Pressure-Driven Motion of Drops and Bubbles through Cylindrical

measurements of the terminal velocity of drops and bubbles are reported for a wide range of drop sizes in a variety of two-phase systems, and the stea...
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Ind. Eng. Chem. Res. 1998, 37, 3748-3759

Pressure-Driven Motion of Drops and Bubbles through Cylindrical Capillaries: 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 vertical cylindrical capillaries in the presence of an imposed pressure-driven flow. Experimental measurements of the terminal velocity of drops and bubbles are reported for a wide range of drop sizes in a variety of two-phase systems, and the steady drop shapes are quantitatively characterized using digital image analysis. In contrast to the pressure-driven motion of neutrallybuoyant drops, the relative mobility of a buoyant drop is not a monotonically increasing function of capillary number. The relative mobility is enhanced as the buoyancy force becomes more dominant compared to surface tension, or as the drop fluid becomes less viscous relative to the suspending fluid. However, there is a limiting value of the Bond number beyond which the relative mobility becomes insensitive to the value of the Bond number. Similarly, the thickness of the liquid film surrounding large drops increases rapidly with increasing Bond number, but eventually approaches a constant value as the Bond number exceeds the limiting value. This limiting 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 is found 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 arises in many technical applications and can also serve as a model problem for studying the pore-scale dynamics of two-phase flow through porous media. A great deal of theoretical and experimental research has been focused on this topic, with the special case of motion through a cylindrical capillary tube receiving particular attention due to its geometric simplicity (see, for example, Clift et al., 1978; Olbricht, 1996). In many cases, the Reynolds number for the motion of the dispersed phase remains small so that inertial effects can be neglected. For example, in the case of two-phase flow through porous media, such conditions arise due to the small length scales associated with the confining pores, while in solvent extraction processes, the small dispersedphase velocities designed to increase the contact time between the two phases is responsible for the creeping flow conditions. For the motion of a viscous drop, or a gas bubble, of equivalent spherical radius a under the influence of an imposed pressure-driven flow in a cylindrical tube of radius R, the important parameters governing the dynamics of the drop under conditions of negligible inertia include the dimensionless drop size, κ ) a/R, the drop-to-suspending fluid viscosity ratio, λ, the corresponding 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 surface tension between the two phases, respectively, µ is the viscosity of the suspending fluid, V represents the mean velocity of the imposed flow, and g is the magnitude of the gravitational acceleration. Previous theoretical studies of this problem accounting for finite drop * To whom correspondence should be addressed. Tel.: (814) 865-7847. Fax: (814) 865-7846. E-mail: [email protected].

deformations have considered the effects of most of these parameters in the case of neutrally-buoyant drops (Bo ) 0, γ ) 1), as summarized by Olbricht (1996) and Borhan and Mao (1992). On the experimental side, there have been numerous investigations of the motion of very large (κ . 1) drops and bubbles resembling slugs that are separated from the tube wall by a very thin layer 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 experiments is the observation that the film thickness and the drop speed are independent of the drop size κ, in good qualitative agreement with the theoretical predictions. For drop sizes comparable to the tube diameter (κ ∼ O(1)), previous experimental investigations have focused on the motion of neutrally-buoyant drops through horizontal capillary tubes for which buoyancy effects have been negligible (Ho and Leal, 1975; Aul and Olbricht, 1991; Olbricht and Kung, 1992). Ho and Leal (1975) measured the drop speed and the contribution of the drop to the pressure loss through the tube for small capillary numbers (Ca ∼ O(10-2)), while Aul and Olbricht (1991) examined the coalescence of neutrallybuoyant drops over a similar range of capillary numbers. More recently, Olbricht and Kung (1992) performed similar experiments at larger capillary numbers (up to Ca ∼ O(1)) in order to examine the deformation and breakup of neutrally-buoyant drops. For low-viscosityratio drops, these authors observed steady drop shapes containing a small region of negative curvature in the vicinity of the rear stagnation point, qualitatively similar to the shapes predicted numerically for large capillary numbers (Chi, 1986; Martinez and Udell, 1990; Borhan and Mao, 1992; Tsai and Miksis, 1994). They also reported that, upon increasing the capillary number

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Ind. Eng. Chem. Res., Vol. 37, No. 9, 1998 3749 Table 1. Two-Phase Systems Used in the Experimentsa

system

bulk fluid

drop fluid

viscosity of bulk fluid (mPa‚s)

