This work is motivated by the recent experimental development of microfluidic flow-focusing devices that produce highly monodisperse simple or compound drops. Using finite elements with adaptive meshing in a diffuse-interface framework, we simulate the breakup of simple and compound jets in coflowing conditions, and explore the flow regimes that prevail in different parameter ranges. Moreover, we investigate the effects of viscoelasticity on interface rupture and drop pinch-off. The formation of simple drops exhibits a dripping regime at relatively low flow rates and a jetting regime at higher flow rates. In both regimes, drops form because of the combined effects of capillary instability and viscous drag. The drop size increases with the flow rate of the inner fluid and decreases with that of the outer fluid. Viscoelasticity in the drop phase increases the drop size in the dripping regime but decreases it in the jetting regime. The formation of compound drops is a delicate process that takes place in a narrow window of flow and rheological parameters. Encapsulation of the inner drop depends critically on coordination of capillary waves on the inner and outer interfaces.

1.
H. A.
Stone
and
S.
Kim
, “
Microfluidics: Basic issues, applications, and challenges
,”
AIChE J.
47
,
1250
(
2001
).
2.
H. A.
Stone
,
A. D.
Stroock
, and
A.
Ajdari
, “
Engineering flows in small devices: Microfluidics toward a lab-on-a-chip
,”
Annu. Rev. Fluid Mech.
36
,
381
(
2004
).
3.
J.
Thilmay
, “
Think small: Lab-on-a-chip technology shrinks the biological laboratory to the micro scale and expands the potential for future applications
,”
EMBO Rep.
6
,
913
(
2005
).
4.
T. M.
Squires
and
S. R.
Quake
, “
Microfluidics: Fluid physics at the nanoliter scale
,”
Rev. Mod. Phys.
77
,
977
(
2005
).
5.
J.
Atencia
and
D. J.
Beebe
, “
Controlled microfluidic interfaces
,”
Nature (London)
437
,
648
(
2005
).
6.
M.
Joanicot
and
A.
Ajdari
, “
Droplet control for microfluidics
,”
Science
309
,
887
(
2005
).
7.
D. J.
Laser
and
J. G.
Santiago
, “
A review of micropumps
,”
J. Micromech. Microeng.
14
,
R35
(
2004
).
8.
M. R.
Bringer
,
C. J.
Gerdts
,
H.
Song
,
J. D.
Tice
, and
R. F.
Ismagilov
, “
Microfluidic systems for chemical kinetics that rely on chaotic mixing in droplets
,”
Philos. Trans. R. Soc. London, Ser. A
362
,
1087
(
2004
).
9.
G.
Lu
,
Z. H.
An
,
C.
Tao
, and
J. B.
Li
, “
Microcapsule assembly of human serum albumin at the liquid/liquid interface by the pendent drop technique
,”
Langmuir
20
,
8401
(
2004
).
10.
D.
Dendukuri
,
K.
Tsoi
,
T. A.
Hatton
, and
P. S.
Doyle
, “
Controlled synthesis of nonspherical microparticles using microfluidics
,”
Langmuir
21
,
2113
(
2005
).
11.
A.
Fernandez-Nieves
,
G.
Cristobal
,
V.
Garces-Chavez
,
G. C.
Spalding
,
K.
Dholakia
, and
D. A.
Weitz
, “
Optically anisotropic colloids of controllable shape
,”
Adv. Mater. (Weinheim, Ger.)
17
,
680
(
2005
).
12.
P.
Becher
,
Emulsions: Theory and Practice
(
Oxford University Press
,
Oxford, U.K.
,
2001
).
13.
D. R.
Link
,
S. L.
Anna
,
D. A.
Weitz
, and
H. A.
Stone
, “
Geometrically mediated breakup of drops in microfluidic devices
,”
Phys. Rev. Lett.
92
,
054503
(
2004
).
14.
S.
Okushima
,
T.
Nisisako
,
T.
