Modeling and simulation of droplet dynamics for microfluidic applications 6831ef11

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2. Ph.D. Thesis
Modeling and simulation of droplet dynamics for
microfluidic applications
Graduate School of Yeungnam University
Department of Mechanical Engineering
Major in Mechanical Engineering
Van Thanh Hoang
Advisor: Professor Jang Min Park, Ph.D.
August 2019

4. Ph.D. Thesis
Advisor: Professor Jang Min Park, Ph.D.
Presented as Ph.D. Thesis
Graduate School of Yeungnam University
Van Thanh Hoang
August 2019
Department of Mechanical Engineering
Major in Mechanical Engineering
Modeling and simulation of droplet dynamics for
microfluidic applications

6. I
ACKNOWLEDGMENTS
I would like to dedicate this thesis for my late father who highly encouraged me to
pursue a master and a doctoral program when he left this world almost nine years
ago. The thesis also is dedicated to the author’s mother who is seventy six years
old and living far from me now.
I really would like to express my deepest gratitude to my thesis advisor, Professor
Jang Min Park for dedicated help, valuable and devoted instructions, and
everything he has done for me in academic direction and in my life as well over the
last three years of my doctoral program.
I am so grateful to the committee members, Prof. Jiseok Lim, Prof. Jungwook Choi,
Prof. Kisoo Yoo, and Prof. Kyoung Duck Seo for attending my presentation as well
as providing pieces of advice for my doctoral thesis completion.
During my doctoral program, I wish to express my thanks to the Yeungnam
University for supporting the scholarship and providing an excellent academic
environment.
I also thank all of Lab. members, Mr. Gong Yao, Mr. Liu Wankun, Mr. Wu Yue,
Mr. Heeseung Lee, Mr. Seung-Yeop Lee, always gave me encouragement and
support during my doctoral program. Finally, I would like to thank my family,
especially my wife for their constant support and encouragement.
Date: May 15th
, 2019
Van Thanh Hoang (호앙반탄)
Multiphase Materials Processing Lab., ME/YU

7. II
ABSTRACT
Design of microchannel geometry plays a key role for transport and
manipulation of liquid droplets and contraction microchannel has been widely used
for many applications in droplet-based microfluidic systems. This study first aims
to investigate droplet dynamics in contraction microchannel for more details and
then to propose a simplified model used for microfluidic systems to describe
droplet dynamics. In particular, for contraction microchannel, three regimes of
droplet dynamics, including trap, squeeze and breakup are characterized, which
depends on capillary number (Ca) and contraction ratio (C). Theoretical models
have been also proposed to describe transitions from one to another regime as a
function of capillary number and contraction ratio. The critical capillary number of
transition from trap to squeeze has been found as a function of contraction ratio
expressed as CaIc=a(CM
-1), whereas critical capillary number CaIIc = c1C-1
depicts
the transition from squeeze to breakup. Additionally, the deformation, retraction
and breakup along downstream of the contraction microchannel have been
explored for more details.
To describe dynamics of droplet in microfluidic system, one-dimensional
model based a Taylor analogy has been proposed to predict droplet deformation at
steady state and transient behavior accurately. The characteristic time for droplet
reaching steady state is dependent on viscosity ratio and the droplet deformation at
steady state is significantly influenced by viscosity ratio of which the order of

8. III
magnitude ranges from -1 to 1. Finally, theoretical estimation of condition for
droplet breakup was also proposed in the present study, which shows a good
agreement with experimental result in the literature.
Keywords: Droplet dynamics, Microfluidics, Contraction microchannel,
Numerical simulation, Taylor analogy model.

9. IV
TABLE OF CONTENTS
ACKNOWLEDGMENTS .......................................................................................I
ABSTRACT........................................................................................................... II
TABLE OF CONTENTS......................................................................................IV
LIST OF FIGURES ..............................................................................................VI
NOMENCLATURES ........................................................................................VIII
CHAPTER 1. INTRODUCTION ...........................................................................1
1.1. Droplet-based microfluidic system..............................................................1
1.2. Contraction microchannel in microfluidic system.......................................2
1.3. Dynamics of droplet in contraction microchannel.......................................2
1.4. Droplet dynamics in extensional flow .........................................................4
1.5. Problem statement........................................................................................5
1.6. Dissertation overview ..................................................................................7
CHAPTER 2. PROBLEM DESCRIPTION ...........................................................8
2.1. Problem description of contraction microchannel .......................................8
2.2. Problem description for proposed model.....................................................9
2.3. Dimensionless numbers .............................................................................11
CHAPTER 3. TAYLOR ANALOGY MODELING............................................12
3.1. Damped spring-mass model.......................................................................12
3.2. Taylor analogy breakup (TAB) model.......................................................13
3.3. Proposed model..........................................................................................15
3.4. Condition for droplet breakup....................................................................17
CHAPTER 4. COMPUTATIONAL MODEL AND VALIDATION ..................18
4.1. Computational model and methods ...........................................................18
4.2. Computational domain of contraction microchannel.................................19
4.3. Computational domain for the proposed model.........................................22
4.4. Validation of simulation results in planar extensional flow ......................25
CHAPTER 5. RESULTS AND DISCUSSIONS..................................................27
5.1. Droplet dynamics in the contraction microchannel ...................................27
5.1.1. Three regimes of the droplet dynamics...............................................27

