Study of Immiscible Liquid-Liquid Microfluidic Flow using

SPH-based Explicit Numerical Simulation

Hamideh Elekaei Behjati

A thesis submitted for the degree of Doctor of Philosophy

School of Chemical Engineering

The University of Adelaide

Australia

November 2015

i

Declaration

I certify that this work contains no material which has been accepted for the award of any

other degree or diploma in my name, in any university or other tertiary institution and, to

the best of my knowledge and belief, contains no material previously published or written

by another person, except where due reference has been made in the text. In addition, I

certify that no part of this work will, in the future, be used in a submission in my name, for

any other degree or diploma in any university or other tertiary institution without the prior

approval of the University of Adelaide and where applicable, any partner institution

responsible for the joint-award of this degree.

I give consent to this copy of my thesis when deposited in the University Library, being

made available for loan and photocopying, subject to the provisions of the Copyright Act

1968. The author acknowledges that copyright of published works contained within this

thesis resides with the copyright holder(s) of those works.

I also give permission for the digital version of my thesis to be made available on the web,

via the University’s digital research repository, the Library Search and also through web

search engines, unless permission has been granted by the University to restrict access for

a period of time.

Hamideh Elekaei Behjati

22 November 2015

ii

Abstract

Microfluidic devices are utilized in a wide range of applications, including micro-

electromechanical devices, drug delivery, biological diagnostics and micro-fuel cell

systems. Of particular interest here are liquid-liquid microfluidic systems; which are used

in drug discovery, food and oil industry amongst others. Increased understanding of the

fundamentals of flows in such devices and an improved capacity to design them can come

from modelling. In the case of liquid-liquid flows in microfluidic systems, it is necessary

to explicitly model the behaviour of the individual liquid phases. Such explicit numerical

simulation (ENS) as it is termed requires advanced numerical methods that are able to

evaluate flow involving multiple deforming fluid domains within often complex

boundaries. Smoothed Particle Hydrodynamics (SPH), a Lagrangian meshless method, is

particularly suitable for such problems. This use of a CFD allows determination of

parameters that are difficult to determine experimentally because of the challenges faced in

microfabrication. The study reported in this thesis addresses these concerns through

development of a new SPH-based model to correctly capture the immiscible liquid-liquid

interfaces in general and for a microfluidic hydrodynamic focusing system in particular.

The model includes surface tension to enforce immiscibility between different liquids

based on a new immiscibility model, enforces strict incompressibility, and allows for

arbitrary fluid constitutive models. This work presents a detailed study on the effects of

various flow parameters including flowrate ratio, viscosity ratio and capillary number of

each liquid phase, and geometry characteristics such as channel size, width ratio, and the

angle between the inlet main and side channels on the flow dynamics and topological

changes of the multiphase microfluidic system. According to our findings, both flowrate

quantity and flowrate ratio affect the droplet length in the dripping regime and a large

viscosity ratio imposes an increase in the flowrate of the continuous phase with the same

capillary number of the dispersed phase to attain dripping regime in the outlet channel.

Also, increasing the side channel width causes longer droplets, and the right-angled design

makes the most efficient focusing behaviour. This study will provide great insights in

designing microfluidic devices involving immiscible liquid-liquid flows.

iii

Achievements

Three following papers were achieved from this work:

1) Elekaei Behjati, H., Navvab Kashani, M., Biggs, M. J., Modelling of Immiscible

Liquid-Liquid Systems by Smoothed Particle Hydrodynamics, to be submitted to

Computational Mechanics, 2015.

2) Elekaei Behjati, H., Navvab Kashani, M., Zhnag, H., Biggs, M. J., Smoothed

Particle Hydrodynamics-based Simulations of Immiscible Liquid-Liquid

Microfluidics in a Hydrodynamic Focusing Configuration, to be submitted to

Physics of Fluids, 2015.

3) Elekaei Behjati, H., Navvab Kashani, M., Zhnag, H., Biggs, M. J., Effects of Flow

Parameters and Geometrical Characteristics on the Dynamics of Two Immiscible

Liquids in a Hydrodynamic Focusing Microgeometry, to be submitted to Physics

of Fluids, 2015.

iv

Acknowledgements

Over the past four years, I was inspired and supported by many people without whom I

would never have been able to finalize this dissertation. This section is therefore dedicated

to all those people who have shared their time, enthusiasm and expertise with me,

admitting that I will never be able to thank enough those whom I am indebted the most. I

would like to express my deepest gratitude and appreciation to all of them.

I would like to start by thanking my principal supervisor, Prof Mark J. Biggs for his

ongoing advice and support during the entire course of my PhD study. Mark, your

expertise and detailed critical comments on every part of my research, pushed me to think

deeper about my research and shape my ideas. I am also very grateful to the other members

of the supervision team, A/Prof Zeyad Alwahabi and Dr Hu Zhang for their valuable

feedback and supports.

During the four years of my PhD, I have been lucky to discuss my research and collaborate

with many bright people. In particular, I wish to extend my appreciation to Dr Milan

Mijajlovic. Milan, thanks for your occasional support in computer coding and

programming.

I am more thankful than I can say to Mrs Jacqueline Cookes, Mrs Marilyn Seidel and Mr

Ron Seidel. Mrs Jacqui, you have been always nice and kind to me, that made me feel at

home. You have been a cheerful caring friend for me. Your honesty, experience and

kindness make you a person with whom talking is nothing else but a pleasure. I have learnt

a lot from you that will never forget. Marilyn and Ron, I am not able to describe your big

hearts in a few words, I can just wholeheartedly thank you both for all the endless support I

have received from you. You all made my PhD journey much more enjoyable. Thanks for

everything.

The Government of Australia and the University of Adelaide are gratefully acknowledged

for providing me Adelaide Scholarships International (ASI) for my study in Australia. The

cluster super-computational facilities and high-performance computing (HPC) system for

this work were provided by eResearch SA organization, which is warmly appreciated.

v

I am deeply indebted to my loving parents, who have always been encouraging me to

continue my education and broaden my mind. Dad, thank you for your infinite support.

Mum, you created a home environment that encouraged me to study and learn. Words

cannot describe how grateful I am for your never ending love and care and all you have

done for me. Without your support, I would never achieve what I have today. I am also

thankful to my caring brothers, Majid, Habib and Mehdi, and my lovely sisters, Zahra and

Shabnam, for all their support, accompany, and encouragement during my PhD study

overseas. Likewise, I would like to thank my parents-in-law for their support, inspiration

and influence.

Finally, and above all, I cannot begin to express my unfailing gratitude and love to my best

friend, my dear husband, Dr Moein N. Kashani. Moein, you have always supported me

through every step of this long journey, and have constantly encouraged me when the tasks

seemed arduous and insurmountable. Without you, I would never have had the inspiration

and motivation to complete this dissertation. Thank you for your warm presence in my life.

Hamideh Elekaei Behjati

November 2015

vi

List of Figures

Figure 2-1 Examples of liquid-liquid microfluidics; (a) integrated microfluidic liquid-

liquid extraction [92] and (b) water-in-oil-in-water (W/O/W) emulsion in microchannels

[93] .........................................................................................................................................9

Figure 2-2 Possible flow patterns in immiscible-liquid microfluidics at (a) cross-junction

[14], the flow regimes from top to bottom show threading, jetting, dripping, tubing, and

viscos displacement regimes, (b) T-junction [94], the flow patterns from top to bottom

show slugs, monodispersed droplet, droplet populations, parallel flows, parallel flows with

wavy interface, chaotic thin striations flow .........................................................................10

Figure 2-3 Typical Ca-based flow map for immiscible-liquid two-phase microfluidics [14]

..............................................................................................................................................11

Figure 2-4 Computational mesh close to the interface for a rising droplet [159] and (b)

adaptive mesh refinement for a droplet impact on a Liquid-Liquid interface [160] ............18

Figure 3-1 An illustration of an SPH weighting function with compact support sited on

particle-i (shown in red) that leads to the particle interacting with all other particles-j

within the support of size O(h). ...........................................................................................35

Figure 3-2 Typical variation through time, 𝒕 ∗= 𝒕𝜸𝝆𝒉𝟑, of the circumference of an

initially square neutrally buoyant droplet suspended in a second continuous phase and the

associated change in pressure difference, ∆𝑷 ∗= ∆𝑷𝒉𝜸, across the interface between the

two liquids. Snapshots of the droplet along the transformation pathway are shown at

various points; the continuous phase is not shown for simplicity. .......................................41

Figure 3-3 Variation of dimensionless pressure drop across the droplet interface with the

inverse of its dimensionless radius, 𝑹 ∗= 𝑹𝒉, for 𝜸 = 𝟎. 𝟎𝟏𝟒𝟐 N/m (obtained using

𝜺 = 𝟏𝟔 N/m2). ......................................................................................................................42

Figure 3-4 Variation of the interfacial tension, 𝜸, with the interaction model parameter that

induces immiscibility here, 𝜺. ..............................................................................................42

Figure 3-5 Droplet shape and velocity field obtained from a typical SPH simulation with

the key characteristics shown. ..............................................................................................43

Figure 3-6 Variation of velocity field and droplet morphology with time for 𝐂𝐚 = 𝟎. 𝟓,

𝐑𝐞 = 𝟎. 𝟏, 𝝀 = 𝟏,and 𝑫𝑯 = 𝟎. 𝟓. .......................................................................................44

vii

Figure 3-7 Variation of velocity field and droplet morphology with time for 𝐂𝐚 = 𝟎. 𝟐,

𝐑𝐞 = 𝟎. 𝟏, 𝝀 = 𝟏,and 𝑫𝑯 = 𝟎. 𝟓. .......................................................................................45

Figure 3-8 Variation of the deformation parameter as defined in Equation (24) with

Capillary Number as predicted by SPH with Np=7701 (solid circles with linear fit shown

as a line; R^2=0.99), SPH with Np=4961 (open circles) Taylor theory as defined by

Equation (25) (solid line) [244], and experimental data (open squares) [245]. ...................46

Figure 3-9 Variation of the droplet tilt angle with Capillary Number as predicted by SPH

with Np=4961(open circles), SPH with Np=7701 (solid circles), Taylor theory as defined

by Equation (25) (solid line) [244], and the experimental values of Sibillo [246] (open

squares). ...............................................................................................................................47

Figure 3-10 Variation of the droplet descent speed with time as predicted by SPH (dash

line) and Han and Tryggvason [247] (solid line) for 𝛈 = 𝟏. 𝟏𝟓, 𝛌 = 𝟏, 𝐄𝐨 = 𝟏𝟎, 𝐎𝐡𝒅 =

𝟎. 𝟐𝟒 and 𝐌𝐨 = 𝟎. 𝟎𝟒. Snapshots of the deforming droplet are shown at various times....48

Figure 3-11 (a) variation of droplet descent speed with time as predicted by SPH for

𝜼 = 𝟏. 𝟏𝟓, 𝑶𝒉𝒅 = 𝟎. 𝟐𝟑, 𝑶𝒉𝒐 = 𝟏. 𝟐𝟓, 𝑬𝒐 = 𝟐𝟒 (solid line), and 𝑬𝒐 = 𝟏𝟒𝟒 (dash line).

