboer, Gupta, and Manica Phys. Fluids 26, 032107 (2014)

The flux across any x-section of the film must be constant (in the bubble frame of reference)

h∫0

udy = 1

μ

dp

dx

[y3

6− y2

2h

]h

0

− Uh = Ub. (7)

The last equality follows from the fact that for x → −∞, dp/dx = 0 (since the flat film velocity iszero in the laboratory frame of reference, it must be −U in the bubble frame of reference and thusthe flux is Ub). This will lead to dp

dx = −3μU h−bh3 . This equation is valid in both the flat film and the

transition region. The pressure will decay very rapidly in front of the bubble due to the 1/h2 behaviorof dp/dx.

Thus, finally the pressure buildup in the transition region can be obtained from lubricationtheory as

dp

dx= −3μU

h − b

h3= −σ

d3h

dx3. (8)

In the last equality, we have replaced the pressure using Eq. (3). Following Bretherton,9 a universalversion of the last two terms of Eq. (8), containing no free parameters, can be obtained by introducingthe following dimensionless scaling for h, x, and p:

h = bη, x = b (3Ca)−1/3 ξ, p = σ

b(3Ca)2/3 p∗. (9)

Equation (8) can then be rewritten as

d3η

dξ 3= η − 1

η3. (10)

This equation is also often referred to as the Landau-Levich equation.35 It does not have ananalytical solution, however it must have the asymptotic solution η = 1 (or h = b) for ξ → −∞(constant film thickness in the film region). At the front of the bubble, the asymptotic solution willbe η = 1/2P(ξ − ξ 0)2 + R for ξ → ∞ (a parabolic profile corresponding to a zero pressure). Indimensional coordinates, this corresponds to

h = 1/2P (3Ca)2/3 (x − x0)2 /b + bR. (11)

This is just part of a parabolic approximation of a sphere with radius bP (3Ca)−2/3, which must

join up with the front of the Taylor bubble. With the curvature at the front being κ = 2/RF, weget: 1

RF= P

b (3Ca)2/3, where P is called the Landau-Levich constant36 and has been calculatedby Bretherton9 to be P = 0.643 while the value of R was found by Bretherton9 to be R = 2.79.Rearranging we get

b

RF= P (3Ca)2/3 . (12)

This is almost the same equation as Eq. (17) in Bretherton,9 but with the subtle, yet importantdifference that Bretherton used “r” (the tube radius) instead of “RF,” the Laplace radius as shownin Figure 1. Equation (12) is a more accurate representation of the physics involved, especially forhigher Ca numbers for which the assumption of RF ∼r is not true. For very low Ca numbers, RF

reverts back to r, since the film thickness b tends towards zero.We now enter a crucial point in the derivation, which was not used by Bretherton.9 Ideally, the

bubble should “fit” in the tube as illustrated in Figure 2, thus, h(x = x0) + RF = r, with h(x = x0) =bR from Eq. (11). That is,

h (x = x0) + RF = bR + RF = r (13)

with R = 2.79. Equation (13) will give an additional equation for the unknown Laplace radius.Equations (12) and (13) have two unknowns, b and RF. By eliminating b we obtain the followingrelationship:

2.79 (3Ca)2/3 P RF + RF = RF[2.79 (3Ca)2/3 P + 1

] = r. (14)

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032107-5 Klaseboer, Gupta, and Manica Phys. Fluids 26, 032107 (2014)

FIG. 2. The “tube fit” condition illustrated graphically. If the spherical solution of the front of the bubble with radius RF isextended towards the film region, a thickness bR remains. Ideally, RF+bR should fit inside the tube and thus be equal to r,for a consistent theory.

FIG. 3. (a) W and (b) film thickness b/r as a function of Ca for the various models; the current model, the Bretherton model,and the Aussillous-Quere12 fit. Also shown are the experiments of Taylor.7 Note that b/r is approximately 1/3 of the tuberadius for Ca∼2 and the same trend is observed (b/r has been obtained from the experimentally measured W values ofTaylor7). The model also reverts back to the original Bretherton results for low Ca numbers.

Thus resulting in the following equation for the Laplace radius RF:

RF = r

1 + 2.79P (3Ca)2/3 . (15)

Combining Eqs. (12) and (15) gives an expression for b (with P = 0.643)

b

r= P (3Ca)2/3

1 + 2.79P (3Ca)2/3 . (16)

This answer is quite close to the fit of Aussillous and Quere12 (see Figure 3(b)) and the experimentaldata of Taylor.7 The constant 2.79 originates from R = 2.79 of Eq. (11). Note that P × 32/3 = 0.643× 32/3 = 1.34 (accounting for this factor in Eq. (2)). Our model has a factor R = 2.79 instead of2.50, a discussion on these values for R can be found in Sec. III.

