J.
Fluid
Mech.
(1986),
vol.
167,
pp.
241-283
Printed in
&eat
Britain
24
1
An experimental investigation of drop
deformation and breakup in steady,
two-dimensional linear
flows
By B.
J.
BENTLEYt
AND
L.
G.
LEAL
Chemical Engineering Department, California Institute of Technology,
Pasadena,
CA
91 125, USA
(Received
11
April 1985 and in revised form 21 December 1985)
We consider the deformation and burst of small fluid droplets in steady linear,
two-dimensional motions of
a
second immiscible fluid. Experiments using a computer-
controlled, four-roll mill to investigate the effect of flow type are described, and the
results compared with predictions of several available asymptotic deformation and
burst theories, as well
as
numerical calculations. The comparison clarifies the range
of validity of the theories, and demonstrates that they provide quite adequate
predictions over a wide range
of
viscosity ratio, capillary number, and flow type.
1.
Introduction
In this paper, we discuss the behaviour ofa fluid drop, freely suspended in a second,
immiscible, viscous fluid which
is
undergoing a general linear two-dimensional flow.
The flow-induced stress on the drop surface tends to deform the drop, and the
interfacial tension between the phases resists this deformation. Under some conditions,
the interfacial forces are insufficient to balance the viscous stresses, and the drop
bursts. The problem is of both practical and academic interest, and has thus received
considerable attention in the fluid-mechanics literature over the past fifty years. Our
particular contribution lies in a systematic investigation of the effect of vorticity in
the imposed flow for so-called ‘strong’ flows, where the magnitude of the strain rate
exceeds that of the vorticity.
In most practical applications, the objective is to disperse one fluid phase in
another, either to form an emulsion,
or
to increase the surface area between the two
phases for more efficient heat and/or mass transfer. In these cases, determination of
flow conditions resulting in drop burst is of paramount importance. Examples include
dispersion of anti-static or anti-soiling agents, dispersion of colour concentrates, and
blending of immiscible polymer systems to form two-phase structures of unique
properties (Grace
1971).
Even when the drop does not burst, the distortion produced by a given flow is of
interest in understanding the rheological behaviour of flowing emulsions. Emulsions
are known to exhibit such non-Newtonian characteristicsas shear-dependent viscosity,
viscoelasticity
,
and normal stress differences in rectilinear flow, even when the
concentration of the dispersed phase is small (Frankel
&
Acrivos
1970
;
Barthks-Biesel
&
Acrivos
19733).
From
a
knowledge of the deformation of the drops forming the
dispersed phase and of the disturbance flow in their vicinity, a constitutive equation
can be developed (at least in principle) for the emulsion.
t
Current address: Dynamic Solutions, Inc., 2355 Portola Road, Ventura,
CA
93003,
USA.
242
B.
J.
Bentley and
L.
G.
Leal
From a theoretical point of view, the drop-deformation problem is extremely
difficult. The equations of motion must be solved for the flow both inside and outside
the drop, with boundary conditions applied on its surface. However, the shape of the
drop is not known
a
priori,
but must be determined
as
part of the solution.
To
date,
no general solution has been found, but progress has been made through asymptotic
analysis for slightly deformed drops, a slender-body theory for highly elongated drops
which applies when the viscosity of the drop is small compared with that of the
suspending fluid and the velocity gradient is large, and numerical analyses for selected
intermediate cases. Recent review articles by Acrivos
(1983)
and Rallison
(1984)
describe these efforts in considerable detail. A brief summary is included in
$4
below.
On the experimental side, a relatively large number of studies of drop deformation
and breakup have been reported. These date back to the pioneering work of
G. I. Taylor
(1934).
In this early work, Taylor investigated drop behaviour experi-
mentally in two flow fields, simple shear flow, where the magnitudes of the vorticity
and strain rate are equal, and two-dimensional pure-straining flow. The former flow
was generated in
a
parallel-band apparatus, and the four-roll mill was evidently
invented to produce the latter. Although few in number, Taylor’s experiments
uncovered most of the qualitative aspects of the drop deformation and burst process,
including the following general conclusions
:
1.