10. V
5.1.2. Droplet dynamics along downstream of contraction microchannel ...34
5.2. Performance of the proposed model ..........................................................41
5.2.1. Steady behavior of droplet deformation .............................................42
5.2.2. Transient behavior of droplet deformation .........................................44
5.2.3. Critical capillary number for droplet breakup ....................................45
CHAPTER 6. CONSCLUSIONS AND RECOMMENDATIONS......................47
6.1. Conclusions................................................................................................47
6.2. Recommendations......................................................................................48
REFERENCES .....................................................................................................50
요약.......................................................................................................................59
CURRICULUM VITAE.......................................................................................61

11. VI
LIST OF FIGURES
Fig. 1.1. Overview of the dissertation.....................................................................7
Fig. 2.1. Geometry of the contraction microchannel: (a) a full geometry and
symmetric domain for computational model, which is illustrated by the grey color
and (b) a view from top of the contraction microchannel [52]...............................9
Fig. 2.2. Illustration of (a) a droplet suspending in a planar extensional flow and (b)
a description of the droplet magnified at XY plane [53]. ......................................10
Fig. 4.1. Schematic diagram of T-junction used in validation: (a) a full geometry
and (b) side view of the geometry. Dimensions unit is in micrometer. Wc and Wd
are the inlet widths for the continuous phase and dispersed phase, respectively (WT
= Wc = Wd) and LT is droplet length in the downstream.......................................20
Fig. 4.2. Three regimes of droplet generation. (1) Experimental results from Li et
al., (2012) and (2) present simulation. (a) vct=0.83mm/s, vd=0.69mm/s, (b)
vct=3.47mm/s, vd=0.28mm/s, (c) vct=10.0mm/s, vd=5.0mm/s, (d) vct=20.0mm/s,
vd=10.0mm/s, where vct and vd represent the continuous phase inlet velocity and the
dispersed phase inlet velocity, respectively..........................................................21
Fig. 4.3. Dimensionless droplet length as a function of Ca for two different flow
rates (8.06μL/h and 20μL/h) of the dispersed phase. Li et al.’s experiment 1 and
present simulation 1 applied the disperse phase flow rate of 8.06μL/h, and Li et
al.’s experiment 2 and present simulation 2 applied the disperse phase flow rate of
20μL/h...................................................................................................................22
Fig. 4.4. A one-eighth of the full model used for the computational domain in planar
extensional flow. [53]. ..........................................................................................23
Fig. 4.5. Mesh convergence test for λ=1 and Ca=0.067; (a) steady state, (b)
transient behavior of the droplet deformation.......................................................24
Fig. 4.6. Comparison of droplet deformations at steady state [53] between
computational simulation and experiments [37,40]..............................................26
Fig. 4.7. Comparison of droplet deformation at transient behaviors between
computational simulation and experiments [40] when the viscosity ratio of unity
[53]........................................................................................................................26
Fig. 5.1. Droplet cross-section snapshots at the symmetric plane viewed from top:
(a) trap behavior, (b) squeeze behavior, and (c) breakup behavior [52]...............28
Fig. 5.2. Droplet dynamics described by a map of three regimes depending on C
and CaI, CaII: (a) trap and squeeze regimes and (b) squeeze and breakup regimes.
The results of trap (■), squeeze (○) and breakup (▲) are plotted by symbols, and
transition models of trap-to-squeeze (―) and squeeze-to-breakup (---) are plotted
by fitted curves [52]..............................................................................................30
Fig. 5.3. Description of trap mechanism of the droplet at the early stage of the
contraction microchannel [52]..............................................................................32

12. VII
Fig. 5.4. Two types of droplet breakup: (a) the first type of droplet breakup as CaII
> 2.4, tail formation (1), initial breakup (2), breakup into small pieces (3), the entire
droplet breakup (4) and (b) the second type of breakup as CaII < 2.4, back-interface
collapse and neck formation (1), initial tearing (2), tearing growth (3) and breakup
(4)-(5) [52]. ...........................................................................................................35
Fig. 5.5. Positions of front (Zf) and back (Zb) interfaces and length (Ld) of the
droplet in contraction microchannel [52]..............................................................36
Fig. 5.6. The relationship between normalized droplet length (Ld/D) and
normalized droplet position (Zd/D) for various values of C and CaII [52]............36
Fig. 5.7. Droplet position at steady state as a function of contraction C and capillary
CaII. Simulation results are plotted by the symbols, CaII = 0.1 (■); CaII = 0.3 (○);
CaII = 0.5 (▲), and predicted model is plotted by curves, CaII = 0.1 (―); CaII = 0.3
(---); CaII = 0.5 (-·-) [52].......................................................................................40
Fig. 5.8. Initial topology change position (Zd/D) of the droplet in the contraction
microchannel as a function of CaII [52]................................................................41
Fig. 5.9. Droplet deformations at steady state depending on viscosity ratio and
capillary number [53]............................................................................................43
Fig. 5.10. Steady behavior of droplet deformations obtained by the numerical
simulation and the proposed model for various viscosity ratios [53]...................43
Fig. 5.11. The droplet deformation at transient behaviors for different viscosity
ratios as Ca=0.085 [53].........................................................................................44
Fig. 5.12. The verification of the proposed model by the numerical simulation for
the droplet deformation at transient behavior when the viscosity ratio of unity [53].
...............................................................................................................................45
Fig. 5.13. Critical capillary number for droplet breakup as a function of viscosity
ratio .......................................................................................................................46

