Pressure-Driven Motion of Drops and Bubbles through Cylindrical Capillaries: Effect of Buoyancy

Pressure-Driven Motion of Drops and Bubbles through CylindricalCapillaries: Effect of Buoyancy

Ali Borhan* and Jayanthi Pallinti

Department of Chemical Engineering, The Pennsylvania State University,University Park, Pennsylvania 16802

We examine the motion and deformation of air bubbles and viscous drops through verticalcylindrical capillaries in the presence of an imposed pressure-driven flow. Experimentalmeasurements of the terminal velocity of drops and bubbles are reported for a wide range ofdrop sizes in a variety of two-phase systems, and the steady drop shapes are quantitativelycharacterized using digital image analysis. In contrast to the pressure-driven motion of neutrally-buoyant drops, the relative mobility of a buoyant drop is not a monotonically increasing functionof capillary number. The relative mobility is enhanced as the buoyancy force becomes moredominant compared to surface tension, or as the drop fluid becomes less viscous relative to thesuspending fluid. However, there is a limiting value of the Bond number beyond which therelative mobility becomes insensitive to the value of the Bond number. Similarly, the thicknessof the liquid film surrounding large drops increases rapidly with increasing Bond number, buteventually approaches a constant value as the Bond number exceeds the limiting value. Thislimiting value of the Bond number is found to be a decreasing function of capillary number.When buoyancy and pressure forces act in the same direction, increasing the Bond number isfound to delay the formation of a re-entrant cavity at the trailing end of the drop.

Introduction

The motion of drops and bubbles through tubes arisesin many technical applications and can also serve as amodel problem for studying the pore-scale dynamics oftwo-phase flow through porous media. A great deal oftheoretical and experimental research has been focusedon this topic, with the special case of motion through acylindrical capillary tube receiving particular attentiondue to its geometric simplicity (see, for example, Cliftet al., 1978; Olbricht, 1996). In many cases, the Rey-nolds number for the motion of the dispersed phaseremains small so that inertial effects can be neglected.For example, in the case of two-phase flow throughporous media, such conditions arise due to the smalllength scales associated with the confining pores, whilein solvent extraction processes, the small dispersed-phase velocities designed to increase the contact timebetween the two phases is responsible for the creepingflow conditions.

For the motion of a viscous drop, or a gas bubble, ofequivalent spherical radius a under the influence of animposed pressure-driven flow in a cylindrical tube ofradius R, the important parameters governing thedynamics of the drop under conditions of negligibleinertia include the dimensionless drop size, κ ) a/R, thedrop-to-suspending fluid viscosity ratio, λ, the corre-sponding density ratio, γ, the capillary number, Ca )(µV)/σ, and the Bond number, Bo ) (∆FgR2)/σ), where∆F and σ denote the density difference and the surfacetension between the two phases, respectively, µ is theviscosity of the suspending fluid, V represents the meanvelocity of the imposed flow, and g is the magnitude ofthe gravitational acceleration. Previous theoreticalstudies of this problem accounting for finite drop

deformations have considered the effects of most of theseparameters in the case of neutrally-buoyant drops (Bo) 0, γ ) 1), as summarized by Olbricht (1996) andBorhan and Mao (1992). On the experimental side,there have been numerous investigations of the motionof very large (κ . 1) drops and bubbles resembling slugsthat are separated from the tube wall by a very thinlayer of the suspending fluid (Fairbrother and Stubbs,1935; Marchessault and Mason, 1960; Bretherton, 1961;Prothero and Burton, 1961; Taylor, 1961; Cox, 1962;Goldsmith and Mason, 1963; Schwartz et al., 1986;Chen, 1986). An important result of these experimentsis the observation that the film thickness and the dropspeed are independent of the drop size κ, in goodqualitative agreement with the theoretical predictions.For drop sizes comparable to the tube diameter (κ ∼O(1)), previous experimental investigations have focusedon the motion of neutrally-buoyant drops throughhorizontal capillary tubes for which buoyancy effectshave been negligible (Ho and Leal, 1975; Aul andOlbricht, 1991; Olbricht and Kung, 1992). Ho and Leal(1975) measured the drop speed and the contributionof the drop to the pressure loss through the tube forsmall capillary numbers (Ca ∼ O(10-2)), while Aul andOlbricht (1991) examined the coalescence of neutrally-buoyant drops over a similar range of capillary numbers.More recently, Olbricht and Kung (1992) performedsimilar experiments at larger capillary numbers (up toCa ∼ O(1)) in order to examine the deformation andbreakup of neutrally-buoyant drops. For low-viscosity-ratio drops, these authors observed steady drop shapescontaining a small region of negative curvature in thevicinity of the rear stagnation point, qualitativelysimilar to the shapes predicted numerically for largecapillary numbers (Chi, 1986; Martinez and Udell, 1990;Borhan and Mao, 1992; Tsai and Miksis, 1994). Theyalso reported that, upon increasing the capillary number

* To whom correspondence should be addressed. Tel.: (814)865-7847. Fax: (814) 865-7846. E-mail: [email protected].

