Self-diffusion coefficients that describe cross-flow migration of non-Brownian drops in a dilute sheared emulsion were obtained by trajectory calculations. A boundary integral formulation was used to describe pairwise interactions between deformable drops; interactions between undeformed drops were described with mobility functions for spherical drops. The results indicate that drops have large anisotropic self-diffusivities which depend strongly on the drop viscosity and modestly on the shear-rate. Pairwise interactions between drops in shear-flow do not appreciably promote drop breakup

Hinch, E. J.

Loewenberg, Michael

English

NASA Technical Reports Server

SELF-DIFFUSION OF DROPS IN A DILUTE SHEARED EMULSION
M. Loewenberg t & E.J. Hinch*
tDepartment of Chemical Engineering
Yale University
New Haven, Connecticut 06520-8286
*Department of Applied Mathematics and Theoretical Physics
University of Cambridge
Cambridge CB3 9EW
ABSTRACT
j
Self-diffusion coefficients that describe cross-flow migration of nonBrownian drops in a dilute sheared emulsion
were obtained by trajectory calculations. A boundary integral formulation was used to describe pairwise
interactions between deformable drops; interactions between undeformed drops were described with mobility
functions for spherical drops. The results indicate that drops have large anisotropic self-diffusivities which
depend strongly on the drop viscosity and modestly on the shear-rate. Pairwise interactions between drops
in shear-flow do not appreciably promote drop breakup.
INTRODUCTION
Shear-induced self-diffusion of nonBrownian drops in emulsions is important because it facilitates mixing. In
the production of specialized polymer blends, macroscopic properties such as the mechanical strength rely
on effective mixing of the emulsified melt to produce a uniform microstructure. In the absence of interracial
tension, the drop size in an efficient mixing device will continuously decrease until Brownian motion can
effectively homogenize the fine droplets throughout the continuous phase.
However, interracial tension places a lower bound on the drop size that can be obtained by mixing.
For drops in shear-flow, significant deformation and breakup generally occur when the capillary number,
Ca - tt'Ya is O(1), where _ is the imposed shear-rate, a is the undeformed drop radius, p is the continuous-
a
phase viscosity, and a is the interracial tension. The numerical value of the critical capillary number depends
on the dispersed- to continuous-phase viscosity ratio, A, and the dispersed-phase volume fraction, ¢. For
isolated drops with 0 < A < 4, the criterion Ca = O(1) provides a reasonable estimate of the drop size
produced by a given shear-rate (Hineh & Acrivos 1980; Grace 1982). The effect of drop interactions on
the critical capillary number has not been studied thus the effect of volume fraction is unknown. Simple
scaling arguments indicate that typical mixing operations generate nonBrownian deformable drops; thus,
self-diffusion of deformable drops is an important mixing mechanism in most emulsification processes.
The O(a) displacements and O(X/a) relative velocities between interacting particles or drops produce
self-diffusivities that are O('_a2). Shear-induced self-diffusion of rigid spherical particles in has been exper-
imentally observed (Eckstein et al. 1977; Leighton & Acrivos 1987) and numerically simulated (Brady &
Bossis 1987). Pairwise interactions do not induce a net cross-flow displacement of smooth spheres; cross-flow
self-diffusion results from multi-particle interactions. Wang et al. (1996) calculated the O(¢_'_a 2) cross-flow
self-diffusivity of smooth spheres in a sheared dilute suspension. Net cross-flow displacements can result
from pairwise interactions between nonspherical particles or drops. Thus, nonspherical particles and drops
have larger O(¢')a 2) self-diffusivities in dilute suspensions that result from pair interactions, da Cunha &
Hinch (1996) computed the self-diffusivity of rough spheres in a dilute suspension but there have been no
studies on the self-diffusion of drops.
The aim of this article is to compute cross-flow self-diffusion coefficients of nonBrownian deformable
drops in a sheared dilute emulsion. The effects of the capillary number and viscosity ratio were the focus
