Paper No 7th International Conference on Multiphase Flow
ICMF 2010, Tampa, FL USA, May 30June 4, 2010
Numerical Simulation on the Mechanism of the Drop Deformation
and Breakup in Shear Flow
S.L. Chen, C.Z. Lin and L.J. Guo
State Key Laboratory of Multiphase Flow in Power Engineering, Xi'an Jiaotong University
Xi'an, 710049,China
liguo@tmail.xitu.edu.cn
Keywords: Simple shear flows; diffuse interface method; critical capillary number; viscosity ratio
Abstract
This paper presents the deformation and breakup of an isolated drop immersed in immiscible liquid phase undergoing shear
flow, using the diffuse interface method. The interface between the drop and the fluid is tracked by an order parameter, namely
the mass concentration. Two dimensional NavierStokes equations for an incompressible fluid are solved by a projection
method on a fixed Cartesian grid and surface tension effects are incorporated into the model through a modified stress. In the
paper, the critical capillary number was plotted as a function of viscosity ratios with the method of approximation. Small
deformation and breakup of the droplets was investigated. Breakup of the drop occurred by three mechanisms, namely,
necking, end pinching, and capillary instability. The distribution of drop breakup mechanism at a given viscosity ratio was also
simulated numerically. In the end, velocity field was analysed to investigate the mechanism of drop deformation and breakup.
Good agreement was found between numerical simulations and the experimental results, which indicate that the diffuse
interface method can successfully capture the main behavior of the drop deformation and breakup.
Introduction
The deformation and breakup of an isolated drop immersed
in an immiscible liquid phase undergoing shear flow has
been the subject of a number of studies. They are ubiquitous
in the oil recovery, material processing, medicine, paints and
cosmetics industries (Mason & Bibette, 1996). In such areas,
the size distribution of drops can affect the efficiency of the
industrial process, and the deformation and breakup of a
droplet have a direct effect on the interface area and hence
mass transfer. In order to optimize and control the
performance, it is necessary to understand the mechanism of
drop deformation and breakup.
Drop dynamics are mainly characterized by the capillary
number Ca (where the capillary number Ca =yrla/a) and
viscosity ratio 2. Many experimental studies of drop
deformation and breakup have been reported (Mason &
Bibette, 1996). Since Tayor's pioneer work in the 1930s
(Taylor, 1934), there have many valuable results on the
deformation drops in welldefined flow fields such as the
steady drop shape at small deformation (Guido &
Villone,1982), the critical condition for breakup (Grace,
1982;Bently & Leal, 1986), breakup of threads in a
quiescent matrix (Tomotika, 1935;Stone & Leal, 1989), and
quasiequilibrium breakup(Torza et al.,1972;Janssen &
Meijer, 1993). Reviews (Stone, 1994; Rallison, 1984) gave
useful summaries of those topics. Elemans et al.(1993)
found that drop deforms affinely when Cal>2Ca, for a
Newtonian system of X=0.135 under a constant shear rate.
Tsakalos et al.(1998) observed viscous elastic drop breakup
due to both end pinching and capillary instability at
Ca,>>2Ca,. They found that the capillary instability starts
to develop at a constant thread diameter which does not
depend on the initial drop diameter and is inversely
proportional to the shear rate.
However, the complexity of this three dimensional free
surface problem has limited the investigation mostly to
experiments, and the existing theoretical studies are carried
out primarily using simplified treatments. From a
fundamental viewpoint, the key to investigate the drop
dynamics is modelling the moving interface for it is
previously unknown and may undergo severe deformations.
Therefore, the choice of an interface tracking method is
critical for a successful simulation. A conceptually
straightforward way of handling the moving interfaces is to
employ a mesh that has grid points on the interfaces, and
deforms according to the flow on both sides of the boundary.