13. VIII
NOMENCLATURES
Ca Capillary number
CaI Capillary number at large microchannel
CaIc Critical capillary number for transition from trap to squeeze
CaII Capillary number at contraction microchannel
CaIIc Critical capillary number for transition from squeeze to breakup
Cac Critical capillary number for droplet breakup
Re Reynolds number
We Weber number
B The half breadth of droplet in Y-direction
C Contraction ratio
Cf Friction factor
C1, Ck, CF, Cd, Q Dimensionless constants
c1, c2, M, N Dimensionless constants
D Diameter of droplet
Df Parameter of droplet deformation
Ds Droplet deformation at steady behavior
Δph Hydrostatic pressure
Δpb Laplace pressure at back interface of droplet
Δpf Laplace pressure at front interface of droplet

14. IX
Δp Net Laplace pressure
μm Medium viscosity
μd Droplet viscosity
ρm Medium density
ρd Droplet density
σ Surface tension between medium and droplet phases
𝜀̇ Extensional rate
λ Viscosity ratio
к Density ratio
ξ Damping ratio
v Characteristic velocity
vi Velocity in large microchannel
vc Velocity in contraction microchannel
vX Velocity component in X direction
vY Velocity component in Y direction
vZ Velocity component in Z direction
d Damping coefficient
F External force

15. X
Fc Capillary force
Fd Viscous drag force
k Spring coefficient
L The half length of droplet in X-direction
Ld Droplet length in microchannel
Li Large microchannel length
m Mass of droplet
R Radius of droplet
Rb Back interface radius of droplet
Rf Front interface radius of droplet
x Dimensional displacement of droplet equator
y Dimensionless displacement of droplet equator
ys Dimensionless displacement at steady state of droplet equator
Zb Position of back interface of droplet in microchannel
Zf Position of front interface of droplet in microchannel
Zd Position of droplet in microchannel
Zs Steady state position of droplet in microchannel
W Width of contraction microchannel

16. 1
CHAPTER 1. INTRODUCTION
In the first chapter, introduction to microfluidics and droplet-based microfluidic
systems will be presented first. Then applications and droplet dynamics in
contraction microchannel, and extensional flow are reviewed in details. Finally,
problem statements which include objectives of the research will be stated.
1.1. Droplet-based microfluidic system
Microfluidics is a terminology divided into Micro and Fluidics, where fluid
and/or gas are introduced into small size on the order of microliters or nanoliters.
Microfluidics is well known for simple high-performance biochemical analysis.
There has many advantages such as precise control and manipulation, fast
processing, small amounts of samples and reagents, and so forth. Microfluidics has
been employed for development of Lab-on-a-Chip and micro total analysis systems
for applications in pharmaceutical, biomedical, chemistry and life science [1]. The
flow regime in a microfluidic system is described by laminar flow and with
assumption of no slip boundary condition, so it can control the flow for
manipulating chemicals and reagents precisely [2]. Microfluidic system is first
employed to produce droplets for materials processing applications. Later, droplet-
based microfluidic devices have been applied for development of chemical and
biological analysis, and droplets can be considered as micro-reactors with small
volume [3-6].

17. 2
1.2. Contraction microchannel in microfluidic system
In droplet-based microfluidic devices, contraction microchannel is typically
employed to generate extensional flow with high strain rates [7]. This configuration
microchannel has many applications. For instance, Anna et al. [8] used contraction
microchannel to generate water droplets which are suspended in continuous phase
of oil. Zhu et al. [9] experimentally studied droplet breakup in expansion-
contraction microchannel. Large DNA molecules are controlled and stretched
continuously for optical detection and genes analysis by using a hyperbolic
contraction microchannel [10]. In addition, rheological properties of polymeric
materials can be measured by employing contraction microchannel [11].
1.3. Dynamics of droplet in contraction microchannel
Droplet dynamics in contraction microchannel has been investigated recently
via numerical methods, experiments and some theoretical analysis. In relation to
numerical studies is concerned, nearly previous researches were carried out with a
two-dimensional case. For example, the effects of viscoelasticity on drop and
medium were explored in 5:1:5 planar contraction-expansion microchannel via
applying a finite element method [12-14]. Entrance effects of contraction geometry
and rheology on the droplet behavior were studied by using numerical method [15].
The impact of shear and elongation on the droplet deformation was also
numerically and experimentally examined by using a hyperbolic convergent-