3748 Ind. Eng. Chem. Res. 1998, 37, 3748-3759

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beyond a critical value, the observed re-entrant cavityat the trailing end of the drop eventually became theleading edge of a viscous jet of suspending fluid thatpenetrated the drop along its axis and led to dropbreakup.

Olbricht and Leal (1982) considered the creepingmotion of buoyant drops in pressure-driven flow througha horizontal cylindrical capillary and examined theeffect of small density differences between the twophases (|γ - 1| < 0.04) on drop shape and mobility, aswell as on the extra pressure loss due to the presenceof the drop. They showed that even a small densitydifference can lead to a qualitatively different depen-dence of the measured quantities on drop size. Gold-smith and Mason (1962) also examined the shapes ofbuoyant drops with small density differences, but theirexperiments were limited to very small drop sizes (κ e0.07). In this study, we focus on the pressure-drivenmotion of air bubbles and buoyant drops throughvertical cylindrical capillaries at low Reynolds numbers.We consider the two cases in which buoyancy andpressure forces act in the same, or in opposite, direc-tions. Experimental measurements of the relativemobility of drops and their shape deformations will bepresented. The observed drop shapes are quantitativelyexamined using digital image analysis in order to

extract useful geometric information such as the axialand radial dimensions of the drop profile, and theaverage thickness of the liquid film separating largedrops from the capillary wall. The effects of Bondnumber, capillary number, and viscosity ratio on themeasured quantities will be reported over a wide rangeof drop sizes.

Experimental Procedures

A schematic illustration of the experimental setup isshown in Figure 1. The experimental apparatus wassimilar to that used by Borhan and Pallinti (1995) tostudy the buoyancy-driven motion of drops in cylindricalcapillaries. It consisted of a vertical precision-bore glasscapillary tube enclosed by a Plexiglas chamber of squarecross-section containing an aqueous solution of sodiumiodide. The refractive index of the sodium iodidesolution was matched with that of the glass capillary

Table 1. Two-Phase Systems Used in the Experimentsa

system bulk fluid drop fluid

viscosity ofbulk fluid(mPa‚s)

viscosity ofdrop fluid(mPa‚s)

density ofbulk fluid

(kg/m3)

density ofdrop fluid

(kg/m3)

interfacialtension

(N/m × 103)

GW1 glycerol-water (96.2 wt %) silicon oil 427 238 1250 967 25.3GW2 glycerol-water (96.2 wt %) DC510-100 427 105 1250 990 26.8GW3 glycerol-water (96.2 wt %) UCON-1145 427 528 1250 995 11.6GW4 glycerol-water (96.2 wt %) UCON-50HB55 427 83 1250 970 3.5GW5 glycerol-water (96.2 wt %) UCON-50HB100 427 97 1250 950 6.5GW6 glycerol-water (96.2 wt %) air 427 0 1250 1 42.0GW7 glycerol-water (96.2 wt %) DC510-50 427 64 1250 986 26.5GW8 glycerol-water (96.2 wt %) DC510-500 427 607 1250 994 26.0DEG1 diethylene glycol silicon oil 23 238 1110 967 8.5DEG6 diethylene glycol air 23 0 1110 1 34.5DEG7 diethylene glycol DC510-50 23 64 1110 986 9.6DEG8 diethylene glycol DC510-500 23 607 1110 994 9.6DEG9 diethylene glycol DC550 23 166 1110 1059 7.6WG1 glycerol-water (84.2 wt %) silicon oil 198 238 1228 967 25.3WG2 glycerol-water (84.2 wt %) DC510-100 80 105 1212 990 26.8CW3 corn syrup-water (98.5 wt %) UCON-1145 2444 528 1375 995 14.3CW4 corn syrup-water (98.5 wt %) UCON-50HB55 2444 83 1375 970 6.0CW5 corn syrup-water (98.5 wt %) UCON-50HB100 2444 97 1375 950 8.0CW6 corn syrup-water (98.5 wt %) air 2444 0 1375 1 80.8

a Note: All physical properties were measured at 25 °C.

Figure 1. Schematic illustration of the experimental setup.

