2750
Znd. Eng. Chem. Res. 1996,34, 2750-2761
Buoyancy-Driven Motion of Viscous Drops through Cylindrical Capillaries at Small Reynolds Numbers Ali Borhan* and Jayanthi Pallinti Department of Chemical Engineering, The Pennsylvania State University, University Park, Pennsylvania 16802
We examine the buoyancy-driven motion and deformation of viscous drops through vertical cylindrical capillaries under conditions where interfacial tension effects are important while inertial effects are negligible. Experimental measurements of the terminal velocity of the drops are reported for a range of drop sizes in a variety of two-phase systems, and the drop shapes are quantitatively characterized using digital image analysis. The retarding effect of the capillary wall is found to decrease 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, for a given viscosity ratio, there is a limiting value of the Bond number beyond which the wall effect remains unchanged with further increases in the Bond number. The thickness of the liquid film surrounding large drops is found to be rather insensitive to the value of the viscosity ratio. Theoretical predictions based on the numerical solution of the creeping flow equations using the boundary integral method are shown to be in good agreement with the experimental measurements.
Introduction The motion of drops and bubbles in confined domains is encountered in a variety of natural phenomena and technical applications such as underground transport of pollutants, intrusion of water into oil-bearing strata, solvent extraction processes, and penetration of air into water-saturated networks of fibers in papermaking. In addition, drop motion through capillaries remains a problem of considerable fundamental importance as a pore-scale model for studying the dynamics of two-phase flow through porous media. Consequently, 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 t o its geometric simplicity. A general review of the previous work on this problem is given by Clift, Grace, and Weber (1978). In many situations, 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 smalllength scales associated with the confining geometries, while, in solvent extraction processes, the small dispersed phase velocities designed to increase the contact time between the two phases is responsible for the creeping flow conditions. Thus, the low Reynolds number motion of drops and bubbles through cylindrical tubes is of considerable interest, particularly since it also provides opportunities for theoretical analysis. In the case of the motion of a viscous drop or a gas bubble of volume V in a vertical cylindrical capillary of radius R under the action of gravity, the important parameters governing the dynamics of the system under conditions of negligible inertia include the ratio of the equivalent spherical drop radius to capillary radius, K =(~VI~JCR the~ )drop ~ ’ ~to, suspending fluid viscosity ratio, A, the corresponding density ratio, y , and a Bond number defined as Bo = AQgR21a, with AQ and u denoting the density difference and surface tension between the two phases, respectively, and g the magnitude of the gravitational acceleration. The effect of these parameters on the creeping motion of drops and bubbles within cylindrical capillaries has been the
subject of several theoretical studies, ranging from asymptotic analyses and approximate solutions for small spherical or elongated cylindrical drop shapes (Haberman and Sayre, 1958; Bretherton, 1961; Goldsmith and Mason, 1962; Hetsroni et al., 1970; Hyman and Skalak, 1972; Coutanceau and Thizon, 1981) to computational studies involving elongated gas bubbles (Reinelt, 1987). A brief review of this work is provided in a recent paper by Pozrikidis (1992b), who has also performed a detailed numerical study of the buoyancydriven motion of a periodic train of viscous drops with ;1 = 1 a t low Reynolds numbers. The latter is the only theoretical study of this problem which accounts for finite deformations of viscous drops. On the experimental side, most previous investigations