There is a difference between the speed of the bubble U and the average speed of the liquidfar from the bubble, U0. This difference is expressed as: U−U0 = U*W. Thus, W = 1−U0/U.Conservation of mass leads to Uπ (r − b)2 = U0πr2 = (U − U W )πr2. This leads to the followingrelationship between W and the film thickness b:

W = 1 − (1 − b/r )2 . (17)

In Figure 3(a), we have plotted W of Eq. (17), with b/r according to Eq. (16). The original Brethertonmodel is only accurate within 10% for Ca < 5 × 10−3. In Figure 3(b), b/r as a function of Ca is

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032107-6 Klaseboer, Gupta, and Manica Phys. Fluids 26, 032107 (2014)

plotted. We see a similar trend as in Figure 3(a). Again, a remarkable improvement with respect to theBretherton model is observed. The results are in good agreement with those obtained experimentallyby Taylor.7

For the pressure jump across the front bubble surface we get, using Eq. (15)

2σ

RF= 2σ

r

[1 + 2.79 (3Ca)2/3 P

]. (18)

The additional pressure jump, obtained by subtracting the “static” value 2σr from Eq. (18), is now

�p = 2σ

r2.79 (3Ca)2/3 P = 2σ

r1.79 (3Ca)2/3 = 3.58

σ

r(3Ca)2/3 (19)

which is identical to the “dynamic pressure” expression found by Bretherton9 (p. 172). �p isessentially the additional pressure difference across the front of the bubble due to the motion of thebubble (it is not to be confused with the pressure drop across the whole bubble). Bretherton9 statesthat Eq. (19) is only valid for values of Ca approaching zero, but the above derivation suggests thatthis equation is valid for much higher Ca numbers. To the best knowledge of the authors, there is nodirect way to measure this pressure jump experimentally.

III. DISCUSSION

The factor R = 2.79 was obtained by Bretherton9 by matching a parabola to the front profile ofthe bubble. A careful numerical analysis reveals that R converges very slowly to a final value of R =2.90, when the parabola is matched at η = 1 × 106. For example, fixing the parabola at η = 1000,will give R = 2.88 and using η = 1 × 105 will result in R = 2.89. However, for Ca∼1, the filmthickness actually occupies about one third of the tube radius. If the parabola is matched at a morephysical height of η = 50, we will get R = 2.5 (as in the Aussillous and Quere12 fit). This probablyexplains the slight difference in value found for R by Bretherton9 and Aussillous and Quere.12 In ourwork, we have kept the Bretherton value of R = 2.79. The factor P = 0.643 can be calculated withmuch more accuracy (since it can also be obtained from integrating the dimensionless pressure, thus−∞∫∞

η−1η3 dξ = P). Landau and Levich2 and Levich37 found a slightly different value of P = 0.63.

It is illustrative to plot the pressure in the transition region. This is done in Figure 4. Indimensionless form, it starts at zero and builds up to a value of P (the horizontal axis is arbitrary). Aquestion we could ask ourselves is how long the bubble needs to be in order to be considered “long”for the Bretherton model to be applicable. According to Figure 4, when investigating the pressure,the dimensionless transition region is about 10 dimensionless units wide. Expressed in dimensionalform this leads to (using Eqs. (9) and (16)): L = b (3Ca)−1/3 10 = 10r P(3Ca)1/3

1+2.79P(3Ca)2/3 . Thus, if the bubbleis much longer than “L,” the bubble can be considered long, such that entry and exit effects are nolonger important (we have assumed here that the transition region at the back of the bubble is ofsimilar order of magnitude).

Careful comparison of the theory developed here and the original one by Bretherton9 showsthat the only differences are the introduction of the Laplace radius RF instead of the tube radius rand the introduction of the “tube fit criterion” in Eq. (13) to determine this Laplace radius. Thesetwo modifications are sufficient to predict the film thickness for relatively large Ca numbers up toCa∼2 (a factor of 400 improvement over the original Bretherton model which only worked for Ca< 5 × 10−3). Bretherton9 thus used r instead of RF in Eq. (12), but actually reintroduced RF withthe �p term of Eq. (19). As it turns out, both approaches give the same value for �p (the pressuredrop across the front of the bubble).

