7th International Conference on Multiphase Flow
ICMF 2010, Tampa, FL USA, May 30June 4, 2010
Monodisperse droplet formation in microchannel
Jing Lou*, Jinsong Huat, and Baili Zhang*
Institute of High Performance Computing, 1 Fusionopolis Way, #1616, Singapore
t Institute for Energy Technology, NO2027 Kjeller, Norway
Keywords: Multiphase flow, Front tracking method, Droplet dynamics, Microfluidics
Abstract
It has been confirmed that microdroplets can be generated by injecting a disperse liquid via a capillary nozzle into another
immiscible coflowing fluid in microchannel. The mode of droplet formation depends on liquid flow rates of the inner
disperse phase and outer continuous phase, liquid viscosity, nozzle dimensions, and interface tension force. In this study, we
numerically simulate such a coflowing system using front tracking method to investigate the drop formation. The flow
condition and its effect on droplet generation and size distribution has been investigated. The numerical method solves one set
of NavierStokes equations for both liquid phases on a fixed Eulerian twodimensional cylindrical coordinate mesh to account
for the flow dynamics. Front tracking method is applied to track the movement of the interface between the two immiscible
liquids as well as the surface tension force. In this study, the effects of flow inertial, capillary, viscous, and gravitational
forces are all accounted to explore the droplet formation modes and dynamics in coflowing system. The simulation results
show that the process of droplet formation in a coflowing immiscible liquid is reasonably predicted by the present numerical
simulation method. In addition, the effects of the continuous phase flow speed, viscosity and the interface tension force on
droplet formation are investigated.
Introduction
Droplets of one fluid in a second immiscible fluid are useful
in a wide range of applications, particularly for the micro
and nanodroplet. With new advances in the production of
microcapsules, the demand of monodisperse emulsions has
been rising. Some emerging technologies and processes in
making the monodispersed droplets with controllable size
have been reviewed by Barasan (2002).
Anna et al. (2003) presented the droplet formation in a
microfluidic flowfocusing device of rectangular
crosssection in which the inner and outer flow streams
were water and oil respectively. Xu and Nakajima (2' 4)
presented another configuration of "flowfocusing"
microchannel to generate highly monodisperse droplet
with diameter much smaller than the width of the channel
junction. Through control of the flow rate ratios, the
eventual breakup of the dispersed liquid phase occurs as
the flow is focused to a critical break up width. Utada et al
(2005) proposed a method to produce double emulsion
using multiple streams of coaxial flows through
microcapillary nozzles in micro channel. In all these state
of art fabrication technologies, the production of highly
monodisperse droplet is the core technical in improving the
reproducibility of microsystem and the uniformity of
microstructure of particles. The fundamental understandings
of the hydrodynamic mechanism governing the breakup of
the continuous liquid stream into monodisperse droplets
under the confinement of micro flow channels are essential
to the success of applying these fabrication technologies to
industrial applications.
Although the breakup of capillary streams in open
macroscopic environments has been studied well (Zhang &
Stone, 1997; Zhang, 1999), the breakup of stream in confined
microscopic environments is still not understood. Cramer and
Windhab (2'k14) have conducted experiments of droplet
formation at a capillary tip in a coflowing fluid. The droplet
formation is found to be affected by the flow rates of the
continuous and the disperse phases, fluid viscosity and
interface tension. The dynamics of satellite drop formation
was also observed. Because the drop formation mechanism in
a coflowing system is affected by a number of parameters of
liquid flow and fluid property such as flow speed, viscosity,
capillary force and gravitational force, and the microchannel
geometry, it is difficult to obtain a complete solution for such
a complex problem if solely relying on experimental
investigation and theoretical analysis. Herein, the numerical
simulations do provide a suitable alternative approach to
tackle this kind of problem. The development of numerical
model will not only help us to understand the dominant effects
on the drop formation mechanism, but also provide a tool for
the microdevice system design.
Zhang (1999) has conducted a numerical simulation of a
viscous drop generation at the tip of vertical, circular tube with
an ambient viscous fluid. A volumeoffluid (VOF) method
was used to predict the evolution of drop shape and its
breakup. Wilkes et al. (1999) also performed numerical study
of drop formation from a capillary tube into ambient gas phase
by using a finite element method in an axisymmetric
domain. Numerical simulation of droplet formation using
flow focusing in microchannels can also be found in the
work of Davidson et al. (2005) by using VOF method to
track the interface, solves incompressible NavierStokes
equation for flow dynamics.