Table 2. Range of Dimensionless Parameters

system λ γ Bo κ Re Ca*

GW1-1 0.56 0.77 1.7 0.45-1.32 0.03-0.21 0.05-0.31GW2-1 0.25 0.79 1.5 0.45-1.57 0.03-0.21 0.05-0.30GW3-1 1.24 0.80 3.4 0.45-1.32 0.00-0.17 0.01-0.54GW4-1 0.19 0.78 12.6 0.45-1.57 0.02-0.31 0.26-3.30GW5-1 0.23 0.76 7.0 0.45-1.32 0.01-0.28 0.06-1.60GW6-1 0.00 0.00 4.6 0.45-1.32 0.08-0.24 0.11-0.39GW1-2 0.56 0.77 0.3 0.65-2.39 0.05-0.27 0.16-0.89GW3-2 1.18 0.80 0.7 0.65-2.39 0.04-0.25 0.30-0.93GW4-2 0.18 0.78 2.4 0.65-2.39 0.07-0.24 1.92-6.44GW5-2 0.22 0.76 1.4 0.65-2.39 0.06-0.37 0.84-5.33GW6-2 0.00 0.00 0.9 0.65-2.39 0.04-0.37 0.08-0.68GW7-2 0.15 0.79 0.3 0.65-2.39 0.06-0.32 0.19-0.99GW8-2 1.43 0.79 0.3 0.65-2.84 0.05-0.31 0.16-0.47GW4-3 0.18 0.78 0.7 1.05-2.06 0.03-0.04 5.23-5.42GW5-3 0.22 0.76 0.4 1.79-2.58 0.07-0.11 2.32-3.92DEG1-1 10.35 0.87 2.6 0.61-1.32 0.66-8.63 0.01-0.12DEG7-1 2.77 0.89 2.0 0.61-1.32 0.73-8.70 0.01-0.11DEG8-1 26.40 0.90 1.8 0.61-1.32 0.49-8.13 0.01-0.10DEG9-1 7.20 0.95 1.0 0.61-1.32 0.38-8.51 0.01-0.14WG1-1 1.20 0.79 1.6 0.58-1.32 0.13-1.02 0.04-0.32WG2-1 1.30 0.82 1.3 0.58-1.32 0.13-0.25 0.07-0.25CW3-1 0.22 0.72 4.1 0.58-1.57 0.01-0.05 0.66-3.97CW4-1 0.03 0.71 10.4 >0.58 breakup breakupCW5-1 0.04 0.69 8.2 0.58-1.05 0.01-0.03 0.26-0.48

>1.05 breakup breakupCW6-1 0.00 0.00 2.6 0.58-1.97 0.01-0.06 0.05-0.30

Ind. Eng. Chem. Res., Vol. 37, No. 9, 1998 3749

to minimize optical distortions due to the refraction oflight at the outer wall of the capillary. Three differentcapillary tubes were used to cover a wide range of dropsizes and Bond numbers. They were all 120-cm longand were labeled 1, 2, and 3 according to their innerdiameters of 0.796, 0.347, and 0.186 cm, respectively.The suspending fluids used in these experiments con-sisted of 99% pure diethylene glycol (DEG), a 98.5 wt% corn syrup-water mixture (CW), and various aqueousglycerol solutions (GW and WG). A variety of DowCorning fluids and UCON oils, silicon oil, and air wereused as drop fluids. The various two-phase systemsused in these experiments and their relevant physicalproperties are listed in Table 1. To facilitate thepresentation of the experimental results, each two-phasesystem in this table is designated by a symbol identify-ing the suspending fluid, followed by a number thatspecifies the drop fluid. The fluid properties shown inTable 1 were all measured at a temperature of 25 °C.However, for each set of experiments with the same two-phase system, all fluid properties were measured againat the actual temperature of the experiments (typicallyin the range 24.3-25.9 °C), which was determined by a

digital thermometer connected to thermocouples placednear the inlet and outlet regions of the capillary tube.The viscosities of all liquids were measured usingthermostated capillary viscometers, and the interfacialtension between the two phases was obtained using aring tensiometer.

For each set of experiments with the same two-phasesystem, the suspending fluid was pumped through thecapillary tube at a known flow rate. In most experi-ments, the suspending fluid flowed from the bottom tothe top of the capillary and the desired volume of thedrop fluid was injected at the symmetry axis of thecapillary near the inlet region (at the bottom of the

Figure 2. Typical images of the steady profiles of bubbles andviscous drops for Ca ) 0.14 in the (i) GW6-1 system and (ii) GW3-1system: (a) κ ) 0.73, (b) κ ) 0.92, (c) κ ) 1.05, (d) κ ) 1.15, and (e)κ ) 1.32.

Figure 3. Variations of the dimensionless geometric parameterswith drop size for systems with Bo ) 1.0-1.5 at Ca ) 0.07: (a)the ratio of the perimeter of the deformed drop profile in themeridional plane to that of a spherical drop with the same volume,D; (b) drop length, LA; (c) maximum equatorial dimension, LR.

3750 Ind. Eng. Chem. Res., Vol. 37, No. 9, 1998

capillary) using a micrometer syringe. The capillarytube and the syringe were thoroughly cleaned withdistilled water, benzene, and acetone and then dried inair before each new set of experiments. For all two-phase systems considered in our experiments, the dropfluid had a lower density than the suspending fluid andthe drop traveled upward along the axis of symmetryof the capillary. In a few experiments, the imposed flowof the suspending fluid was reversed (i.e., flowing fromthe top to the bottom of the capillary) in order toexamine the case in which pressure and buoyancy forcesact in opposite directions. The motion of the dropthrough the capillary was recorded using a CCD cameraconnected to a video recorder capable of frame-by-frame

playback. The video camera was mounted on a movingplatform whose speed was adjusted by a controller tofollow the vertical motion of the drop, thereby allowingthe drop to be monitored as it passed through the entirelength of the capillary.