This has been implemented in boundary integral methods
(Cristini et al., 1998) and the front tracking method
(Tryggvason et al., 2001). These methods have been
successfully used for the simulations of complex multiphase
flows. However, these interface descriptions break down
when interfaces undergo severe deformation because
significant deformations may cause loss of simulation
accuracy and singularities in the solution. Thus, these
methods have been applied mostly to relatively mild
deformations. As an alternative, fixedgrid methods that
regularize the interface have been highly successful in
Paper No
treating deforming interfaces. These include the
volumeoffluid (VOF) method (Li & Renardy 2000a) and
the levelset method (Chang et al. 1996). These methods
introduce the interfacial tension as a body force and a single
set governing equation is solved. The interface is tracked by
an artificial colour function. The disadvantage of the
levelset method is that mass is not conversed and the
disadvantage of the VOF method is that it is difficult to
compute accurate local curvatures from the discontinuous
volume fractions. In the diffuse interface method, the
interface between two immiscible fluids is considered to
have a small but finite thickness (Lowengrub J &
Truskinovsky L., 1998). Various variables change
continuously over this interfacial region. The advantage of
the diffuse interface method is that the explicit tracking of
the interface is unnecessary and changes in the interface
topology are easily handled.
The present work focuses on the small deformation and
transient breakup of a drop in simple shear flow. The study
involves numerical modelling based on the diffuse interface
method. The next section contains computational techniques.
These are followed by a result and discussion section and a
conclusion.
Nomenclature
Ca Capillary number
a Radius (m)
Greek letters
y Shear rate ( s')
17 Viscosity (Pas)
a Interfacial tension (mNm )
A Viscosity ration
p Density(gm11)
Subsripts
c Critical
i Initial
Numerical Scheme
2.1 Problem statement
We considered an incompressible and immiscible drop with
volume 47a3 and viscosity A/, in a fluid of viscosity p. The
drop subjected to simple shear flow generated by the motion
of top and bottom walls, as shown in Fig 1, where the upper
walls moved to the right with constant velocity U and the
lower wall moved to the opposite direction with the constant
velocity of U. The domain rage in the x and y directions is
Lx and Ly, respectively. Besides, the average shear rate was
y=2 UILy, and a was interfacial tension.
U
0 )72
U
Figurel: The diagram of the drop immersed in shear flow
7th International Conference on Multiphase Flow
ICMF 2010, Tampa, FL USA, May 30June 4, 2010
2.2 CahnHilliard equation
The dynamics of fluid at hand was modelled using the
diffuse interface method to mimic the experimental results
obtained through visualization. The governing equations of
the model are described by NavierStokesCahnHilliard
equations. The surface tension is incorporated into the
model though a modified stress. The uniform staggered
Cartesian grid is used (Lowengrub & Truskinovsky, 1998;
Anderson et al., 1998; Verschueren, 1999; Kim,2002).
p(u,+u.Vu)=Vp+V.[7(c)(Vu+Vu')]+Fr (1)
Vu=0 (2)
c, +u Vc= V (M (c) V) (3)
dF(c) _2Ac
dc (4)
where M(c)=c(1c) is the mobility, p/ is the generalized
chemical potential, F(c)=0.25c(1c) is the Helmholtz free
energy, r(c) is the dimensionless viscosity, Pe is the
diffusional Peclet number, and C is the Cahn number
respectively. Pe is the ratio between convective and
diffusive mass transport. C is a measure of the thickness of
the interface. The term Fs is the body force arising from
interfacial tension.
A mass concentration field c(x,y) was introduced to denote
the mass ratio of one of the components in the mixture of
two fluids. The transition of c(x,y) across the interface was
smooth in the interfacial region. For simplification, the drop
was located in the center of the domain region. The initial
velocity was equal to zero and the initial concentration was
(Jacqmin, 2000):
1 1 Lx 2 L]2
1 ltn 2 r (x Lv )
c'(x,y) = 1+tanh 24
2 2 rE
(5)
2.3 Projection method
An effective solver for the continuity and momentum
equation is the approximate projection method (Almgren et
al., 1998). The timestepping procedure is based on a
CrankNicholson type method. The advection term in
equation (8) is solved by second order essentially
nonoscillatory (ENO) method (Harten et al., 1987).The first
step is to calculate an intermediate velocity u', which
satisfies:
u u }
At (uVu)
At
1 1
nV n+
VP 2 +F] 2
+ V (Vu+ (Vu)T (6)
+ V [q"+(Vu+(Vu+l))]
2Re Id
In general, u is not divergence free. Next, solve the
pressure correction 0 from the Poission equation:
A = V (7)
At )
Then update the new velocity u"+'2 at the time n+l, which
satisfies Vu"+=0.
u" = u AtVq (8)
Paper No
and pressure p
1 1
P 2 =P2+ (9)
p =p + (9)
The resulting discrete equations are solved using a
multigrid method.