GW1 GW2 GW3 GW4 GW5 GW6 GW7 GW8 DEG1 DEG6 DEG7 DEG8 DEG9 WG1 WG2 CW3 CW4 CW5 CW6

glycerol-water (96.2 wt %) glycerol-water (96.2 wt %) glycerol-water (96.2 wt %) glycerol-water (96.2 wt %) glycerol-water (96.2 wt %) glycerol-water (96.2 wt %) glycerol-water (96.2 wt %) glycerol-water (96.2 wt %) diethylene glycol diethylene glycol diethylene glycol diethylene glycol diethylene glycol glycerol-water (84.2 wt %) glycerol-water (84.2 wt %) corn syrup-water (98.5 wt %) corn syrup-water (98.5 wt %) corn syrup-water (98.5 wt %) corn syrup-water (98.5 wt %)

silicon oil DC510-100 UCON-1145 UCON-50HB55 UCON-50HB100 air DC510-50 DC510-500 silicon oil air DC510-50 DC510-500 DC550 silicon oil DC510-100 UCON-1145 UCON-50HB55 UCON-50HB100 air

427 427 427 427 427 427 427 427 23 23 23 23 23 198 80 2444 2444 2444 2444

a

viscosity of drop fluid (mPa‚s)

density of bulk fluid (kg/m3)

density of drop fluid (kg/m3)

interfacial tension (N/m × 103)

238 105 528 83 97 0 64 607 238 0 64 607 166 238 105 528 83 97 0

1250 1250 1250 1250 1250 1250 1250 1250 1110 1110 1110 1110 1110 1228 1212 1375 1375 1375 1375

967 990 995 970 950 1 986 994 967 1 986 994 1059 967 990 995 970 950 1

25.3 26.8 11.6 3.5 6.5 42.0 26.5 26.0 8.5 34.5 9.6 9.6 7.6 25.3 26.8 14.3 6.0 8.0 80.8

Note: All physical properties were measured at 25 °C. Table 2. Range of Dimensionless Parameters system

Figure 1. Schematic illustration of the experimental setup.

beyond a critical value, the observed re-entrant cavity at the trailing end of the drop eventually became the leading edge of a viscous jet of suspending fluid that penetrated the drop along its axis and led to drop breakup. Olbricht and Leal (1982) considered the creeping motion of buoyant drops in pressure-driven flow through a horizontal cylindrical capillary and examined the effect of small density differences between the two phases (|γ - 1| < 0.04) on drop shape and mobility, as well as on the extra pressure loss due to the presence of the drop. They showed that even a small density difference can lead to a qualitatively different dependence of the measured quantities on drop size. Goldsmith and Mason (1962) also examined the shapes of buoyant drops with small density differences, but their experiments were limited to very small drop sizes (κ e 0.07). In this study, we focus on the pressure-driven motion of air bubbles and buoyant drops through vertical cylindrical capillaries at low Reynolds numbers. We consider the two cases in which buoyancy and pressure forces act in the same, or in opposite, directions. Experimental measurements of the relative mobility of drops and their shape deformations will be presented. The observed drop shapes are quantitatively examined using digital image analysis in order to

λ

γ

Bo

κ

GW1-1 0.56 0.77 1.7 0.45-1.32 GW2-1 0.25 0.79 1.5 0.45-1.57 GW3-1 1.24 0.80 3.4 0.45-1.32 GW4-1 0.19 0.78 12.6 0.45-1.57 GW5-1 0.23 0.76 7.0 0.45-1.32 GW6-1 0.00 0.00 4.6 0.45-1.32 GW1-2 0.56 0.77 0.3 0.65-2.39 GW3-2 1.18 0.80 0.7 0.65-2.39 GW4-2 0.18 0.78 2.4 0.65-2.39 GW5-2 0.22 0.76 1.4 0.65-2.39 GW6-2 0.00 0.00 0.9 0.65-2.39 GW7-2 0.15 0.79 0.3 0.65-2.39 GW8-2 1.43 0.79 0.3 0.65-2.84 GW4-3 0.18 0.78 0.7 1.05-2.06 GW5-3 0.22 0.76 0.4 1.79-2.58 DEG1-1 10.35 0.87 2.6 0.61-1.32 DEG7-1 2.77 0.89 2.0 0.61-1.32 DEG8-1 26.40 0.90 1.8 0.61-1.32 DEG9-1 7.20 0.95 1.0 0.61-1.32 WG1-1 1.20 0.79 1.6 0.58-1.32 WG2-1 1.30 0.82 1.3 0.58-1.32 CW3-1 0.22 0.72 4.1 0.58-1.57 CW4-1 0.03 0.71 10.4 >0.58 CW5-1 0.04 0.69 8.2 0.58-1.05 >1.05 CW6-1 0.00 0.00 2.6 0.58-1.97

Re

Ca*

0.03-0.21 0.03-0.21 0.00-0.17 0.02-0.31 0.01-0.28 0.08-0.24 0.05-0.27 0.04-0.25 0.07-0.24 0.06-0.37 0.04-0.37 0.06-0.32 0.05-0.31 0.03-0.04 0.07-0.11 0.66-8.63 0.73-8.70 0.49-8.13 0.38-8.51 0.13-1.02 0.13-0.25 0.01-0.05 breakup 0.01-0.03 breakup 0.01-0.06