Droplet deformation for (a) 𝑶𝒉𝒅 = 𝟎. 𝟐𝟑, 𝑶𝒉𝒐 = 𝟏. 𝟐𝟓, 𝑬𝒐 = 𝟐𝟒, and (c) 𝑶𝒉𝒅 = 𝟎. 𝟐𝟑,

𝑶𝒉𝒐 = 𝟏. 𝟐𝟓, 𝑬𝒐 = 𝟏𝟒𝟒 . ...................................................................................................49

Figure 3-12 (a) variation of droplet descent speed with time as predicted by SPH for

𝜼 = 𝟏. 𝟏𝟓, 𝑶𝒉𝒅 = 𝟏. 𝟏𝟔, 𝑬𝒐 = 𝟐𝟒, 𝑶𝒉𝒄 = 𝟎. 𝟎𝟓 (solid line), and 𝑶𝒉𝒄 = 𝟏. 𝟐𝟓 (dash

line). Droplet deformation for 𝑶𝒉𝒅 = 𝟏. 𝟏𝟔, 𝑬𝒐 = 𝟐𝟒, (b) 𝑶𝒉𝒄 = 𝟎. 𝟎𝟓 and 𝑴𝒐 =

𝟎. 𝟎𝟎𝟎𝟏𝟓, and (c) 𝑶𝒉𝒄 = 𝟏. 𝟐𝟓 and 𝑴𝒐 = 𝟓𝟖 ...................................................................49

Figure 4-1 (a) schematic of cross-junction flow focusing geometry considered in detail in

[14] and here, where 𝑸𝒊, 𝑪𝒂𝒊, 𝝁𝒊 and 𝑼𝒊 = 𝑸𝒊𝑯𝟐 are the flow rate, Capillary Number,

viscosity and superficial velocity of liquid Li, respectively, and 𝜸𝟏𝟐 is the interfacial

tension for the liquid pair; (b) phase morphology map based on the Capillary Numbers of

the liquid L1 (𝑪𝒂𝟏) and the focusing liquid L2 (𝑪𝒂𝟐): (a) threading; (b) jetting; (c)

dripping; (d) tubing; and (e) viscous displacement. The numbers on the map refer to the

conditions that were considered in the study reported on here. ...........................................56

Figure 4-2 Representative outcome for threading regime at 𝐂𝐚𝟏 = 𝟎. 𝟓 and 𝐂𝐚𝟐 = 𝟎. 𝟎𝟐,

which corresponds to point 1 on Figure 4-1(b). ...................................................................60

viii

Figure 4-3 Dimensionless thickness of threads as a function of flowrate ratio as evaluated

using the SPH model (open circles) and equation (1) (solid line). ......................................61

Figure 4-4 Representative outcome of the SPH model for dynamics of the droplet

formation in dripping regime at 𝐂𝐚𝟏 = 𝟎. 𝟏 and 𝐂𝐚𝟐 = 𝟎. 𝟎𝟎𝟑, which corresponds to point

2 on Figure 4-1(b). The snapshots belong to (a) 𝐭 ∗= 𝟎. 𝟎𝟕 , (b) 𝐭 ∗= 𝟎. 𝟐 , (c) 𝐭 ∗= 𝟎. 𝟑𝟖 ,

and (d) 𝐭 ∗= 𝟎. 𝟒 , where 𝐭 ∗= 𝐔𝟏𝐭𝐇 ...................................................................................63

Figure 4-5 Dimensionless length of droplet as estimated by the SPH model (open circles)

and equation (2) (solid line) .................................................................................................63

Figure 4-6 Representative outcome of the SPH model for dynamics of the jetting regime at

𝑪𝒂𝟏 = 𝟎. 𝟑 and 𝑪𝒂𝟐 = 𝟎. 𝟎𝟎𝟓, which corresponds to point 3 on Figure 4-1(b)). The

snapshots belong to (a) 𝒕 ∗= 𝟎. 𝟒𝟐 , and (b) 𝒕 ∗= 𝟎. 𝟒𝟒 , where 𝒕 ∗= 𝑼𝟏𝒕𝑯 .....................64

Figure 4-7 Dimensionless (a) droplet diameter (equation (3)), and (b) thread length

(equation (4)) in dripping regime as evaluated by the SPH model (open circles) and the

related correlations (solid line).............................................................................................65

Figure 4-8 Representative outcome of the SPH model for the tubing regime at 𝑪𝒂𝟏 = 𝟏. 𝟎

and 𝑪𝒂𝟐 = 𝟎. 𝟎𝟎𝟔, which corresponds to point 4 on Figure 4-1(b). ...................................65

Figure 4-9 Representative outcome of the SPH model for the tubing regime at 𝑪𝒂𝟏 =

𝟎. 𝟏𝟗 and 𝑪𝒂𝟐 = 𝟎. 𝟎𝟎𝟎𝟐, which corresponds to point 16 on Figure 4-1(b). ....................66

Figure ‎4-10 Representative outcome for threading regime at (a) 𝐂𝐚𝟏 = 𝟎. 𝟓 and 𝐂𝐚𝟐 =

𝟎. 𝟎𝟎𝟖, which corresponds to point 12 on Figure 4-1(b), (a) 𝐂𝐚𝟏 = 𝟎. 𝟒 and 𝐂𝐚𝟐 = 𝟎. 𝟎𝟏,

which corresponds to point 11 on Figure 4-1(b), and (a) 𝐂𝐚𝟏 = 𝟎. 𝟑 and 𝐂𝐚𝟐 = 𝟎. 𝟎𝟐𝟓,

which corresponds to point 10 on Figure 4-1(b). .................................................................67

Figure ‎4-11 Representative outcome for dripping regime at (a) 𝐂𝐚𝟏 = 𝟎. 𝟎𝟖𝟓 and 𝐂𝐚𝟐 =

𝟎. 𝟎𝟎𝟐 which corresponds to point 15 on Figure 4-1(b), (a) 𝐂𝐚𝟏 = 𝟎. 𝟏 and 𝐂𝐚𝟐 = 𝟎. 𝟎𝟎𝟏,

which corresponds to point 8 on Figure 4-1(b), and (a) 𝐂𝐚𝟏 = 𝟎. 𝟎𝟓 and 𝐂𝐚𝟐 = 𝟎. 𝟎𝟎𝟎𝟔

which corresponds to point 7 on Figure 4-1(b). ...................................................................68

Figure ‎4-12 Representative outcome for jetting regime at (a) 𝐂𝐚𝟏 = 𝟎. 𝟏𝟗 and 𝐂𝐚𝟐 =

𝟎. 𝟎𝟎𝟓 which corresponds to point 15 on Figure 1(b), (a) 𝐂𝐚𝟏 = 𝟎. 𝟏𝟕 and 𝐂𝐚𝟐 = 𝟎. 𝟎𝟎𝟖,

which corresponds to point 14 on Figure 1(b), and (a) 𝐂𝐚𝟏 = 𝟎. 𝟐 and 𝐂𝐚𝟐 = 𝟎. 𝟎𝟎𝟖

which corresponds to point 13 on Figure 1(b). ....................................................................69

Figure 5-1 Schematic of cross-junction flow focusing geometry ........................................76

ix

Figure 5-2 Effect of flowrate ratio on the flow pattern at (a) q=0.1, (b) q=0.5, (c) q=1, (d)

q=2, (e) q=3, (f) q=5, (g) q=10, and (h) q=20, with a fixed viscosity ratio of 𝛍𝟏𝛍𝟐 = 𝟗𝟒78

Figure 5-3 Flow regime map in terms of 𝐂𝐚𝟏, 𝐂𝐚𝟐 and flowrate ratio. Dripping (solid

square), displacement (solid circles), tubing (solid triangle), threading (solid diamond), and

jetting flow regime (solid pentagon). ...................................................................................79

Figure 5-4 Effect of the flowrate quantities on the droplet size for the constant flowrate

ratio of q=2 and at (a) 𝐐𝟏 = 𝟏𝟎 µ𝐥𝐦𝐢𝐧 , 𝐂𝐚𝟏 = 𝟎. 𝟎𝟒, 𝐂𝐚𝟐 = 𝟎. 𝟎𝟎𝟎𝟖𝟔 , (b) 𝐐𝟏 =

𝟐𝟓 µ𝐥𝐦𝐢𝐧 , 𝐂𝐚𝟏 = 𝟎. 𝟏, 𝐂𝐚𝟐 = 𝟎. 𝟎𝟎𝟐𝟐 and (c) 𝐐𝟏 = 𝟒𝟎 µ𝐥𝐦𝐢𝐧 , 𝐂𝐚𝟏 = 𝟎. 𝟏𝟔𝟖,

𝐂𝐚𝟐 = 𝟎. 𝟎𝟎𝟑𝟔 , (d) dimensionless droplet diameter vs. flowrate of 𝐐𝟏 .........................80