pertaining t o the creeping motion of viscous drops through capillary tubes have focused on pressure-driven motion. A summary of these studies is provided elsewhere (e.g.,Borhan and Mao, 1992;Park, 1992; Olbricht and Kung, 1992). Aside from the recent work of Coutanceau and Texier (19861, experimental studies of buoyancy-driven motion at small Reynolds numbers have been mainly restricted to the case of gas bubbles, particularly large bubbles (with bubble lengths that are several times the capillary diameter) which are separated from the capillary wall by only a thin layer of suspending fluid (e.g., Barr, 1926; Bretherton, 1961; White and Beardmore, 1962; Goldsmith and Mason, 1962; Zukoski, 1966; Coutanceau and Thizon, 1981). Coutanceau and Texier measured the terminal settling velocity of water and glycerine drops in a vertical tube filled with an otherwise quiescent silicone fluid. They considered drop sizes comparable to the tube diameter (0.2 < K < 1.2) under conditions of negligible inertial and interfacial tension effects (corresponding to small Reynolds numbers and large Bond numbers, respectively). However, the three systems used in their experiments were all characterized by extremely low viscosity ratios (1 < 0.04) reminiscent of gas bubbles. Hence, they found the evolution of the drop shape with drop size K to be similar to that for air bubbles. They also attributed the observed differences in the thickness of the liquid film surrounding large drops to the varia-
~ a ~ a - ~ a a ~ 1 ~ ~ 1 ~ ~ ~ ~ -o~ 1995 ~ ~ oAmerican $ o 9 . oChemical o/o Society
Ind. Eng. Chem. Res., Vol. 34, No. 8, 1995 2751 Table 1. Two-Phase Systems Used in the Experiments'
system
suspending fluid
drop fluid
GW1 GW2 GW3 GW4 GW5 GW6 GW7 GW8 cw3 cw4
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 %) corn syrup-water (98.5 wt %) corn syrup-water (98.5 wt %) corn syrup-water (98.5 wt %) corn syrup-water (98.5 wt %) di(ethy1ene glycol) di(ethy1ene glycol) di(ethy1ene glycol) di(ethylene glycol) di(ethy1ene glycol) glycerol-water (89.4 wt %) glycerol-water (84.2 wt %) glycerol-water (82.0 wt %) glycerol-water (82.48 wt %)
silicon oil DC510-100 UCON-1145 UCON-50HB55 UCON-5OHB 100 air DC510-50 DC510-500 UCON-1145 UCON-50HB55 UCON-50HB100 air silicon oil air DC510-50 DC510-500 DC550 silicon oil DC510-100 UCON-50HB55 UCON-50HB100
cw5
CW6 DEGl DEG6 DEG7 DEG8 DEG9 WG1 WG2 WG4 WG5 a
viscosity of viscosity of density of density of interfacial suspending fluid drop fluid suspending fluid drop fluid tension (mPad (mPad (kg/m3) (kg/m3) (N/m) x lo3 427 427 427 427 427 427 427 427 2444 2444 2444 2444 23 23 23 23 23 198 80 67 77
238 105 528 83 97 0 64 607 528 83 97 0 238 0 64 607 166 238 105 83 97
1250 1250 1250 1250 1250 1250 1250 1250 1375 1375 1375 1375 1110 1110 1110 1110 1110 1228 1212 1206 1207
967 990 995 970 950 1.3 986 994 995 970 950 1.3 967 1.3 986 994 1059 967 990 970 950
25.3 26.8 11.6 3.5 6.5 42.0 26.5 26.0 14.3 6.0 8.0 80.8 8.5 34.5 9.6 9.6 7.6 25.3 26.8 3.5 6.5
Note: All physical properties were measured at 25 "C.
tions in the density ratio y (which was in the range 1.02 1.22 0.3 0.97 > 1.22 0.2 1.79 0.7 1.79 0.4 1.79 0.3 1.79 2.6 0.40-1.56 2.0 0.45-1.32 1.8 0.49-1.62 1.0 0.49-1.15 0.3 1.79 0.1 1.79 0.1 1.79 0.05 1.79 1.6 0.58-1.32 1.3 0.58-1.32 10.4 0.58-1.32 6.1 0.58-1.32 4.1 0.58-1.57 10.4 0.49-1.57 8.2 0.58-1.57 2.6 0.58-1.97
0.77 1.7 0.79 1.5 3.3 0.80 0.78 12.6 0.76 7.0 0.001 4.6 0.3 0.77 0.8 0.7
Re x 103 Cu x 103 4-42 1-30 5-19 4-14 17-54 5-17 300-380 29-36 160-190 28-34 44-78 40-70 0 0-1 0 1-2 0-1 2-4 0 0-1 0 0-1 0 0 0 0 0 170-1700 150-1700 480-1200 56-680 0 0 0 0 5-62 1-300 800-960 700-760 1-2 1-2 1-2 1-2
0 0-2 0 27-49 2-11 0-1 0 0-1 0 0-1 0 0 0 0 0 0-2 2-21 1-15 0-11 0 0 0 0 1-20 0-15 220-270 130-140 44-94 270-310 230-260 18-22
specified vertical distances 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 for the drop shapes at axial positions within those regions were 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 the requirement that each measurement have a variation of less than 5% from the reported mean value.