579
https://ntrs.nasa.gov/search.jsp?R=19970000449 2020-06-18T00:06:33+00:00Z
of our study. Neutrally buoyant drops, a monodisperse drop size distribution, and low Reynolds number
conditions are assumed. Marangoni stresses on the drop interfaces that result from surfactant concentration
gradients are neglected. The numerical procedure and numerical results are presented in the next two
sections. Concluding remarks are made in the final section.
CALCULATION PROCEDURE
The nonsingular bound-
ary integral formulation
developed by Loewenberg
& Hinch (1996) was used
to compute the interac-
tions and trajectories of
deformable drops (Ca >
0) in shear-flow. For
the limiting case Ca --,
0, spherical drop trajec-
tories were obtained from
Zinchenko (1983, 1984).
Figure 1: Schematic of
relative trajectory be-
tween pair of deformable
interacting drops in shear
flOW.
X 2
A
"'"'"'""X
X 3
(x
Self-diffusivities-- The self-diffusion tensor is defined as half the rate-of-change of the random walk variance:
D = liE 1 _t (t-_2 AA) , (1)
where the cross-flow random walk displacement, A has zero mean. The cross-flow displacement, A =
(A2, A3), is defined in fig. 1. For given values of Ca and A, the cross-flow displacement of a drop resulting
from its pairwise interaction with another identical drop depends on the initial offset of the drop trajectories,
(x °, x°), at large separations in the flow direction, xl. Only pairwise interactions between identical drops are
considered on the assumption that the emulsion is dilute and monodisperse.
The rate of interactions that result in a particular cross-flow displacement, A(x °, x0; Ca, A), is
n'_ Ix01 dx ° dx °, where n is the number density of drops: ¢ = -_a3n. By definition (1), the dimensionless
self-diffusion coefficient for drops in a dilute emulsion is (da Cunha & Hinch 1996):
fa(Ca, A) - Da 3 f0°_0c¢ 2 0 dxOdx o (2)¢-_a 2 - 2r A_x2 '
where a = 2 or 3 for cross-flow self-diffusion in the velocity gradient or the vorticity direction (cf. fig. 1).
The result exploits the four-fold symmetry of integration in the x ° - x ° plane.
For deformable drops, the double integral of formula (2) was evaluated with about 1% error using 32
integrand evaluations (trajectory calculations) on a truncated square domain and extrapolating the result
to infinity using the far-field form of the pair interaction (Kim & Karrila 1991). The calculations presented
herein were obtained with a 4 x 4 domain. A comparison to the values obtained with an 8 x 8 domain
indicated that the truncation error is 1-2%.
580
Ax 2
2.0
1.5
1.0
0.5
(a) N = 18o_
320\\
0.96
0.94
0.92
A2
0.90
(b)
0.88
00. ' ' ' 086. .....
10 15 20 25 30 0 .002 .004 .006
t 1/N
Figure 2: Relative trajectory of two interacting drops for Ca = 0.3, A = 1; initially, one drop centered at
origin and other drop at (-10,0.5,0). a) Cross-flow separation (velocity gradient direction) versus time;
results obtained obtained using N = 180 (dotted curve), 320 (dashed-dotted curve), 720 (dashed curve), and
1280 (solid curve) boundary elements, b) Cross-flow displacement versus reciprocal number of boundary
dements, 1/N.
_ q 15'1/3Unstabilized hydrodynamically interacting spherical drops will coalesce for R < "-_12 , where R =
¢(x°) 2 + (xj°) 2 is the magnitude of the initial trajectory offset and En is the "collision efficiency" (Wang et
al. 1994). Herein, spherical drops were stabilized against coalescence by supplementing the hydrodynamic
interaction with a singular, infintesimally short-range repulsion (Zinchenko 1984). For spherical drops with
_ 9_ '1/3short-range repulsion, A(x °,x °) is nonzero only within the circular domain R < "_12 - The mobility
functions were tabulated for improved computational speed. For spherical drops formula (2) was evaluated
with about 0.1% error using 1225 trajectory calculations.
Deformable drops can also be stabilized with a short-range repulsive interaction (Loewenberg & Hinch
1996) however, this was implemented only for 2 < £ < 5 and for A < 0.1 on the most closely-spaced
trajectories (R < 1). Very small surface separations occur under these conditions. For £ = 1, we confirmed
that the short-range repulsion had a negligible effect on the trajectories of deformable drops.
Cross-flow trajectory offsets were obtained with one drop initially centered at the origin and the second