Figure 5-5 Effect of viscosity ratio of (a) λ=1484 and Ca2=0.0036, (b) λ=264 and

Ca2=0.02, and (c) λ=24 and Ca2=0.2, on flow pattern at q=10 and Ca1=0.5 .......................81

Figure 5-6 Effect of viscosity ratio of (a) λ=1484 and Ca2=0.00054, (b) λ=264 and

Ca2=0.003, and (c) λ=24 and Ca2=0.033 on flow pattern at q=8 and Ca1=0.1 ....................82

Figure 5-7 Effect of viscosity ratio of (a) λ=1484 and Ca2=0.0009, (b) λ=264 and

Ca2=0.005 and (c) λ=24 and Ca2=0.054, on flow pattern at q=4.4 and Ca1=0.3 .................83

Figure 5-8 Effect of width ratio of (a) 𝛇 = 𝟎. 𝟓 and Ca2=0.0036, (b) 𝛇 =1 and Ca2=0.0009,

and (c) 𝛇 =2 and Ca2=0.00026 on droplet length at 𝐪 = 𝟐 and 𝐂𝐚𝟏 = 𝟎. 𝟎𝟒 ...................84

Figure 5-9 Normalized droplet length in terms of the width ratio 𝛇 and the flowrate ratio q,

SPH simulation (solid diamond) and Equation (1) (open diamond) for𝛇 = 𝟎. 𝟓, SPH

simulation (solid circle) and Equation (1) (open circle) for 𝛇 = 𝟏, and SPH simulation

(solid square) and Equation (1) (open square) for 𝛇 = 𝟐 .....................................................85

Figure 5-10 Effect of an inlet angle of (a) θ=45⁰, (b) θ=90⁰ and (c) θ=135⁰ on the droplet

length at q=2, Ca1=0.1 and Ca2=0.002 .................................................................................86

x

List of Tables

Table ‎4-1: Properties of liquid pair considered ....................................................................59

xi

Abbreviations

Ca Capillary Number

CAD Computer Aided Design

CFD Computational Fluid Dynamics

DNS Direct Numerical Simulation

DPD Dissipative Particle Dynamics

ENS Explicit Numerical Simulation

Eo Eotvos Number

FDM Finite Difference Method

FEM Finite Element Method

FVM Finite Volume Method

HPC High-Performance Computing

LBM Lattice Boltzmann Method

LGA Lattice Gas Automata

LOC Lab-On-a-Chip

LoG Laplacian of Gaussian

LSM Level Set Method

MD Molecular Dynamics

Mo Morton Number

NNPS Nearest Neighbouring Particle Searching

Oh Ohnesorge Number

PPE Pressure Poisson Equation

Re Reynolds Number

SD Standard deviation

SPH Smoothed Particle Hydrodynamics

VOF Volume of Fluid

µTAS Micro Total Analysis System

3D Three-dimensional

2D Two-dimensional

xii

Nomenclatures

a [m] Length of the major axis of the ellipsoid

A [m2] Area

b [m] Length of the minor axis of the ellipsoid

c [ - ] Colour function

CD [ - ] Drag coefficient

g [m/s2] Gravitational acceleration

h [m] Cut off distance

I [ - ] Unit tensor

L0 [m] Initial distance between particles

m [kg] Mass of particle

Np [ - ] Number of SPH particles

P [Pa] Pressure

∆𝑃 [Pa] Pressure drop

r [m] Position

R [m] Radius of droplet

𝑟𝑖𝑗 [m] Distance between particles i and 𝑗

t [s] Time

t [ - ] Time step

∆𝑡 [s] Time step size

u [m/s] Volume averaged fluid velocity

U [m/s] Superficial velocity

𝑈𝑚𝑎𝑥 [m/s] Maximum characteristic velocity

V [m3] Particle volume

V [m/s] Velocity vector

𝑊 [m-3

] Smoothing kernel

x [m] Distance vector

[ - ] Interfacial tension

[ - ] Dirac delta function

[ - ] Thread thickness

𝜀 [ - ] Immiscibility model parameter

[ - ] Density ratio

λ [ - ] Viscosity ratio

µ [Pa.s] Dynamic viscosity

𝜌 [kg/m3] Density

σ N/m2 Stress

τ N/m2 Shear stress

𝜉 [ - ] Channel width ratio

Gradient operator

xiii

Subscript

c Value for continuous phase

i Value for particle of interest

j Value for neighbouring particles

j Value for jetting

d Value for droplet

p Value for particle

w Related to wall

α -coordinate direction

β β-coordinate direction

* Intermediate state

Superscript

α Number of dimensions

β Number of dimensions

* Dimensionless value

xiv

Table of contents

Declaration…. ....................................................................................................................... i

Abstract……. ....................................................................................................................... ii

Achievements ...................................................................................................................... iii

Acknowledgements ............................................................................................................. iv

List of Figures ..................................................................................................................... vi

List of Tables ........................................................................................................................x

Abbreviations ..................................................................................................................... xi

Nomenclatures ................................................................................................................... xii

Table of contents .............................................................................................................. xiv

Introduction ....................................................................................................1 Chapter 1:

Literature Review...........................................................................................6 Chapter 2:

2.1 Microfluidics ................................................................................................................6

A brief history of microfluidics .............................................................................6 2.1.1

Advanced techniques to make microfluidics .........................................................7 2.1.2

Brief overview of current microfluidic flows ........................................................7 2.1.3

2.2 Multiphase microfluidics ..............................................................................................8

2.3 Immiscible Liquid-Liquid Microfluidics ......................................................................8

Experimental study of immiscible liquid-liquid microfluidic system .................12 2.3.1

2.4 Numerical study of immiscible liquid-liquid microfluidic system ............................15

2.5 Smoothed Particle Hydrodynamics (SPH) Method ....................................................21

General SPH Formulation ....................................................................................23 2.5.1

Neighbour searching algorithm ...........................................................................25 2.5.2

SPH Algorithm ....................................................................................................26 2.5.3

2.6 Conclusion ..................................................................................................................26

Modelling of Immiscible Liquid-Liquid Systems by Smoothed Particle Chapter 3:

Hydrodynamics ..................................................................................................................28

3.1 Introduction ................................................................................................................31

3.2 The Method ................................................................................................................34

Governing Equations ...........................................................................................34 3.2.1

xv

SPH Discretization ...............................................................................................35 3.2.2

Immiscibility Model ............................................................................................37 3.2.3

Solution Algorithm ..............................................................................................38 3.2.4

3.3 Results and Discussion ...............................................................................................40

Young-Laplace and Parameterization of Immiscibility Model ...........................40 3.3.1

Deformation of a Confined Droplet in a Linear Shear Field ...............................42 3.3.2

Deformation of Free-Falling Droplet ...................................................................47 3.3.3

3.4 Conclusions ................................................................................................................50

Smoothed Particle Hydrodynamics-based Simulations of Immiscible Chapter 4:

Liquid-Liquid Microfluidics in a Hydrodynamic Focusing Configuration ..................51

4.1 Introduction ................................................................................................................54

4.2 Model and Study Details ............................................................................................58

Model ...................................................................................................................58 4.2.1

Study Details ........................................................................................................59 4.2.2

4.3 Results and Discussion ...............................................................................................59

4.4 Conclusion ..................................................................................................................66

4.5 Supplementary Information ........................................................................................67

Effects of Flow Parameters and Geometrical Characteristics on the Chapter 5:

Dynamics of Two Immiscible Liquids in a Hydrodynamic Focusing Microgeometry 70

5.1 Introduction ................................................................................................................73

5.2 Model Details .............................................................................................................75

5.3 Results and Discussion ...............................................................................................76

Effects of flowrate and flowrate ratio ..................................................................76 5.3.1

The effect of the viscosity ratio ...........................................................................80 5.3.2

Effect of geometry ...............................................................................................83 5.3.3

5.3.3.1 Width ratio effect ..........................................................................................83

5.3.3.2 Inlet angle effect ............................................................................................85

5.4 Conclusion ..................................................................................................................86

Conclusion .....................................................................................................88 Chapter 6:

References… .......................................................................................................................90

1

Introduction Chapter 1:

Microfluidics, the study of fluid motion at a micro-scale, is currently a growing research

area. It has practical applications in the design of systems in which a small amount of fluid

in channels with dimensions in the order of tens to hundreds of micrometres is manipulated

and processed [1-3]. Increased understandings and developments in microfluidics will lead

to drastic progress in biotechnology, drug discovery, microelectronics, energy generation,

pharmaceutical, food and many other industries. For instance, microfluidics allows the

breakup of a droplet of blood into thousands of micro-droplets to perform biological tests

in parallel [3] , and this method is far better than those current ones, which require much

larger samples and have slower reaction times.

At micro-scale, a high surface area to volume ratio results in that the interfacial forces,

such as viscous force and surface tension, become dominant and intensely influences the

fluid flow behaviour [4]. The advancement in micro-engineering and the scale-dependence

behaviour of fluids allow us to be able to accurately control multiple fluid interfaces,

which have been utilised in different areas ranging from chemical processing and energy to

medical science and biology [4]. Practically, microfluidics offers a wealth of advantages

including accurate manufacturing of devices, and precise control of interfaces.

Microfluidic technology allows us to manipulate the deformable and moving interfaces in

multiphase systems to achieve the benefits which cannot be obtained from the macro-scale

flow devices.

Of particular interest are such multiphase microfluidic systems, such as micro-extraction

used in drug discovery [5, 6], and micro-emulsification in food, cosmetic, pharmaceutical

and oil industries [7-12]. The multiphase flow and moving/deformed interfaces

unfortunately are far less well developed compared to the single-phase counterpart.