0.05-0.31 0.05-0.30 0.01-0.54 0.26-3.30 0.06-1.60 0.11-0.39 0.16-0.89 0.30-0.93 1.92-6.44 0.84-5.33 0.08-0.68 0.19-0.99 0.16-0.47 5.23-5.42 2.32-3.92 0.01-0.12 0.01-0.11 0.01-0.10 0.01-0.14 0.04-0.32 0.07-0.25 0.66-3.97 breakup 0.26-0.48 breakup 0.05-0.30

extract useful geometric information such as the axial and radial dimensions of the drop profile, and the average thickness of the liquid film separating large drops from the capillary wall. The effects of Bond number, capillary number, and viscosity ratio on the measured quantities will be reported over a wide range of drop sizes. Experimental Procedures A schematic illustration of the experimental setup is shown in Figure 1. The experimental apparatus was similar to that used by Borhan and Pallinti (1995) to study the buoyancy-driven motion of drops in cylindrical capillaries. It consisted of a vertical precision-bore glass capillary tube enclosed by a Plexiglas chamber of square cross-section containing an aqueous solution of sodium iodide. The refractive index of the sodium iodide solution was matched with that of the glass capillary

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

to minimize optical distortions due to the refraction of light at the outer wall of the capillary. Three different capillary tubes were used to cover a wide range of drop sizes and Bond numbers. They were all 120-cm long and were labeled 1, 2, and 3 according to their inner diameters of 0.796, 0.347, and 0.186 cm, respectively. The suspending fluids used in these experiments consisted of 99% pure diethylene glycol (DEG), a 98.5 wt % corn syrup-water mixture (CW), and various aqueous glycerol solutions (GW and WG). A variety of Dow Corning fluids and UCON oils, silicon oil, and air were used as drop fluids. The various two-phase systems used in these experiments and their relevant physical properties are listed in Table 1. To facilitate the presentation of the experimental results, each two-phase system in this table is designated by a symbol identifying the suspending fluid, followed by a number that specifies the drop fluid. The fluid properties shown in Table 1 were all measured at a temperature of 25 °C. However, for each set of experiments with the same twophase system, all fluid properties were measured again at the actual temperature of the experiments (typically in the range 24.3-25.9 °C), which was determined by a

Figure 3. Variations of the dimensionless geometric parameters with 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 the meridional plane to that of a spherical drop with the same volume, D; (b) drop length, LA; (c) maximum equatorial dimension, LR.

digital thermometer connected to thermocouples placed near the inlet and outlet regions of the capillary tube. The viscosities of all liquids were measured using thermostated capillary viscometers, and the interfacial tension between the two phases was obtained using a ring tensiometer. For each set of experiments with the same two-phase system, the suspending fluid was pumped through the capillary tube at a known flow rate. In most experiments, the suspending fluid flowed from the bottom to the top of the capillary and the desired volume of the drop fluid was injected at the symmetry axis of the capillary near the inlet region (at the bottom of the

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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.

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

capillary) using a micrometer syringe. The capillary tube and the syringe were thoroughly cleaned with distilled water, benzene, and acetone and then dried in air before each new set of experiments. For all twophase systems considered in our experiments, the drop fluid had a lower density than the suspending fluid and the drop traveled upward along the axis of symmetry of the capillary. In a few experiments, the imposed flow of the suspending fluid was reversed (i.e., flowing from the top to the bottom of the capillary) in order to examine the case in which pressure and buoyancy forces act in opposite directions. The motion of the drop through the capillary was recorded using a CCD camera connected to a video recorder capable of frame-by-frame

playback. The video camera was mounted on a moving platform whose speed was adjusted by a controller to follow the vertical motion of the drop, thereby allowing the drop to be monitored as it passed through the entire length of the capillary. To study drop deformations, the recorded images of the drop profile were played back frame-by-frame and the signal from the video recorder was digitized using a computer equipped with a frame-grabber board. A stop-motion filter was applied to the digitized images to remove any jittering caused by the motion of the drop, and the Bioscan Optimas image analysis software was then used to quantitatively characterize the drop shapes by measuring various geometric features such as the perimeter of the drop profile in the meridional plane and its maximum axial and radial dimensions. For each experiment, the volume of the axisymmetric drop was also determined using Optimas to ensure the accuracy of the drop size measurement based on the micrometer syringe reading. The terminal velocity of the drop was determined by measuring the time required for the drop to travel a specified vertical distance between selected markers on the capillary wall. For each experiment, three independent velocity measurements over different regions of the capillary tube were made to check for any unsteadiness in the motion of the drop. The geometric information obtained from the drop shape at various axial positions was also used to detect unsteady drop behavior in the experiments and to ensure that velocity measurements were being made in regions sufficiently far from the inlet of the capillary to avoid entrance effects. Finally, each experiment was repeated to ensure reproducibility of the results. In all cases, the reported terminal velocity represents the average of six velocity measurements, with each measurement having a variation of less than 5% from the reported mean value. Results and Discussion In this section, we present the experimental results in terms of the effects of Bond number, viscosity ratio, and capillary number on the steady shape of the drop and its terminal velocity over a wide range of drop sizes. A list of parameter values for each set of experiments with the same two-phase system in a particular capillary tube is shown in Table 2. In this table, each set of experiments is labeled by a symbol identifying the two-