The results are valid as long as the film is not so thin that disjoining pressures (such as electricaldouble layer and Van der Waals) become important. Usually these effects appear for a film thicknessthinner than 0.1 μm.

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032107-7 Klaseboer, Gupta, and Manica Phys. Fluids 26, 032107 (2014)

FIG. 4. The film profile and pressure given in the dimensionless variables η and p*, see Eq. (9) as a function of ξ . Thepressure builds up in front of the bubble (right hand side of the graph). For ease of understanding, the pressure and filmprofile at the back of the bubble are also given (left hand side of the graph). Only for Ca ∼ 0 the pressure at the back willtend towards zero again (as in the above graph, which corresponds to Bretherton’s Ca ∼ 0 case). The profile at the front ofthe bubble is universal and thus valid for larger Ca numbers as well. Note that the dotted line represents the fact that the frontand back solutions were obtained separately.

IV. CONCLUSIONS

It has been shown that the film thickness around a Taylor bubble can be obtained analyticallyfor relatively large Ca numbers and corresponds well to the experimental results of Taylor7 by acombination of lubrication and surface tension effects. This was done by extending the Brethertonmodel, using the Laplace radius and requiring that the bubble “fits” in the tube. It does show thetenacity of the Reynolds film lubrication theory, which usually can give very accurate results. Theresults could also have implications in areas closely related to Taylor bubbles, such as in the studyof foams38 (which are still not yet fully understood), or applications in physiological flows.39

1 O. Reynolds, “On the theory of lubrication and its applications to Mr. Beauchamp’s experiments, including an experimentaldetermination of the viscosity of olive oil,” Philos. Trans. R. Soc. 177, 157–234 (1886), see also Papers on Mechanical andPhysical Subjects by Osborne Reynolds (Cambridge University Press, 1901), Vol. II, pp. 228–310.

2 L. Landau and B. Levich, “Dragging of a liquid by a moving plate,” Acta Physicochim. URSS XVII, 42 (1942).3 H. Moes, “The air gap between tape and drum in a video recorder,” J. Magn. Magn. Mater. 95, 1 (1991).4 D. Y. C. Chan, E. Klaseboer, and R. Manica, “Film drainage and coalescence between deformable drops and bubbles,”

Soft Matter 7, 2235 (2011).5 M. H. W. Hendrix, R. Manica, E. Klaseboer, D. Y. C. Chan, and C. D. Ohl, “Spatiotemporal evolution of thin liquid films

during impact of water bubbles on glass on a micrometer to nanometer scale,” Phys. Rev. Lett. 108, 247803 (2012).6 R. Manica, M. H. W. Hendrix, R. Gupta, E. Klaseboer, C. D. Ohl, and D. Y. C. Chan, “Effects of hydrodynamic film

boundary conditions on bubble-wall impact,” Soft Matter 9, 9755–9758 (2013).7 G. I. Taylor, “Deposition of a viscous fluid on the wall of a tube,” J. Fluid Mech. 10, 161 (1961).8 F. Fairbrother and A. E. Stubbs, “Studies in electro-endosmosis. Part VI. The bubble-tube method of measurement,” J.

Chem. Soc. 1, 527 (1935).9 F. P. Bretherton, “The motion of long bubbles in tubes,” J. Fluid Mech. 10, 166 (1961).

10 A. de Ryck, “The effect of weak inertia on the emptying of a tube,” Phys. Fluids 14, 2102 (2002).11 M. Heil, “Finite Reynolds number effects in the Bretherton problem,” Phys. Fluids 13, 2517 (2001).12 P. Aussillous and D. Quere, “Quick deposition of a fluid on the wall of a tube,” Phys. Fluids 12, 2367 (2000).13 A. Gunther and K. F. Jensen, “Multiphase microfluidics: From flow characteristics to chemical and material synthesis,”

Lab Chip 6, 1487 (2006).14 A. Gunther, S. A. Khan, M. Thalmann, F. Trachsel, and K. F. Jensen, “Transport and reaction in microscale segmented

gas-liquid flow,” Lab Chip 4, 278 (2004).15 M. Muradoglu, A. Gunther, and H. A. Stone, “A computational study of axial dispersion in segmented gas-liquid flow,”

Phys. Fluids 19, 072109 (2007).16 V. Hessel, S. Hardt, and H. Lowe, Chemical Micro Process Engineering: Fundamentals, Modelling and Reactions (Wiley-

VCH, Weinheim, 2004).17 S. 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).18 J. Ratulowski and H. C. Chang, “Transport of gas bubbles in capillaries,” Phys. Fluids A 1, 1642 (1989).