Although a variety of experimental studies have been done so far on the droplet-based

microfluidics [13-21], they are still sporadic, because of only limited material used for

fabrication, a small range of flow parameters, and few configurations and geometries

tested during measurement. Importantly, these reports are more focused on the observed

phenomena, and the physics for such phenomena is not well clarified. Therefore, a full

2

understanding of the fundamental physics of multiphase liquid flow dynamics and the

effects of geometry in the microfluidics and process parameters on the moving interfaces

are essentially needed. Experimental studies for microchannel multiphase flows are quite

challenging and expensive, while analytical solutions at such small scales are limited to

very few simple cases. Consequently, modelling and numerical studies are often required

to address complex flows, which are poorly understood, and to facilitate developments in

microfluidics. To this end, this project aims to develop a numerical model for the

immiscible liquid-liquid phase in microfluidics, which can be used as foundation for

building a design capability for multichannel microfluidics devices.

Due to complex features of multiphase flow in microchannel devices including moving

interfaces, a large ratio of the surface to volume and micro-mechanics such as capillary

forces and potential intricate boundaries (e.g. fluid-fluid interface dynamics), numerical

simulation of multiphase flows in microfluidics is not straightforward. Computational

Fluid Dynamics (CFD) has been extensively used to simulate fluid flows in micro-fluidic

devices as indicated by literature [22, 23]. These have been, however, largely attributed to

a single phase systems. In the case of a multiphase system, the dearth of studies is due to

challenges in explicitly tracking the individual phases. It is not possible to simply adopt a

multi-fluid modelling approach for the immiscible-liquid multiphase microfluidic system,

which treats the immiscible liquids as a homogeneous mixture without any interfaces and

is typically applied for macro-scale counterparts [24].

Explicit numerical simulation (ENS) is a general concept where one attempts to model all

the significant phenomena explicitly rather than in a mean field or average way as is done

in many engineering models; in this sense, direct numerical simulation (DNS) of

turbulence is a class of ENS [25]. Explicit simulation of the multiphase system allows us to

track individual phases, deal with complex phase geometries and topologies, and consider

multiple time and/or length scales in one simulation [26]. In most of the cases in

microchannels, it is become important to explicitly model the behaviour of the individual

phases and the way they have interaction in multiphase flow. In addition to that, based on

the application of microfluidic devices, we are interested in the shape of dispersed phase

rather than the other things. It means that, in addition to the continuous fluid phase,

disperse phase must be explicitly modelled.

3

In the explicit simulation, modelling and tracking the interface with large deformation and

topological changes is the main challenge of traditional CFD methods which need high

resolution meshes, mesh refinement, proper mesh aspect ratios and intricate discretization

techniques. Among those challenging issues, mesh generation and regeneration is a

problematic task. It is a time-consuming and computationally-expensive process for

problems with a complex geometry. Constructing a regular mesh for an irregular or

intricate geometry may carry more computational load than solving the problem itself.

Equally challenging tasks are to determining accurate locations of the free surfaces,

deformable boundaries and moving interfaces within the frame of the fixed grid. This

mesh-based numerical simulation is an expensive operation, which also bears another

source for modelling errors and numerical instability [27-30]. Therefore, simulation of

multiphase microfluidic systems using mesh-based ENS methods is computationally

expensive and inefficient. Therefore, algorithms based on the Lagrangian meshless method

should be considered in these problems.

Meshless methods have been developed over the past few decades with more efficient

computational schemes designed for more complicated problems. Regarding dynamic

simulations of continuum materials experiencing large deformations, the mesh creation

processes in conventional mesh-based methods are involved by inevitability re-meshing

the domain with the intention to keep away from weaken accuracy of the results owing to

reduced aspect ratios of the mesh and termination of the simulation because of mesh

complication and inversion. In this context, meshless particle methods have no inherent

restrictions on the extent of deformation of the considered domain, since the connectivity

between particles is updated for each computation step. Therefore, particle method holds

advantages for large deformation simulations as mesh-based methods require constant re-

meshing to obtain well-behaved elements. Moreover, the method is capable to be

connected with a CAD database much simpler than mesh-based approaches due to the fact

that this method does not need to generate an element grid. Meshless particle methods also

produce more accurate results. Discretization techniques in meshless particle methods

allow prices depiction of geometric object in compared with traditional mesh-based

methods [27, 31, 32].

4

One preferable meshless particle method is Smoothed Particle Hydrodynamics (SPH) [33]

which is a fully Lagrangian meshless method. For computational fluid dynamics problems

governed by the Navier-Stokes equations, SPH is a powerful solution technique. In the

SPH method, a set of particles that have material properties and interact with each other

under the area delimited by a weight function or smoothing function, demonstrate the state

of a system [34]. With the aid of these discrete particles, the governing equations are

discretised and a diversity of particle-based formulations is used to calculate the

acceleration, velocity and local density of the fluid. Afterwards through an appropriate

equation of state, fluid pressure is calculated and hence the particle acceleration will be

achieved using the pressure gradient and density. Since the movement of the fluid is

illustrated by the movement of the particles, there is no extra procedure for interface

tracking in multiphase flows [35].

Compared to the long-established mesh-based numerical methods, SPH has some

particular benefits. Firstly, SPH is a meshless particle method with Lagrangian nature,

which can record temporal profile of the material particles and so flow of the system can

be accordingly obtained. Secondly, by positioning appropriate particles at particular

locations at the early stage in the process of the simulation, free surfaces, interfaces and

moving boundaries can be traced. As a consequence, interface tracking is explicit through

capturing the locations of the particles, much simpler than Eulerian methods. Therefore,

SPH is an ideal candidate for modelling free surface and interfacial flow problems.

Thirdly, SPH is also appropriate for situations where the object of interest is not a

continuum, such as micro-scale and nano-scale simulation in bio-engineering and nano-

engineering. In addition, SPH is reasonably less complicated in numerical implementation,

especially in comparison with mesh-based methods. Therefore, the SPH will be chosen to

apply to the immiscible liquid-liquid flow in microchannel system in this project.

There are only a limited number of numerical studies with regard to immiscible-liquid

multiphase microfluidics in literature [23, 35-37]. Among them, lattice Boltzmann method

(LBM) has been mostly employed for simulation of droplet-based microfluidic systems

[32, 38, 39], however, LBM-based simulation suffer from stability problems [40].

This project will be one of the first to apply SPH, a powerful meshless particle method, to

explicitly modelling immiscible liquid-liquid flows in microchannels. This novel

5

modelling will enable us to better understand the effect of flow parameters, microfluidic

geometry, and interfacial phenomena in such microfluidic devices. Moreover, this work

will provide great insights in designing microfluidic devices involving immiscible liquid-

liquid flows.

The aim of the work reported in this thesis was directed towards developing the application

of meshless method to the explicit numerical simulation of immiscible- liquid flows in

microfluidic systems so as to achieve an increased understanding of fluid dynamics and

phase morphologies within the phases in such systems. The study reported in this thesis

addresses these concerns through two related developments. ENS of immiscible liquid-

liquid flows based on incompressible SPH method with particular regard to correctly

capturing the liquid-liquid interface represents the first development. This goal was

achieved through modelling of the system, code development, and validation of the model

by applying it to a range of interfacial problems (Objective I). The second development of

the work has been divided into two parts: the first involved the application of the proposed

model for the study of hydrodynamic focusing of a liquid flowing in a deep microchannel

by a second immiscible liquid introduced through identical but symmetrically opposed

microchannels (Objective II). The second involved building a basis for optimal design of

the microfluidic system through determining the effects of various flow parameters and

geometry characteristics on the flow dynamics and topological changes of the multiphase

flow system (Objective III).

The thesis is laid out as follows. Chapter 2 provides a literature review that identifies the

strength and weak points of the different CFD methods, some research gaps that support

the aim of this thesis, and the technical background of the method used. Chapters 3, 4 and

5 report the work undertaken to achieve Objectives I, II and III, respectively; these

chapters are papers, all of which will be submitted to the Physics of Fluids. The final

chapter draws together the work reported here to make overall conclusions and suggest

future work.

6

Literature Review Chapter 2:

2.1 Microfluidics

A brief history of microfluidics 2.1.1

Microfluidics is defined as the study of the behaviour of fluid flow geometrically confined

to a micro scale with dimensions varying roughly between 100 nm and 100 µm [1, 2]. The

potential for microfluidics to revolutionize the energy, chemical processing and medical

fields has attracted significant interest and research effort over the past few decades. In the

1980s, research on microfluidic devices appeared due to the availability of the

micromechanics technology [41]. With the development of the concept of “Micro Total

Analysis System” (µTAS) [42-46], and subsequently Lab-on-a-Chip (LOC) [47-50],

micro-fabrication techniques are capable of making microchannels with complex structures

from resistant and inactive substrate materials which are appropriate for dealing with

chemicals and biological samples [43-45, 51, 52].

Characteristically, a microfluidic system is composed of microfluidic channels, micro-

reservoirs and reaction chambers at micro-meter length scales. It is possible to accomplish

the unit operations including mixing, separation, reaction, detection and analysis, through

one integrated micro-device [53, 54]. There have been several attempts for integration and

parallelization of microfluidic devices. Best examples include Quake et al [48] for

biological systems, Wei et al [55] and Torii et al [56] on droplet generation, and Tetradis-

Meris et al [57] for emulsion formation. Applications of microfluidics have been explored

in colloid science [58], plant biology [59] and process intensifications [60-62]. In

particular, process intensifications involve the micro-chemical engineering technology [63-

67], which can lead to improved quality products, increasing yields, reduction in

investment costs, lower energy consumption, reduced environmental and safety risks [68],

and control of extreme reactions [66, 69-71].

7

Advanced techniques to make microfluidics 2.1.2

Since the beginning of the microfluidic era, the techniques of fabrication of microfluidic

devices have been in constant improvement. Examples of the fabrication techniques

include soft-lithography [72], laser ablation [73], and X-ray lithography [74]. The main

challenges of the common microfabrication techniques are connected with the construction

of open microchannels and assembly of machined substrates to create an integrated

structure. Open microchannels can be made through etching process which is expensive,

and may involve aggressive materials [75].