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

phase system followed by a number specifying the size of the capillary tube. The Reynolds numbers reported in Table 2 are based on the actual terminal velocity of the drop, U, rather than the mean velocity, V, of the imposed flow, that is Re ) (FUR)/µ. The overall range of Reynolds numbers associated with these experiments was between 0.0 and 8.7. Aside from the motion of small drops in diethylene glycol (DEG) systems, the Reynolds numbers for the motion of the dispersed phase remained in the Stokes regime. The range of measured drop velocities for each set of experiments is also shown in the last column of Table 2 in the form of a capillary number defined as Ca* ) (µU)/σ.

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) maximum equatorial dimension, LR.

a. Drop Shape. In all of the experiments reported here, the shapes of drops and bubbles remained axisymmetric as they passed through the capillary tube. Typical profiles (in the meridional plane) of the steady shapes observed experimentally for air bubbles and viscous drops are shown in Figure 2 for various drop sizes. The steady shape of air bubbles approaches an elongated ellipsoid as the bubble size increases, with a slight loss of fore and aft symmetry due to the flattening of the trailing end caused by the imposed flow. As the interior phase becomes more viscous relative to the suspending fluid, a slightly more tapered drop shape

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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) as a function of capillary number.

results without significantly affecting the qualitative features of the leading end of the drop. To quantitatively characterize the steady shapes of drops and bubbles, a deformation parameter D is defined as the ratio of the perimeter of the deformed drop profile in the meridional plane to that of a spherical drop with the same volume. This quantity is easily obtained from digitized images of drops and bubbles using the Optimas image analysis software. Figure 3 demonstrates the typical dependence of the deformation parameter on the dimensionless drop size. Also shown in this figure are the geometric parameters LA and LR, representing the maximum axial and radial dimensions of the steady drop profile relative to the capillary radius, respectively. The dashed lines in this figure denote the values of the geometric parameters for spherical drops, while the dotted lines represent those corresponding to a stagnant cylindrical drop with hemispherical ends. For a given two-phase system (with fixed Bond number and viscosity ratio), both the radial and axial dimensions of the drop initially grow almost linearly as a function of drop size, with the first shape transition occurring at differ-

ent drop sizes, depending on the value of the capillary number. It is clear from Figure 3 that, for fixed capillary and Bond numbers, drop deformation is only slightly affected by large variations in the viscosity ratio. While the deformation parameter D shows a weak dependence on the viscosity ratio, the values of LA and LR are virtually the same for systems with widely different values of λ. This is consistent with the qualitative observations of a slightly more tapered drop shape with nearly the same drop length at larger viscosity ratios. The weak dependence of D on the viscosity ratio disappears as the capillary number becomes large. The effect of Bond number on drop shape is shown in Figure 4 for systems with nearly constant values of other parameters. It is evident that increasing the Bond number leads to larger deformations in the form of axial elongation of the drop, particularly for larger drops. Once the transition to a cylindrical shape occurs, increases in the Bond number lead to faster rates of elongation as a function of drop size, as well as smaller radii (limiting values of LR) for the cylindrical section of the drop or, equivalently, thicker liquid films between the cylindrical drops and the capillary wall. However, when the Bond number exceeds a limiting value, the value of LR remains unchanged with further increases in the Bond number. This limiting value of the Bond number decreases as the capillary number is increased. This is evident from the experimental data in Figure 5 which show the measured liquid film thickness δ (made dimensionless with capillary radius R), for the largest drops in the low-viscosity-ratio systems. The solid symbols in this figure represent measurements of the film thickness in the case of buoyancy-driven motion (i.e., for Ca ) 0), while the open symbols represent the corresponding measurements in the presence of pressure-driven flow with Ca ) 0.14. For small Bond numbers, the values of the film thickness in the presence of imposed flow are larger than those corresponding to Ca ) 0. However, for large Bond numbers, the limiting value of the film thickness is not significantly

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.

