Dynamics of drops in cavity flows: Aggregation of high
viscosity ratio drops
Michael Mangaa)
Department of Geology and Geophysics, University of California, Berkeley, California 94720-4767
~Received 17 January 1996; accepted 26 March 1996!
The interactions of deformable drops in cavity flows is studied numerically in the limit of low
Reynolds numbers for a two-dimensional model. Flow in a square cavity is driven by the steady
motion of one of the walls. Deformable drops will migrate across streamlines until they reach an
equilibrium trajectory or equilibrium position; the rate and direction of migration depend on both the
viscosity ratio and capillary number. High viscosity ratio deformable drops have a tendency to
aggregate and form clusters. The presence of a deformable dispersed phase results in an elastic
behavior of the suspension. © 1996 American Institute of Physics. @S1070-6631~96!02507-X#
I. INTRODUCTION
The motion of a dispersed phase in cellular, cavity, or
other simple flows is often studied as a model of mixing
processes.1 The goal of such studies is to determine the relationship between characteristics of the flow, the rate of
mixing, and the evolution and distribution of the dispersed
phase. The limit in which the dispersed phase exists as an
immiscible fluid has motivated a large number of studies
focusing on the deformation of single drops,2,3 breakup,4 and
migration across streamlines due to deformation.5
Studies of single deformable particles in very dilute systems can describe certain aspects of the microstructure, e.g.,
shape and orientation. In non-dilute systems, interactions between the particles will affect the microstructure, which in
turn affects macroscopic rheological properties of the fluid.
Understanding and characterizing the effects of particleparticle interactions thus motivates the study of the interactions between two particles in simple flows, e.g., Ref. 6, and
the numerical simulation of suspensions in shear and
pressure-driven flows, e.g., Refs. 7–9. One approach to
studying complex, time-dependent flows is to introduce
models for particle deformation and orientation,10 and then
calculate the interaction and evolution of the microstructure
and the flow.11,12
Here we study numerically the evolution of a deformable
dispersed phase in a cavity flow driven by the steady motion
of one the boundaries. Compared with other frequently studied flows, such as shear flows and pressure-driven flow
through pipes and channels,8,13,9,14 cavity flows contain stagnation points, spatially variable shear rates, and curved
streamlines. We begin, in Sec. III, by considering certain
aspects of the microstructure including drop deformation,
distribution, and migration.
In Sec. IV we focus on the interactions between deformable drops and demonstrate that for sufficiently large viscosity ratios and capillary numbers drops tend to aggregate. The
segregation of immiscible fluids is of considerable interest
because of possible engineering application to separation
processes. For example, at moderate to high Reynolds numa!
Now at Department of Geological Sciences, University of Oregon, Eugene,
Oregon 97403.
1732
Phys. Fluids 8 (7), July 1996
bers, mixtures of two fluids with differing viscosities are
observed to separate when forced through tubes, with the
less viscous fluid surrounding the more viscous fluid, leading
to ‘‘lubricated pipelining.’’15 Flow-induced segregation due
to viscosity contrasts may also occur in nature, such as in
certain magmatic systems.16 Separation or aggregation occurs in other situations; for example, rigid particles are
sometimes observed to form fractal aggregates in unsteady
cavity flows if two particles adhere once they touch each
other.17
It is well known that suspensions of deformable drops
exhibit a non-Newtonian behavior.18 In Sec. V we show that
drop deformation also produces an elastic response in cavity
flows.
II. MODEL
The geometry of the model problem for a single drop is
shown in Fig. 1a. Two-dimensional drops are studied since
comparable numerical computations for three-dimensional
drops in bounded systems are not feasible. Thus the results
presented here may differ from three-dimensional systems,
although a comparison of two-dimensional calculations13 and
three-dimensional analytical results5 indicates that the qualitative behavior of two- and three-dimensional drops is similar. The two-dimensional approximation prevents us from
studying drop breakup due to a capillary instability, although
drops may still become highly elongated in a local extensional flow.
Inside and outside the drop, fluid motions satisfy Stokes
equations
¹•T1 r g5 m ¹ 2 u2¹p50
and
¹•u50,
~1!
where u is velocity, m is viscosity, p is the dynamic pressure, and T is the stress tensor. Across the surface of the
drop, there is a stress jump due to interfacial tension,
v n•Tb 5 s n¹•n,
~2!
where s is the interfacial tension which is assumed constant,
n is a unit normal directed outward from the drop, and we
assume the drop and suspending fluid have the same density.
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FIG. 2. Center-of-mass trajectories for a single drop. Viscosity ratio and
capillary number are labeled on the figure. A few drop shapes are shown for
reference.
