Modeling and simulation of droplet dynamics for microfluidic applications

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 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

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Ph.D Thesis

Modeling and simulation of droplet dynamics for

microfluidic applications

Graduate School of Yeungnam University

Department of Mechanical EngineeringMajor in Mechanical Engineering

Van Thanh Hoang

Advisor:Professor Jang Min Park, Ph.D.

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

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(C M-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

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

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

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

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] 9Fig 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 W c and W d

are the inlet widths for the continuous phase and dispersed phase, respectively (W T

= W c = W d ) and L T is droplet length in the downstream 20Fig 4.2 Three regimes of droplet generation (1) Experimental results from Li et

al., (2012) and (2) present simulation (a) v ct =0.83mm/s, v d=0.69mm/s, (b)

v ct =3.47mm/s, v d =0.28mm/s, (c) v ct =10.0mm/s, v d =5.0mm/s, (d) v ct=20.0mm/s,

v d =10.0mm/s, where v ct and v d represent the continuous phase inlet velocity and the dispersed phase inlet velocity, respectively 21Fig 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 22Fig 4.4 A one-eighth of the full model used for the computational domain in planar extensional flow [53] 23Fig 4.5 Mesh convergence test for λ=1 and Ca=0.067; (a) steady state, (b) transient behavior of the droplet deformation 24Fig 4.6 Comparison of droplet deformations at steady state [53] between computational simulation and experiments [37,40] 26Fig 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] 30Fig 5.3 Description of trap mechanism of the droplet at the early stage of the contraction microchannel [52] 32

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] 41Fig 5.9 Droplet deformations at steady state depending on viscosity ratio and capillary number [53] 43Fig 5.10 Steady behavior of droplet deformations obtained by the numerical simulation and the proposed model for various viscosity ratios [53] 43Fig 5.11 The droplet deformation at transient behaviors for different viscosity ratios as Ca=0.085 [53] 44Fig 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] 45Fig 5.13 Critical capillary number for droplet breakup as a function of viscosity ratio 46

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

D f Parameter of droplet deformation

D s Droplet deformation at steady behavior

Δp h Hydrostatic pressure

Δp b Laplace pressure at back interface of droplet

Δp f Laplace pressure at front interface of droplet

Δp Net Laplace pressure

v i Velocity in large microchannel

v c Velocity in contraction microchannel

v X Velocity component in X direction

v Y Velocity component in Y direction

v Z Velocity component in Z direction

d Damping coefficient

F External force

F c Capillary force

F d Viscous drag force

k Spring coefficient

L The half length of droplet in X-direction

L d Droplet length in microchannel

L i Large microchannel length

m Mass of droplet

R Radius of droplet

R b Back interface radius of droplet

R f Front interface radius of droplet

x Dimensional displacement of droplet equator

y Dimensionless displacement of droplet equator

y s Dimensionless displacement at steady state of droplet equator

Z b Position of back interface of droplet in microchannel

Z f Position of front interface of droplet in microchannel

Z d Position of droplet in microchannel

Z s Steady state position of droplet in microchannel

W Width of contraction microchannel

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]

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 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]

expansion-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-

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

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

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

dynamics in geometry of contraction microchannel should be studied by 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

three-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

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

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 L i 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)

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

extensional flow is expressed by Equation (2.1) where 𝜀̇ is extension rate and

velocity components in X, Y and Z directions are termed V X , V Y and V Z, 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]

2.3 Dimensionless numbers

The droplet and medium viscosities are denoted as μ d and μ m respectively The

droplet and medium densities are denoted as ρ d and ρ m 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 =𝜇𝑚 𝑣𝑖

𝜎 is

used for the large microchannel within the length of L i , where the inlet velocity (v i)

is considered as the characteristic velocity, whereas the average velocity in the

contraction microchannel (v c) is utilized to defined the capillary number CaII

defined as 𝐶𝑎𝐼𝐼 =𝜇𝑚 𝑣 𝑐

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]

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 − 𝑑𝑥 ̇ (3.1)

According to the damped spring-mass system, there can be three different cases of motions depending on damping ratio ξ =2𝑚√𝑘/𝑚𝑑 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= −2𝑚𝑑 + √(2𝑚𝑑 )2−𝑚𝑘 , 𝑟2 = −2𝑚𝑑 − √(2𝑚𝑑 )2−𝑚𝑘, α =2𝑚𝑑 ,

𝜔 = √𝑚𝑘 − (2𝑚𝑑 )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 → ∞) =𝑅1𝐹𝑘

𝑦(𝑡) = 𝑅1𝐹𝑘+ 𝑏1𝑒𝑟1𝑡+ 𝑏2𝑡𝑒𝑟1𝑡 , when ξ = 1 (3.2) 𝑦(𝑡) = 𝑅1𝐹𝑘+ 𝑏1𝑒𝑟1𝑡+ 𝑏2𝑒𝑟2𝑡 , when ξ > 1 (3.3) 𝑦(𝑡) = 𝑅1𝐹𝑘+ 𝑒−𝛼𝑡(𝑏1cos 𝜔𝑡 + 𝑏2sin 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

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 C F , C k , and C d are

the dimensionless constants and v is the relative velocity between the droplet and

where We is the Weber number defined as We= 𝜌𝑚 𝑣2𝑅

𝜎 , y0 and 𝑦̇0 are initial displacement and velocity, respectively, which are assumed to be zero in the TAB

model [55]

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 = 𝜌𝑑43𝜋𝑅3

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

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

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= 𝐶𝑑𝜇𝑑𝑄𝜇𝑚1−𝑄

𝜌𝑑𝑅 2 (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

Equation (3.15), r1 and r2 are defined in the section 3.1, and Ca is defined as Ca=

previous experimental data in the literature [37,40], the droplet deformation D f

shown in Equation (2.2) should be evaluated It can be seen that there is only one

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 D f is rewritten

as Equation (3.16) The droplet deformation at steady state D f, which occurs at an

infinite time, is denoted as D s

𝐷𝑓(𝑡) =(𝑦(𝑡)+1)(𝑦(𝑡)+1)22−1+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 C b, 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 1.53𝜆+2𝜆+1 𝐶𝑎−𝐶1 ≥ 𝐶𝑏 (3.17)

𝐶𝑎𝑐 = 𝐶𝑘

(1

𝐶𝑏+𝐶1)1.5

3𝜆+2 𝜆+1

(3.18)

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

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

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 W c and W d

are the inlet widths for the continuous phase and dispersed phase, respectively (W T

= W c = W d ) and L T is droplet length in the downstream

Fig 4.2 Three regimes of droplet generation (1) Experimental results from Li et

al., (2012) and (2) present simulation (a) v ct =0.83mm/s, v d=0.69mm/s, (b)

v ct =3.47mm/s, v d =0.28mm/s, (c) v ct =10.0mm/s, v d =5.0mm/s, (d) v ct=20.0mm/s,

v d =10.0mm/s, where v ct and v d represent the continuous phase inlet velocity and the dispersed phase inlet velocity, respectively

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

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]