Theoretical Analysis Consider the steady buoyancy-driven motion of a drop of volume V and density yg along the axis of a cylindrical capillary of radius R containing an otherwise quiescent suspending fluid of density g, as shown in Figure 1.The exterior and interior bulk phases denoted by Q and Q', respectively, are incompressible and Newtonian with corresponding viscosities ,LA and Ap. The shape of the drop is assumed to remain axisymmetric as the drop moves through the capillary, with a constant surface tension o acting at the surface of the drop denoted by Sa. In the analysis that follows, all lengths are made dimensionless with R, all velocities with Ub = AggR2/p, and all stresses with ,uUdR. When the Reynolds number for flow in both phases is small so that inertial effects can be neglected, the flow
Ind. Eng. Chem. Res., Vol. 34,No. 8,1995 2753
(4
(1))
(d)
(C)
(r)
(0)
Figure 3. Digitized images of the steady drop shapes for the CW3-1 experiments: (a) K = 0.58, (b) K = 0.73,(c) ( e )K = 1.15,(Q K = 1.32.
K
= 0.92, (d) K = 1.05,
fields in the two phases are governed by the continuity and Stokes equations which, in dimensionlessform, can be written as
v-~=o
vp = v2u in Q
(1)
with u and p representing the velocity and dynamic pressure fields in the suspending fluid, respectively, and all quantities with primes referring to those associated with the drop phase. In a reference frame attached to the center of mass of the drop and moving with its terminal velocity U , the boundary conditions are given by
u = -Ue,
for
x E St + Si+ So
(3)
lAu(x)
for
where the surface force f is defined as f = IIm, x is a fured point, y is a variable point on the domain of integration, and +
where e, is the unit vector in the axial direction, II and II’represent the Newtonian stress tensors based on the dynamic pressure in Q and Q’, respectively, n is the unit normal vector on S d directed into Q, and V, = (I n n ) V is the surface gradient operator. The Bond number, Bo, appearing in eq 6 is a measure of the importance of the buoyancy force relative to the interfacial tension force. In eq 3, Si and So represent inflow and outflow planes, respectively, located far away from the drop, and St denotes the section of the tube wall bounded by those planes (see Figure 1). It should be noted that conditions 4-6 on the surface of the drop do not overspecifythe problem since the shape of the drop is unknown a priori and must be determined as part of the solution. Using the classical procedures for the boundary integral formulation of Stokes flow problems described elsewhere (e.g., Rallison and Acrivos, 1978, Tanzosh et al., 1992, and Pozrikidis, 1992a),the disturbance velocity field created by the motion of the drop can be expressed as
x E Q’
(y-x)(y-x) IY - Xl3
]
(8)
Hence, the velocity a t any point can be obtained from a knowledge of the surface velocity and force distributions on S d and St, respectively, by evaluating the requisite surface integrals. The unknown surface velocity and force distributions themselves can be obtained from a solution of vector integral eqs 7a and 7c. For the misymmetric drop shapes considered in this study, the surface integrals in eq 7 are reduced to line integrals by performing the azimuthal integrations analytically. Starting with a spherical drop, or a steady drop shape obtained in a previous computation, integral eqs 7a and 7c are then solved using a standard boundary elementcollocation method. Details of the numerical procedure are described by Borhan and Mao (1992). Once the surface velocities on S d are determined, the kinematic condition (eq 5 ) is integrated to update the drop shape, and the calculations are repeated until a steady drop shape is obtained.