Among the materials used for fabrication of the microfluidic chips, polydimethylsiloxane

(PDMS) is of particular interest, since it is transparent, flexible, replicable, and low-cost,

however, it swells and deforms in the presence of organic oils. Therefore, more robust

materials such as glass, metal, silicon, and thiolene are needed to be applied for droplet-

based microfluidic systems.

Moreover, significant efforts have to be done to develop functional parts of the

microfluidic system, including microvalves for isolation purposes, and digital pumps for

precise control of flow rates [76]. These difficulties associated with the microfabrication

techniques and materials restrict the experimental research studies. Numerical studies,

therefore, should be called to facilitate building of understanding of flow behaviour in such

systems.

Brief overview of current microfluidic flows 2.1.3

Single phase microfluidic systems are based on continuous liquid flows through

microchannels and are efficiently used for trapping, focusing and/or separation of solid

particles [77], cells [78], and single DNA molecules [79]. Protein analysis through

immobilized microfluidic enzymatic reactors is also another application of the single phase

microfluidics [80]. Single phase microfluidic systems, however, are less suitable for tasks

requiring a high degree of flexibility.

In contrast, multiphase flows provide different mechanisms for improving the performance

of single phase systems. For example, immiscible multiphase microfluidic systems enable

manipulating of discrete volumes of fluids in immiscible phases that benefit large

8

interfacial areas, fast mixing and reduced mass transfer limitations [81]. Examples include

emulsions and droplet-based microfluidics with a wide range of applications from

chemical and material processing to biology and medicine [4].

2.2 Multiphase microfluidics

Fluids of interest in many engineering and industrial applications involve more than a

single phase. Multiphase microfluidic systems play a significant role in drug discovery,

biomedical purposes and many other industrial fluid dynamical problems involving fluid

transport and interfacial phenomena [43]. For instance, biomedical examples in Lab on

Chip devices include emulsions of bacterial cells (solid-liquid-liquid flow) and analysis of

whole blood samples (solid-liquid flow) [49, 82]. Other applications of multiphase micro-

flow systems are synthesis of a small-amount of fine chemicals or active pharmaceutical

ingredients in chemical and pharmaceutical industries, where solid catalysts are used for

gas or liquid reactants [44, 83, 84] . In this case, the hydrodynamics of multiphase flow has

the most important effect on the reactor performance and consequently in the design of a

reactor [44]. Some investigation has been done on the design of microchannel mixers in

order to enhance the advection phenomena in liquid-liquid mixing processes, such as the

micro-mixer designed by Bringer et al [85] for dilute solutions in aqueous drops dispersed

in an oily continuous phase. Liquid-Liquid phase interactions have also been observed in

mono-dispersed emulsions such as water-oil two phase flows in microfluidic devices [11,

57].

2.3 Immiscible Liquid-Liquid Microfluidics

Of particular interest in this project is multiphase flow of immiscible liquids in

microfluidics. Through interaction of immiscible-liquid micro-flows, the flow and

capillary instabilities emerge and threads, jets or monodispersed droplets form. This is the

physical basis for producing emulsions and dispersion of one liquid phase in the other one

[86, 87]. Such microfluidic systems are of relevance to emulsifications used in food [8, 9],

cosmetics [7], pharmaceutical [6, 12] and oil industries [10, 11, 88], plus liquid-liquid

extraction used in drug discovery [5, 89-91]. Figure 2-1 displays a schematic illustration of

9

(a) integrated microfluidic liquid-liquid extraction [5, 92] and (b) water-in-oil-in-water

(W/O/W) emulsion [93].

(a) (b)

Figure 2-1 Examples of liquid-liquid microfluidics; (a) integrated microfluidic liquid-liquid

extraction [92] and (b) water-in-oil-in-water (W/O/W) emulsion in microchannels [93]

Immiscible liquid interfaces are important throughout many areas, particularly chemistry and

biology, because of their unique chemical and interfacial behaviour. There is a capillary

pressure across a curved liquid interface at equilibrium, which is described by Young-

Laplace equation as:

∆𝑃𝑐𝑎𝑝 = 𝛾𝑘 (1)

Where, 𝛾 and 𝑘 are interfacial tension and local mean curvature at the interface, respectively.

According to the equation, the variation of the pressure drop across the interface is a function

of the interface curvature. For instance, the capillary pressure across the interface of a

spherical water droplet with a radius of 1 µm in air is approximately 1000 times, larger than

that of the capillary pressure of the same droplet with a radius of 1 mm [81]. When two

immiscible liquids are brought into contact with each other, various flow patterns and phase

10

distributions can occur due to the interplay between various physical forces such as

interfacial, gravitational, viscous and inertial forces. Figure 2-2 shows the different flow

patterns such as threading, jetting, dripping, tubing, and viscous displacement [14] in the

immiscible-liquid microchannel flow. At micro-scale, because of the laminar nature of the

flow, the flow pattern is dictated by the competition between viscous forces and capillary

force at the interface. The former tends to drag the liquids along the microchannel, while the

latter tends to minimize the interface of the two liquids.

The competition of the various forces is affected by flow parameters, microchannel geometry

and surface chemistry. These effects can be captured by a number of dimensionless numbers

expressing the relative importance of the competing forces [2, 81, 86]. Scaling these forces

based on flow conditions and fluid characteristics helps predict multiphase microfluidic

behaviour, and introduce design guidelines for microfluidic devices.

(a) (b)

Figure 2-2 Possible flow patterns in immiscible-liquid microfluidics at (a) cross-junction [14],

the flow regimes from top to bottom show threading, jetting, dripping, tubing, and viscos

displacement regimes, (b) T-junction [94], the flow patterns from top to bottom show slugs,

monodispersed droplet, droplet populations, parallel flows, parallel flows with wavy interface,

chaotic thin striations flow

One of the most important dimensionless numbers is capillary number (Ca) which is defined

as the relative effect of viscous force to interfacial tension acting across the interface between

two media:

11

𝐶𝑎 =𝜇 𝑈

𝛾 (2)

Where 𝜇, 𝑈 and 𝛾 denote viscosity, velocity and interfacial tension. Capillary number is

calculated for both dispersed phase 𝐶𝑎𝑑 and continuous phase 𝐶𝑎𝑐 . According to literature in

the microfluidic context, Capillary numbers are usually less than 1.0 which means that

capillary forces (or interfacial forces) play a more important role in the flow behaviour than

viscous forces. When Ca is very small around zero, the viscous shearing effect can be

neglected and the interfacial tension is enough to keep the interface away from any

deformation, however by increasing the Ca, deformation of the interface increases and may

lead to formation of droplets and threads. Figure 2-3 shows a typical flow map for an

immiscible-liquid pair [14] based on Ca values of the dispersed and continuous phases. The

flow map reveals how changes in capillary numbers can influence the flow patterns and

instabilities. Five typical flow patterns are indicated by the map including threading, jetting,

dripping, tubing, and displacement. The threading flow pattern refers to a long core thread of

the main channel liquid. The jetting flow pattern refers to a thin thread that emits droplets

smaller than size of the channel at a distance from the junction. In dripping flow regime, the

droplet forms near the junction. The tubing flow pattern is viscous stress controlled and refers

to a viscous core. At low momentum of the continuous phase viscous displacement occurs

[14].

Figure 2-3 Typical Ca-based flow map for immiscible-liquid two-phase microfluidics [14]

12

In addition, the influence of flow rates on the flow pattern and topological changes can be

determined from Reynolds number (Re), which represents the ratio of inertia to viscous

forces:

𝑅𝑒 =𝜌𝑈𝑑

𝜇 (3)

Where ρ and d are fluid density and hydraulic diameter of microchannel, respectively.

Literature shows that the flow rate of about 1 l/s is typically applied in microchannels with

a dimension in the order of 100 m [14, 95]. Using these data, Re can be estimated in the

orders of 0.1 to 1. At a small Reynolds Number (Re < 1), the flow is governed by viscous

stresses and pressure gradients, and inertial effects can be omitted, which reflects the typical

microfluidic characteristics. The other important dimensionless numbers in multiphase

microfluidic systems include Weber number, which implies the influence of inertia with

respect to surface forces (i.e. interfacial tension), and Eotvos number which refers to the ratio

of gravity force to surface forces.

Experimental study of immiscible liquid-liquid microfluidic system 2.3.1

Nakajima et al. [17, 96] and Knight et al. [97] were pioneers for creating monodisperse

emulsions in a microfluidic device, the former through an array of cross-flow microchannel

network and the latter using a hydrodynamic focusing microfluidic system. Similarly,

Thorsen et al [15] employed a T-junction microfluidic configuration for generating dispersed

water droplets in oil emulations, and they demonstrated how the interface instability evolves

due to a competition between shear forces and surface tension. Flow instability usually causes

the formation of monodispersed droplets in microfluidic devices, whereas, a capillary

instability is anticipated to create segmented flows with identical droplets in liquid-liquid

systems [98, 99]. Stone et al. [100] experimentally investigated formation of liquid droplets

in an immiscible liquid through a flow-focusing microfluidic device. They concluded a phase

diagram that illustrates the drop size as a function of flowrate and flowrate ratio of two

liquids. Dummann et al. [101] developed a capillary-microreactor concept and

experimentally studied nitration of a single ring aromatic in an exothermic liquid-liquid

capillary-microreactor which is proposed for intensification of heat and mass transfer.

Dreyfuss et al. [102] used a microfluidic cross-junction for their experiments to examine the

13

role of wetting properties in controlling flow patterns. Moreover, they demonstrated that

phase distribution is influenced by superficial velocities of both continuous and dispersed

phases in immiscible-liquid microchannel flows. Cramer et al. [103] investigated dripping

and jetting flow regimes in co-flowing streams. Furthermore, Okushima et al. [104] reported

a new technique for producing double emulsions (water-in-oil-in-water dispersions) using a

two-step droplet formation in T-shaped microchannels. Over the past decade, different

microchannel configurations including T-junctions [35, 105], Y-junctions [18], cross-

junctions [20, 38], flow focusing [14, 16, 100, 106-109], concentric injection [110] and

membrane emulsification [111-113] have been reported for droplet formation process.