This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP:

185.41.20.13 On: Wed, 02 Apr 2014 17:58:36

032107-8 Klaseboer, Gupta, and Manica Phys. Fluids 26, 032107 (2014)

19 D. Halpern and D. P. Gaver, “Boundary element analysis of the time-dependent motion of a semi-infinite bubble in achannel,” J. Comput. Phys. 115, 366 (1994).

20 V. Suresh and J. B. Grotberg, “The effect of gravity on liquid plug propagation in a two-dimensional channel,” Phys. Fluids17, 031507 (2005).

21 Y. Zheng, H. Fujioka, and J. B. Grotberg, “Effects of gravity, inertia and surfactant on steady plug propagation in atwo-dimensional channel,” Phys. Fluids 19, 082107 (2007).

22 M. T. Kreutzer, F. Kapteijn, J. A. Moulijn, C. R. Kleijn, and J. J. Heiszwolf, “Inertial and interfacial effects on pressuredrop of Taylor flow in capillaries,” AICHE J. 51, 2428 (2005).

23 P. Angeli and A. Gavriilidis, “Hydrodynamics of Taylor flow in small channels: A review,” Proc. IMechE: Part C: J. Mech.Eng. Sci. 222, 737 (2008).

24 R. Gupta, D. F. Fletcher, and B. S. Haynes, “Taylor flow in microchannels: A review of experimental and computationalwork,” J. Comput. Multiphase Flows 2, 1–32 (2010).

25 T. Abadie, J. Aubin, D. Legendre, and C. Xuereb, “Hydrodynamics of gas-liquid Taylor flow in rectangular microchannels,”Microfluid Nanofluid 12, 355 (2012).

26 T. Cubaud and C. M. Ho, “Transport of bubbles in square microchannels,” Phys. Fluids 16, 4575 (2004).27 M. Fischer, D. Juric, and D. Poulikakos, “Large convective heat transfer enhancement in microchannels with a train of

coflowing immiscible or colloidal droplets,” J. Heat Transfer 132, 112402 (2010).28 R. Gupta, S. S. Y. Leung, D. F. Fletcher, and B. S. Haynes, “Hydrodynamics of liquid-liquid Taylor flow in microchannels,”

Chem. Eng. Sci. 92, 180 (2013).29 M. Suo and P. Griffith, “Two-phase flow in capillary tubes,” J. Basic Eng. 86, 576 (1964).30 M. D. Giavedoni and F. A. Saita, “The axisymmetric and plane cases of a gas phase steadily displacing a Newtonian liquid:

A simultaneous solution of the governing equations,” Phys. Fluids 9, 2420 (1997).31 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).32 E. Klaseboer, J. Ph. Chevaillier, C. Gourdon, and O. Masbernat, “Film drainage between colliding drops at constant

approach velocity: Experiments and modeling,” J. Colloid Interface Sci. 229, 274 (2000).33 R. Manica, J. N. Connor, R. R. Dagastine, S. L. Carnie, R. G. Horn, and D. Y. C. Chan, “Hydrodynamic forces involving

deformable interfaces at nanometer separations,” Phys. Fluids 20, 032101 (2008).34 D. Y. C. Chan, E. Klaseboer, and R. Manica, “Theory of non-equilibrium force measurements involving deformable drops

and bubbles,” Adv. Colloid Interface Sci. 165, 70 (2011).35 M. Muradoglu and H. A. Stone, “Motion of large bubbles in curved channels,” J. Fluid Mech. 570, 455 (2007).36 M. Taroni, C. J. W. Breward, P. D. Howell, and J. M. Oliver, “Boundary conditions for free surface inlet and outlet

problems,” J. Fluid Mech. 708, 100 (2012).37 V. G. Levich, Physicochemical Hydrodynamics (Prentice-Hall, 1962), pp. 674–683.38 N. D. Denkov, V. Subramanian, D. Gurovich, and A. Lips, “Wall slip and viscous dissipation in sheared foams: Effect of

surface mobility,” Colloid Surf., A 263, 129 (2005).39 D. Halpern, O. E. Jensen, and J. B. Grotberg, “A theoretical study of surfactant and liquid delivery into the lung,” J. Appl.

Physiol. 85, 333 (1998).

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