Results and Discussion In this section, we present the experimental results in terms of the effects of viscosity ratio, Bond number, and drop size on the steady shape of the drop and its
2754 Ind. Eng. Chem. Res., Vol. 34,No. 8, 1995
(a)
(b)
(c)
(d)
(e)
(f)
Figure 4. Digitized images of the steady drop shapes for the GW5-1 experiments: (a) K = 0.58, (b) K = 0.73, ( c ) K = 0.92, (d) K = 1.05, (e) K = 1.15, (f) K = 1.32.
terminal velocity. 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, with each experiment being labeled by a symbol identifying the two-phase system followed by a number specifying the size of the capillary tube. The Reynolds numbers reported in this table are based on the actual drop speeds rather than the characteristic velocity ub,that is Re = @RIP. The overall range of Reynolds numbers associated with the experiments is between 1.8 x and 1.7. Aside from the WG4 and WG5 experiments, and the motion of small drops in di(ethy1ene glycol) (DEG) systems, the Reynolds numbers for the motion of the dispersed phase remained in the creeping flow regime. The range of measured terminal 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 = pula. In all experiments listed in Table 2, the shapes of the drops and bubbles remained axisymmetric as they passed steadily through the capillary tube, consistent with the assumptions of our theoretical analysis. Digitized images of typical steady drop shapes observed experimentally are shown in Figures 3-8 for various drop sizes, while the corresponding profiles for air bubbles (A = 0) are shown in Figure 2. As the size of an air bubble increases, it deforms from a spherical shape into an elongated ellipsoid, more or less maintaining its fore and aft symmetry. A viscous drop, on the other hand, loses its fore and aft symmetry with the trailing end flattening while the curvature of the leading end increases. This difference in shape evolution is borne out by a comparison of the shapes of bubbles and drops of the same size a t nearly the same Bond number shown in Figures 2 and 3, respectively. The drop profiles shown in Figures 3-5 demonstrate the sensitivity of the shape of viscous drops to the value of the Bond number, keeping the other parameters nearly constant. It is evident that increasing the Bond number leads to elongation of the drop in the axial direction, with the increase in the length of the drop becoming more pronounced as the drop size increases. At the same time, the curvature of the trailing end of the drop seems to increase as the Bond number becomes larger. Thus, increasing the Bond number tends to restore the fore and aft symmetry of the drop shape, particularly for large drops. The same effect of the Bond number is evident in Figure 9 for two-phase systems with vanishing viscosity ratios but vastly different density ratios, similar to the systems considered by Coutanceau and Texier (1986). However, whereas these authors concluded that a decrease in the density ratio
(4
(b)
(c)
(d)
(e)
(f)
Figure 5. Digitized images of the steady drop shapes for the GW4-1 experiments: (a) K = 0.58, (b) K = 0.73, (c) K = 0.92, (d) K = 1.05, (e) K = 1.15, (f) K = 1.32.
Figure 6. Digitized images of the steady drop shapes for the WG2-1 experiments: (a) K = 0.58, (b) K = 0.73, ( c ) K = 0.92, (d) K = 1.15, (e) K = 1.32.
Figure 7. Digitized images of the steady drop shapes for the DEG7-1 experiments: (a) K = 0.58, (b) K = 0.73, (c) K = 0.92, (d) K = 1.05, (e) K = 1.15, (f) K = 1.32.
y would lead to more elongated drop shapes, the drop profiles in this figure clearly show the opposite trend. It seems that Coutanceau and Texier misinterpreted their experimental results by neglecting interfacial tension effects altogether, thereby attributing the observed increase in the length of their drops to a decrease in the density ratio rather than to an increase in the Bond number. Their low-y systems were most probably characterized by larger Bond numbers than their large-y systems, in contrast to the situation represented by the systems in Figure 9. Figures 6-8 qualitatively demonstrate the effect of the viscosity ratio on the drop shape. As the drop phase becomes more viscous relative to the suspending fluid, a more elongated shape seems to develop without significantly affecting the qualitative features of the leading and trailing ends of the drop. For large drops, this should result in the formation of thicker liquid films between the drop and the capillary wall as the viscosity
Ind. Eng. Chem. Res., Vol. 34,No. 8, 1995 2755
a
3.0
4
ow5.l CW3.1
A
2.0
0
CW4-2
0
Gwz-I 8
A
8
Q
o
8
80
m A
.
1.o
(b)
(a)
(e)
(d)
(e)
(f)
Figure 8. Digitized images of the steady drop shapes for the DEG8-1 experiments: (a) K = 0.58, (b) K = 0.73, (c)K = 0.92, (d) K = 1.05, (e) K = 1.15,(0 K = 1.32.