18. 3
divergent microchannel [16]. Christafakis and Tsangaris [17] investigated the
effects of capillary number (Ca), Reynolds number (Re), Weber number (We) and
viscosity ratio (λ) on the droplet dynamics in a two-dimensional contraction
microchannel. Harvie et al., [18-20] studied the influence of Reynolds number,
capillary number, and viscosity ratio on droplet dynamics in an axisymmetric 4:1
microfluidic contraction. In regard to three-dimensional numerical studies of
droplet dynamics in contraction microchannel, there are only few studies in the
literature. Zhang et al. [21,22] performed three-dimensional numerical
investigations on the deformation of droplet in different three-dimensional
contractions. In the meantime, as far as experimental study is concerned, there are
not many researches in the literature. Droplet deformation and breakup in a planar
hyperbolic contraction microchannel were experimentally examined by Mulligan
and Rothstein [23,24]. Chio et al., [25] studied influence of transient pressure,
bubble deformation and bubble length on clogging pressure in microchannel
contraction. Faustino et al., [26] explored the deformability of red blood cells
undergoing extensional and shear flow generated in hyperbolic microchannel with
low aspect ratio. Carvalho et al. [27] proposed an aqueous fluid containing GUVs
to mimic the rheological behavior of blood by using hyperbolic extensional flow.
Regarding theoretical studies, Jensen et al., [28] provided a theoretical and
numerical research of large wetting bubbles in contraction microchannel for
minimizing the clogging pressure.

19. 4
1.4. Droplet dynamics in extensional flow
Dynamics of droplet in microfluidic systems is controlled by the strength of the
flow type which is extension or shear [23]. Planar extension is a typical flow
selected to describe droplet dynamics in microfluidic systems. Planar extensional
flow is well known for several practical applications in materials processing and
microfluidics in this study. The planar extensional flow is first well known for
charactering emulsions, polymers and obtaining droplet viscosity by measuring the
drop deformation [29]. In droplet-based microfluidic systems, extensional flow is
usually used for generation, trap, mixing and manipulation of liquid droplets with
small volume [2]. There are also several investigations about cells or vesicles which
undergo the extensional flow. For example, planar extensional flow is employed to
trap and manipulate cells for long time scales [30], and to measure cellular
mechanical behavior [31]. A microfluidic cross-slot device generating planar
extensional flow is used to study dynamics of vesicles [32]. The mechanical
damage of cells in bioreactors was quantitatively assessed via planar extensional
flow [33].
There are several experimental, numerical and theoretical investigations
reported previously. The deformation and breakup under shear and extensional
flow were first presented by Taylor [34]. Similar to the Taylor’s research, the
experimental studies on the droplet dynamics were carried out for a wide range of
flow conditions [34-37] and for the details of three-dimensional droplet shapes at

20. 5
steady and transient states [38]. In addition, non-Newtonian impact of the droplet
and medium were explored due to the complication of rheological properties in
polymer processing [39,40]. Later, the effects of rheological properties of the
droplet and/or medium on the droplet dynamics of deformation and breakup were
studied [41-43]. Also, droplet position and difference of flow rates in axisymmetric
extensional flow were taken into account for study on the asymmetric breakup [44].
In order to explore droplet deformation in extensional flow, there have been
some theories studied. A theory of small deformation was suggested by Taylor [34]
for prediction of the droplet deformation at steady state at low capillary number
flow. Later, theoretical models for transient behavior of droplet were investigated
by Cox [45] and Barthès-Biesel and Acrivos [46]. Another approximate theory was
developed to depict the breakup of a slender droplet at large deformation [47].
Droplet deformation in three-dimensional shape for arbitrary flow was described
by a phenomenological model [48].
1.5. Problem statement
It is noted that according to fabricated methods, the practical microfluidic
systems have microchannel geometry of rectangular cross-section. It is necessary
to have a clearer understanding about droplet dynamics in such microchannel.
However the previous studies on droplet behavior in contraction microchannel are
limited to two-dimensional model. Therefore, for a detailed analysis droplet

21. 6
dynamics in geometry of contraction microchannel should be studied by three-
dimensional model. Up to our review, it has not been found a full guideline in the
literature for designing a contraction microchannel for droplet manipulation. In this
regard, the first objective of this study is to investigate droplet dynamics in details
in contraction microchannel through three-dimensional numerical simulation and
theoretical modeling.
In droplet-based microfluidic systems, dynamics of droplet in microfluidic
systems is determined by the strength of the flow type which is extension or shear
[23]. In practical cases, it is impossible to perform a three-dimensional simulation
for the whole microfluidic systems due to high computational cost. Therefore,
theoretical models for prediction of droplet deformation should be encouraged in
this case. As far as the previous research are concerned, the theoretical models are
quite complicated [45-47]. In the present study, the second objective is to propose
a simplified model for description of droplet deformation in the microfluidic
systems. The approach is based on an analogy between a droplet dynamics and a
damped spring-mass system. Particularly, the external force and damping force are
developed for investigating droplet dynamics in low Reynolds number and
capillary number. The proposed model has been examined the accuracy relying
upon an extensive computational simulation. Additionally, theoretical estimation
of critical capillary number for droplet breakup has been proposed in this study.