In the present work, we use a hybrid approach of the front
capturing and front tracking technique proposed by Unverdi
and Tryggvason (1992) to investigate the microdroplet
formation in a coflowing. A set of adaptive elements on the
front is used to mark the interface. This method is based on
solving a single set of governing equations (NavierStokes
equation) for the whole computational domain, while
treating the different phases as one fluid with variable
material properties. The fluid properties such as density and
viscosity are calculated based on the position of the
interface. Interfacial source terms like interface tension are
computed on the interface between two phases and
transferred to the fixed grid using a diracdelta function. The
transient incompressible NavierStokes equations are solved
numerically using SIMPLE algorithm (Hua & Lou, 2007).
The interface or front is tracked explicitly by extracting the
advection velocity from the fixed grid. The advection of
fluid properties is then achieved based on the new position
of the immiscible fluid interface.
In this study, the front tracking method is applird to simulate
the microdroplet formation in a coflowing viscous fluid.
We also study the effect of liquid flow and fluid property on
the drop formation dynamics.
Numerical Methods
Problem description
7th International Conference on Multiphase Flow
ICMF 2010, Tampa, FL USA, May 30June 4, 2010
the system. Such a system can be well represented by an
axisymmetric model.
Governing equations and boundary conditions
For an incompressible multifluid system, the velocity field in
each phase should be continuous that satisfies,
[u].n=0, (1)
in which the square brackets represent the jump across the
interface, u is the fluid velocity and n represents the unit
normal vector to the interface.
Secondly, a stress balance on the interface may be expressed
[_p +/ (Vu + Vu)] n = c n
[ (Vu+VTU)]t = 0
where p is the pressure in the fluid domain, a is the surface
tension, K is the curvature of the interface, t the unit tangent
vector of the interface, and p the fluid viscosity.
In this study, it is reasonable to treat the liquid phase as
incompressible fluid. Hence, the mass is conserved in the
whole domain,
V u=0.
The NavierStokes equation can be expressed as,
(u) + V puu = Vp + V [p(Vu + VTu)]
at
+ o in (x x,)+( (P p)g
Fluid 2 go I
Fluid 1 
Fluid 2 Q 4I IR,
Figure 1: Schematic of droplet formation from a capillary
and breakup into a coflowing viscous liquid.
A coflowing system of two viscous immiscible liquids is
illustrated in Figure 1. The disperse liquid (Fluid 1) is
injected into the system through a capillary nozzle with a
radius of R into the coflowing liquid (Fluid 2) in a coaxial
cylindrical tube with an inner radius of Ro. The gravitational
force aligns with the axis of the tube and nozzle. The
disperse fluid is injected into the system continuously at a
constant flow rate of Q,, and the continuous phase at a
constant flow rate of Qo. The fluid properties such as
density and viscosity for both phases are assumed to be
constant. As the two fluid phases are immiscible, an
axissymmetric interface is formed between two streams.
The interface evolves, collapses and forms droplets when
the two immiscible liquid phases are flowing through into
where, S(xxf) is a diracdelta function defined at the
interface, g is the gravitational acceleration, p refers to the
density of fluids, and po the density of continuous liquid
phase.
The above equations are solved for the drop formation
beginning at the instant at which the free surface of drop is
flat and situates at the tip of the inner capillary tube. In this
study, we assume that the thickness of capillary tube is
sufficiently thin comparing to the tube diameter. Hence, the
effect of wetting of fluid on the wall surface and contact
angle of interface to wall can be neglected.
On all the solid walls, noneslip boundary condition is
applied. Symmetric boundary condition is applied along the
central axis. Outflow boundary condition is applied at the
right most boundary of the solution domain. The length in x
direction is set to be sufficiently longer than the estimated
drop break off distance.
We introduce the following dimensionless characteristic
variables,
x. u
X =gR,, 2 T
R, (gR,)112
t
R, g/2
7th International Conference on Multiphase Flow
ICMF 2010, Tampa, FL USA, May 30June 4, 2010
p /i. K
p 1,_
PigR, Pi R,'
where, the radius of the capillary nozzle for the disperse
phase, R,, is used as characteristic length, and the gravity
based characteristic velocity (gR,)05 is used to normalize
the NavierStokes equation. The nondimensional equation
is then reexpressed as,
(p u) +Vpuu
9t t
= Vp + V [p (Vu + VTu)]
Re
+ K cn(x xf)+ (p 1)g
Bo
in which the superscript is omitted for convenience.
The nondimensional Reynolds number and Bond number
are defined as follows,
Re = ,12R2 ; Bo= pgR, (7)
P, 0T
The problem of a microdroplet formation in a coflowing
liquid could then be characterized by the following
parameters of the density ratio (pOp,) and viscosity ratio
(pup,) of two fluids, Reynolds number, Bond number,
velocity ratio (u/u,), geometry ratio (i1'.. i,' I as well as the
two liquid phase inlet average flow rate (Q,, Qo).