'
0.0
1
0.0
0.6
1.2
1.8
2.4
1.8
2.4
K
J (a)
(b)
10
0
(C)
Figure 9. Comparison of the steady shapes for a drop of size K = 0.92 in systems with vanishing viscosity ratios: (a) CW4-1,(b) CW5-1, (c) GW6-1.
8
0
o A
.:A A8
5 1
ratio is increased. However, this trend is not immediately apparent from a qualitative comparison of the shapes of large drops in Figures 6-8. In order 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 to that of the equivalent spherical drop. This quantity is easily obtained from digitized images of drops and bubbles using the OPTIMAS image analysis software. Figures 10-14 demonstrate the dependence of the deformation parameter on the dimensionless drop size. Also shown in these figures are LAand LR,respectively representing the maximum axial and radial dimensions of the steady drop profiles relative to the capillary radius. The dashed lines in these figures represent the values of the geometric parameters for spherical drop shapes (for K < l), while the dotted curves show the corresponding values for large ( K > 1)stagnant drops consisting of two hemispherical caps of radius R connected by a cylindrical midsection. These curves provide lower bounds for the values of D and LAwhich are helpful in identifjlng the contribution of drop motion to deformations of its shape. As expected, increasing the drop size leads to larger values of the deformation parameter and drop length as the shape of the drop changes from spherical to ellipsoidal, eventually evolving into a nearly cylindrical form with ellipsoidal caps at the leading and trailing ends. Initially, both the radial and axial dimensions of the drop grow almost linearly as a hnction of drop size, as shown by the dashed lines in Figures 10-14. However, once the first shape transition is approached (roughly at K = 0.7), the growth rate in the radial direction diminishes and LR eventually tends to a constant value corresponding to
A
L
0.6
0.0
1.2
K
c
2.0
; ;
b 1 0
, I
1
'
O
0
0
0
0
I
1.5
1.0
I
I
I
,, I
0.5
,
I
I
0.0 0.0
0.6
1.2
1.8
2.4
K
Figure 10. Variations of the geometric parameters with drop size for systems with I = 0.2; (a) deformation parameter D,(b) drop length LA, (c) maximum equatorial dimension LR.
the radius of the resulting cylindrical drop shape. The transition to a cylindrical shape occurs typically for K > 1.2, which approximately coincides with the point at which the length of the drop exceeds 3 times the radius
2756 Ind. Eng. Chem. Res., Vol. 34, No. 8, 1995
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2.0
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0
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6
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c
c
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I
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3
'.
oi
1.0
8
1.5
3
1.0
('0
0.5
0.0 0.0 1(
0.4
0.8
I .2
1.6
K
Figure 11. Variations of the geometric parameters with drop size for systems with izz 0.0: (a) deformation parameter D,(b) drop length LA,(c) maximum equatorial dimension LR.
Figure 12. Variations of the geometric parameters with drop size
of the capillary tube. Further evolution of the drop shape beyond this point is characterized by a faster rate of elongation of the drop (with increasing drop size) than that observed in the initial phase of deformation.
The experimental data presented in Figures 10-12 also confirm our earlier qualitative observations of the larger drop deformations associated with increases in the Bond number, particularly for large drops. Once
for systems with ip. 1.2; (a) deformation parameter D,(b) drop length LA,(c) maximum equatorial dimension LR.
Ind. Eng. Chem. Res., Vol. 34, No. 8, 1995 2757
a
a
2.0
1
2.5
I
'A
e
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CI
CI
1.0
0.5 0.5 -
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.
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.
b
0
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e
e
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=
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.
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0.0 0.0
0.4
0.8
1.2
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0.0
0.6
1.2
1.8
2.4
K
Figure 13. Variations of the geometric parameters with drop size for systems with Bo =i 1.7-2: (a) deformation parameter D, (b) drop length LA, (c) maximum equatorial dimension LR.