FIG. 1. ~a! Geometry of the model problem. ~b! Calculated streamlines and
flow field in the absence of a drop.
The relative importance of viscous stresses, proportional to
m U/L, compared with interfacial tension stresses is characterized by the capillary number,
Ca5
m Ua
.
sL
~3!
Equations ~1! can be recast in the form of integral equations. The appropriate form of the integral equations for N
neutrally buoyant drops with viscosity l m is
N
2 ~ 12l !
2
(
j51
E
Sj
N
n–K–u j dS2
E
n•TG •JdS2
5
5
G
E
G
(
j51
E
Sj
v n•Tb j •JdS
n–K–uG dS
u~ x! ,
xPsuspending fluid,
lui ~ x! ,
xPdrop i,
11l
ui ~ x! ,
2
xPS i ,
1
u~ x! ,
2
xPG,
III. MIGRATION OF DROPS
~4!
where the subscript i indicates drop number i, x is a position
vector, and J and K are the Green’s functions for velocity
and stress, respectively.19 On the upper surface uG 5(U,0),
and uG 50 on the other three sides of the cavity. The position
Phys. Fluids, Vol. 8, No. 7, July 1996
of fluid-fluid interfaces is determined by solving the kinematic equation, dx/dt5ui for x P S i , using a second-order
Runge-Kutta method.
The steady base flow in the absence of a dispersed phase
is shown in Fig. 1b. Streamlines are calculated using a
second-order Runge-Kutta method and a time step twice as
large as that used in calculations including drops.
In the calculations presented below, drops and cavity
walls are described by 60 and 240 collocation points, respectively. The drop shape is described by cubic splines parameterized in terms of arc length. The unknown velocities are
assumed to vary linearly, and the tractions along the wall are
assumed to be constant, between collocation points. The singularities in the kernels are subtracted and integrated
analytically.20 Integrations are performed using eight point
Gauss-Legendre quadrature. In all cases, we assume
a/L50.05. The calculations presented in Figs. 2a, 2c, 4 and
5 took 16, 51, 333 and 44 CPU hours on a Sparc server 1000,
respectively. Time is normalized by the convective timescale
L/U, and a typical turnover time is 3-6 dimensionless time
units.
Analytical results,5 experimental observations,21 and numerical calculations,14 indicate that deformable drops in
shear flows will migrate away from walls for all viscosity
ratios. However, in a Poiseuille flow, Chan and Leal found
that the direction of migration depends on the viscosity ratio:
for 0.5,l,10 drops migrate towards the wall, and otherwise towards the centerline of the flow. Laboratory
experiments22 and two-dimensional numerical calculations13
indicate that for moderate viscosity ratios an equilibrium position will be reached at some distance between the wall and
flow centerline. The specific values of the viscosity ratio
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FIG. 3. ~a! Equilibrium trajectory, characterized by y c /L, as a function of
the viscosity ratio l; Ca50.5. ~b! Definition of y c and an example of a
trajectory for l50.2 and Ca50.5 which reaches an equilibrium trajectory.
dictating these different behaviors differs between the twodimensional calculations and three-dimensional analytical results.
Here we study the effect of the viscosity ratio and deformation on the migration of a single drop. The term migration
describes motion across streamlines which arises due to deformation. In Fig. 2 we show calculated trajectories of the
center-of-mass of deformable drops for various viscosity ratios and capillary numbers. In all cases in Fig. 2 the drops
migrate towards to center of the flow. Comparing the trajectories in Figs. 2abc, we observe that the rate of migration
increases as the viscosity ratio decreases. In particular, the
rate of migration for the l510 drop becomes very small,
and the drop appears to reach an ‘‘equilibrium trajectory.’’
However, we will see in Fig. 3 that only the drop with
l50.2 moves to an equilibrium trajectory whereas the other
drops migrate to the center of the flow ~which is defined as
the point of zero velocity; see Fig. 1b!. Comparing the trajectories in Figs. 2cd, we observe that the less deformable,
Ca50.25, drop migrates towards the center of the flow more
rapidly than the more deformable, Ca50.5, drop. Owing to
the reversibility of Stokes flows, there will be no migration
across streamlines in the limit that Ca→0; thus, the maximum rate of migration occurs for a finite value of the capillary number.
In order to characterize the equilibrium trajectory, we
measure the vertical distance y c between the upper surface
and the equilibrium trajectory when the center of mass of the
drop is at horizontal position x/L50.5: here, x and y are the
horizontal and vertical positions defined with (x,y)5(0,0) in
the upper left-hand corner of the box and (L,L) in the lower
right-hand corner. The equilibrium position, y c /L, as a function of the viscosity ratio is shown in Fig. 3a for Ca50.5.