2.4 Surface tension force
The flow field depends on the concentration field c(x,y)
through the introduction of extra stresses that model the
surface tension between the two fluid components. The
effects of surface tension are included in the computational
model through an external forcing term added to the
momentum equation (10). The surface tension force is
based on a continuum surface force formulation and
introduced according to Kim as:
F, = V V VcI
ReCa c (10)
To match the surface tension of the sharp interface model,
a must satisfy:
ea J (c) dx 1 (11)
From reference (Jacqmin, 2000), we obtain a = 6, .
2. 5 Numerical process
The outline of the simulation was as follows. For each nth
time step, n = 1, 2,...
(1) To initialize c(x,y,0) to be the locally equilibrated
concentration profile.
(2) To solve the CahnHilliard equation with nonlinear full
approximation storage multigrid method to obtain c"+ and
pn+1. While p7Vc was calculated by using a secondorder
ENO scheme (Harten et al.,1987).
(3) Using c' 2=(3c" c"')/2 to compute the surface force
n+1/2
(4) To solve the NavierStokes equations by an approximate
projection method to obtain p/ /2 and p+1, While uVu was
calculated by using a secondorder ENO scheme .
(5) To update the time and repeat steps 24.
Results and Discussion
It is well known that, in simple shear flows, for a given
viscosity ratio 2, if the strain rate is sufficiently small, the
isolated drop immersed in an immiscible liquid phase
becomes deformation and will attain equilibrium shape but
never break up. However, as the strain increases, initial
capillary number exceeds a critical value, and the drop will
begin to break up. Such a critical value is known as critical
capillary number, described as Ca,. And while the initial
capillary number of the drop exceeds Ca, as the increase of
Ca,, the breakup mechanisms of the drop present variance.
1. Steady shape
We study the deformation of a viscous drop for subcritical
capillary numbers, where the drop is stretched to an
approximately ellipsoidal shape. We have studied the case
which has been most analyzed in the literature with the
boundary integral method and VOF method, which is in the
Stokes flow regime with A=1. Numerical simulations have
7th International Conference on Multiphase Flow
ICMF 2010, Tampa, FL USA, May 30June 4, 2010
been conducted for capillary numbers Ca,, 0.2, 0.3, and 0.4.
For deformation parameter calculation, the computational
domain is a box of dimension 8a x 4a and the computations
have been done with the mesh128x64. The deformation
parameter D for the steady state solution is shown in Fig.2.
By comparing our results to the previous results in the
literature, we can judge the accuracy of our method. In our
simulations, the Reynolds number is 0.0625 which is
different from the boundary integral method simulation
which has been done for Stokes flow. The values of
deformation parameter are only slightly larger than the
values obtained by small deformation theory (C. E.Chaffey
& H. Brenner,1967)and the boundary integral method
(J.M.Rallison & A.Acrivos, 1981)for Stokes flow, and
consistent with the results from the VOF method (J. Li &
Y.Y.Renardy, 2000) for low Reynolds number flow and the
trend of the experimental results(FD.Rumscheidt &
S.GMason, 1961). From Fig.2, we can conclude that the
deformation parameter increases with Ca. The capillary
number is the ratio between the viscous shear stress which
deforms the drop and the capillary pressure which resists the
deformation. When Ca, is under the critical value, the
capillary pressure is dominant, so that the drop will not
breakup. As Ca, increases, more viscous shear stress is
imposed. The longer the drop will be stretched, and the
larger the curvature at the ends of the drop.
A Present simulation
 Theory calculation
SVOF simulation a
SExperiment observation A
A BIM simulation
1^^
Capillarynuber
Figure 2: Steady state drop deformation parameter D
under various capillary numbers
2. The relationship between 2 and Cac
In this section, we presented the result of the relationship
between viscosity ratio 2 and the critical capillary number
Ca,, as depicted in figure 3. Drop breakup was reached for
different viscosity ratios between 0.013.78, corresponding
to Ca, between 0.41.3. Comparison of the results literature
data had also been depicted. In the figure, the solid line
represents fit equation in the shear flow by Marks (1998);
the dash line pictures the theoretical value by BarthesBiesel
and Acrivos(1973); X and O represents the critical
Capillary number by Grace(1982) and Torza et al.(1972)
respectively, using pseudosteady state experiment. A
depicted the numerical results in the paper. The critical
Capillary numbers Ca, are found to be higher at lower and
higher viscosity. When the viscosity ratio is close to 0.5, the
critical Capillary number reaches the minimum. This
tendency is expected for the comparison of experimental
and theoretical results in the literature.