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

affected (within experimental error) by the addition of pressure-driven flow. The limiting value of Bond number beyond which the film thickness remains nearly constant decreases from approximately 12 to 2 as the capillary number increases from 0 to 0.14. Hence, the effect of Bond number on drop deformation eventually disappears as the capillary number becomes large. The effect of capillary number on the steady drop shape is illustrated in Figure 6. Clearly, the radial dimension of a viscous drop of constant volume decreases, and its axial length grows, with increasing capillary number. It can also be seen from this figure that the transition from a spherical to an elongated shape occurs at smaller drop sizes as the capillary number increases. The additional viscous stresses generated by the imposed flow can promote shape deformations even for small drops which remain spherical in the absence of imposed flow. The increase in drop length is accompanied by an increase in the thickness of the liquid film surrounding the drop. However, there is a limiting value of capillary number beyond which the drop shape becomes relatively insensitive to the value of the capillary number, as shown in Figure 7. For the two-phase system in this figure, increasing the capillary number from 0.3 to 0.9 does not lead to any significant changes in LR or, equivalently, the thickness of the liquid film surrounding large drops. The effect of capillary number on the film thickness, δ, is more

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

clearly demonstrated in Figure 8. The values of δ in this figure have been scaled by the film thickness δ0 in the absence of imposed flow (at Ca ) 0). The film thickness becomes larger as the capillary number increases, and eventually approaches a limiting value at sufficiently large values of Ca. The limiting value of capillary number beyond which the film thickness becomes independent of Ca is smaller for systems with larger Bond numbers. The effect of capillary number on drop deformation also diminishes as the viscosity ratio tends to zero. For example, for the air bubbles in the GW6-1 system, the bubble length becomes independent of the capillary number for Ca g 0.02, whereas the length of viscous drops in the CW3-1 system (with nearly the same Bond number as the GW6-1 system) remains sensitive to the value of Ca for capillary numbers as large as 0.75. As the capillary number increases, there is a notable reduction in the curvature of the trailing end of the drop, as illustrated by the sequence of images in Figure 9 for a drop of size κ = 0.91 in the CW3-1 system. For sufficiently large capillary numbers, the drop can even develop a region of negative curvature at its trailing end. For the steady drop shape corresponding to Ca ) 0.75 in Figure 9, a re-entrant cavity was actually observed at the trailing interface, though it is difficult to detect its presence in the meridional profile of the drop. The

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

qualitative features of the trailing interface in the CW3-1 system are similar to those observed experimentally by Olbricht and Kung (1992) and predicted numerically by other investigators (cf. Chi, 1986; Martinez and Udell, 1990; Borhan and Mao, 1992; Tsai and Miksis, 1994), for neutrally-buoyant drops in pressuredriven flow through a cylindrical capillary. However, the capillary number at which the trailing end of the drop first exhibited a region of negative curvature was found to be larger than that predicted numerically for neutrally-buoyant drops. For example, for a drop of size κ = 0.73 in a two-phase system with nearly the same viscosity ratio as the CW3 system, Martinez and Udell (1990) predict the first appearance of a re-entrant cavity at Ca ) 0.75, whereas a similar response was first observed at Ca ) 1.21 in our experiments. This difference can be attributed, at least partially, to the nonzero Bond numbers in the CW3-1 experiments, in contrast to the zero Bond numbers associated with the neutrallybuoyant drops considered in the computational studies. In the case of pressure-driven motion of neutrallybuoyant drops, the drop always moves slower than the suspending fluid elements near the capillary axis, leading to the formation of stagnation rings on the surface of the drop. The resulting flow field in the vicinity of the trailing end of the drop is similar to an axisymmetric stagnation flow with the suspending fluid

Figure 13. The drop speed as a function of drop size for the GW4-1 system (Bo ) 12.6, λ ) 0.19). The solid curves represent the best fit to the experimental data, while the dotted curves show the corresponding asymptotic predictions of Hetsroni et al. (1970).

elements on the capillary axis moving toward the trailing interface with a relative velocity (2V - U) > 0. For low-viscosity-ratio systems (such as CW3), increasing the capillary number produces larger stagnation pressures within the suspending fluid at the rear stagnation point of the drop, without significantly affecting the pressure inside the drop. This eventually leads to the appearance of negative curvature at the trailing interface once the flow-induced normal stresses in the suspending fluid exceed those within the lowviscosity-ratio drop. For a buoyant drop, however, the stagnation flow in the vicinity of the trailing end of the drop is weakened when the buoyancy force acts in the direction of the imposed flow. This is mainly due to the larger terminal velocity of the buoyant drop at the same capillary number, resulting in a smaller relative velocity (2V - U) for the stagnation flow at the trailing end of the drop. Consequently, a larger value of capillary number will be required to produce the same stagnation pressure at the rear stagnation point of the drop. Since the terminal velocity of the drop is a monotonically increasing function of Bond number when the buoyancy force is aligned with the imposed flow (as will be shown later), it is reasonable to expect that increasing the Bond number would delay the formation of a region of negative curvature at the back of the drop until larger values of capillary number are reached. When the Bond number is so large that the drop moves faster than the maximum velocity of the imposed flow on the capillary axis, the stagnation rings on the surface of the drop disappear altogether and the flow field resembles that