Interface treatment and tracking
Solving the above equations requires the distribution of
fluid properties. Although the density and viscosity of
each fluid is constant, their abrupt jump across an interface
may lead to numerical diffusion and instability. The front
tacking method treats the interface at a finite thickness of
the order of the background mesh size instead of zero
thickness. Across the interface, the fluid properties change
smoothly and continuously from the value on one side of
the interface to the value on the other side. The material
property in the domain may be reconstructed using an
indicator functionl(x, t), which has the value of one in the
one lqiud phase and zero in another liquid phase at a given
time t
b(x,t)=b, +(bb b,)I(x,t) ;
I(x,=t) ,f (x x')dv'
Q)(t)
in which b stands for either fluid density or viscosity. The
indicator function can be written in the form of an integral
over the whole domain Q(t) bounded by the phase interface
F(t). 6(xx') is a delta function that has a value of one at
x'=x and zero everywhere else. In this study, the delta
function is approximated as suggested by Peskin (1993).
Since the fluid velocity is updated on the fixed grid, the
node moving velocity on the front should be computed by
interpolating from the fixed background grid to ensure that
the front moves at the same velocity as the surrounding
fluids. A similarly interpolation from the stationary fixed
grid to the front can be used,
u = ZD(x1 x)u(x).
Then, the front is advected along its normal direction in a
Lagrangian fashion,
xf xf =At u, n
After the position of the front is updated, the distance of some
of the front marker node may have been modified. The front
elements should then be adapted to maintain the element
quality.
Numerical solution
A projection method for the integration of the NavierStokes
equations (5) was used by Tryggvason et al (1992). It is found
that a large density or/and viscosity ratio may lead to difficulty
in convergence. In order to overcome this problem in solving
the pressure equation, an alternative approach is implemented
in the present work. Here, the coupling fluid velocity and
pressure is updated by solving the momentum equations and
continuity equation using SIMPLE scheme. In the multifluid
system, since there is density jump over the phase interface,
mass flux conservation in the control volume crossing the
front interface is not valid. Hence, volume flux conservation
is adopted to modify the SIMPLE algorithm.
Results and Discussion
Effect of the flow rate of continuous liquid phase
The flow speed of continuous liquid phase has significant
effect on the droplet formation as shown from the simulation
results. Figure 2 shows that the smaller sized droplet is formed
when the speed of continuous liquid phase is increased. This
is because the higher speed of the continuous liquid phase
induces higher velocity gradient on the liquid interface, and
the higher viscous drag force, which drive the breakup of
liquid thread into smaller droplets.
x
: = 5.0
0 1
o 3 6 9 12 15 18 21 24 27
x
3
= 105.0
o 3 6 9 12 15 18 21 24 27
X
u o 10.0
3 OF 31 . O 0
O 3 6 9 12 15 18 21 24 27
(b)
(c)
Figure 2: Effect of the flow speed of continuous liquid
phase on droplet formation.
Figure 3 shows the effects of the continuous phase flow rate
on the size of droplet formed in the coflowing system. A
sharp transition of droplet forming mode is observed when
the continuous phase liquid flow speed is about 9.5. During
the transition of droplet formation mode there is a sudden
drop of the droplet size. The sharp transition of the droplet
generation mode from dripping and jetting by adjusting the
liquid flow rate has also been observed and reported in the
experimental works of Cramer et al I,2 '4) and Utada et
al.(2005). Based on the simulation results, the dependence of
droplet size on the continuous phase flow speed can be
correlated using curve fitting, and a power law correlation
rd oc (U0*)048 is obtained for the dripping mode, and rd*c
(uo*) 73 for the jetting mode.
2.0
ell
1.X
a
OA
0A
O 5 10 15 20 25
Flow speed of connous phaue (o7)
Figure 3. Effect of the flow speed of continuous liquid
phase on droplet size formed in the coflowing system.
7th International Conference on Multiphase Flow
ICMF 2010, Tampa, FL USA, May 30June 4, 2010
1)
a o o0 o0
*= 4.0
0 3 6 9 12 15 18 21 24 27
X
b) 2.0
0 = L 0
O 3 6 9 12 15 18 21 24 27
X
c 0.5
0 =0.5
O 3 6 9 12 15 18 21 24 27
X
Figure 4: The effect of viscosity ratios on droplet
formation.
1.1
1A I
0 1 2 3 4 5 6
Vtscosity rateo (')
Effect of liquid viscosity ratio Figure 5. The effect of viscosity ratios on droplet size.