However, the deviation of the obtained results from
presented by Grace and Torza needs attention. The
experiment data presented by Grace and Torza used
pseudosteady method, while we used approximation
method while we used approximation method in our
Paper No
simulation. Possible reasons for systematic deviation maybe
the assumption that the liquids in the simulation were
treated as Newtonian, while no absolutely Newtonian
liquids exist in the nature.
2.4X Fl 1962
..... x 9
2.2 1 3o a a t. J..1972
1.2 
X
108 XX
0.4 
0.2
1E.3 0.01 B.1 1 10
Viscosity ratio
Figure 3: The critical Capillary number as a function of
viscosity ratios and the comparison of the results with
literature date depicted in the figure
3. Drop breakup mechanisms with different initial
capillary number
The deformation and breakup of the drop with different
Capillary number (1.01 Ca, 1.1 Ca,, 1.3 Ca,, 1.98 Ca,) is
showed in the figure 4. In the simulation, the drop and the
liquid had the same viscosity ratio of 0.5. As the figure
depicted, different breakup mechanisms was presented with
different initial Capillary number. First of all, the initial
shape of the drop was spherical. And the drop had the same
viscosity ratio of 0.5.
0
S
a
a
lop
(a): Ca = 1.O1Ca,, = 4.6s1, a,
O
t=0
t =0.109
t=0217
t= 0.435 a
t=0j652
t=0269 s
t=1D87 s
t=1304 a
t=1.521 5
t =1.739
t=1956 s
0.582
t=0
t= 0.782 s
t =1.172 j
t =1953 s
i =3.516 j
t = 5.079 s
t =6641 j
= 7.423 5
7th International Conference on Multiphase Flow
ICMF 2010, Tampa, FL USA, May 30June 4, 2010
(1)While the initial capillary number of the droplet slightly
bigger than Ca, (t=0.217s) under shear flow, the process of
the drop deformation and breakup presented in the figure.
4(a). As the time went on, the drop kept stretching, and a
neck formed in the middle of the drop (t=0.625s). The drop
became thinner and thinner, and broke up to two daughter
drop eventually (t= 1.962). The two daughter drops had the
same size and opposite flow direction. In the same time,
there were smaller satellite drops between the daughter
drops. After that, the distance between the drops became
farer and farer. This process is named necking mechanism.
(2)As slightly larger capillary number with Ca,=1. Ca,, see
figure.4(b), at first, initially spherical drop deformed to an
ellipsoid (t=0.263s) under shear flow. As keeping stretching,
the drop became dumbbell shaped. The drop kept the shape
in the end and became thinner and thinner in the middle
(t=2.105). In the end, it broke up to two daughters, with a
bigger satellite drop between them (t=2.368s).
(3)Figure.4(c) depicted the drop deformation and breakup
with a capillary of 1.3Cac. The shear flow filed deformed
the initially spherical drop to an ellipsoid (t=0.782s).As the
time went on, the drop was stretched into a long thread its
end bulb up, and a bridge formed between the bulbous end
and the uniform central thread (t=5.079s). The bridge
continued to be thinner which lead to the pinch off of a
daughter drop in the end (t=6.641s). Similar processes
repeated on the remaining part of the thread as time went on
(t=7.423s). This phenomenon was called the end pinching.
0
*
p
____ p
t=0Z
t=0263 s
t= 0.789 s
t=1f053
t=1316
t=1579
t=1842 s
S=2.105 j
t= 2.368
t= 2.632 s
t =2.895
=0.8
t=0
t = 0.0556 s
=0.111
= 0223
=0334
=0.445
= 0556
= 0668
=0.779
=0290
=1000
t=7314 j. ..
S t =8.204 5 ....