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Figure 14. The drop speed as a function of drop size for the GW6-1 system (Bo ) 4.6, λ ) 0.0). The solid curves represent the best fit to the experimental data shown by the open symbols, while the dotted curves show the corresponding asymptotic predictions of Hetsroni et al. (1970).

in the case of buoyancy-driven motion (i.e., surface flow from the front stagnation point to the rear stagnation point of the drop). In this case, one would expect the development of a region of negative curvature at the back of the drop to be inhibited for all capillary numbers. Indeed, no indentation at the trailing end of lowviscosity-ratio drops was observed in the experiments of Borhan and Pallinti (1995) for the case of buoyancydriven motion, and the curvature of the trailing interface of large drops was found to be an increasing function of Bond number in those experiments. b. Drop Mobility. We now consider the effects of capillary number, Bond number, and viscosity ratio on the terminal velocity of drops and bubbles. The typical dependence of the terminal velocity on drop size is shown in Figure 10 where the velocity U* is made dimensionless with (∆FgR2)/µ). The solid curves in this figure represent the best fit to the experimental data, taking into account the fact that U* must approach the dimensionless centerline velocity of the imposed flow as κ tends to zero. The dotted curves represent the corresponding theoretical predictions of Hetsroni et al. (1970) for slightly deformed drops in each system. For systems with O(1) values of Ca/Bo, the terminal velocity decreases rapidly with increasing drop size as the retarding 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 other hand, the terminal velocity initially increases with drop size before reaching a plateau for large drop sizes. The

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

experimentally measured drop speeds deviate substantially from the asymptotic predictions of Hetsroni et al. as the drop size is increased beyond κ ) 0.5. This is not surprising since the asymptotic predictions are valid for small drops which remain nearly spherical, whereas significant drop deformations are typically observed in the experiments for κ g 0.5. Hence, the shape of the drop significantly affects its mobility and finite drop deformations must be taken into account in order to accurately predict the drop speed. The relative mobility of the drop, defined as the ratio of the drop speed to the average velocity of the imposed flow, is plotted in Figures 11 and 12 as a function of drop size. To identify the contribution of the imposed flow to the mobility of the drop, the quantity (U - U0)/V is also plotted in each of these figures, where U0 represents the terminal velocity in the absence of pressure-driven flow (i.e., at Ca ) 0). Figure 11 shows the effect of the viscosity ratio on the relative mobility of the drop. For a given drop size, the relative mobility is reduced as the viscosity ratio is increased while keeping Ca and Bo constant. However, there is a limiting value of λ beyond which the relative mobility becomes independent of the viscosity ratio. This limiting value of λ becomes larger as the Bond number increases and is more easily detected in the low Bond number systems. For example, there is very little

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

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

difference between the relative mobilities of drops in the DEG8-1 and DEG9-1 systems (shown in Figure 11) which are characterized by widely different viscosity ratios. The dashed and dotted curves in Figure 11 represent the asymptotic predictions of Hetsroni et al. (1970) for the parameter values corresponding to the GW2-1 and DEG8-1 systems, respectively. Clearly, the experimentally observed drop mobilities for κ > 0.5 are not accurately predicted by the asymptotic analysis, due to significant deformations of large drops. The effect of Bond number on the relative mobility is presented in Figure 12. As the Bond number increases for fixed values of κ, Ca, and λ, the relative mobility of the drop is enhanced. This is consistent with our earlier discussion of the effect of Bond number on drop shape since the more elongated drop shapes formed at larger Bond numbers are localized near the symmetry axis of the capillary and experience a weaker wall effect. Similar to the effect of Bond number on drop shape, increasing the Bond number beyond a limiting value does not significantly affect the relative mobility. The results of the asymptotic analysis, shown by the dotted curves in Figure 12, predict the quantity (U - U0)/V to be independent of Bond number, whereas the experimental data clearly indicate a strong Bond number dependence for large drops. Again, this discrepancy can

be attributed to the effect of Bond number on shape deformations of large drops. The effects of capillary number on the terminal velocity and relative mobility of drops are shown in Figures 13 and 14 for large and small Bond numbers, respectively. As expected, the drop moves faster as the capillary number increases, due to the larger velocities associated with the imposed flow as well as the more elongated drop shapes which tend to reduce the retarding effect of the capillary wall by localizing the drop near the centerline of the capillary. The former effect is dominant for small drops (which remain nearly spherical) and is also reflected in the asymptotic predictions shown by the dotted curves in Figures 13 and 14, while the latter effect becomes significant for κ > 0.4 which also represents the point at which the experimental measurements begin to deviate substantially from the asymptotic predictions. In contrast to the pressuredriven motion of neutrally-buoyant drops (cf. Ho and Leal, 1975), however, the relative mobility of the drop is not a monotonically increasing function of the capillary number. For small capillary numbers, the relative mobility is a decreasing function of Ca despite the fact that the actual drop speed U increases with capillary number. The relative mobility seems to attain a local minimum as the capillary number is increased, before becoming relatively insensitive to the value of the