Torii
, and
T.
Higuchi
, “
Controlled production of monodisperse double emulsions by two-step droplet breakup in microfluidic devices
,”
Langmuir
20
,
9905
(
2004
).
15.
A. S.
Utada
,
E.
Lorenceau
,
D. R.
Link
,
P. D.
Kaplan
,
H. A.
Stone
, and
D. A.
Weitz
, “
Monodisperse double emulsions generated from a microcapillary device
,”
Science
308
,
537
(
2005
).
16.
J. L.
Grossiord
and
M.
Seiller
,
Multiple Emulsion: Structure, Properties and Applications
(
Editions de Sante
,
Paris
,
1998
).
17.
C.
Goubault
,
K.
Pays
,
D.
Olea
,
P.
Gorria
,
J.
Bibette
,
V.
Schmitt
, and
F.
Leal-Calderon
, “
Shear rupturing of complex fluids: Application to the preparation of quasi-monodisperse water-in-oil-in-water double emulsions
,”
Langmuir
17
,
5184
(
2001
).
18.
L.
Olivieri
,
M.
Seiller
,
L.
Bromberg
,
M.
Besnard
,
T.-N.-L.
Duong
, and
J.-L.
Grossiord
, “
Optimization of a thermally reversible w/o/w multiple emulsion for shear-induced drug release
,”
J. Controlled Release
88
,
401
(
2003
).
19.
P. B.
Umbanhowar
,
V.
Prasad
, and
D. A.
Weitz
, “
Monodisperse emulsion generation via drop break off in a coflowing stream
,”
Langmuir
16
,
347
(
2000
).
20.
M. R.
Davidson
,
D. J. E.
Harvie
, and
J. J.
Cooper-White
, “
Flow focusing in microchannels
,”
ANZIAM J.
46E
,
C47
(
2005
).
21.
A.
Groisman
,
M.
Enzelberger
, and
S. R.
Quake
, “
Microfluidic memory and control devices
,”
Science
300
,
955
(
2003
).
22.
V.
Cristini
and
Y. C.
Tan
, “
Theory and numerical simulation of droplet dynamics in complex flows—A review
,”
Lab Chip
4
,
257
(
2004
).
23.
J. J.
Feng
,
C.
Liu
,
J.
Shen
, and
P.
Yue
, “
An energetic variational formulation with phase field methods for interfacial dynamics of complex fluids: Advantages and challenges
,” in
Modeling of Soft Matter
, edited by
M.-C. T.
Calderer
and
E.
Terentjev
(
Springer
, New York,
2005
).
24.
P.
Yue
,
C.
Zhou
,
J. J.
Feng
,
C. F.
Ollivier-Gooch
, and
H. H.
Hu
, “
Phase-field simulations of interfacial dynamics in viscoelastic fluids using finite elements with adaptive meshing
,”
J. Comput. Phys.
(to be published).
25.
P.
Yue
,
J. J.
Feng
,
C.
Liu
, and
J.
Shen
, “
A diffuse-interface method for simulating two-phase flows of complex fluids
,”
J. Fluid Mech.
515
,
293
(
2004
).
26.
P.
Yue
,
J. J.
Feng
,
C.
Liu
, and
J.
Shen
, “
Transient drop deformation upon startup of shear in viscoelastic fluids
,”
Phys. Fluids
17
,
123101
(
2005
).
27.
L. A.
Freitag
and
C. F.
Ollivier-Gooch
, “
Tetrahedral mesh improvement using swapping and smoothing
,”
Int. J. Numer. Methods Eng.
40
,
3979
(
1997
).
28.
J.
Eggers
, “
Nonlinear dynamics and breakup of free-surface flows
,”
Rev. Mod. Phys.
69
,
865
(
1997
).
29.
E. D.
Wilkes
,
S. D.
Phillips
, and
O. A.
Basaran
, “
Computational and experimental analysis of dynamics of drop formation
,”
Phys. Fluids
11
,
3577
(
1999
).