To study drop deformations, the recorded images ofthe drop profile were played back frame-by-frame andthe signal from the video recorder was digitized usinga computer equipped with a frame-grabber board. Astop-motion filter was applied to the digitized imagesto remove any jittering caused by the motion of the drop,and the Bioscan Optimas image analysis software wasthen used to quantitatively characterize the drop shapesby measuring various geometric features such as theperimeter of the drop profile in the meridional planeand its maximum axial and radial dimensions. For eachexperiment, the volume of the axisymmetric drop wasalso determined using Optimas to ensure the accuracyof the drop size measurement based on the micrometersyringe reading. The terminal velocity of the drop wasdetermined by measuring the time required for the dropto travel a specified vertical distance between selectedmarkers on the capillary wall. For each experiment,three independent velocity measurements over differentregions of the capillary tube were made to check for anyunsteadiness in the motion of the drop. The geometricinformation obtained from the drop shape at variousaxial positions was also used to detect unsteady dropbehavior in the experiments and to ensure that velocitymeasurements were being made in regions sufficientlyfar from the inlet of the capillary to avoid entranceeffects. Finally, each experiment was repeated toensure reproducibility of the results. In all cases, thereported terminal velocity represents the average of sixvelocity measurements, with each measurement havinga variation of less than 5% from the reported meanvalue.

Results and Discussion

In this section, we present the experimental resultsin terms of the effects of Bond number, viscosity ratio,and capillary number on the steady shape of the dropand its terminal velocity over a wide range of drop sizes.A list of parameter values for each set of experimentswith the same two-phase system in a particular capil-lary tube is shown in Table 2. In this table, each set ofexperiments is labeled by a symbol identifying the two-

Figure 4. Variations of the dimensionless geometric parameterswith drop size for systems with λ ) 0.0 at Ca ) 0.09: (a)deformation parameter, D; (b) drop length, LA; (c) maximumequatorial dimension, LR.

Figure 5. The thickness of the liquid film surrounding large (κ> 1.1) drops with λ = 0.0-0.2 as a function of Bond number at Ca) 0.14 and Ca ) 0.0.


phase system followed by a number specifying the sizeof the capillary tube. The Reynolds numbers reportedin Table 2 are based on the actual terminal velocity ofthe drop, U, rather than the mean velocity, V, of theimposed flow, that is Re ) (FUR)/µ. The overall rangeof Reynolds numbers associated with these experimentswas between 0.0 and 8.7. Aside from the motion ofsmall drops in diethylene glycol (DEG) systems, theReynolds numbers for the motion of the dispersed phaseremained in the Stokes regime. The range of measureddrop velocities for each set of experiments is also shownin the last column of Table 2 in the form of a capillarynumber defined as Ca* ) (µU)/σ.

a. Drop Shape. In all of the experiments reportedhere, the shapes of drops and bubbles remained axi-symmetric as they passed through the capillary tube.Typical profiles (in the meridional plane) of the steadyshapes observed experimentally for air bubbles andviscous drops are shown in Figure 2 for various dropsizes. The steady shape of air bubbles approaches anelongated ellipsoid as the bubble size increases, with aslight loss of fore and aft symmetry due to the flatteningof the trailing end caused by the imposed flow. As theinterior phase becomes more viscous relative to thesuspending fluid, a slightly more tapered drop shape

Figure 6. Dimensionless geometric parameters for the GW2-1system (Bo ) 1.5, λ ) 0.15) at various capillary numbers: (a)deformation parameter, D; (b) drop length, LA; (c) maximumequatorial dimension, LR. Figure 7. Dimensionless geometric parameters for the GW4-1

system (Bo ) 12.6, λ ) 0.19) at various capillary numbers: (a)deformation parameter, D; (b) drop length, LA; (c) maximumequatorial dimension, LR.


results without significantly affecting the qualitativefeatures of the leading end of the drop. To quantita-tively characterize the steady shapes of drops andbubbles, a deformation parameter D is defined as theratio of the perimeter of the deformed drop profile inthe meridional plane to that of a spherical drop withthe same volume. This quantity is easily obtained fromdigitized images of drops and bubbles using the Optimasimage analysis software. Figure 3 demonstrates thetypical dependence of the deformation parameter on thedimensionless drop size. Also shown in this figure arethe geometric parameters LA and LR, representing themaximum axial and radial dimensions of the steadydrop profile relative to the capillary radius, respectively.The dashed lines in this figure denote the values of thegeometric parameters for spherical drops, while thedotted lines represent those corresponding to a stagnantcylindrical drop with hemispherical ends. For a giventwo-phase system (with fixed Bond number and viscos-ity ratio), both the radial and axial dimensions of thedrop initially grow almost linearly as a function of dropsize, with the first shape transition occurring at differ-

ent drop sizes, depending on the value of the capillarynumber. It is clear from Figure 3 that, for fixedcapillary and Bond numbers, drop deformation is onlyslightly affected by large variations in the viscosity ratio.While the deformation parameter D shows a weakdependence on the viscosity ratio, the values of LA andLR are virtually the same for systems with widelydifferent values of λ. This is consistent with thequalitative observations of a slightly more tapered dropshape with nearly the same drop length at largerviscosity ratios. The weak dependence of D on theviscosity ratio disappears as the capillary numberbecomes large.