3758 Ind. Eng. Chem. Res., Vol. 37, No. 9, 1998

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

capillary number. The minimum in the relative mobility occurs at smaller capillary numbers in systems with lower Bond numbers or larger viscosity ratios, as shown in Figures 15 and 16. To identify the contribution of shape deformations to the mobility of a drop as the capillary number is increased, the quantity (U - U0)/V is also plotted in Figures 15 and 16. If the increase in the drop speed with increasing capillary number were solely due to the larger mean velocity of the imposed flow in an experiment, the quantity (U - U0)/V would be expected to remain nearly constant as the capillary number is increased for a given drop size. This is indeed the behavior predicted by the asymptotic analysis of Hetsroni et al. (shown by the dashed and dotted lines in Figures 15 and 16) for small drops which experience negligible shape deformations. For most two-phase systems considered here, however, this quantity initially decreases as the capillary number is increased from zero. This is an indication that the addition of the pressure-driven flow initially results in drop deformations that adversely affect the mobility of the drop in these systems. As the capillary number is further increased, the stronger imposed flow eventually leads to drop deformations which enhance drop mobility, and (U - U0)/V becomes a weakly increasing function of capillary number, in line with the experimental results of Ho and Leal (1975), and the numerical results of Martinez and Udell (1990), for pressure-driven motion of neutrally-buoyant drops through cylindrical capillaries. Hence, as expected, the dynamics of a buoyant drop approaches that of a neutrally-buoyant drop described by Ho and Leal (1975) once the capillary number becomes sufficiently large. Figure 17 illustrates the typical results for experiments in which the pressure and buoyancy forces acted in opposite directions. In these experiments, the flow

of the suspending fluid was established from the top to the bottom of the capillary tube, but the imposed flow was sufficiently weak to allow the drop to rise vertically within the capillary. The drop speed decreases as the magnitude of the capillary number is increased (i.e., as the imposed flow opposing the buoyancy-driven motion of the drop becomes stronger). However, the reduction in the mobility of the drop is not entirely due to the larger mean velocities of the opposing pressure-driven flow. For very small capillary numbers, the introduction of pressure-driven flow indirectly affects the mobility of the drop through subtle changes in drop shape. The steady drop shape becomes more flattened at the leading edge, and shorter in length, compared to the corresponding drop shapes at zero capillary number. As the capillary number increases, the drop shrinks in the axial direction, simultaneously decreasing the thickness of the liquid film surrounding it and increasing the retarding effect of the capillary wall. This response is exactly the opposite of that described earlier for the case in which buoyancy and pressure forces acted in the same direction. The retarding effect of flow-induced deformations can be identified by examining the dependence of (U U0)/V on capillary number in Figure 17. For large Bond number systems, this quantity is found to be a strong function of capillary number (in contrast to the asymptotic predictions shown by the dotted curve), indicating the significant contribution of drop deformations to the reduction in drop mobility. We conclude by presenting a sequence of images showing the typical behavior of the trailing end of large drops in the CW4 and CW5 systems which were characterized by large Bond numbers and vanishing viscosity ratios. As shown in Figure 18, an unstable pointed tail formed at the trailing end of the drops in these systems and subsequently disintegrated into a

Ind. Eng. Chem. Res., Vol. 37, No. 9, 1998 3759

stream of small satellite drops. For large (κ > 1.6) drops, this tail-streaming behavior was observed even in the absence of imposed flow (i.e., at Ca ) 0). Increasing the capillary number led to more intense streaming at the trailing end of a drop of fixed size and shifted the onset of tail streaming to smaller drop sizes. The behavior of the trailing interface of the drop in the CW4 and CW5 systems is qualitatively similar to the tip-streaming phenomenon observed in some previous experiments involving low-viscosity-ratio (λ < O(0.1)) drops in shear flows (cf. Milliken and Leal, 1991; de Bruijn, 1993). The onset of tip-streaming in lowviscosity-ratio systems is believed to be due to the presence of surface-active species (de Bruijn, 1993; Stone, 1994). Although the experimental systems used in this study were carefully cleaned to avoid contamination, the presence of surface-active components in corn syrup may have been responsible for the observed tailstreaming phenomenon. This explanation is further supported by the experimental observations of diminishing streaming intensity at the trailing end of the drop as the drop passed through the capillary. In some cases, tail streaming disappeared altogether and the trailing interface achieved a steady shape near the top of the capillary, as shown in Figure 18a. While the reduction in drop volume due to tail-streaming could have, in principle, resulted in a subcritical drop size as the drop passed through the capillary tube, experimental measurements of the drop volume from digitized images of the steady drop profile near the top of the capillary tube did not support this hypothesis. The actual change in drop volume was too small to explain the disappearance of tail-streaming in the experiments, particularly for large drops (κ > 1.6) in capillary 1 where tail-streaming was observed even in the absence of imposed flow. A plausible explanation for the observed behavior is the reduction in the surface concentration of the surfaceactive species through the formation of satellite drops, consistent with the mechanism suggested by de Bruijn (1993). The role of surface-active impurities in the dynamics of the observed tail-streaming behavior warrants further study. Acknowledgment Acknowledgment is made to the donors of the Petroleum Research Fund, administered by the American Chemical Society, for partial support of this research. This work was also supported by the National Science Foundation under Grant CTS-9110470. Literature Cited Aul, R. W.; Olbricht, W. L. Coalescence of Liquid Drops Suspended in Flow through a Small Pore. J. Colloid Interface Sci. 1991, 145, 478.