. .... t=8.790s j a** .. *
(c): Ca = 1.3Ca ,y = 2.7s5, a, =1.31 (d): Ca = 1.98Ca, = 9s1, a, = 0.61
Figure 4: the numerical result of different processes of drop breakup under shear flow
(b): Ca = 1.Ca,y= 3.8s 1,a
p
Paper No
(4)When the capillary increased to 1.98Cac, initially
spherical drop deformed to an ellipsoid (t=0.0556s) under
shear flow. As time went on, the drop behaved as the same
as the condition of the capillary number 1.3 Cac. However,
the capillary instability grew on the central part of the thread
and the thread broke into a line of uniform sized daughter
drops finally.
initial capillary
number (C,
a.,
go,
ffll.
end pinching mechanism
necking mechanism
capillary instability mechanism
* no breakup
'
I I
19 The number
Figure 5: Distribution of drop breakup mechanism at a
given viscosity ratio
Good agreement was found between numerical simulations
and former experimental results (Lin & Guo, 2007). As the
initial capillary number changing from Ca, to 2Ca,, the
deformation and breakup of the drop presents three
mechanisms: necking mechanism, the end pinching; the
capillary instability, which was the same as the results of
the experimental results. However, the time used for drop
breakup in the simulation results was far less than that in
the experiment results. As mentioned before, in our paper,
the velocity field at the beginning was given as the stability
of a shear flow. We can get from the figure that the critical
capillary number gotten in step flow is lower than
pseudosteady state flow. The time used for making the
fluid to a shear flow field seems much more than making
the drop breakup. Another reason may be that the drop in
the simulation was treated as Newtonian, while no
absolutely Newtonian liquids exist in the experiment. From
the comparison, we also can see the advantage of the
numerical simulation method in the studies of drop
deformation and breakup as numerical simulation closer to
theoretical value.
In the paper, distribution of drop breakup mechanism at a
given viscosity ratio was simulated numerically, as figure 5
shown. From numerical results above, the breakup of a
drop immersed in another liquid occurs by three
mechanism: when Ca, slightly bigger than critical capillary
number, the drop breakup via necking mechanism; As fig.5
depicted in the figure, for Cal
dominant drop breakup mechanism; for Ca, near or bigger
than 1.8Ca,, the capillary instability and end pinching are
dominant drop breakup mechanism. In the process of the
breakup, satellite drops come into being with daughter
drops. They are symmetrical to the center of the drop and
have the opposite flow velocities.
4. The velocity fields of drop breakup
Further insights into the process of the drop breakup can be
gained by examining the velocity fields at different stages
7th International Conference on Multiphase Flow
ICMF 2010, Tampa, FL USA, May 30June 4, 2010
of the drop breakup. The velocity fields in the xz plane
through the centre of drop during the breakup are
presented in Figure 6. There existed a vortical motion
inside the bulb which was created by the competition of
viscous shear stress and surface tension, except near the
neck. The surface tension force stressed the flow faster
toward the bulbs end while in the waist near the centre the
flow was much slower, which induced the thinning of the
bridge and finally pinched to generate a daughter drop.
(a) Elongation
(b) Evolving
c) Breakup
Figure 6: Velocity fields in xz plane through the center
Conclusions
The deformation and breakup of a droplet immersed in
another liquid was investigated in the paper, using the
diffuse interface method. The problem we concerned was
the mechanism of the drop deformation and breakup. In
view of the results presented in this work following
conclusions can be drawn:
1. For a given viscosity ratio 2, if the strain rate is
sufficiently small, the isolated drop immersed in an
immiscible liquid phase becomes deformation and will
attain equilibrium shape but never break up. As Cai
increases, more viscous shear stress is imposed. The longer
the drop will be stretched, and the larger the curvature at the
ends of the drop.
2.The breakup occurs by three mechanisms:
for Cai~l.OCac,necking mechanism;
for Cai <=1.8 Cac end pinching;
for Cai >1.8 Cac capillary instability mechanism and end
pinching.
3. The diffuse interface method can numerically simulate
the deformation and breakup of the drop immersed in
another immiscible liquid. Using such method, we can
simulate more complex flow field that experiment can not
solve easily.
4. In the simulation, drop and the fluid were treated as
Newtonian, while no absolutely Newtonian liquids exist in
the experiment. In our later work, we will concern the
NonNewtonian fluids to our method.
f*
Paper No
Acknowledgements
The authors are grateful to the National Science Foundation
of China (Contract No.50823002 and No.50536020) for
financial support of this work. We thank the referees of this
paper for their valuable suggestions to improve the quality
of this paper.
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