The effect of Bond number on drop shape is shown inFigure 4 for systems with nearly constant values ofother parameters. It is evident that increasing the Bondnumber leads to larger deformations in the form of axialelongation of the drop, particularly for larger drops.Once the transition to a cylindrical shape occurs,increases in the Bond number lead to faster rates ofelongation as a function of drop size, as well as smallerradii (limiting values of LR) for the cylindrical sectionof the drop or, equivalently, thicker liquid films betweenthe cylindrical drops and the capillary wall. However,when the Bond number exceeds a limiting value, thevalue of LR remains unchanged with further increasesin the Bond number. This limiting value of the Bondnumber decreases as the capillary number is increased.This is evident from the experimental data in Figure 5which show the measured liquid film thickness δ (madedimensionless with capillary radius R), for the largestdrops in the low-viscosity-ratio systems. The solidsymbols in this figure represent measurements of thefilm thickness in the case of buoyancy-driven motion(i.e., for Ca ) 0), while the open symbols represent thecorresponding measurements in the presence of pres-sure-driven flow with Ca ) 0.14. For small Bondnumbers, the values of the film thickness in the pres-ence of imposed flow are larger than those correspond-ing to Ca ) 0. However, for large Bond numbers, thelimiting value of the film thickness is not significantly

Figure 8. The thickness of the liquid film surrounding large (κ> 1.1) drops with λ ) 0.2 (normalized by its value at Ca ) 0) asa function of capillary number.

Figure 9. Comparison of the steady shapes for a drop size of κ = 0.91 in the CW3-1 system (Bo ) 4.1, λ ) 0.22) at (a) Ca ) 0.0, (b) Ca) 0.27, (c) Ca ) 0.54, and (d) Ca ) 0.75.


affected (within experimental error) by the addition ofpressure-driven flow. The limiting value of Bond num-ber beyond which the film thickness remains nearlyconstant decreases from approximately 12 to 2 as thecapillary number increases from 0 to 0.14. Hence, theeffect of Bond number on drop deformation eventuallydisappears as the capillary number becomes large.

The effect of capillary number on the steady dropshape is illustrated in Figure 6. Clearly, the radialdimension of a viscous drop of constant volume de-creases, and its axial length grows, with increasingcapillary number. It can also be seen from this figurethat the transition from a spherical to an elongatedshape occurs at smaller drop sizes as the capillarynumber increases. The additional viscous stressesgenerated by the imposed flow can promote shapedeformations even for small drops which remain spheri-cal in the absence of imposed flow. The increase in droplength is accompanied by an increase in the thicknessof the liquid film surrounding the drop. However, thereis a limiting value of capillary number beyond whichthe drop shape becomes relatively insensitive to thevalue of the capillary number, as shown in Figure 7.For the two-phase system in this figure, increasing thecapillary number from 0.3 to 0.9 does not lead to anysignificant changes in LR or, equivalently, the thicknessof the liquid film surrounding large drops. The effectof capillary number on the film thickness, δ, is more

clearly demonstrated in Figure 8. The values of δ inthis figure have been scaled by the film thickness δ0 inthe absence of imposed flow (at Ca ) 0). The filmthickness becomes larger as the capillary numberincreases, and eventually approaches a limiting valueat sufficiently large values of Ca. The limiting value ofcapillary number beyond which the film thicknessbecomes independent of Ca is smaller for systems withlarger Bond numbers. The effect of capillary numberon drop deformation also diminishes as the viscosityratio tends to zero. For example, for the air bubbles inthe GW6-1 system, the bubble length becomes indepen-dent of the capillary number for Ca g 0.02, whereas thelength of viscous drops in the CW3-1 system (withnearly the same Bond number as the GW6-1 system)remains sensitive to the value of Ca for capillarynumbers as large as 0.75.

As the capillary number increases, there is a notablereduction in the curvature of the trailing end of the drop,as illustrated by the sequence of images in Figure 9 fora drop of size κ = 0.91 in the CW3-1 system. Forsufficiently large capillary numbers, the drop can evendevelop a region of negative curvature at its trailing end.For the steady drop shape corresponding to Ca ) 0.75in Figure 9, a re-entrant cavity was actually observedat the trailing interface, though it is difficult to detectits presence in the meridional profile of the drop. The

Figure 10. The dimensionless drop velocity as a function of dropsize for (a) systems with O(1) values of Ca/Bo and (b) systems withsmall values of Ca/Bo (∼0.02). The solid curves represent the bestfit to the experimental data, while the dotted curves show thecorresponding asymptotic predictions of Hetsroni et al. (1970).