22. 7
1.6. Dissertation overview
This dissertation has been divided into 6 chapters. The research framework is
shown in Fig. 1.1. Chapter 1 introduces the related works, motivations and
objectives of this study. Chapter 2 presents a problem description of the study. In
Chapter 3, the Taylor analogy is briefly presented, and details of the proposed
model are described. Chapter 4 is a presentation of three-dimensional
computational model and validation also. The results and discussions are provided
in Chapter 5. Finally, conclusions and recommendations are given in Chapter 6.
Fig. 1.1. Overview of the dissertation

23. 8
CHAPTER 2. PROBLEM DESCRIPTION
2.1. Problem description of contraction microchannel
A contraction microchannel geometry is shown in Fig. 2.1, where a droplet of
diameter D suspended in a medium fluid. The droplet is initially located in large
microchannel which has a length and width of Li and 2D, respectively. The droplet
then transports into contraction microchannel which has a length of 15D and width
of W. To completely capture droplet dynamics and to eliminate effect of outlet
boundary, the length of the contraction microchannel of 15D was used in this study.
The depth of the whole microchannel of 2.5D was utilized so that effect of walls at
top and bottom of the microchannel on droplet dynamics can be neglected [49-51].
A dimensionless number of contraction level is defined as C=D/W for studying
effect of the width of the microchannel on the droplet dynamics. The initial droplet
diameter D is always larger than the microchannel width W, and the contraction
value C ranges from 1.11 to 2.5. For saving computation time, a symmetric model
which corresponds to a quarter of the full three-dimensional geometry was used as
shown by the grey color in Fig. 2.1 (a).

24. 9
Fig. 2.1. Geometry of the contraction microchannel: (a) a full geometry and
symmetric domain for computational model, which is illustrated by the grey color
and (b) a view from top of the contraction microchannel [52].
2.2. Problem description for proposed model
In this research, planar extensional flow was selected to study droplet dynamics
in microfluidic systems relying on a proposed theoretical model and simulation
data. Fig. 2.2 illustrates a droplet having radius R suspending in a medium fluid
undergoing planar extensional flow. The velocity field used to describe the planar

25. 10
extensional flow is expressed by Equation (2.1) where 𝜀̇ is extension rate and
velocity components in X, Y and Z directions are termed VX, VY and VZ, respectively.
𝑣𝑋 = 𝜀̇𝑋, 𝑣𝑌 = −𝜀̇𝑌, 𝑣𝑍 = 0 (2.1)
Fig. 2.2 (a) is an illustration of the droplet at XY plane, and a magnification of
the droplet is shown in Fig. 2.2 (b). The parameters of droplet deformation are
defined in terms of L and B, where L is the half length of droplet in X-direction and
B is the half breadth of droplet in Y-direction. The displacement of droplet equator
in X-direction is defined as x = L – B. It is assumed that droplet deformation is
ellipsoidal in all times, so droplet deformation parameter is commonly defined by
a dimensionless parameter D as Equation (2.2) [37,48]:
𝐷𝑓 = (𝐿 − 𝐵)/(𝐿 + 𝐵) (2.2)
Fig. 2.2. Illustration of (a) a droplet suspending in a planar extensional flow and
(b) a description of the droplet magnified at XY plane [53].

26. 11
2.3. Dimensionless numbers
The droplet and medium viscosities are denoted as μd and μm respectively. The
droplet and medium densities are denoted as ρd and ρm respectively. The denotation
of σ is the surface tension coefficient between the droplet and medium phases. The
droplet dynamics is characterized by dimensionless parameters. A capillary
number (Ca) is defined as Ca = 𝜇𝑚𝜀̇𝑅/𝜎 where the extension rate is defined as
𝜀̇ = 𝑣/𝑅 [23], v is a characteristic velocity. Depending on droplet position in the
contraction microchannel, two kinds of the characteristic velocities were employed
to define capillary numbers. The capillary number CaI defined as 𝐶𝑎𝐼 =
𝜇𝑚𝑣𝑖
𝜎
is
used for the large microchannel within the length of Li, where the inlet velocity (vi)
is considered as the characteristic velocity, whereas the average velocity in the
contraction microchannel (vc) is utilized to defined the capillary number CaII
defined as 𝐶𝑎𝐼𝐼 =
𝜇𝑚𝑣𝑐
𝜎
in the contraction microchannel within the length of 15D.
Values of viscosity, velocity and interfacial tension can be determined thanks
to the definitions of capillary number above. Reynolds number (Re) is defined
as Re = 𝜌𝑚𝜀̇𝑅2
/𝜇𝑚. A viscosity ratio (λ) is defined as 𝜆 = 𝜇𝑑/𝜇𝑚, and λ of 0.15
was employed to study droplet dynamics in contraction microchannel. In proposed
model based on Taylor analogy, droplet dynamics was investigated for a wide
range of viscosity ratio and capillary number. A density ratio is defined as 𝜅 =
𝜌𝑑/𝜌𝑚 which is fixed as unity in this study [37].