Figure 14. Variations of the geometric parameters with drop size for systems with Bo = 1.0-1.6: (a) deformation parameter D, (b) drop length LA, (c) maximum equatorial dimension LR.
the transition to a cylindrical shape has occurred, increases in the Bond number lead to faster rates of elongation as a function of drop size, as well as thicker liquid films between the cylindrical drops and the
capillary wall (i.e., smaller limiting values of LR). However, this effect becomes less pronounced as the Bond number increases. This is evident from Figure 15 which shows the measured liquid film thickness, Q,
2758 Ind. Eng. Chem. Res., Vol. 34,No. 8, 1995 0.4
_ 0_ _ _ _ 8_ _ _ - - - - - - - _0.3
0.2
0.1
-
0
a 1.2-1.3
8
k g 0.2 a-0.0
0.0
I
0
4
8
12
16
Bo Figure 15. The thickness of the liquid film surrounding large drops as a function of Bond number: (. *) asymptotic predictions of Bretherton for A = 0; (- - -1 numerical predictions of Reinelt for iz = 0; (- - -) best visual fit of the experimental data for A = 0; (-1 numerical predictions of this study for A = 1.0.
for the largest drops in systems with nearly the same viscosity ratio. The dotted and dot-dashed curves in this figure respectively represent the asymptotic results of Bretherton (1961) and the numerical predictions of Reinelt (1987) for the case of a gas bubble (A = 0), while the solid curve shows our numerical predictions for A = 1.0. The film thickness initially grows rapidly with increasing Bond number but then approaches a limiting value as the Bond number becomes large, in line with the theoretical predictions of Reinelt (1987) and Pozrikidis (1992). It is interesting to note, however, that drop breakup through "tipstreaming" was eventually observed for large drops ( K > 1.6) in the CW4 and CW5 systems which were characterized by large Bond numbers and vanishing viscosity ratios. For these systems, a small unstable tail that formed a t the trailing end of large drops (see Figure 16) disintegrated into a stream of small satellite drops as the drop passed through the midsection of the capillary tube. The onset of tipstreaming in low viscosity ratio systems is believed to be due to the presence of surface-active impurities (cf. de Bruijn, 1993; Stone, 1994). Although the experimental systems used in this study were carefully cleaned to avoid contamination, the presence of minor components in corn syrup may have been responsible for the observed tipstreaming phenomenon. Once the surface contamination was sufficiently reduced through the formation of satellite drops, the unstable tail eventually disappeared and the trailing end of the drop achieved a steady shape near the top of the capillary tube, as shown in the sequence of images in Figure 16. Based on the experimental data in Figure 15, the film thickness seems to be rather insensitive to the value of the viscosity ratio, at least within the experimental error involved in the measurements. However, a comparison of the geometric parameters for systems with different values of A, shown in Figures 13 and 14, reveals a weak dependence of the drop shape on viscosity ratio that is qualitatively similar to its dependence on the Bond number. In particular, increasing the viscosity ratio seems to lead to slightly larger values of D and LAbut smaller values of LR. The latter observation suggests
Figure 16. Sequence of images of the unstable tail section of a large drop as it rises through the midsection of the capillary tube: (a-c) CW5-1,(d-f) CW4-1.
a slight increase in the film thickness with increasing viscosity ratio which is also consistent with the numerical predictions shown in Figure 15. In comparing the experimental data for the DEG systems with those for other systems in Figures 13 and 14, one should realize that the Reynolds numbers associated with the DEG experiments are in the range 0.06 < Re < 1.0 (typically 1-3 orders of magnitude larger than the corresponding values for other experiments),where inertial effects may have to be considered. Inertial effects have been found (e.g., Zukoski, 1966) to rapidly reduce the thickness of the liquid film around large bubbles. Hence, it is reasonable to assume that the unusually large values of LRfor the DEG systems in Figures 13 and 14 are due to inertial effects. A similar effect is also present in the geometric measurements for the WG4 and WG5 systems shown in Figure 12. We conclude by considering the effects of Bond number and viscosity ratio on the terminal rise velocity of viscous drops. The dimensionlessterminal velocities shown in Figure 17 demonstrate the two types of dependence on drop size observed in our experiments. For systems with small Bond numbers, a local maximum in the terminal velocity is reached for small drop sizes ( K < 1)as the retarding effect of the capillary wall begins to overwhelm the increase in the buoyancy force with increasing drop size. In contrast, the terminal velocity in large Bond number systems seems to be a monotonically increasing function of drop size. This difference can be attributed to the enhanced deformability of drops at large Bond numbers which, for a given drop size, translates into a lower drag on the drop for the same buoyancy driving force. In both cases, the terminal velocity eventually approaches a limiting value which remains unchanged with further increases in the
Ind. Eng. Chem. Res., Vol. 34,No. 8, 1995 2759
n
2 H
3
0 ' 0.0
1.o
0.5
2.0
1.5
0.20
.a
1.00
0.60
1
1.80
K
K
Figure 17. Dimensionless rise velocity as a function of drop size. All curves represent best fits to the experimental data.