Figure 11. The relative mobility of the drop as a function of dropsize for systems with Bo = 1.0-1.6 at Ca ) 0.08. The solid curvesrepresent the best fit to the experimental data. The dashed anddotted curves represent the asymptotic predictions of Hetsroni etal. (1970) corresponding to the GW2-1 and DEG8-1 systems,respectively.


qualitative features of the trailing interface in theCW3-1 system are similar to those observed experimen-tally by Olbricht and Kung (1992) and predicted nu-merically by other investigators (cf. Chi, 1986; Martinezand Udell, 1990; Borhan and Mao, 1992; Tsai andMiksis, 1994), for neutrally-buoyant drops in pressure-driven flow through a cylindrical capillary. However,the capillary number at which the trailing end of thedrop first exhibited a region of negative curvature wasfound to be larger than that predicted numerically forneutrally-buoyant drops. For example, for a drop of sizeκ = 0.73 in a two-phase system with nearly the sameviscosity ratio as the CW3 system, Martinez and Udell(1990) predict the first appearance of a re-entrant cavityat Ca ) 0.75, whereas a similar response was firstobserved at Ca ) 1.21 in our experiments. This differ-ence can be attributed, at least partially, to the nonzeroBond numbers in the CW3-1 experiments, in contrastto the zero Bond numbers associated with the neutrally-buoyant drops considered in the computational studies.

In the case of pressure-driven motion of neutrally-buoyant drops, the drop always moves slower than thesuspending fluid elements near the capillary axis,leading to the formation of stagnation rings on thesurface of the drop. The resulting flow field in thevicinity of the trailing end of the drop is similar to anaxisymmetric stagnation flow with the suspending fluid

elements on the capillary axis moving toward thetrailing interface with a relative velocity (2V - U) > 0.For low-viscosity-ratio systems (such as CW3), increas-ing the capillary number produces larger stagnationpressures within the suspending fluid at the rearstagnation point of the drop, without significantlyaffecting the pressure inside the drop. This eventuallyleads to the appearance of negative curvature at thetrailing interface once the flow-induced normal stressesin the suspending fluid exceed those within the low-viscosity-ratio drop. For a buoyant drop, however, thestagnation flow in the vicinity of the trailing end of thedrop is weakened when the buoyancy force acts in thedirection of the imposed flow. This is mainly due to thelarger terminal velocity of the buoyant drop at the samecapillary number, resulting in a smaller relative velocity(2V - U) for the stagnation flow at the trailing end ofthe drop. Consequently, a larger value of capillarynumber will be required to produce the same stagnationpressure at the rear stagnation point of the drop. Sincethe terminal velocity of the drop is a monotonicallyincreasing function of Bond number when the buoyancyforce is aligned with the imposed flow (as will be shownlater), it is reasonable to expect that increasing the Bondnumber would delay the formation of a region ofnegative curvature at the back of the drop until largervalues of capillary number are reached. When the Bondnumber is so large that the drop moves faster than themaximum velocity of the imposed flow on the capillaryaxis, the stagnation rings on the surface of the dropdisappear altogether and the flow field resembles that

Figure 12. The relative mobility of the drop as a function of dropsize for systems with λ = 1.2-1.4 at Ca ) 0.14. The solid curvesrepresent the best fit to the experimental data while the dottedcurves show the corresponding asymptotic predictions of Hetsroniet al. (1970) for small drops with λ ) 1.3.

Figure 13. The drop speed as a function of drop size for theGW4-1 system (Bo ) 12.6, λ ) 0.19). The solid curves representthe best fit to the experimental data, while the dotted curves showthe corresponding asymptotic predictions of Hetsroni et al. (1970).


in the case of buoyancy-driven motion (i.e., surface flowfrom the front stagnation point to the rear stagnationpoint of the drop). In this case, one would expect thedevelopment of a region of negative curvature at theback of the drop to be inhibited for all capillary numbers.Indeed, no indentation at the trailing end of low-viscosity-ratio drops was observed in the experimentsof Borhan and Pallinti (1995) for the case of buoyancy-driven motion, and the curvature of the trailing inter-face of large drops was found to be an increasingfunction of Bond number in those experiments.

b. Drop Mobility. We now consider the effects ofcapillary number, Bond number, and viscosity ratio onthe terminal velocity of drops and bubbles. The typicaldependence of the terminal velocity on drop size isshown in Figure 10 where the velocity U* is madedimensionless with (∆FgR2)/µ). The solid curves in thisfigure represent the best fit to the experimental data,taking into account the fact that U* must approach thedimensionless centerline velocity of the imposed flow asκ tends to zero. The dotted curves represent thecorresponding theoretical predictions of Hetsroni et al.(1970) for slightly deformed drops in each system. Forsystems with O(1) values of Ca/Bo, the terminal velocitydecreases rapidly with increasing drop size as theretarding effect of the wall becomes more pronounced,and eventually approaches a limiting value for κ > 1.For systems with small values of Ca/Bo, on the otherhand, the terminal velocity initially increases with dropsize before reaching a plateau for large drop sizes. The

experimentally measured drop speeds deviate substan-tially from the asymptotic predictions of Hetsroni et al.as the drop size is increased beyond κ ) 0.5. This isnot surprising since the asymptotic predictions are validfor small drops which remain nearly spherical, whereassignificant drop deformations are typically observed inthe experiments for κ g 0.5. Hence, the shape of thedrop significantly affects its mobility and finite dropdeformations must be taken into account in order toaccurately predict the drop speed.