At low flow strengths, drops of all viscosity ratios deform into prolate spheroids.
The longest axis
of
the drop
is
initially aligned with the principal axis of strain for
both irrotational and simple shear flows.
2.
When the drop viscosity is low compared with that of the suspending fluid, the
shear rate required
for
burst becomes quite large, and the drops attain highly
deformed steady shapes with pointed ends. Under some conditions, small drops are
ejected from these pointed ends, a phenomena which has come to be called ‘tip
streaming
’.
3.
When the ratio of drop to suspending fluid viscosity
is
large, drop behaviour
is
qualitatively different in simple shear and irrotational flow fields. In irrotational
flows, burst occurs at low strain rates. In simple shear, on the other hand, viscous
drops assume slightly deformed shapes which are unaffected by further increases in
the shear rate, and drop burst becomes impossible beyond a certain critical viscosity
ratio.
The dramatic qualitative difference in drop burst between pure-straining flow and
simple shear flow furnishes
a
motivation for study of the drop-deformation and burst
process in flows of intermediate vorticity. While such flows can be generated in the
four-roll mill (Giesekus
1962;
Fuller
&
Leal
1981),
experimental difficulties in
controlling the drop at the centre stagnation point
of
such flows have, until now,
prevented studies of their effect on drop behaviour. In the limiting case when there
is no vorticity (the case considered by Taylor
1934),
the control problem is simplified
because the dividing streamlines are at right angles to the roller geometry, and hand
control is possible, though only at the cost of fairly large variations in the flow with
time.
For
other strong flows, however, the dividing streamlines are
at
angles to the
roller geometry, and the complications are too severe for successful manual control.
For this reason, almost all
of
the drop-deformation and burst experiments that
followed those of Taylor
(1934)
(e.g. Rumscheidt
&
Mason
1961
;
Torza, Cox
&
Mason
1972
;
Grace
1971)
have been restricted
to
simple shear and/or two-dimensional
pure-straining flow.
A
notable exception
is
the work of Hakimi
&
Schowalter
(1980),
who studied drop deformation in the flow produced in an orthogonal rheometer. In
this device, flows of varying
vorticity-to-strain-rate
ratio can be generated, but the
Drop
deformation
and
breakup
243
flows are always ‘weak
’,
meaning that the magnitude of the vorticity is always larger
than that of the strain rate. Thus, the experiments were limited to small deformations
(Dp
<
0.2),
and only one viscosity ratio
(A
=
0.09).
In order to investigate experimentally the problem of drop deformation and burst
for flows between simple shear and hyperbolic extension, we developed a computer-
based control system for the four-roll mill. The drop position
was
sensed using a
digital television camera, and the speeds of stepping motors driving the rollers were
regulated to maintain the drop at the centre of the device.
A
detailed description of
the apparatus and control system is reported in Bentley
&
Leal
(1986).
Using this
device, we systematically investigated the effect of flow type on the deformation and
burst of drops in Newtonian fluids, covering
a
wide range of viscosity and strain-
rate-to-vorticity ratios. Computer control of the experiment not only allowed
us
to
study intermediate flows that had not been previously investigated, but also resulted
in drop deformation and burst data of considerably improved quality for the
irrotational flow limit. In the present paper, we report the results of this experimental
investigation, including detailed comparisons with the predictions of available drop
deformation and burst theories.
2.
Problem statement
We consider the behaviour of
a
drop of volume
$xu3,
viscosity
p‘
and density
p’,
that
is
freely suspended in an infinite bath of
a
second fluid of viscosity
p
and density
p.
The interfacial tension between the two immiscible fluids is
u.
The interface is
assumed to transmit tangential stresses undiminished
;
thus other possible surface
effects such as interfacial viscosity and interfacial tension gradients are neglected.
Far from the drop, the suspending fluid undergoes
a
steady linear flow. The situation
is illustrated schematically in figure
1.
Both fluids are Newtonian and incompressible,
so
that the governing equations are the NavieI-Stokes equations and the continuity
equation, applied inside and outside the drop. At the drop surface, the velocity fields
satisfy the conditions of continuity of velocity and tangential stress, and the normal
stress suffers
a
jump due to the interfacial tension. In
our
experiments in the four-roll
mill, we generate an approximation to the idealized linear flow in a bath of finite size.