For l.1 the equilibrium position is the center of the flow.
The shape of high viscosity ratio drops at the center of the
flow is not steady, and the drops undergo an evolution analogous to the tank-treading motion of high viscosity ratio drops
in shear flows.
These results are qualitatively consistent with previous
analytical, experimental, and numerical results, although the
flow studied here is more complicated than the linear and
parabolic flows studied in the work discussed above. For
example, Zhou and Pozrikidis13 found that low viscosity ratio drops will migrate to the centerline in a pressure-driven
flow in a channel; here we find that low viscosity ratio drops
migrate to trajectories where the curvature of the velocity
field is small ~see Fig. 1b!. Thus, in a suspension, high viscosity ratio drops are expected to migrate towards the center
of the flow, whereas low viscosity ratio drops will tend to
some equilibrium off-center position, in both cases leading to
the eventual segregation of a dispersed phase. These results
apply to the limit of dilute suspensions so that drop-drop
interactions are not important; next we consider the behavior
of more concentrated suspensions.
In Fig. 4, we show the results of a calculation with 7
drops for l51 and Ca50.25. We characterize the distribution of drops by calculating the average distance, D 2 , between the center-of-mass of each drop, xcm
i , and the center
of the flow, xo :
N
D 25
1
~ xcm 2xo ! • ~ xcm
i 2xo ! .
N i51 i
(
~5!
As the drops migrate towards the center of the flow, D 2
decreases, and an equilibrium distribution is reached after
t'80.
IV. AGGREGATION OF HIGH VISCOSITY RATIO
DROPS
FIG. 4. Evolution of the distribution of drops in a calculation with 7 drops;
l51, Ca50.25. The initial distribution of drops is the same as in Fig. 4.
D is the average distance between the center-of-mass of each drop and the
center of the flow, and is defined by ~5!. Time is normalized by L/U.
1734
Phys. Fluids, Vol. 8, No. 7, July 1996
In Fig. 5 we show two sequences of drop shapes at different times for calculations with seven drops. The value of
the Capillary number was chosen so that the magnitude of
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can be provided. Consider the problem of two drops approaching each other in a shear flow, as illustrated in Fig. 6.
The l51 drops pass over and around each other. However,
the high viscosity ratio drops form a doublet and rotate
around each other before eventually separating. For viscosity
ratios l.4, Taylor23 recognized that drops will rotate more
rapidly than they are stretched ~simple shear consists of
equal parts strain-rate, which leads to stretching, and vorticity, which leads to rotation!; for sufficiently viscous drops,
the rate of deformation is slowed so that the drop rotates
before significant deformation occurs. Once the separation
distance between the high viscosity ratio drops becomes
small and the flattened area between the drops becomes
large, the drops behave essentially as a single drop and thus
rotate as a single larger drop. The gap between the drops
must be sufficiently small that large lubrication forces prevent the drops from reseparating as they rotate around each
other. In Fig. 6c we show two high viscosity ratio drops
approaching each other with l55 and Ca51. The finite
Capillary number results in smaller deformation, and the
drops pass over each other. Thus, in flows with velocity gradients, deformable high viscosity ratio drops which approach
each other may tend to aggregate, as observed in Figs. 5 and
6. The time required for drops to pass over and around each
other increases as the viscosity ratio increases ~compare the
times in Fig. 6! so that high viscosity ratio drops will also
spend more time close to each other.
The aggregation process observed in Fig. 5, however, is
not entirely due to the processes illustrated in Fig. 6, but also
depends on the concentration of the drops. In Fig. 7 we show
a calculation with only two drops, drops 1 and 2 from Fig. 5.
In the calculations with seven drops, drops 1 and 2 stay in
close proximity to each during the whole calculation with
drops l510. However, in the two drop calculation drops 1
and 2 separate. Thus, the concentration of drops plays a role
in the aggregation process, presumably by limiting the freedom of drops to move around each other. Higher concentrations constrain drops to interact more closely, thus allowing
the processes illustrated in Fig. 6 to occur.