27. 12
CHAPTER 3. TAYLOR ANALOGY MODELING
Taylor [54] first used an analogy between a spring-mass system and droplet to
investigate droplet deformation in a high-speed air flow. As stated by this model,
the spring force is analogous to the surface tension force and the pressure drag force
on the droplet represents an external force. In regard to the analogy, next, O’Rourke
and Amsden [55] introduced a damping component for describing the viscous
behavior of the droplet and damped spring-mass system was employed to calculate
droplet breakup in a spray at high Reynolds number. The model was called Taylor
Analogy Breakup (TAB) model [55]. In the present study, Taylor analogy will be
used to depict the droplet dynamics in planar extensional flow at low Re regime.
Specifically, viscous drag force will be operated as an external force term, and
damping component will be empirically considered to capture the droplet dynamics
for a wide range of capillary number and viscosity ratio.
3.1. Damped spring-mass model
A simple oscillatory system consists of a mass, as spring and a damper. The
damped spring-mass model is expressed in Equation (3.1) where x is the
displacement of the spring, m is the mass, F is the external force, k is the spring
coefficient, d is the damping coefficient.
𝑚𝑥̈ = 𝐹 − 𝑘𝑥 − 𝑑𝑥̇ (3.1)

28. 13
According to the damped spring-mass system, there can be three different
cases of motions depending on damping ratio ξ =
𝑑
2𝑚√𝑘/𝑚
. When ξ = 1, the system
is critical damping, so any slight decrease in the damping force leads to oscillatory
motion. When (ξ > 1), the system is overdamping, in this case the damping
coefficient d is large in comparing with the spring constant k. When (0 < ξ < 1),
the system is underdamping, the damping coefficient is small in comparing with
the spring constant. The solutions for each case can be shown as Equations (3.2),
(3.3), and (3.4), where the displacement x of spring is non-dimensionalized by
setting 𝑦 = 𝑥/𝑅, 𝑟1 = −
𝑑
2𝑚
+ √(
𝑑
2𝑚
)
2
−
𝑘
𝑚
, 𝑟2 = −
𝑑
2𝑚
− √(
𝑑
2𝑚
)
2
−
𝑘
𝑚
, α =
𝑑
2𝑚
,
𝜔 = √𝑘
𝑚
− (
𝑑
2𝑚
)
2
, and b1, b2 are constants defined based on initial conditions [56].
It can be seen that the displacement at steady behavior is given as y (𝑡 → ∞) =
1
𝑅
𝐹
𝑘
.
𝑦(𝑡) =
1
𝑅
𝐹
𝑘
+ 𝑏1𝑒𝑟1𝑡
+ 𝑏2𝑡𝑒𝑟1𝑡
, when ξ = 1 (3.2)
𝑦(𝑡) =
1
𝑅
𝐹
𝑘
+ 𝑏1𝑒𝑟1𝑡
+ 𝑏2𝑒𝑟2𝑡
, when ξ > 1 (3.3)
𝑦(𝑡) =
1
𝑅
𝐹
𝑘
+ 𝑒−𝛼𝑡
(𝑏1 cos 𝜔𝑡 + 𝑏2 sin 𝜔𝑡), when 0 < ξ < 1 (3.4)
3.2. Taylor analogy breakup (TAB) model
Taylor analogy breakup (TAB) model was developed to depict the droplet
breakup in a spray model. This TAB model was found to have advantages in terms
of simplicity and accuracy, so it has been used in several applications [57-61].

29. 14
According to the TAB model, the displacement x in Equation (3.1) corresponds to
the displacement of the droplet equator x described in Fig. 2.2(b), m is the mass of
the droplet, F is the pressure drag force, k is the surface tension component, d is
the viscosity component. More specifically, the physical coefficients in Equation
(3.1) can be expressed as Equations (3.5), (3.6), and (3.7), where CF, Ck, and Cd are
the dimensionless constants and v is the relative velocity between the droplet and
the medium.
𝐹
𝑚
= 𝐶𝐹
𝜌𝑚𝑣2
𝜌𝑑𝑅
(3.5)
𝑘
𝑚
= 𝐶𝑘
𝜎
𝜌𝑑𝑅3
(3.6)
𝑑
𝑚
= 𝐶𝑑
𝜇𝑑
𝜌𝑑𝑅2
(3.7)
In the engine sprays, the surface tension is more dominant than the viscosity, it
means the damping ratio ξ is less than unity, thus it can be regarded as an
underdamped case. Thus, the solution of Equation (3.1) in the TAB model can be
written as Equation (3.8).
𝑦(𝑡) =
𝐶𝐹
𝐶𝑘
We + 𝑒−α𝑡
[{𝑦0 −
𝐶𝐹
𝐶𝑘
We} cos𝜔𝑡 +
1
𝜔
{𝑦0
̇ + α (𝑦0 −
𝐶𝐹
𝐶𝑘
We)} sin 𝜔𝑡] (3.8)
where We is the Weber number defined as We =
𝜌𝑚𝑣2𝑅
𝜎
, y0 and 𝑦̇0 are initial
displacement and velocity, respectively, which are assumed to be zero in the TAB
model [55].