The relative mobility of the drop, defined as the ratioof the drop speed to the average velocity of the imposedflow, is plotted in Figures 11 and 12 as a function ofdrop size. To identify the contribution of the imposedflow to the mobility of the drop, the quantity (U - U0)/Vis also plotted in each of these figures, where U0represents the terminal velocity in the absence ofpressure-driven flow (i.e., at Ca ) 0). Figure 11 showsthe effect of the viscosity ratio on the relative mobilityof the drop. For a given drop size, the relative mobilityis reduced as the viscosity ratio is increased whilekeeping Ca and Bo constant. However, there is alimiting value of λ beyond which the relative mobilitybecomes independent of the viscosity ratio. This limit-ing value of λ becomes larger as the Bond numberincreases and is more easily detected in the low Bondnumber systems. For example, there is very little

Figure 14. The drop speed as a function of drop size for theGW6-1 system (Bo ) 4.6, λ ) 0.0). The solid curves represent thebest fit to the experimental data shown by the open symbols, whilethe dotted curves show the corresponding asymptotic predictionsof Hetsroni et al. (1970).

Figure 15. The relative mobility of the drop as a function ofcapillary number for systems with Bo = 1.0-1.7 and differentviscosity ratios. U0 represents the terminal velocity in the absenceof pressure-driven flow. The dashed and dotted curves representthe asymptotic predictions of Hetsroni et al. (1970) for the systemscorresponding to the open squares and open circles, respectively.


difference between the relative mobilities of drops in theDEG8-1 and DEG9-1 systems (shown in Figure 11)which are characterized by widely different viscosityratios. The dashed and dotted curves in Figure 11represent the asymptotic predictions of Hetsroni et al.(1970) for the parameter values corresponding to theGW2-1 and DEG8-1 systems, respectively. Clearly, theexperimentally observed drop mobilities for κ > 0.5 arenot accurately predicted by the asymptotic analysis, dueto significant deformations of large drops.

The effect of Bond number on the relative mobility ispresented in Figure 12. As the Bond number increasesfor fixed values of κ, Ca, and λ, the relative mobility ofthe drop is enhanced. This is consistent with our earlierdiscussion of the effect of Bond number on drop shapesince the more elongated drop shapes formed at largerBond numbers are localized near the symmetry axis ofthe capillary and experience a weaker wall effect.Similar to the effect of Bond number on drop shape,increasing the Bond number beyond a limiting valuedoes not significantly affect the relative mobility. Theresults of the asymptotic analysis, shown by the dottedcurves in Figure 12, predict the quantity (U - U0)/V tobe independent of Bond number, whereas the experi-mental data clearly indicate a strong Bond numberdependence for large drops. Again, this discrepancy can

be attributed to the effect of Bond number on shapedeformations of large drops.

The effects of capillary number on the terminalvelocity and relative mobility of drops are shown inFigures 13 and 14 for large and small Bond numbers,respectively. As expected, the drop moves faster as thecapillary number increases, due to the larger velocitiesassociated with the imposed flow as well as the moreelongated drop shapes which tend to reduce the retard-ing effect of the capillary wall by localizing the drop nearthe centerline of the capillary. The former effect isdominant for small drops (which remain nearly spheri-cal) and is also reflected in the asymptotic predictionsshown by the dotted curves in Figures 13 and 14, whilethe latter effect becomes significant for κ > 0.4 whichalso represents the point at which the experimentalmeasurements begin to deviate substantially from theasymptotic predictions. In contrast to the pressure-driven motion of neutrally-buoyant drops (cf. Ho andLeal, 1975), however, the relative mobility of the dropis not a monotonically increasing function of the capil-lary number. For small capillary numbers, the relativemobility is a decreasing function of Ca despite the factthat the actual drop speed U increases with capillarynumber. The relative mobility seems to attain a localminimum as the capillary number is increased, beforebecoming relatively insensitive to the value of the

Figure 16. The relative mobility of the drop as a function ofcapillary number for systems with λ = 0.0 and different Bondnumbers. U0 represents the terminal velocity in the absence ofpressure-driven flow. The dashed and dotted curves represent theasymptotic predictions of Hetsroni et al. (1970) for the systemscorresponding to the open squares and open circles, respectively.

Figure 17. The drop speed as a function of drop size in the GW5-1system (Bo ) 7.0, λ ) 0.23) when buoyancy and pressure forcesare acting in opposite directions. U0 represents the terminalvelocity in the absence of pressure-driven flow. The dotted curvesshow the corresponding asymptotic predictions of Hetsroni et al.(1970).


capillary number. The minimum in the relative mobil-ity occurs at smaller capillary numbers in systems withlower Bond numbers or larger viscosity ratios, as shownin Figures 15 and 16.