In order to consider the evolution of the drop distribution
in a more quantitative manner, we calculated the evolution of
the average minimum separation distance between drops,
i.e.,
FIG. 5. Interactions between seven drops for ~a! l51, Ca50.5 and ~b!
l510, Ca52. Time is normalized by the convective timescale L/U.
drop deformation in both calculations was similar. In the
calculation with l51 and Ca50.5 the drops remain well
separated and show no tendency to aggregate, even over timescales much longer than shown in Fig. 4a. However, for
l510 and Ca52, the drops tend to form doublets and remain in close proximity to each other for long periods of
time. For example, compare the relative motions of drops 1,
2 and 7 in the two simulations, and in particular, notice that
drops 1 and 2 remain close to each other throughout the
entire simulation in Fig. 5b.
A straightforward explanation for the different dynamics
Phys. Fluids, Vol. 8, No. 7, July 1996
N
d̄5
1
cm
minimumu xcm
i 2x j u ,
N i51
(
j51→N,
iÞ j,
~6!
is the position of the center-of-mass of drop i.
where xcm
i
The interaction of deformable drops results in irreversible
trajectories and hydrodynamic self-diffusion;9 for example,
the vertical offset between two drops in Fig. 7 increases as
the drops pass over and around each other. As a result, d̄
increases at early times.
In the previous section, we found that a single drop with
l51 migrates towards the center of the cavity flow. At long
times, d̄ decreases for the l51 simulation as the drops migrate towards the center of the cavity flow (d̄ continues to
decrease until t'80). The high viscosity ratio drops, howMichael Manga
1735
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FIG. 6. Interaction of two deformable drops in a shear flow: ~a! l51, Ca50.25, ~b! l55, Ca5`, and ~c! l55, Ca51. Time is normalized by the shear
rate G.
ever, migrate only very slowly across streamlines ~see Fig.
2!, and the decrease of d̄ in Fig. 8 is due to the aggregation.
Due to numerical limitations we are only able to consider a small number of drops, and only observed a pair-wise
aggregation. It has not been verified that the processes described here, which results in the tendency of high viscosity
ratio and deformable drops to form doublets ~e.g., Figs. 5
and 6!, will lead to the formation of clusters containing many
drops.
The aggregation process depends on the close approach
of drops and thus should also occur in three-dimensional
flows.
V. MACROSCOPIC PROPERTIES OF CAVITY FLOWS
In the previous sections, we considered the evolution of
the microstructure, i.e., the deformation, interaction, and migration of drops. Here we consider a macroscopic property of
the flow, and calculate the force on the moving upper boundary due to the presence of a dispersed phase. The force on
the upper surface diverges since stresses decay as 1/r away
from the corner so that the force diverges logarithmically.24
In a real problem, the force will not be infinite but will depend on details of the flow in the corner. In order to avoid
numerical problems associated with the corners, we only integrate the normal deviatoric stresses on the upper surface,
i.e.,
N5
1
mU
E
W
~ n•T1 pn! y ds,
~7!
where W is the upper moving surface, the subscript y denotes components in the vertical direction, and N is normal-
FIG. 7. Evolution of two drops, corresponding to drops number 1 and 2 in
Fig. 5, for l510 and Ca52. Time is normalized by L/U.
1736
Phys. Fluids, Vol. 8, No. 7, July 1996
FIG. 8. Evolution of the average minimum separation distance between
drops for the two calculations shown in Fig. 5; l51, Ca50.5 ~solid curve!
and l510, Ca52 ~dashed curve!. Time is normalized by L/U.
Michael Manga
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O’Connell provided computer time. This work benefitted
from comments and suggestions by C. Pozrikidis and H. A.
Stone.
1
J. M. Ottino, The Kinematics of Mixing: Stretching, Chaos and Transport
~Cambridge University Press, Cambridge, 1989!.
A. Acrivos, ‘‘The breakup of small drops and bubbles in shear flows,’’ 4th
International Conference on Physicochemical Hydrodynamics, Ann. N.Y.
Acad. Sci. 404, 1 ~1983!.
3
J. M. Rallison, ‘‘The deformation of small viscous drops and bubbles in
shear flows,’’ Annu. Rev. Fluid Mech. 16, 45 ~1984!.
4
H. A. Stone, ‘‘Dynamics of drop breakup in viscous fluids,’’ Annu. Rev.
Fluid Mech. 26, 65 ~1994!.
5
P. H. Chan and L. G. Leal, ‘‘The motion of a deformable drop in a
second-order fluid,’’ J. Fluid Mech. 92, 131 ~1979!.
6
A. Z. Zinchenko and R. H. Davis, ‘‘Collision rates of spherical drops or
particles in a shear flow at arbitrary Peclet numbers,’’ Phys. Fluids 7, 2310
~1995!.
7
P. R. Nott and J. F. Brady, ‘‘Pressure-driven flow of suspensions: Simulation and theory,’’ J. Fluid Mech 275, 157 ~1994!.