30. 15
3.3. Proposed model
This section presents a theoretical model to describe the droplet dynamics at
low Reynolds regime by using Taylor analogy. Theoretical models for the external
force (F) and the damping coefficient (d) in Equation (3.1) have been proposed,
while the surface tension force component is assumed to be the same with TAB
model as shown in Equation (3.6). Effect of Reynolds number was neglected in this
theoretical models.
At the low Re regime, the drag is dominated by viscous friction. The viscous
drag force applying on a liquid droplet is given as Equation (3.9) [62]. In addition,
the viscous drag force is dependent on the droplet shape which is changed during
deformation in the present case, so the external force can be proposed as Equation
(3.10) where C1 is a constant. The mass of droplet, m is given as 𝑚 = 𝜌𝑑
4
3
𝜋𝑅3
.
Therefore, F/m is given as Equation (3.11). It can be noted that the present model
including Equations (3.6) and (3.11) assumes at low Ca and low Re regimes.
𝐹𝑑 = 2𝜋𝜇𝑚𝑣𝑅
3𝜆+2
𝜆+1
(3.9)
𝐹 = 2𝜋𝜇𝑚𝑣𝑅
3𝜆+2
𝜆+1
(1 + 𝐶1𝑦) (3.10)
𝐹
𝑚
= 1.5
3𝜆+2
𝜆+1
(1 + 𝐶1𝑦)
𝜇𝑚𝑣
𝜌𝑑𝑅2 (3.11)
For the effect of damping coefficient (d) in the TAB model, it should be realized
that only the droplet viscosity is taken into account because the air viscosity is
negligible. However, in the present study, both viscosities of droplet and medium

31. 16
are dominant and they should be considered in viscosity effect. Hence, the
component d/m is empirically proposed as Equation (3.12), where Q is a constant.
𝑑
𝑚
= 𝐶𝑑
𝜇𝑑
𝑄
𝜇𝑚
1−𝑄
𝜌𝑑𝑅2
(3.12)
Finally, by substituting Equations (3.6), (3.11), (3.12) into Equation (3.1) and
by using a dimensionless displacement as 𝑦 = 𝑥/𝑅, the Equation (3.1) can be non-
dimensionalized as Equation (3.13).
𝑦̈ = 1.5
3𝜆+2
𝜆+1
(1 + 𝐶1𝑦)
𝜇𝑚𝑣
𝜌𝑑𝑅3
− 𝐶𝑘
𝜎
𝜌𝑑𝑅3
𝑦 − 𝐶𝑑
𝜇𝑑
𝑄
𝜇𝑚
1−𝑄
𝜌𝑑𝑅2
𝑦̇ (3.13)
In this study, the viscosity is much more dominant than the surface tension, it
means the damping ratio ξ is larger than unity, therefore the system can be operated
as an overdamped case. Initial conditions of the position y0 and the velocity 𝑦̇0 are
set at zero and the solution of Equation (3.13) can be expressed as Equation (3.14),
where 𝑦𝑠is the displacement of the droplet equator at steady state and defined as
Equation (3.15), r1 and r2 are defined in the section 3.1, and Ca is defined as Ca =
𝜇𝑚𝑣/𝜎 [52].
𝑦(𝑡) = 𝑦𝑠 +
𝑦𝑠
𝑟2−𝑟1
(𝑟1𝑒𝑟2𝑡
− 𝑟2𝑒𝑟1𝑡) (3.14)
𝑦𝑠 =
1
𝐶𝑘
1.5
3𝜆+2
𝜆+1
𝐶𝑎
−𝐶1
(3.15)
The performance of the proposed model was verified by comparing with the
previous experimental data in the literature [37,40], the droplet deformation Df
shown in Equation (2.2) should be evaluated. It can be seen that there is only one
parameter of one-dimensional displacement y in the present model, it is assumed

32. 17
that the cross-sectional area of the droplet at XY plane keeps constant during the
droplet deformation. Thus, it should be noted this assumption is acceptable for low
Capillary number flow. Then, the deformation parameter of droplet Df is rewritten
as Equation (3.16). The droplet deformation at steady state Df, which occurs at an
infinite time, is denoted as Ds.
𝐷𝑓(𝑡) =
(𝑦(𝑡)+1)2−1
(𝑦(𝑡)+1)2+1
(3.16)
3.4. Condition for droplet breakup
Droplet dynamics of breakup is one of the main objectives of this study. The
purpose of this part is to Figure out critical capillary number for droplet breakup
and critical droplet deformation which are a function of viscosity ratio. Droplet will
be broken as long as the dimensionless displacement 𝑦𝑠is larger than a critical
coefficient Cb, and the condition is expressed as inequation (3.17). Therefore, at
the critical limit, the critical capillary number Cac for breakup of droplet can be
derived as Equation (3.18).
1
𝐶𝑘
1.5
3𝜆+2
𝜆+1
𝐶𝑎
−𝐶1
≥ 𝐶𝑏 (3.17)
𝐶𝑎𝑐 =
𝐶𝑘
(
1
𝐶𝑏
+𝐶1)1.5
3𝜆+2
𝜆+1
(3.18)

33. 18
CHAPTER 4. COMPUTATIONAL MODEL AND VALIDATION
4.1. Computational model and methods
Stokes flow arises from Navier-Stokes equations where inertial forces are
assumed to be negligible. In microfluidic system, laminar flow is applied and
droplet dynamics is governed by the mathematical models of the Stokes flow that
consists of conservation of momentum and conservation of mass. The medium and
the droplet phases are assumed incompressible Newtonian liquids. A volume of
fluid (VOF) model is used to capture the interface between the droplet and medium
phases [63], and the continuum surface force model is employed to handle the
surface tension as a body force [64]. No slip condition is used at the microchannel
wall, the droplet is assumed to be not wetting on the wall, so the contact angle
between droplet phase and wall is 180o
.
Tool of ANSYS Fluent is used for the numerical simulation of droplet
deformation in the contraction microchannel and planar extensional flow. The
solution methods include the coupled scheme for pressure-velocity, a second order
upwind scheme for momentum conservation equation, the PRESTO! scheme for
pressure interpolation, and the Geo-Reconstruct scheme for interface interpolation.
For the time discretization, a variable time step method is employed for run
calculation. The Courant number of 0.05 was used to capture the transient behavior
accurately and a larger Courant number of 0.25 can be used for recording droplet
deformation at steady state.