Understanding of flow dynamics and droplet formation mechanisms in the proposed

microfluidic configurations has also been well progressed to facilitate a smart design of

liquid-liquid microfluidic devices. In this context, Garstecki et al. [21] studied the effects of

flow rates and material parameters on the mechanism of droplet formation and breakup in

liquid-liquid emulsions using a microfluidic T-junction. Constraining the liquid-liquid flow

between microchannel walls influences the rupture of an annular liquid core flow into

droplets. At a small Capillary number, less than 10-2

, the shear stress forced on the interface

of the appearing droplet becomes so inadequate that the force is not able to deform

significantly the interface. Thus, the droplet blocks approximately the whole cross-section of

the microchannel and confines the continuous flow to the walls. Garstecki et al. [21]

concluded that increasing Ca in a liquid-liquid system may lead to three different patterns for

the formation of droplets including squeezing, dripping and jetting. Droplet generation in the

squeezing regime is stimulated with pressure instabilities during the breakup process.

A successful example of the microfluidic configuration for producing monodispersed droplets

is microfluidic hydrodynamic focusing wherein a centrally located flow is narrowed or

hydrodynamically focused by two neighbouring flows through cross-like microchannels. For

example, Xu et al. [16] set up a three-stream laminar flow configuration to hydrodynamically

focus a more viscous liquid surrounded by two inviscid liquids to facilitate droplet formation

through control of the flow rate ratio between the continuous phase and the dispersed phase.

The efficiency of droplet formation in microchannels is mainly affected by the nature of

liquids (viscosity and interfacial tension), flow parameters (flowrate ratio, capillary number),

and geometrical characteristics. In this respect, Cubaud et al. [14] performed a

14

comprehensive experimental study on a few Newtonian immiscible liquids in a microfluidic

hydrodynamic focusing device. They investigated the competition between viscosity and

interfacial tension effects on the formation of different flow regimes using hydrodynamic

focusing over a range of viscosities and interfacial tensions at a fixed geometry. Although

their experiments cover a useful range of flowrate ratios, viscosity ratios and Ca, their results

provide an insight to the flow behaviour in the conditions limited to the examined cases. We

meant to develop a reliable simulation to cover a wide range of parameters in order to

increase our predictive capacity for a smart design of immiscible-liquid microfluidic devices.

Despite the various studies on droplet formation and breakup process through different

microfluidic junctions have been reported so far, a detailed study on the effects of

geometrical characteristics and junction/inlet design of microchannels is still missing. The

flow and junction geometry, the cross-section of the channels, the dimensions and the angle

made by the inlet channels all can be influential on flow pattern. There are only a few

researches on the geometry effects reported by literature. For instance, Ménétrier-Deremble

et al. [114] found out the controlling effect of flow geometry in breakup process through

their experiments related to droplet breakup in different-angled microfluidic T-junction

configuration. Also, Tan et al. [115] considered a microfluidic system through a trifurcating

junction to illustrate the effect of junction geometry on droplet coalescence. They deduced

that wider junction lessens the shear force and aligns the droplets to the core section of the

channel and let droplets to create a zigzag pattern resulting in a lateral coalescence. Abate et

al. [116] demonstrated that droplet formation through a flow-focusing device occurs at

higher capillary numbers than to that of a T-shaped microfluidic junction. Furthermore, Priest

et al. [117] designed a Y-shaped microchip for the recovery of copper from leach solutions

in which two liquid phases (organic and aqueous) are contacted in parallel through the main

channel and separated at a Y-junction. They also concluded that the proposed microchip has

greater potential for recycling of volatile liquids. More recently, Li et al. [19] focused on the

droplet formation mechanism in an immiscible liquid-liquid system with T-junction

distributor and noticed that smart geometric design of micro-fluidic devices can affect

controlling the droplet formation on demand. Also, Kunstmann-Olsen et al. [108] studied

experimentally and theoretically a two-dimensional microfluidic cross-flow and showed that

the local geometry of the microchannel-junction affects the hydrodynamic focusing

15

behaviour. We aimed to figure out these effects through our simulation and present a robust

model to predict behaviour of immiscible-liquid flows under the different conditions of

process, material and geometry in a microfluidic hydrodynamic focusing system.

2.4 Numerical study of immiscible liquid-liquid microfluidic system

There is an increasing need to understand and exploit the link between liquid-liquid phase

morphology and fluid dynamics within the phases on the one hand and the processing

conditions that produce them on the other. For example, the droplet size distribution of an

emulsion produced through agitation is a function of the balance between droplet breakup

and coalescence, which can be controlled by the surfactant and stabilizer concentration and

relative velocities of the phases [118-120]. Microfluidic production of encapsulates

provides a further example: in this case, encapsulate morphology can be varied through the

nature of the flow-focusing in the microchannels and, amongst other things, the

continuous-to-dispersed phase flow rate ratio [121, 122]. The challenges faced in

experimentally elucidating these types of relationships are significant, however. For

example, the visualization of the morphologies of the phases and flow fields therein are

still very much in their infancy [123-126]. Models that treat the phases and interfaces

between them explicitly have, therefore, an important role to play in building

understanding of liquid-liquid systems and exploiting this understanding in a systematic

way.

Models of liquid-liquid systems in which the individual phases and interfaces between

them are treated explicitly are long-standing [127-130]. The earliest models, which focus

on droplets in a continuous phase, include those of Taylor [131], Mason and co-workers

[132, 133], Cox and co-workers [134, 135], and Acrivos and co-workers [136-138]

amongst others. Whilst these models were important in building understanding, they are

limited by a good number of the simplifying assumptions [139], including negligible

inertial effects (i.e. small Reynolds numbers), small viscosity ratio ranges, and regular

droplet shapes (e.g. spheres; ellipsoids). Indeed, it was shown by Mason et al. [135] that no

wholly theoretical approach can encompass the entire range of process parameters

adequately. Phenomenological approaches have been adopted to overcome some

limitations associated with wholly analytical models. For example, Maffettone et al. [87]

16

used such an approach, determining its parameters by ensuring it matches analytical model

results in appropriate limits. Such models are, however, also limited by, for example,

underpinning assumptions about droplet shape and the absence of any connection between

the model parameters and underlying fundamentals.

Adoption of wholly numerical approaches can, in principle, overcome limitations faced by

analytical and related models. Some of the earliest numerical models include those of

Acrivos and co-workers [139-141], who studied the deformation and breakup of a viscous

droplet freely suspended in another liquid within extensional and shear flow fields. More

recent examples include the works of Loewenberg, Hinch and Davis [142-148], which are

based on a boundary-integral approach. Afterwards, interface tracking techniques such as

Volume of Fluid (VOF) [130, 149, 150] were used for simulation of immiscible fluids. For

instance, Richards et al. [151] simulated formation of immiscible liquid jets and their

transient to breakup into drops numerically based on the VOF method. Although this

model represent acceptable mass conservation properties, the geometrical characteristics of

the interface, the maintenance of a sharp boundary between the different fluids, and the

surface tension computation are cumbersome to be implemented by VOF, since the volume

fractions are non-smooth functions [149]. As a remedy, a number of techniques such as

Level Set method (LSM) [152, 153] were developed to improve the interface resolution

and include the surface tension computations.

Level Set method set up a smooth distant function in which zero value displays the

position of the interface. As LSM is based on a stream-function formulation and needs to

use a vector potential and solve three elliptic equations, therefore, it becomes more

expensive in three dimensions [154]. Unlike the Level Set method, improved VOF

methods usually involve an efficient interface reconstruction step based on the local

volume fraction and orientation of its gradient. However, it is worthwhile adding that

three-dimensional reconstruction process and as discussed before tracking of curved

interfaces by this method is truly problematic [155]. Alternatively, some techniques made

use of markers to follow the interfaces. Marker methods set up a secondary moving grid

aimed to present precisely the interface location and curvature. Particle-in-Cell

methods[156] which pack the area close to the interface with markers moving with the

flow are another example of this category. The downside of these methods, however, is

17

that they can become rather expensive as the number of markers increases. In addition,

they may suffer from difficulties in controlling the addition or deletion of markers when

the interface is extended or condensed by the flow.

As all these methods involve the use of computational meshes, mesh adjustment is

required as the phases deform. Tracking the interface with substantial deformation and

topological changes is the main challenge of the traditional methods which need high

resolution meshes, mesh refinement, proper mesh aspect ratios and intricate discretization

techniques. Generally, in traditional mesh-based methods, the finite element and finite

volume methods solve the Navier-Stokes equations by integrating the equations over a

mesh of finite elements or volumes. To make the set of nonlinear equations linear,

unknowns are substituted with finite difference equations, and subsequently the linear

equations are solved through an iterative method. To solve the governing equations, the

flow domain has to be properly discretised into tiny volumes or elements resulting in the

solution of a partial differential equation. This process is referred to as mesh or grid

generation. If a proper grid or mesh is not generated, geometric reliability problems will

occur regarding the representation of the computational domain. Furthermore, if a proper

grid is not produced, instabilities will happen when trying to solve the governing

equations. Figure 2-4 illustrates this difficulty with meshing and re-meshing process in

different microfluidic systems. Once a solution has been formed, the flow parameters can

be extracted for each individual cell or element for analysis. These various operations are

complex, computationally expensive and prone to numerical instability [27]. Treatment of

coalescence and rupture of interfaces between phases is extremely challenging if not

impossible in many of these techniques [157]. In conclusion, simulation of immiscible

liquid-liquid systems using mesh-based Computational Fluid Dynamics (CFD) methods

will be computationally very expensive and a formidable task to undertake [27].

