Droplets resulting from liquid filament contraction have been widely used in industrial processes. However, detailed investigations of liquid compound filament contraction processes are lacking in the literature. Therefore, this study provides a numerical investigation of the contraction of a two-layered compound filament. The simulations are based on an axisymmetric front-tracking method. It is found that because of the interfacial tension force, the initially long cylindrical filament contracts to a compound droplet without any breakup or breaks up into smaller droplets during contraction. Unlike simple filaments, the presence of the inner filament inside the compound filament results in a more complicated compound filament breakup process with various droplet types, e.g., simple droplets, single-core compound droplets, and multi-core compound droplets. We find that the inner filament breaks up into droplets, while the outer does not induce breakup. Such a breakup mode produces a multi-core compound droplet after contraction. In some cases, while the inner filament only contracts to a single droplet, its enclosing filament breaks up to produce simple droplets at each end. We also find a breakup mode that combines these two modes, where both the inner and outer filaments perform breakup. In addition, the breakup of the compound filament occurs via one of two mechanisms: end-pinching and necking. These breakup modes and mechanisms are affected by various parameters such as the inner and outer aspect ratios, the Ohnesorge number, the interfacial tension ratio, and the viscosity ratios. Based on these parameters, various regime diagrams of breakup and non-breakup are proposed.

1.
Abate
,
A. R.
,
Thiele
,
J.
, and
Weitz
,
D. A.
, “
One-step formation of multiple emulsions in microfluidics
,”
Lab Chip
11
,
253
258
(
2011
).
2.
Abate
,
A. R.
and
Weitz
,
D. A.
, “
High-order multiple emulsions formed in poly(dimethylsiloxane) microfluidics
,”
Small
5
,
2030
2032
(
2009
).
3.
Anthony
,
C. R.
,
Kamat
,
P. M.
,
Harris
,
M. T.
, and
Basaran
,
O. A.
, “
Dynamics of contracting filaments
,”
Phys. Rev. Fluids
4
,
093601
(
2019
).
4.
Bhagat
,
K. D.
,
Vu
,
T. V.
,
Wells
,
J. C.
,
Takakura
,
H.
,
Kawano
,
Y.
, and
Ogawa
,
F.
, “
Production of hollow germanium alloy quasi-spheres through a coaxial nozzle
,”
Jpn. J. Appl. Phys., Part 1
58
,
068001
(
2019
).
5.
Castrejon-Pita
,
A. A.
,
Castrejón-Pita
,
J. R.
, and
Hutchings
,
I. M.
, “
Breakup of liquid filaments
,”
Phys. Rev. Lett.
108
,
074506
(
2012
).
6.
Chauhan
,
A.
,
Maldarelli
,
C.
,
Papageorgiou
,
D. T.
, and
Rumschitzki
,
D. S.
, “
The absolute instability of an inviscid compound jet
,”
J. Fluid Mech.
549
,
81
98
(
2006
).
7.
Chen
,
Y.
,
Liu
,
X.
, and
Shi
,
M.
, “
Hydrodynamics of double emulsion droplet in shear flow
,”
Appl. Phys. Lett.
102
,
051609
(
2013
).
8.
Contò
,
F. P.
,
Marín
,
J. F.
,
Antkowiak
,
A.
,
Castrejón-Pita
,
J. R.
, and
Gordillo
,
L.
, “
Shape of a recoiling liquid filament
,”
Sci. Rep.
9
,
15488
(
2019
).
9.
Cuellar
,
I.
,
Ravazzoli
,
P. D.
,
Diez
,
P. D.
, and
González
,
A. G
, “
Drop pattern resulting from the breakup of a bidimensional grid of liquid filaments
,”
Phys. Fluids
29
,
102103
(
2017
).
10.
Driessen
,
T.
,
Jeurissen
,
R.
,
Wijshoff
,
H.
,
Toschi
,
F.
, and
Lohse
,
D.
, “
Stability of viscous long liquid filaments
,”
Phys. Fluids
25
,
062109
(
2013
).
11.
Dziedzic
,
A.
,
Nakrani
,
M.
,
Ezra
,
B.
,
Syed
,
M.
,
Popinet
,
S.
, and
Afkhami
,
S.