Also, surface impurities present in the real system may affect the behaviour of the
fluid-fluid interface. We assume that these are small effects.
When the governing equations are put in dimensionless form, with the undeformed
radius of the drop
a as
the characteristic lengthscale, the inverse of the magnitude
of the velocity-gradient tensor,
G-I,
as the characteristic timescale, and
aG
as the
velocity scale, the following dimensionless parameters appear
:
(1)
PI
h
=
-
(viscosity ratio),
P
(Capillary number),
pG2a
P
R=--
(Reynolds number),
(3)
(4)
P’
K
=
-
(density ratio).
P
In our experiments, we restricted our attention to cases where viscous effects
dominated,
so
that the Reynolds number based on the drop size was always negligible.
244
B.
J.
Bentley and L.
G.
Leal
a,
G
FIGURE
1.
Schematic
of
problem.
We also considered only neutrally buoyant drops,
K
=
1.
Under these conditions, the
evolution of the drop shape depends only on the viscosity ratio, the capillary number,
and the nature of the applied flow. In the flows which can be (approximately)
generated in the four-roll mill, the form of the velocity-gradient tensor is characterized
by
a
single parameter
a,
which specifies the relative strength of the strain rate and
vorticity in the flow. This parameter is defined by:
The ratio of the magnitude of the rate-of-strain tensor to that
of
the vorticity in such
flows can be expressed as
:
magnitude of strain rate
-
1
+a
--
magnitude of vorticity
1
-a'
In particular,
u
=
+
1
for pure-straining
flow,
a
=
0
for simple shear flow, and a
=
-
1
for
purely rotational flow. Streamlines for thc positive values of
CY
are shown in
figure
2.
For any
a,
the vorticity vector is in the negative z-direction, and the principal
axes of the rate-of-strain tensor are in the x- and y-directions. The angle
Be
between
the x-axis and the linear exit streamline
is
given by
u-1
sin
(28,)
=
-
a+l'
(7)
In our experiments, photographs of the drop were taken with the camera mounted
perpendicular to the plane of the flow, yielding a projection of the drop in the
(5,
y)-plane. For convenient comparisons to theoretical predictions, two distinct
scalar measures of drop deformation were determined. These are illustrated in
figure
3.
The first defines
a
deformation parameter
D,
in terms of the longest and
shortest semi-axes of the drop cross-section
(L
and
B
respectively), following Taylor
(1934)
:
L-B
D
---
f-~+~'
and is strictly applicable only for elliptically deformed drops, though
it
is used in
practice whenever the deformation is small. This parameter is zero for spherical drops
and asymptotically approaches unity
as
the drops become infinitely extended. When
LIB
is large (highly deformed drops), however,
D,
changes very little with increasing
deformation, and in this case a different measure
of
the deformation
is
more
appropriate. We follow the precedent of previous studies and choose the ratio of the
Drop
deformation
and
breakup
245
a
=
0.4
a
=
0.2
a
=
0.0
FIQURE
2.
Streamlines of
flow
field
of equation
(1)
for
a
2
0.
FIQURE
3.
Scalar
measurements
of
deformation.
half-length of the deformed drop to the undeformed radius
Lla.
In all cases, the
orientation angle of the drop (the angle between the longest axis of the drop and the
major principal axis of the rate-of-strain tensor)
was
also measured.
In the experiments reported here, we focused our attention on two aspects of drop
behaviour. First, we investigated the equilibrium deformation and orientation of
drops in steady flows as a function
of
C,
for various values of viscosity ratio and flow
type
:
(9)
Second, we investigated the critical capillary number required for drop burst,
as
a
function of flow type and viscosity ratio:
D,
=
D,(C;
A,
a)
or
Lla
=
L/a(C;
A,
a)
and
0
=
0(C;
A,
a).
c,
=
C,(A;a).
(10)
When burst was observed, the maximum stable deformation (the deformation
at
C
just below
C,)
and corresponding orientation angle were measured
:
DfC
=
DfC(A;a)
and@,
=
@,(A;a).
(11)