Above and beyond the mentioned strengths and weaknesses, modelling of multiphase

micro-flows is particularly challenging because it is not possible to simply adopt a multi-

fluid modelling approach– which treats the immiscible liquids as a homogeneous mixture

without any interfaces –as is typically done for macro-scale counterparts [24, 158]. At

micro-scale, the flow is predominantly laminar, the discrete phases are controllable and of

the order of the size of the geometry. Existence of movable and complex boundaries at the

18

liquid-liquid interface, and liquid-solid contact line dynamics makes the simulation to be

problematic. In such scales, the surface to volume ratio is extremely high and notable,

constitutive relations of liquids are mainly under the influence of the boundary, and the no-

slip boundary condition is not valid. To overcome these issues, we must explicitly model

the behaviour of the individual phases and their interaction at the interface. Explicit

Numerical Simulation (ENS) of multiphase systems takes advantages of investigating

complex phase geometries and topologies and considering multiple time and length scales

in one simulation [26]. As a result, traditional mesh-based CFD methods cannot be

efficiently applied for simulation of multiphase microfluidics. Thus, for some cases such as

immiscible-liquid flows in microchannels with complicated geometries, a more powerful

approach should be called to remove restrictions on mesh entanglement, complex

geometry, Reynolds number and so forth. This study aims to develop a meshless method

for modelling of such systems.

(a) (b)

Figure 2-4 Computational mesh close to the interface for a rising droplet [159] and (b)

adaptive mesh refinement for a droplet impact on a Liquid-Liquid interface [160]

19

Particle-based approaches have been used as a remedy to the issues faced with traditional

mesh-based approaches. Historically, particle methods present the most primitive efforts of

developing meshless numerical schemes. Particle methods have been demonstrated to be

powerful in numerically treatment of large deformation processes and discrete phenomena

[27]. These methods take the advantage of having no inherent restrictions normally on the

permissible deformation of the considered domain in opposition to mesh-based methods.

As for the former, the connectivity between particles is created as a part of computation

and can vary with time which is not the case for 3D meshes. This positive side of particle

methods is very important for substantial deformation simulations as mesh-based methods

may necessitate constant re-meshing to keep well-behaved elements up. Moreover, the

meshless particle method is capable to be connected with a CAD database more

effortlessly than mesh-based approaches, due to the fact that it is not essential to generate

an element grid. The other considerable aspect of particle methods is that the accuracy and

damage of the components can be simply controlled and meshless discretization supplies

precise depiction of geometric object [27, 31].

Lattice Boltzmann method (LBM) [32, 89] is a mesh-based particle method which was

developed for fluid flow applications. The idea behind this method is to construct

simplified kinetic models integrating the essential physics of microscopic or mesoscopic

systems to facilitate the macroscopic averaged parameters follow the desired macroscopic

equations [32]. LBM has been widely employed in numerical simulation of multiphase

microfluidic systems. For instance, Wu et al. [38] presented a lattice Boltzmann model for

simulation of immiscible two-phase flows in a cross-junction microchannel and also

studied the effect of Ca on droplet formation in such a cross-flow systems. Liu et al. [20]

studied the effect of flowrate ratio and capillary number on droplet formation in a

microfluidic cross-flow configuration using a multiphase lattice Boltzmann model. They

used a limited range of capillary numbers (Ca<0.01) and viscosity ratio (𝛌≤1/50), but a

variety of flowrate ratios in their simulation and recognized three different flow regimes

for their different flowrate ratios at a fixed capillary number. Gupta et al. [161] used LBM

to show how the length of the plugs in the squeezing regime are influenced by the width of

the two channels and the depth of the assembly in a microfluidic T-junction. Additionally,

they examined the channel width ratio effect on the droplet formation process in a T-

20

junction microchannel at high capillary numbers. The benefit of LBM, against the

conventional interface tracking methods, lies in its capability that maintains sharp

interfaces without weighty effort. However, this method as well as its less widely-used

antecedent, lattice-gas automata (LGA) [162, 163], are also not without their problems and

limitations. For example, Galilean invariance is lost in LGA, leading to a spurious density-

dependent factor multiplying the inertial term in the momentum conservation equation

whose effect cannot be removed [164]. On the other hand, LBM-based simulation of

immiscible multiphase fluid flows tend to suffer from stability problems [40]. In the LBM,

the spatial space is discretised in a way that it is consistent with the kinetic equation.

Therefore, it requires not only grid, but also the grid has to be uniform. This actually

causes problems at curved boundaries and so development of irregular grids is needed

[27]. In addition, LBM needs the input thermophysical properties to be adjusted by the

microscopic parameters as a result of microscopic dynamics of distribution functions, and

also due to the kinetic nature of the LB method, hydrodynamic boundary conditions are

difficult to satisfy on a grid point exactly [157]. Therefore, specific techniques are required

to enforce boundary conditions. This work is to develop a robust numerical model for

immiscible liquid microfluidic systems that eliminates the aforementioned needs exist for

LBM simulations of microfluidic systems.

An alternative approach is Dissipative Particle Dynamics (DPD) [164-166], which has

been used to study the rheology of complex fluids [167]. DPD is an off-lattice mesoscopic

simulation technique which involves a set of particles moving in continuous space and

each particle represents a cluster of atoms or molecules, and, as a result of the internal

degrees of freedom associated with each particle, the particle-particle interactions include

random and dissipative contributions [168]. The drawback of DPD is determination of

liquid properties and transport coefficients of the system. Also, because of limitation of

small time steps, DPD is not computationally efficient [169, 170].

One preferable particle method that has a much longer history is Smoothed Particle

Hydrodynamics (SPH) [33, 171-173]. In addition to avoiding all the issues associated with

mesh-based methods, SPH has the advantages that it is derived directly from the Navier-

Stokes equations and, hence, can be easily extended to any non-Newtonian matter. It also

has an innate capacity to adapt through time to better capture steep gradients as they form.

21

Explicit numerical simulation of liquid-liquid micro-flows using SPH can be the most

appropriate approach for capturing liquid-liquid interface deformations [26] thanks to its

advantages over the traditional methods. For example, moving and deformable interfaces

can be easily traced using SPH apart from the complexity of the motion of the particles,

and so numerical investigation of multiphase flows through microfluidic devices with

similar behaviour can be reliably and efficiently relied on the Smoothed Particle

Hydrodynamics Method.

Our findings on the subject of multiphase microfluidic systems are still shallow and need

more attempts, however, experimental studies of such micro-scale flows are quite

challenging and expensive, and the analytical solutions are limited to a few simple cases.

Therefore, development of a powerful numerical technique is highly needed for

comprehensive and effective study of multiphase microfluidic flows with the intention of

achieving a better understanding of dynamics of such flows. The significance of this thesis

lies in the development of a new incompressible SPH-based model to the ENS of

immiscible-liquid microchannel flows with particular regard to correctly capture the

liquid-liquid interface. This work contributes in building of a basis for optimal design of

the microfluidic systems through determining the effects of various flow parameters and

geometry characteristics on the dynamics of the multiphase flow system.

2.5 Smoothed Particle Hydrodynamics (SPH) Method

Smoothed Particle Hydrodynamics, considered as the oldest up to date meshless particle

method, was initially developed by Gingold et al. [174] to solve astrophysical problems in

three dimensional open spaces. They introduced SPH as a technique for finding

approximate numerical solutions of the momentum equations by replacing the fluid with a

set of particles [173]. In this technique, an integral representation of a function, called

kernel approximation, is used to solve the governing partial differential equations using a

Lagrangian approach. A weakly compressible SPH algorithm [175] was developed for

simulation of fluids for which the assumption of slightly compressible is valid such as

coastal applications[176]. The weakly compressible SPH method was also used in some

multiphase flows as dust gas flows [177], interfacial flows [178] and so forth.

22

Morris et al. [179] incorporated the standard SPH formulation with a quasi-incompressible

equation of state to model incompressible flows of low Reynolds number. Later on, Shao

et al. [180] presented the incompressible SPH model to study numerically both Newtonian

and non-Newtonian flows based on the concept of projection SPH [175] and the moving

particle semi-implicit method [181]. In this method, an invariant density equation was used

to calculate the required fluid pressures. Moreover, Hu et al. [182, 183] extended the

incompressible SPH model for simulation of multiphase flows under a wide range of

density ratios. In this approach, the standard SPH formulations were modified to account

for the discontinuity of densities across the interface boundary. Correspondingly, Hu et al.

[29] examined the applicability of SPH for simulation of multiphase flows both at macro-

and meso-scales.

In addition to being used to model multiphase fluid systems via the two-fluid model [177,

184], SPH has been used extensively to model such systems in which the interfaces are

explicitly resolved. Many of these ignore surface tension (e.g. [185-190]) and are, hence,

of limited use for liquid-liquid systems. Others have, however, sought to include surface

tension effects. One broad approach is that first proposed by Morris [191], which

embedded the continuum surface force (CSF) approach [192] within SPH. This approach

was disadvantaged by the need to evaluate interface curvature, a numerically challenging

task that is prone to numerical instability. Adams and co-workers [29] addressed this issue,

although computational expense is still an issue, and the approach has since been extended

to systems involving solid surfaces of desired wettability [29, 193] and, strictly

incompressible systems [182, 183, 194, 195]. A second broad approach sees SPH

combined with the Cahn-Hilliard model [196, 197]; this diffuse interface approach is yet to

see wide use. The third and final broad approach to including surface tension within SPH

is via incorporation of inter-particle forces that lead to phase separation. The earliest

example of this approach was due to Nugent and Posch [198] whose inter-particle force

was inspired by the intermolecular interaction term that arises in the van der Waals

equation of state that was used to close their compressible flow SPH model; this approach

has been widely used since (e.g. [199, 200]) as it is far simpler and cheaper than the CSF-

based approach, although it requires calibration through modelling of the Young-Laplace

problem. A variety of other inter-particle force models have also been proposed since,

23

some were inspired by DPD [170, 201], others are density-dependent [202], whilst various

arbitrary forms have also been used, including trigonometric [203] and inverse square law

[204] forms. In all these cases, however, strict incompressibility has not been imposed; this

has only been done once before [205], using the trigonometric interaction model [203].