30.
P. K.
Notz
,
A. U.
Chen
, and
O. A.
Basaran
, “
Satellite drops: Unexpected dynamics and change of scaling during pinch-off
,”
Phys. Fluids
13
,
549
(
2001
).
31.
Y.
Christanti
and
L. M.
Walker
, “
Surface tension driven jet break up of strain-hardening polymer solutions
,”
J. Non-Newtonian Fluid Mech.
100
,
9
(
2001
).
32.
H. J.
Shore
and
G. M.
Harrison
, “
The effect of added polymers on the formation of drops ejected from a nozzle
,”
Phys. Fluids
17
,
033104
(
2005
).
33.
B.
Ambravaneswaran
,
E. D.
Wilkes
, and
O. A.
Basaran
, “
Drop formation from a capillary tube: Comparison of one-dimensional and two-dimensional analyses and occurrence of satellite drops
,”
Phys. Fluids
14
,
2606
(
2002
).
34.
O. A.
Basaran
, “
Small-scale free surface flows with breakup: Drop formation and emerging applications
,”
AIChE J.
48
,
1842
(
2002
).
35.
P. G.
de Gennes
,
F.
Brochard-Wyart
, and
D.
Quéré
,
Capillarity and Wetting Phenomena: Drops, Bubbles, Pearls, Waves
(
Springer
,
New York
,
2004
).
36.
P.
Yue
,
J. J.
Feng
,
C.
Liu
, and
J.
Shen
, “
Diffuse-interface simulations of drop coalescence and retraction in viscoelastic fluids
,”
J. Non-Newtonian Fluid Mech.
129
,
163
(
2005
).
37.
S. L.
Anna
,
N.
Bontoux
, and
H. A.
Stone
, “
Formation of dispersions using ‘flow focusing’ in microchannels
,”
Appl. Phys. Lett.
82
,
364
(
2003
).
38.
S.
Takeuchi
,
P.
Garstecki
,
D. B.
Weibel
, and
G. M.
Whitesides
, “
An axisymmetric flow-focusing microfluidic device
,”
Adv. Mater. (Weinheim, Ger.)
17
,
1067
(
2005
).
39.
D.
Jacqmin
, “
Contact-line dynamics of a diffuse fluid interface
,”
J. Fluid Mech.
402
,
57
(
2000
).
40.
J. R.
Campanelli
and
X.
Wang
, “
Dynamic interfacial tension of surfactant mixtures at liquid-liquid interfaces
,”
J. Colloid Interface Sci.
213
,
340
(
1999
).
41.
B.
Ambravaneswaran
,
H. J.
Subramani
,
S. D.
Phillips
, and
O. A.
Basaran
, “
Dripping-jetting transitions in a dripping faucet
,”
Phys. Rev. Lett.
93
,
034501
(
2004
).
42.
C.
Cramer
,
P.
Fischer
, and
E. J.
Windhab
, “
Drop formation in a co-flowing ambient fluid
,”
Chem. Eng. Sci.
59
,
3045
(
2004
).
43.
H. A.
Stone
and
L. G.
Leal
, “
Relaxation and breakup of an initially extended drop in an otherwise quiescent fluid
,”
J. Fluid Mech.
198
,
399
(
1989
).
44.
T.
Mikami
and
S. G.
Mason
, “
The capillary break-up of a binary liquid column inside a tube
,”
Can. J. Chem. Eng.
53
,
372
(
1975
).
45.
J. R.
Lister
and
H. A.
Stone
, “
Capillary breakup of a viscous thread surrounded by another viscous fluid
,”
Phys. Fluids
10
,
2758
(
1998
).
46.
G. I.
Taylor
, “
The formation of emulsions in definable fields of flow
,”
Proc. R. Soc. London, Ser. A
146
,
501
(
1934
).
47.
R. A.
de Bruijn
, “
Tipstreaming of drops in simple shear flows
,”
Chem. Eng. Sci.