Figure 19. Retarding effect of the capillary wall as a function of drop size for systems with Bo = 1.7-2.0. The solid and dashed curves represent numerical predictions.
5
4
4
2
3
K
? s 2
/
I 1
0
0
0.2
0.6
1 .o
1.4
1.E
2.2
0.00 0.3
0.7
I .5
1.1
1.9
2.3
K
Figure 18. Retarding effect of the capillary wall as a function of drop size for systems with Bo = 2.5. The solid and dashed curves represent the best visual fits to the experimental data.
drop size. In order to examine the retarding effect of the capillary wall, and its dependence on Bond number and viscosity ratio, the drop speeds in Figures 18-23 are scaled by U,, the dimensionless rise velocity of an equivalent spherical drop of the same fluid in an infinite reservoir of the suspending fluid. Theoretical predictions of the scaled drop speed based on the analysis in the previous section are also shown for selected twophase systems represented by the solid symbols in Figures 19-22. As expected, the scaled velocities are monotonically decreasing functions of drop size due t o the drag exerted on the drops by the capillary wall. In general, the agreement between the theoretical predictions and the experimental measurements seems t o be satisfactory for the entire range of the viscosity ratios considered. A comparison of Figures 18-20 reveals the retarding effect of increasing viscosity ratio (particularly in the range 0 c A c 1)which causes the surface of the drop
K
Figure 20. Scaled drop speed as a function of drop size for systems withBo = 1.0-1.6. The solid and dashed curves represent numerical predictions.
to become more rigid. The velocity measurements in Figures 18 and 19 are presented in terms of U J U to provide a better resolution of the experimental data since the values of UIU, for the experiments represented by those figures were quite small. Note that the variations of the scaled terminal velocity with viscosity ratio in Figures 18-20 are entirely due to interactions of the drop with the capillary wall since the effect of the viscosity ratio on the mobility of the drop in an infinite reservoir has already been removed through the choice of the velocity scale. For a fured value of the viscosity ratio, the retarding effect of the capillary wall is weaker at larger Bond numbers, particularly for large drops (see Figures 21-23). This is due to the enhanced deformation of large drops with increasing Bond numbers which tends to localize the drop near the axis of the capillary and thereby increase the distance between the drop surface and the capillary wall (increased film
2760 Ind. Eng. Chem. Res., Vol. 34,No. 8, 1995 0.40
0.30
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0.30 0.20
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Figure 21. Scaled drop speed as a function of drop size for systems with i G 0.2. The solid and dashed curves represent numerical predictions.
Figure 23. Scaled drop speed as a function of drop size for systems with A = 0.0. The solid and dashed curves represent the best visual fits to the experimental data for the CW4-1 and CW6-1 systems, respectively.
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Figure 22. Scaled drop speed as a function of drop size for systems with A G 1.2. The solid and dashed curves represent numerical predictions.
thickness). However, for a given viscosity ratio, there is a limiting value of the Bond number beyond which the retarding effect of the wall remains unchanged with further increases in the Bond number, as shown in Figure 23. This effect is more clearly demonstrated in Figure 24 which shows the measured drop speed for the largest drops in systems with nearly the same viscosity ratio. As in Figure 15, the dotted and dot-dashed curves in this figure correspond to the asymptotic results of Bretherton (1961) and the numerical predictions of Reinelt (19871, respectively, for the case of a gas bubble (1= 01, while the solid curve represents our numerical predictions for 1 = 1.0. The dimensionless drop speed initially grows rapidly with increasing Bond number but then approaches a nearly constant value for large Bond numbers. This limiting value of the rise speed is found to be about 0.03for , I= 0, in excellent agreement with the experimental results of Goldsmith and Mason
Bo Figure 24. Dimensionless terminal velocity of large drops as a function of Bond number: (. asymptotic predictions of Bretherton for 1= 0; (- - -) numerical predictions of Reinelt for A = 0; (- - -) best visual fit of the experimental data for 1 = 0; (-1 numerical predictions of this study for i = 1.0. e)
(19621, and seems to decrease as the viscosity ratio becomes larger. Also, from the experimental data in Figure 24, it seems that the limiting value of the rise speed is achieved for Bond numbers greater than about 12. Finally, for all systems with Bo < 0.87, the motion of the drop was eventually halted as the drop size was increased, regardless of the value of the viscosity ratio. This critical value of the Bond number is consistent with the value of 0.84reported by Bretherton (1961) for 1 = 0 and seems to be nearly the same for gas bubbles and viscous drops as suggested by Pozrikidis (199213).