, “
Breakup of finite-size liquid filaments: Transition from no-breakup to breakup including substrate effects
,”
Eur. Phys. J. E
42
,
18
(
2019
).
12.
Evangelio
,
A.
,
Campo-Cortés
,
F.
, and
Gordillo
,
J. M.
, “
Simple and double microemulsions via the capillary breakup of highly stretched liquid jets
,”
J. Fluid Mech.
804
,
550
577
(
2016
).
13.
Hertz
,
C. H.
and
Hermanrud
,
B.
, “
A liquid compound jet
,”
J. Fluid Mech.
131
,
271
287
(
1983
).
14.
Ho
,
N. X.
and
Vu
,
T. V.
, “
Numerical simulation of the deformation and breakup of a two-core compound droplet in an axisymmetric T-junction channel
,”
Int. J. Heat Fluid Flow
86
,
108702
(
2020
).
15.
Hoepffner
,
J.
and
Paré
,
G.
, “
Recoil of a liquid filament: Escape from pinch-off through creation of a vortex ring
,”
J. Fluid Mech.
734
,
183
197
(
2013
).
16.
Homma
,
S.
,
Koga
,
J.
,
Matsumoto
,
S.
,
Song
,
M.
, and
Tryggvason
,
G.
, “
Breakup mode of an axisymmetric liquid jet injected into another immiscible liquid
,”
Chem. Eng. Sci.
61
,
3986
3996
(
2006
).
17.
Lao
,
K.-L.
,
Wang
,
K.-L.
, and
Lee
,
G.-B
, “
A microfluidic platform for formation of double-emulsion droplets
,”
Microfluid. Nanofluid.
7
,
709
719
(
2009
).
18.
Liu
,
X.
,
Wu
,
L.
,
Zhao
,
Y.
, and
Chen
,
Y.
, “
Study of compound drop formation in axisymmetric microfluidic devices with different geometries
,”
Colloids Surf., A
533
,
87
98
(
2017
).
19.
Maan
,
A. A.
,
Schroën
,
K.
, and
Boom
,
R.
, “
Spontaneous droplet formation techniques for monodisperse emulsions preparation: Perspectives for food applications
,”
J. Food Eng.
107
,
334
346
(
2011
).
20.
McClements
,
D. J.
, “
Advances in fabrication of emulsions with enhanced functionality using structural design principles
,”
Curr. Opin. Colloid Interface Sci.
17
,
235
245
(
2012
).
21.
Muschiolik
,
G.
, “
Multiple emulsions for food use
,”
Curr. Opin. Colloid Interface Sci.
12
,
213
220
(
2007
).
22.
Notz
,
P. K.
and
Basaran
,
O. A.
, “
Dynamics and breakup of a contracting liquid filament
,”
J. Fluid Mech.
512
,
223
256
(
2004
).
23.
Pan
,
D.
,
Chen
,
Q.
,
Zhang
,
Y.
, and
Li
,
B.
, “
Investigation on millimeter-scale W1/O/W2 compound droplets generation in a co-flowing device with one-step structure
,”
J. Ind. Eng. Chem.
84
,
366
374
(
2020
).
24.
Pierson
,
J.-L.
,
Magnaudet
,
J.
,
Soares
,
E. J.
, and
Popinet
,
S.
, “
Revisiting the Taylor-Culick approximation: Retraction of an axisymmetric filament
,”
Phys. Rev. Fluids
5
,
073602
(
2020
).
25.
Schulkes
,
R. M. S. M.
, “
The contraction of liquid filaments
,”
J. Fluid Mech.
309
,
277
300
(
1996
).
26.
Stone
,
H. A.
,
Bentley
,
B. J.
, and
Leal
,
L. G.
, “
An experimental study of transient effects in the breakup of viscous drops
,”
J. Fluid Mech.
173
,
131
158
(
1986
).
27.
Tryggvason
,
G.
,
Bunner
,
B.
,
Esmaeeli
,
A.
,
Juric
,
D.
,
Al-Rawahi
,
N.
,
Tauber
,
W.
,
Han
,
J.
,
Nas
,
S.
, and
Jan
,
Y.-J.
, “
A front-tracking method for the computations of multiphase flow
,”
J. Comput. Phys.
169
,
708
759
(
2001
).