The viscous drag force on the liquid interface is strongly
depended on the viscosity. A snap shot of droplet formation
with different viscosity ratios is shown in Figure 4. The
viscosity ratio of the continuous phase to the disperse phase
changes from /p* 4.00.5. Other parameters are kept
constant as T =1.0 ; o =7.0 ; Re=0.1 ; Bo=0.001 ;
p* = 0.8; R /R = 3.0. As the Reynolds number defined
in this study only reflects the viscosity of the disperse phase.
Hence, the change of liquid viscosity ratio indicates the
changes of viscosity of continuous liquid phase only.
When the continuous phase has a higher viscosity, this
induces longer liquid thread from the cone shape jet (as
shown in Figure 4a), which breaks into smaller droplets as
well as satellite droplets. On the other hand, when the
continuous phase has lower viscosity, the surface tension
will help to stabilize the droplet growth. When the diameter
of liquid thread is large enough, the viscosity force on the
phase interface will be able to tear away the front part of the
liquid thread to form droplet. Hence, the final droplet will
have relatively larger size.
To correlate the droplet size with viscosity ratio, a series of
simulations are performed by varying the viscosity ratio from
0.5 5. Figure 5 shows the variation of droplet size with the
viscosity ratio. It is noticed that the droplet formation mode
(dripping or jetting mode) can also be achieved by adjusting
the viscosity of the continuous phase liquid. The dependence
of droplet size on the viscosity ratio can be correlated using
power law fitting as rd*o (*)0.345 for the dripping, and rd*X
()0.175 for the jetting.
Effect of interface tension
It is well known that interface tension affects the breakup
behaviour of disperse liquid droplet in micro channels.
Figure 6 illustrates the snap shots of droplet formation under
different interface tension indicated by Bond number
(a) We = 0.0005 ; (b)We= 0.001; (c) We=0.0015, while other
parameters are kept the same as u =1.0; io = 7.0; Re =0.1;
p =0.8; p/=1.5; R/R, =3.0
r=. 6i6W4
rd = 1.162u'0341
A^ r =21AOu, 103
S We= 0.0010
o 0
, I , I l, I I I ,l ,
0 3 6 9 12 1.5 18 21 24 27
X
We= 0.0015
o0 O 0
O 3 6 9 12 15 18 21 24 27
x
Figure 6: The effect of surface tension on droplet
formation.
For small Bound number that corresponds to larger interface
tension, the stronger interface tension will hold the droplet
to the exit of needle. As the liquid front grows, the
interfacial area between the two liquid phases will increased
as well, and higher viscous force will be obtained on the
liquid droplet.
18
02 L
0.00I
00
0.0005 0.0010 0.0015 0.0020 0.0025
Webw Number (r')
Figure 7. The effect of interface tension on droplet size.
Effect of the surface tension on the droplet size was studied
by varying the Weber number within the range 0.0001 < We
< 0.002. The variation of the droplet size with the Weber
number is shown in Figure 7. It is found that the droplet
formation mode can also be affected by the surface tension.
The dependence of droplet size upon the Weber number is
correlated using power law fitting as rd c (We) 043 for the
dripping mode and rd* o (We)0.180for the jetting mode.
Conclusions
The drop formation of a disperse liquid injected into another
coflowing immiscible liquid within a microchannel is
simulated numerically using front tracking method to
investigate the drop formation mechanism. The
understanding of such droplet formation mechanism is very
critical to design of microfluidic devices to generate
monosized droplets in a controllable manner. The key
7th International Conference on Multiphase Flow
ICMF 2010, Tampa, FL USA, May 30June 4, 2010
phenomena in the droplet formation process, such as liquid
drop growth, necking, breakup of primary drop and satellite
drop formation, are reasonably captured. In addition, the
effects of some important parameters of flow conditions and
fluid property, such as continuous liquid phase flow speed,
viscosity and interface tension, on the droplet size formed in
the microchannel were also studied.
The simulation results indicates that the droplet
formation in coflowing system depend strongly on the
balance between the viscous shear force from the continuous
phase flow and interface tense force on the droplet. The
higher flow speed and viscosity normally generates higher
shear viscous drag force on the droplet, this help to droplet
breakup from the capillary exit and form smaller droplets.
On the other hand, the higher interface tension (i.e. lower
Bond number) may delay the droplet breakup from the
capillary exit. And hence, more disperse liquid is
accumulated in the droplet, and the final detached droplet
normally has large size. From this study, it can also be
concluded that the coflowing system in micro channel has
good mechanism to control the size of the droplet produced,
and would produced same size droplet periodically.
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7th International Conference on Multiphase Flow
ICMF 2010, Tampa, FL USA, May 30June 4, 2010
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