34. 19
4.2. Computational domain of contraction microchannel
In order to reduce computational cost, a symmetric contraction microchannel
for simulation is shown by the grey color domain as shown in Fig. 2.1(a). The
computational domain is discretized uniformly by hexahedral elements which have
an element size of W/30 [63]. Up to our knowledge, there was not experimental
data in the literature for verification. Therefore, in this case, the validation of the
computational model was performed for a different problem, i.e. T-junction
microchannel. Schematic diagram of the T-junction microchannel is shown in Fig.
4.1 [63]. In this problem, the study focused on if the computational model can
capture different regimes of droplet generation, and also a quantitative verification
was performed by measuring the droplet length along the microchannel. Fig. 4.2
shows three regimes of droplet generation corresponding to different velocities of
dispersed and medium phases. Fig. 4.3 presents droplet length as a function of
capillary number for two kinds of flow rates. Both qualitative and quantitative
comparisons show that the present simulation results was found to have good
agreement with previous experimental data.

35. 20
Fig. 4.1. Schematic diagram of T-junction used in validation: (a) a full geometry
and (b) side view of the geometry. Dimensions unit is in micrometer. Wc and Wd
are the inlet widths for the continuous phase and dispersed phase, respectively (WT
= Wc = Wd) and LT is droplet length in the downstream.

36. 21
Fig. 4.2. Three regimes of droplet generation. (1) Experimental results from Li et
al., (2012) and (2) present simulation. (a) vct=0.83mm/s, vd=0.69mm/s, (b)
vct=3.47mm/s, vd=0.28mm/s, (c) vct=10.0mm/s, vd=5.0mm/s, (d) vct=20.0mm/s,
vd=10.0mm/s, where vct and vd represent the continuous phase inlet velocity and the
dispersed phase inlet velocity, respectively.

37. 22
Fig. 4.3. Dimensionless droplet length as a function of Ca for two different flow
rates (8.06μL/h and 20μL/h) of the dispersed phase. Li et al.’s experiment 1 and
present simulation 1 applied the disperse phase flow rate of 8.06μL/h, and Li et
al.’s experiment 2 and present simulation 2 applied the disperse phase flow rate of
20μL/h.
4.3. Computational domain for the proposed model
For computation of the droplet dynamics under planar extensional flow, a
computational domain of cube shape is applied. To save computational cost, a
symmetric model which corresponds to a one-eighth of the full three-dimensional
geometry was used as shown in Fig. 4.4. As boundary conditions, surfaces 1, 2 and
3 are applied a given velocity field described as Equation (2.1), while the other
three surfaces are solved as symmetric boundaries as shown in Fig. 4.4. In the
present investigation, the domain edge length is 6 times the droplet radius. With

38. 23
this scale, boundary effect of the computational domain on droplet dynamics can
be negligible [65]. After performing mesh convergence tests, the computational
domain is discretized by 100100100 uniform hexahedral elements. The mesh
convergence tests were carried out by using four types of mesh, including
40×40×40, 75×75×75, 100×100×100, and 150×150×150. In the mesh convergence
tests, the droplet deformation at steady state and transient behavior as shown in Fig.
4.5 and the mesh type of 100×100×100 is found to be reasonable solution with a
reduced computational cost.
Fig. 4.4. A one-eighth of the full model used for the computational domain in
planar extensional flow. [53].
Tải bản FULL (77 trang): bit.ly/2Ywib4t
Dự phòng: fb.com/KhoTaiLieuAZ

39. 24
Fig. 4.5. Mesh convergence test for λ=1 and Ca=0.067; (a) steady state, (b)
transient behavior of the droplet deformation.
Tải bản FULL (77 trang): bit.ly/2Ywib4t
Dự phòng: fb.com/KhoTaiLieuAZ

40. 25
4.4. Validation of simulation results in planar extensional flow
Present simulation of droplet deformation at steady state (Ds) in planar
extensional flow was verified by experimental data [37,40] depending on capillary
number and viscosity ratio, which is shown in Fig. 4.6. It can be seen that the
simulation results is found to have good agreement with the previous experimental
data. Generally, the droplet deformation at steady state increases with the capillary
number and the viscosity ratio. In addition, critical capillary number for the
breakup of droplet decreases when the viscosity ratio increases [37]. In transient
behavior, the droplet deformation is illustrated as Fig. 4.7, where simulation results
was compared to the previous experimental data [40] with various capillary number
for viscosity ratio of unity. The dimensionless time is defined as 𝑡∗
= 𝑡𝜀̇. In Fig.
4.7, the dimensionless characteristic time for droplet reaching steady state is
independent from capillary number. However, in the present research of the
simulation and the proposed theoretical model, it is found that viscosity ratio
strongly affect the dimensionless characteristic time.
6831ef11