48
,
227
(
1993
).
48.
C. D.
Eggleton
,
T.
Tsai
, and
K. J.
Stebe
, “
Tip streaming from a drop in the presence of surfactants
,”
Phys. Rev. Lett.
87
,
048302
(
2001
).
49.
Y. Y.
Renardy
,
M.
Renardy
, and
V.
Cristini
, “
A new volume-of-fluid formulation for surfactants and simulations of drop deformation under shear at a low viscosity ratio
,”
Eur. J. Mech. B/Fluids
21
,
49
(
2002
).
50.
M.
Cloupeau
and
B.
Prunetfoch
, “
Electrohydrodynamic spraying functioning modes—A critical review
,”
J. Aerosol Sci.
25
,
1021
(
1994
).
51.
S.
Tomotika
, “
Breaking up of a drop of viscous liquid immersed in another viscous fluid which is extending at a uniform rate
,”
Proc. R. Soc. London, Ser. A
153
,
302
(
1936
).
52.
J. D.
Sherwood
, “
Tip streaming from slender drops in a nonlinear extensional flow
,”
J. Fluid Mech.
144
,
281
(
1984
).
53.
P.
Garstecki
,
H. A.
Stone
, and
G. M.
Whitesides
, “
Mechanism for flow-rate controlled breakup in confined geometries: A route to monodisperse emulsions
,”
Phys. Rev. Lett.
94
,
164501
(
2005
).
54.
R. H.
Engel
,
S. J.
Riggi
, and
M. J.
Fahrenbach
, “
Insulin: Intestinal absorption as water-in-oil-in-water emulsions
,”
Nature (London)
219
,
856
(
1968
).
55.
N.
Oba
,
H.
Sugimura
,
Y.
Umehara
,
M.
Yoshida
,
T.
Kimura
, and
T.
Yamaguchi
, “
Evaluation of an oleic acid water-in-oil-in-water-type multiple emulsion as potential drug carrier via the enteral route
,”
Lipids
27
,
701
(
1992
).
56.
I. G.
Loscertales
,
A.
Barrero
,
I.
Guerrero
,
R.
Cortijo
,
M.
Marquez
, and
A. M.
Ganan-Calvo
, “
Micro/nano encapsulation via electrified coaxial liquid jets
,”
Science
295
,
1695
(
2002
).
57.
Y.
Yeo
,
A. U.
Chen
,
O. A.
Basaran
, and
K.
Park
, “
Solvent exchange method: A novel microencapsulation technique using dual microdispensers
,”
Pharm. Res.
21
,
1419
(
2004
).
58.
S.
Radev
and
B.
Tchavdarov
, “
Linear capillary instability of compound jets
,”
Int. J. Multiphase Flow
14
,
67
(
1988
).
59.
A.
Chauhan
,
C.
Maldarelli
,
D. T.
Papageorgiou
, and
D.
Rumschitzki
, “
Temporal instability of compound threads and jets
,”
J. Fluid Mech.
420
,
1
(
2000
).
60.
R. B.
Bird
,
R. C.
Armstrong
, and
O.
Hassager
,
Dynamics of Polymeric Liquids, Vol. 1. Fluid Mechanics
(
Wiley
,
New York
,
1987
).
61.
J.
Li
and
M. A.
Fontelos
, “
Drop dynamics on the beads-on-string structure for viscoelastic jets: A numerical study
,”
Phys. Fluids
15
,
922
(
2003
).
62.
T.
Funada
and
D.
Joseph
, “
Viscoelastic potential flow analysis of capillary instability
,”
J. Non-Newtonian Fluid Mech.
111
,
87
(
2003
).
63.
X.
Chen
,
S.
Mandre
, and
J. J.
Feng
, “
An experimental study of the coalescence between a drop and an interface in Newtonian and polymeric fluids
,”
Phys. Fluids
18
,
092103
(
2006
).
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