x ° x °_ Trajectory integration was continued until the second drop passedat a downstream location, (-X1, 2, 31"
the plane xl = +X1. For deformable drops, the initial drop shapes corresponded to the stationary shapes of
isolated drops under the same flow condtions. The error induced by trajectory truncation is O(X{ 3) (Kim
& Karrila 1991). We used X1 = 8 for deformable drops and X1 = 20 for spherical drops. Based on a limited
number of calculations with larger values of X1, we estimate that trajectory truncation error is about 1%
for deformable drops and about 0.1% for spherical drops.
Computation Time and Numerical Convergence-- For deformable drops with Ca = 0.3 and A = 1, self-
diffusivities were obtained in 2 hours on a workstation using N = 720 boundary elements. The calculations
are 3 4 times slower for A -¢ 1 because an iterative solution of the integral equation is required.
The results depicted in fig. 2 demonstrate numerical convergence of the boundary integral calculations
for trajectories of deformable drops. Fig. 2b demonstrates that the total discretization error resulting from
581
1 2 3 4
Figure 3: Sequences (1-4) showing the interaction between two drops in shear flow with Ca = 0.4, )_ = 1.
Initially, one drop centered at origin and other drop at (-10, 0.5, 0).
the curvature calculation, surface integration, and trajectory integration is O(N-1). Herein, self-diffusivities
were computed with N = 720 thus, discretization errors are about 2% which is comparable to the remaining
sources of error discussed above.
Self-diffusivities for spherical drops require less than a minute of CPU time and are accurate to about
0.1%.
NUMERICAL RESULTS
Drop Breakup--An example of a pair interaction between deformable drops in shear-flow is depicted in
fig. 3. The critical capillary number for breakup of an isolated drop in shear flow is lowest for A _ 1; for A = 1,
Ca __ 0.41 is critical (Rallison 1981). The trajectory depicted in fig. 3 demonstrates that even for capillary
numbers very close to critical, strong pairwise interactions do not result in drop breakup. Other calculations
with )_ # 1 also showed no tendency for interaction-enhanced breakup. Apparently pairwise drop interactions
do not induce significantly subcritical capillary number breakup. Thus, the critical capillary number is a
weak function of volume fraction at least in dilute emulsions.
In part, breakup is averted because of the reduced cross-section for interaction between highly deformed
drops. Breakup requires enhanced drop elongation but a detailed inspection of the numerical results revealed
that the extra deformation induced by a pairwise interaction tends to deform drops in the xa-direction. De-
formable drops can easily squeeze past each other which tends to prevent breakup.
Self-diffusivities-- Figs. 4 and 5 depict self-diffusivities as functions of shear-rate (capillary number) and
viscosity ratio. Self-diffusion is only a moderate function of capillary number; figs. 4 and 5 show that self-
diffusivities vary by a factor of about 2 as the capillary number is varied over a wide range. By contrast,
self-diffusivities are a strong function of viscosity ratio.
Anisotropic self-diffusion results from the bias of trajectory displacements in the velocity gradient direc-
tion. Self-diffusion in the velocity gradient direction, f2 is nmch larger than self-diffusion in the vorticity
direction, fa. For stabilized spherical bubbles (A = 0), it can be shown that the ratio f2/f3 is exactly 2. For
deformable bubbles with Ca = 0.3, f2/fa -_ 6. Fig. 5 indicates that anisotropy increases with viscosity ratio
f2/f3 > 20 for A _> 10.
For A = 10, f2 and f3 are approximately equal to the self-diffusivities predicted for spherical particles
with 3% roughness (da Cunha & Hinch 1996). For A < 10, drop diffusivities are considerably larger than
the values for rough spherical particles.
As expected, self-diffusivities vanish for large A because D = 0(¢ 2) for smooth spheres (Wang et al.
1996). The results depicted in fig. 5 indicate that self-diffusion becomes insensitive to capillary number
582
0.05
0.04
0.03
0.02'
0.01
0.00
0.0
T r I
..................: .......
0.1 0.2 0.3 0.4
Ca
Figure 4: Self-diffusion coefficients as a function of