28.
Utada
,
A. S.
,
Lorenceau
,
E.
,
Link
,
D. R.
,
Kaplan
,
P. D.
,
Stone
,
H. A.
, and
Weitz
,
D. A.
, “
Monodisperse double emulsions generated from a microcapillary device
,”
Science
308
,
537
541
(
2005
).
29.
Vu
,
T. V.
,
Homma
,
S.
,
Tryggvason
,
G.
,
Wells
,
J. C.
, and
Takakura
,
H.
, “
Computations of breakup modes in laminar compound liquid jets in a coflowing fluid
,”
Int. J. Multiphase Flow
49
,
58
69
(
2013
).
30.
Vu
,
T. V.
,
Homma
,
S.
,
Wells
,
J. C.
,
Takakura
,
H.
, and
Tryggvason
,
G.
, “
Numerical simulation of formation and breakup of a three-fluid compound jet
,”
J. Fluid Sci. Technol.
6
,
252
263
(
2011
).
31.
Vu
,
T. V.
,
Takakura
,
H.
,
Wells
,
J. C.
, and
Minemoto
,
T.
, “
Production of hollow spheres of eutectic tin-lead solder through a coaxial nozzle
,”
J. Solid Mech. Mater. Eng.
4
,
1530
1538
(
2010
).
32.
Vu
,
T. V.
,
Tryggvason
,
G.
,
Homma
,
S.
, and
Wells
,
J. C.
, “
Numerical investigations of drop solidification on a cold plate in the presence of volume change
,”
Int. J. Multiphase Flow
76
,
73
85
(
2015
).
33.
Vu
,
T.-V.
,
Vu
,
T. V.
,
Nguyen
,
C. T.
, and
Pham
,
P. H.
, “
Deformation and breakup of a double-core compound droplet in an axisymmetric channel
,”
Int. J. Heat Mass Transfer
135
,
796
810
(
2019
).
34.
Vu
,
T. V.
,
Wells
,
J. C.
,
Takakura
,
H.
,
Homma
,
S.
, and
Tryggvason
,
G.
, “
Numerical calculations of pattern formation of compound drops detaching from a compound jet in a co-flowing immiscible fluid
,”
J. Chem. Eng. Jpn.
45
,
721
726
(
2012
).
35.
Wang
,
F.
,
Contò
,
F. P.
,
Naz
,
N.
,
Castrejón-Pita
,
J. R.
,
Castrejón-Pita
,
A. A.
,
Bailey
,
C. G.
,
Wang
,
W.
,
Feng
,
J. J.
, and
Sui
,
Y.
, “
A fate-alternating transitional regime in contracting liquid filaments
,”
J. Fluid Mech.
860
,
640
653
(
2019
).
36.
Wang
,
N.
,
Semprebon
,
C.
,
Liu
,
H.
,
Zhang
,
C.
, and
Kusumaatmaja
,
H.
, “
Modelling double emulsion formation in planar flow-focusing microchannels
,”
J. Fluid Mech.
895
,
A22
(
2020
).
37.
Yang
,
X.
,
Feng
,
J. J.
,
Liu
,
C.
, and
Shen
,
J.
, “
Numerical simulations of jet pinching-off and drop formation using an energetic variational phase-field method
,”
J. Comput. Phys.
218
,
417
428
(
2006
).
38.
Yu
,
C.
,
Wu
,
L.
,
Li
,
L.
, and
Liu
,
M.
, “
Experimental study of double emulsion formation behaviors in a one-step axisymmetric flow-focusing device
,”
Exp. Therm. Fluid Sci.
103
,
18
28
(
2019
).
39.
Zhang
,
T.
,
Zou
,
X.
,
Xu
,
L.
,
Pan
,
D.
, and
Huang
,
W.
, “
Numerical investigation of fluid property effects on formation dynamics of millimeter-scale compound droplets in a co-flowing device
,”
Chem. Eng. Sci.
229
,
116156
(
2021
).
40.
Zhou
,
C.
,
Yue
,
P.
, and
Feng
,
J. J.
, “
Formation of simple and compound drops in microfluidic devices
,”
Phys. Fluids
18
,
092105
(
2006
).
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