Acknowledgment is made to the donors of the Petroleum Research Fund, administered by the American Chemical Society, for partial support of this re-
Ind. Eng.Chem. Res., Vol. 34, No. 8, 1995 2761 search. This work was also supported by the National Science Foundation under Grant CTS-9110470,
Literature Cited Barr, G. The Air-Bubble Viscometer. Philos. Mag. 1926,1, 395. 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. Clift, R.; Grace, J. R.; Weber, M. E. Bubbles, Drops, and Particles; Academic Press: New York, 1978;Chapter 9. Countanceau, M.; Thizon, P. Wall Effects on the Bubble Behavior in Highly Viscous Liquids. J . Fluid Mech. 1981,107,339. Countanceau, M.; Texier, A. Experimental Investigation of the Creeping Motion of a Drop in a Vertical Tube. Exp. Fluzds 1986, 4,241. de Bruijn, R. A. Tipstreaming of Drops in Simple Shear Flows. Chem. Eng. Sci. 1993,48,277. Goldsmith, H. L.; Mason, S. G. The Movement of Single Large Bubbles in Closed Vertical Tubes. J . Fluid Mech. 1962,14,42. 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. Haberman. W. L.: Savre. R. M. Motion of Rieid and Fluid &heres in Stationary and" Moving Liquids insidYe Cylindrical tubes. 1958,David Taylor Model Basin Rep. 1143. Hyman, W. A.; Skalak, R. Viscous Flow of a Suspension of Liquid Drops in a Cylindrical Tube. Appl. Sci. Res. 1972,26,27. 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. Park, C. W. Influence of Soluble Surfactants on the Motion of Finite Bubbles in a Capillary Tube. Phys. Fluids 1992,4,2335.
Pozrikidis, C. Boundary Integral and Singularity Methods for Linearized Viscous Flow; Cambridge University Press: Cambridge, 1992a. Pozrikidis, C. The Buoyancy-Driven Motion of a Train of Viscous Drops within a Cylindrical Tube. J . Fluid Mech. 199213,237, 627. Rallison, J. M.; Acrivos, A. A Numerical Study of the Deformation and Burst of a Viscous Drop in an Extensional Flow. J . Fluid Mech. 1978,89,191. Reinelt, D. A. The Rate at which a Long Bubble Rises in a Vertical Tube. J . Fluid Mech. 1987,175,557. Stone, H. A. Dynamics of Drop Deformation and Breakup in Viscous Fluids. Annu. Rev. Fluid Mech. 1994,26,65. Tanzosh, J.; Manga, M.; Stone, H. A. Boundary Integral Methods for Viscous Free-Boundary Problems: Deformation of Single and Multiple Fluid-Fluid Interfaces. In Boundary Element Technologies; Brebbia, C. A., Ingber, M. S., Eds.; Computational Mechanics Publications and Elsevier Applied Science: Southampton, 1992;pp 19-39. White, E. T.; Beardmore, R. H. The Velocity of Rise of Single Cylindrical Air Bubbles through Liquids Contained in Vertical Tubes. Chem. Eng. Sci. 1962,17,351. Zukoski, E. E. Influence of Viscosity, Surface Tension, and Inclination Angle on Motion of Long Bubbles in Closed Tubes. J . Fluid Mech. 1966,25,821.
Received for review November 15, 1994 Revised manuscript received March 30, 1995 Accepted April 20, 1995@
IE940670B Abstract published in Advance ACS Abstracts, July 1, 1995. @