8
X. Li, H. Zhou, and C. Pozrikidis, ‘‘A numerical study of the shearing
motion of emulsions and foams,’’ J. Fluid Mech. 286, 379 ~1995!.
9
M. Loewenberg and E. J. Hinch, ‘‘A numerical simulation of a concentrated emulsion in shear flow,’’ J. Fluid Mech. ~in press!.
10
W. L. Olbricht, J. M. Rallison, and L. G. Leal, ‘‘Stron flow criteria based
on microstructure deformation,’’ J. Non-Newt. Fluid Mech. 10, 291
~1982!.
11
A. J. Szeri and L. G. Leal, ‘‘Microstructure suspended in threedimensional flows,’’ J. Fluid Mech. 250, 143 ~1993!.
12
A. J. Szeri, S. Wiggins and L. G. Leal, ‘‘On the dynamics of suspended
microstructure in unsteady, spatially inhomogeneous, two-dimensional
fluid flows,’’ J. Fluid Mech. 228, 207 ~1991!.
13
H. Zhou and C. Pozrikidis, ‘‘Pressure-driven flow of suspensions of liquid
drops,’’ Phys. Fluids. 6, 80 ~1994!.
14
M. R. Kennedy, C. Pozrikidis, and R. Skalak, ‘‘Motion and deformation of
liquid drops, and the rheology of dilute emulsions in simple shear flow,’’
Comput. Fluids 23, 251 ~1994!.
15
D. D. Joseph, Fundamentals of Two-Fluid Dynamics ~Springer-Verlag,
Berlin, 1993!.
16
C. R. Carrigan and J. C. Eichelberger, ‘‘Zoning of magmas by viscosity in
volcanic conduits,’’ Nature 343, 248 ~1990!.
17
T. J. Danielson, F. J. Muzzio, and J. M. Ottino, ‘‘Aggregation and structure formation in chaotic and regular flows,’’ Phys. Rev. Lett. 66, 3128
~1991!.
18
W. R. Schowalter, C. E. Chaffey, and H. Brenner, ‘‘Rheological behavior
of a dilute emulsion,’’ J. Colloid Int. Sci. 26, 152 ~1968!.
19
C. Pozrikidis, Boundary Integral and Singularity Methods for Linearized
Viscous Flow ~Cambridge University Press, Cambridge, 1992!.
20
Reference 19, pp. 178–180.
21
J. R. Smart and D. T. Leighton, ‘‘Measurement of the drift of a droplet
due to the presence of a plane,’’ Phys. Fluids A 3, 21 ~1991!.
22
W. L. Olbricht ~personal communication!.
23
G. I. Taylor, ‘‘The formation of emulsions in definable fields of flow,’’
Proc. R. Soc. London Ser. A 146, 501 ~1934!.
24
G. K. Batchelor, An Introduction to Fluid Dynamics ~Cambridge University Press, Cambridge, 1967!, pp. 224–227.
2
FIG. 9. Normal force (N) on the upper surface due to the presence of a
dispersed phase for a calculation with seven drops: l51 and Ca50.25.
Results are shown late in the calculation once a quasi-steady drop distribution is reached. Time is normalized by L/U.
ized by m (U/L)L. N describes the normal stresses on the
upper wall ~N50 in the absence of drops!, and thus is a
measure of the non-Newtonian or elastic behavior of the suspension.
In Fig. 9 we show N as a function of time for a calculation with 7 drops for l51 and Ca50.25. Results are
shown late in the simulation, 90,t,97, once an equilibrium
distribution of drops is reached ~see Fig. 4!. The fluctuations
depend on the positions of the drops. On average N is positive so that the presence of a deformable dispersed phase
results in a force normal to the upper surface and thus an
elastic response of the suspension.
VI. CONCLUDING REMARKS
The numerical results presented here highlight the importance of the viscosity ratio on the evolution of microstructure in suspensions. As noted in previous studies, the large
lubrication forces between high viscosity ratio drops can
change the microstructure at a significant level ~C.
Pozrikidis, personal communication!, and in the problem
studied here, leads to the formation of clusters of drops. Although the processes of breakup and hydrodynamic selfdiffusion tend to disperse drops, the processes of aggregation
and deformation-induced migration can lead to the formation
of structure in suspensions of deformable particles.
ACKNOWLEDGMENTS
Work supported by the Miller Institute for Basic Research in Science and the Petroleum Research Fund. R.J.
Phys. Fluids, Vol. 8, No. 7, July 1996
Michael Manga
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