To identify the contribution of shape deformations tothe mobility of a drop as the capillary number isincreased, the quantity (U - U0)/V is also plotted inFigures 15 and 16. If the increase in the drop speedwith increasing capillary number were solely due to thelarger mean velocity of the imposed flow in an experi-ment, the quantity (U - U0)/V would be expected toremain nearly constant as the capillary number isincreased for a given drop size. This is indeed thebehavior predicted by the asymptotic analysis of Hetsro-ni et al. (shown by the dashed and dotted lines inFigures 15 and 16) for small drops which experiencenegligible shape deformations. For most two-phasesystems considered here, however, this quantity initiallydecreases as the capillary number is increased fromzero. This is an indication that the addition of thepressure-driven flow initially results in drop deforma-tions that adversely affect the mobility of the drop inthese systems. As the capillary number is furtherincreased, the stronger imposed flow eventually leadsto drop deformations which enhance drop mobility, and(U - U0)/V becomes a weakly increasing function ofcapillary number, in line with the experimental resultsof Ho and Leal (1975), and the numerical results ofMartinez and Udell (1990), for pressure-driven motionof neutrally-buoyant drops through cylindrical capillar-ies. Hence, as expected, the dynamics of a buoyant dropapproaches that of a neutrally-buoyant drop describedby Ho and Leal (1975) once the capillary numberbecomes sufficiently large.

Figure 17 illustrates the typical results for experi-ments in which the pressure and buoyancy forces actedin opposite directions. In these experiments, the flow

of the suspending fluid was established from the top tothe bottom of the capillary tube, but the imposed flowwas sufficiently weak to allow the drop to rise verticallywithin the capillary. The drop speed decreases as themagnitude of the capillary number is increased (i.e., asthe imposed flow opposing the buoyancy-driven motionof the drop becomes stronger). However, the reductionin the mobility of the drop is not entirely due to thelarger mean velocities of the opposing pressure-drivenflow. For very small capillary numbers, the introductionof pressure-driven flow indirectly affects the mobilityof the drop through subtle changes in drop shape. Thesteady drop shape becomes more flattened at the leadingedge, and shorter in length, compared to the corre-sponding drop shapes at zero capillary number. As thecapillary number increases, the drop shrinks in the axialdirection, simultaneously decreasing the thickness of theliquid film surrounding it and increasing the retardingeffect of the capillary wall. This response is exactly theopposite of that described earlier for the case in whichbuoyancy and pressure forces acted in the same direc-tion. The retarding effect of flow-induced deformationscan be identified by examining the dependence of (U -U0)/V on capillary number in Figure 17. For large Bondnumber systems, this quantity is found to be a strongfunction of capillary number (in contrast to the asymp-totic predictions shown by the dotted curve), indicatingthe significant contribution of drop deformations to thereduction in drop mobility.

We conclude by presenting a sequence of imagesshowing the typical behavior of the trailing end of largedrops in the CW4 and CW5 systems which werecharacterized by large Bond numbers and vanishingviscosity ratios. As shown in Figure 18, an unstablepointed tail formed at the trailing end of the drops inthese systems and subsequently disintegrated into a

Figure 18. Tail-streaming for a large (κ ) 1.24) drop: (a) in the CW5-1 system and (b) in the CW4-1 system.


stream of small satellite drops. For large (κ > 1.6)drops, this tail-streaming behavior was observed evenin the absence of imposed flow (i.e., at Ca ) 0).Increasing the capillary number led to more intensestreaming at the trailing end of a drop of fixed size andshifted the onset of tail streaming to smaller drop sizes.The behavior of the trailing interface of the drop in theCW4 and CW5 systems is qualitatively similar to thetip-streaming phenomenon observed in some previousexperiments involving low-viscosity-ratio (λ < O(0.1))drops in shear flows (cf. Milliken and Leal, 1991; deBruijn, 1993). The onset of tip-streaming in low-viscosity-ratio systems is believed to be due to thepresence of surface-active species (de Bruijn, 1993;Stone, 1994). Although the experimental systems usedin this study were carefully cleaned to avoid contamina-tion, the presence of surface-active components in cornsyrup may have been responsible for the observed tail-streaming phenomenon. This explanation is furthersupported by the experimental observations of dimin-ishing streaming intensity at the trailing end of the dropas the drop passed through the capillary. In some cases,tail streaming disappeared altogether and the trailinginterface achieved a steady shape near the top of thecapillary, as shown in Figure 18a. While the reductionin drop volume due to tail-streaming could have, inprinciple, resulted in a subcritical drop size as the droppassed through the capillary tube, experimental mea-surements of the drop volume from digitized images ofthe steady drop profile near the top of the capillary tubedid not support this hypothesis. The actual change indrop volume was too small to explain the disappearanceof tail-streaming in the experiments, particularly forlarge drops (κ > 1.6) in capillary 1 where tail-streamingwas observed even in the absence of imposed flow. Aplausible explanation for the observed behavior is thereduction in the surface concentration of the surface-active species through the formation of satellite drops,consistent with the mechanism suggested by de Bruijn(1993). The role of surface-active impurities in thedynamics of the observed tail-streaming behavior war-rants further study.

AcknowledgmentAcknowledgment is made to the donors of the Petro-

leum Research Fund, administered by the AmericanChemical Society, for partial support of this research.This work was also supported by the National ScienceFoundation under Grant CTS-9110470.

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Received for review February 12, 1998Revised manuscript received June 2, 1998

Accepted June 16, 1998

IE980087L


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Pressure-Driven Motion of Drops and Bubbles through Cylindrical Capillaries: Effect of Buoyancy