capillary number for A = 1; solid curve: diffusion
parallel to velocity gradient, ]'2; dashed curve: dif-
fusion parallel to vorticity, f3.
0.07
0.06'
0.05
0.04:
0.03
0.02:
0.01'
0.00
..... I ........ | . , .. _ !
/
0.01 0.1 1 10 20
Figure 5: Self-diffusion coefficients as a function of
viscosity ratio for Ca = 0.3 (A) and Ca = 0 (x);
solid curves: diffusion parallel to velocity gradient,
f2; dashed curves: diffusion parallel to vorticity, f3.
for large viscosity ratios. Closer inspection of the numerical results indicates that self-diffusivities vanish
exponentially for A >> 1.
Fig. 4 shows that f2 exhibits a maximum which may result from a balance between the increase of
far-field (R > 2) trajectory displacements with drop deformation and the decrease of near-field trajectory
displacements with reduced drop cross-section. Under the same conditions, f3 is essentially independent of
Ca. Fig. 4 shows a smooth transition from results for Ca > 0.1 to the results for Ca = 0 that were obtained
using spherical drops stabilized against coalescence by a short-range singular repulsion.
CONCLUSIONS
Cross-flow self-diffusivities have been computed for drops in a dilute emulsion. The results show that self-
diffusivities depend strongly on the viscosity ratio and moderately on the capillary number. Self-diffusivities
are much larger in the velocity gradient direction than in the vorticity direction. Drops have large self-
diffusivities compared to rigid spherical particles.
The interactions between deformable drops were described with boundary intergral calculations; O(1/N)
numerical convergence was obtained, where N is the number of boundary elements. Mobility functions for
spherical drops were used to describe the interactions between drops in the zero deformation limit.
The calculations also indicate that drop interactions do not induce significantly subcritical capillary
number breakup. In dilute emulsions, the critical capillary number is a weak function of volume fraction
because deformable drops can easily squeeze past each other.
583
ACKNOWLEDGEMENTS
The authors thank Dr. Alexander Zinchenko for the use of his FORTRAN subroutines for computing the
mobility functions of spherical drops. This work was supported by a grant from NASA Microgravity Science
and Applications Division, Lewis Research Center.
REFERENCES
Brady, J.F. & Bossis, G. 1987 Self-diffusion of particles in concentrated suspensions under shear. J. Chem.
Phys. 87, 5437-5448.
da Cunha, F.R. & Hinch, E.J. 1996 Shear-induced dispersion in a dilute suspension of rough spheres. J.
Fluid Mech. 309, 211-223.
Eckstein, E.C., Bailey, D.G. & Shapiro, A.H. 1977 Self-diffusion of particles in shear flow of a suspension. J.
Fluid Mech. 79, 191-208.
Grace, H.P. 1971 Dispersion phenomena in high viscosity immiscible fluid systems and application of static
mixers as dispersion devices in such systems. Eng. Found., Res. Conf. Mixing, 3rd, Andover, N.H.
republished 1982 in Chem. Eng. Commun. 14, 225-277.
Hinch, E.J. & Acrivos, A. 1980 Long slender drops in a simple shear flow. J. Fluid Mech. 98, 305-328.
Kim, S. & Karrila, S.J. 1991 Microhydrodynamics: Principles and Selected Applications, Butterworth-
Heinemann.
Leighton, D.T. & Acrivos, A. 1987 Measurement of shear-induced self-diffusion in a concentrated suspension
of spheres. J. Fluid Mech. 177, 109-131.
Loewenberg, M. &: Hinch, E.J. 1996 Numerical simulation of a concentrated emulsion in shear flow. J. Fluid
Mech. In press.
Rallison, J.M. 1981 A numerical study of the deformation and burst of a drop in general shear flows. J.
Fluid Mech. 109, 465-482.
Wang, H., Zinchenko, A.Z. & Davis, R.H. 1994 The collision rate of small drops in linear flow fields. J. Fluid
Mech. 265, 161-188.
Wang, Y., Mauri, R. & Acrivos, A. 1995 Transverse shear-induced diffusion of spheres in a dilute suspension.
J. Fluid Mech. In review.
Zinchenko, A.Z. 1983 Hydrodynamic interaction of two identical liquid spheres in a linear flow field. Prikl.
Mat. Mech. 47, 56-63.
Zinchenko, A.Z. 1984 Effect of hydrodynamic interactions between the particles on the rheological properties
of dilute emulsions. Prikl. Mat. Mech. 48, 198-206.
584


Self-Diffusion of Drops in a Dilute Sheared Emulsion

Abstract

Similar works

Full text

Available Versions

NASA Technical Reports Server