Bretherton, F. P. The Motion of Long Bubbles in Tubes. J. Fluid Mech. 1961, 10, 166. Borhan, A.; Mao, C. F. Effect of Surfactants on the Motion of Drops through Circular Tubes. Phys. Fluids 1992, 4, 2628. Borhan, A.; Pallinti, J. Buoyancy-Driven Motion of Viscous Drops through Cylindrical Capillaries at Small Reynolds Numbers. Ind. Eng. Chem. Res. 1995, 34, 2750. Chen, J. D. Measuring the Film Thickness Surrounding a Bubble Inside a Capillary. J. Colloid Interface Sci. 1986, 109, 341. Chi, B. Ph.D. Thesis, California Institute of Technology, 1986. Clift, R.; Grace, J. R.; Weber, M. E. Bubbles, Drops, and Particles; Academic Press: New York, 1978; Chapter 9. Cox, B. G. On Driving a Viscous Fluid out of a Tube. J. Fluid Mech. 1962, 14, 81. de Bruijn, R. A. Tipstreaming of Drops in Simple Shear Flows. Chem. Eng. Sci. 1993, 48, 277. Fairbrother, F.; Stubbs, A. E. Studies in Electroendosmosis. Part VI. The Bubble-Tube Methods of Measurement. J. Chem. Sci. 1935, 1, 527. Goldsmith, H. L.; Mason, S. G. The Flow of Suspensions through Tubes. I. Single Spheres, Rods, and Discs. J. Colloid Sci. 1962, 17, 448. Goldsmith, H. L.; Mason, S. G. The Flow of Suspensions through Tubes. II. Single Large Bubbles. J. Colloid Sci. 1963, 18, 237. Hetsroni, G.; Haber, S.; Wacholder, E. W. The Flow Fields in and around a Droplet Moving Axially within a Tube. J. Fluid Mech. 1970, 41, 689. Ho, B. P.; Leal, L. G. The Creeping Motion of Liquid Drops through a Circular Tube of Comparable Diameter. J. Fluid Mech. 1975, 71, 361. Marchessault, R. N.; Mason, S. J. Flow of Entrapped Bubbles through a Capillary. Ind. Eng. Chem. 1960, 52, 79. Martinez, M. J.; Udell, K. S. Axisymmetric Creeping Motion of Drops through Circular Tubes. J. Fluid Mech. 1990, 210, 565. Milliken, W. J.; Leal, L. G. Deformation and Breakup of Viscoelastic Drops in Planar Extensional Flows. J. Non-Newtonian Fluid Mech. 1991, 40, 355. Olbricht, W. L. Pore-Scale Prototypes of Multiphase Flow in Porous Media. Annu. Rev. Fluid Mech. 1996, 28, 187. Olbricht, W. L.; Kung, D. M. The Deformation and Breakup of Liquid Drops in Low Reynolds Number Flow through a Capillary. Phys. Fluids 1992, 4 (7), 1347. Olbricht, W. L.; Leal, L. G. The Creeping Motion of Liquid Drops through a Circular Tube of Comparable Diameter: The Effect of Density Differences between the Fluids. J. Fluid Mech. 1982, 115, 187. Prothero, J.; Burton, A. C. The Physics of Blood Flow in Capillaries. J. Biophys. 1961, 2, 525. Schwartz, L. W.; Princen, H. M.; Kiss, A. D. On the Motion of Bubbles in Capillary Tubes. J. Fluid Mech. 1986, 172, 259. Stone, H. A. Dynamics of Drop Deformation and Breakup in Viscous Fluids. Annu. Rev. Fluid Mech. 1994, 26, 65. Taylor, G. I. Deposition of a Viscous Fluid on the Wall of a Tube. J. Fluid Mech. 1961, 10, 161. Tsai, T. M.; Miksis, M. J. Dynamics of a Drop in a Constricted Capillary Tube. J. Fluid Mech. 1994, 274, 197.

Received for review February 12, 1998 Revised manuscript received June 2, 1998 Accepted June 16, 1998 IE980087L