Although there is an increasing interest in meshless methods specifically SPH, only a

limited number of their application in porous media and microfluidics have been reported

so far [157]. In this context, Tartakovsky et al. [206] utilized a SPH-based simulation to

investigate behaviour of immiscible and miscible fluid flows in porous media. They also

explored the influence of pore scale heterogeneity and anisotropy on such flows.

Additionally, they applied SPH for modelling of reactive transport and mineral

precipitation in porous media [207]. Successful examples of SPH in other applications

include free surface flows and simple unsteady flows in microfluidics [208, 209]. In the

following, a brief description of general SPH formulation is given.

General SPH Formulation 2.5.1

In SPH formulation the various influences of the Navier-Stokes equation are simulated by

a set of forces that act on an individual particle at a location r, where these forces are given

by scalar quantities that are interpolated at a location r by a weighted sum of contributions

from all neighbouring particles inside a cut-off distance h in the space. This can be

expressed in integral form as follows[33]:

𝐴𝑖 = ∫𝐴(��)

Ω

𝑊(𝑟 − ��, ℎ)𝑑�� (4)

By approximating the integral interpolant by a summation interpolant, the numerical

equivalent to Equation (4) is obtained:

𝐴𝑖 =∑𝐴𝑗𝑉𝑗𝑊(𝑟𝑖𝑗, ℎ)

𝑗

(5)

where j is iterated over all particles, 𝑉𝑗 the volume attributed implicitly to the particle j,

𝑟𝑖𝑗 = 𝑟𝑖 − 𝑟𝑗, where r is the position of a particle, and finally A is the scalar quantity that is

24

being interpolated. The summation is over particles which lie within the radius of a circle

centered at 𝑟𝑖. The following relation between volume, mass and density applies[33].

𝑉 =𝑚

𝜌 (6)

where m is the mass and 𝜌 is the density. Combining this we get the basic formulation of

the SPH interpolation function.

𝐴𝑖 =∑𝐴𝑗𝑗

𝑚𝑗

𝜌𝑗𝑊(𝑟𝑖𝑗, ℎ) (7)

The function 𝑊(𝑟𝑖𝑗, ℎ) is the smoothing kernel, which is a scalar weighted function. The

function uses a position 𝑟 and a smoothing length h. This radius can be seen as a cut-off for

how many particles will be considered in the interpolation. This cut-off radius sets 𝑊 = 0

for |𝑟𝑖𝑗|> ℎ [33].

In SPH, the derivatives of a function can be obtained by using the derivatives of the

smoothing kernel that results in the Basic Gradient Approximation Formula [172]:

∇𝐴𝑖 =∑𝐴𝑗𝑚𝑗

𝜌𝑗𝑗

∇𝑊(𝑟𝑖𝑗, ℎ) (8)

∇2𝐴𝑖 =∑𝐴𝑗𝑚𝑗

𝜌𝑗𝑗

∇2𝑊(𝑟𝑖𝑗, ℎ) (9)

These formulations can produce spurious results, and several corrected formulations have

been developed. One of these is the Difference Gradient Approximation Formula, which

has the advantage that the force vanishes exactly when the pressure is constant.

∇𝐴𝑖 =1

𝜌𝑖∑𝑚𝑗 (𝐴𝑗 − 𝐴𝑖)

𝑗

∇𝑊(𝑟𝑖𝑗, ℎ) (10)

The forces between two particles must observe Newton’s Third Law that for every action

there is an equal and opposite reaction. Pair-wise forces must be equal in size with

opposite sign (𝑓𝑖 = −𝑓𝑗). This means that the differentials in momentum equations that

25

create these forces must be symmetrized. Monaghan developed a symmetrisation, referred

to as the Symmetric Gradient Approximation Formula [33]. It is commonly used for the

pressure gradient.

∇𝐴𝑖 = 𝜌𝑖∑𝑚𝑗 (𝐴𝑖𝜌𝑖2

−𝐴𝑗

𝜌𝑗2)

𝑗

∇𝑊(𝑟𝑖𝑗, ℎ) (11)

An alternative symmetrical formulation is as Equation (12) [34]:

∇𝐴𝑖 =∑𝑚𝑗

𝜌𝑖 (𝐴𝑖 + 𝐴𝑗)

𝑗

∇𝑊(𝑟𝑖𝑗, ℎ) (12)

The basic formulation of the Laplacian has been found to be somewhat unstable under

certain conditions, and there exist a wide range of possible corrections. Shao and Lo

established a correction well suited for the correction of the Laplacian in the viscous force

[180].

∇. (1

𝜌∇𝐴) =∑𝑚𝑗

8

(𝜌𝑖 + 𝜌𝑗)2(𝐴𝑖 − 𝐴𝑗). ∇𝑊(𝑟𝑖𝑗, ℎ)

|rij|2+ η2

(13)

To remain the denominator non-zero, 𝜂 variable is used, which is equal to 0.1h.

Neighbour searching algorithm 2.5.2

In the SPH method, only a limited number of particles are inside the support domain of

dimension kh for the target particle, which are applied in the particle approximations.

These particles in the support domain are usually mentioned to as nearest neighbouring

particles (NNP) for that target particle. The procedure of detecting the nearest particles is

known as nearest neighbouring particle searching (NNPS). Unlike a grid based numerical

method, where the position of neighbour grid-cells are well defined once the grids are

given, the nearest neighbouring particles in the SPH for a given particle can vary with

time. The different NNPS approaches which can be used in SPH implementations include

all-pair search, linked-list search algorithm, and tree search algorithm [33].

26

SPH Algorithm 2.5.3

The SPH algorithm approximates a function and its spatial derivatives through averaging

or summation over neighbouring particles. The special features in the SPH coding are

generally involved under the main loop of time integration process, including the

smoothing function and derivative calculation, particle interaction calculation, smoothing

length evolution, spatial derivative estimation, boundary treatment, etc. A typical

procedure for SPH simulation includes three main steps and some sub-steps, as below.

I. Initialization step: includes the input of the initial configuration of the problem

geometry (dimensions and boundary conditions), discretization information of the

initial geometry of particles, material properties, time step and other simulation

control parameters.

II. Main SPH step: contains the major parts in the SPH simulation, and is implemented

in the time integration loop. The following steps are required to be considered into

the time integration process:

Nearest neighbouring particle searching (NNPS)

Calculating the smoothing function (for the summation density approach) and its

derivatives from the generated information of interaction particle pairs

Calculating the density

Calculating the stress term (viscous force and pressure gradient), internal forces

arising from the particle- particle interactions, and the body forces

Updating particle momentum, density, particle position, velocity, and checking

the momentum conservation

III. Output step: the time step reaches to a prescribed one or at some interval, the

resultant information is saved for later analyses or post-processing.

2.6 Conclusion

Even with the recent developments being made in the field of immiscible-liquid

microfluidics, an in-depth understanding of flow behaviour in micro-geometry is needed to

27

improve the design and optimization of the microfluidic devices. The studies have been

done so far in this context, are still sporadic due to variations in the material used for

fabrication, flow rates, channel dimensions and geometries. Experimental studies at such

small scales are quite challenging and expensive, therefore numerical studies have to be

called to become complementary to the experiments. On the other hand, the numerical

studies involving microchannel flow simulations using mesh-based methods face a variety

of obstacles. As discussed, the most important problem is the difficulties along with mesh

generation which would be a problematic task, time consuming and an expensive process

for cases with complex geometry. Another challenge in mesh-based simulations is to track

the movement of the interfacial boundaries in multiphase fluid flows using explicit

numerical simulation (ENS) method, which causes another source for modelling error and

numerical instability. However, meshless particle methods such as SPH have been the

focus recently to develop the next generation of more efficient computational schemes

designed for more complex problems. However, there isn’t any report in the literature on

the dynamics of immiscible-liquid microfluidics which have been studied based on SPH

technique.

The space of parameters which can influence the flow pattern of two liquids interacting in

immiscible-liquid microfluidics is very extensive and involves the nature of the liquids, the

details of flow injections, geometry of the channels, surface characteristics and so forth.

For instance, droplet formation and breakup processes depend on capillary number,

superficial velocities of the continuous and dispersed phase, flowrate ratio, viscosity ratio,

interfacial tension and geometry of microchannel. Despite various experimental and

numerical studies have been reported on droplet-based microfluidics so far, the detailed

flow physics and process parameters is still missing and much effort is still required to

accurately control interfaces and flow patterns, make use of their interfacial properties and

also to capture all the key parameters for design development of microfluidic devices.

28

Modelling of Immiscible Liquid-Liquid Chapter 3:

Systems by Smoothed Particle Hydrodynamics

H. Elekaei Behjati1, M. Navvab Kashani

1, M. J. Biggs

1,2*

1. School of Chemical Engineering, the University of Adelaide, SA 5005,

Australia.

2. School of Science, Loughborough University, Loughborough, LE11 3TU,

UK.

* m.biggs@lboro.ac.uk

29

Statement of Authorship

Title of Paper Modelling of Immiscible Liquid-Liquid Systems by Smoothed Particle

Hydrodynamics

Publication Status ☐Published ☐Accepted for Publication

☐Submitted for Publication ☒Publication Style

Publication Details Journal of Computational Mechanics

Author Contributions

By signing the Statement of Authorship, each author certifies that their stated contribution to the

publication is accurate and that permission is granted for the publication to be included in the

candidate’s thesis.

Name of Principal

Author (Candidate) Hamideh Elekaei Behjati

Contribution

Performed modelling and coding for simulation. Processed and

undertook bulk of data interpretation. Prepared manuscript as primary

author.

Signature

Date 23/08/2015

Name of Co-author Moein Navvab Kashani

Contribution Assisted with results analysis, data interpretation and manuscript

preparation.

Signature

Date 23/08/2015

Name of Co-author Mark J. Biggs

Contribution

Supervisor. Advised on experimental design, modelling, and data

interpretation. Advised on manuscript preparation, revised manuscript,

and corresponding author.

Signature Date 23/08/2015

30

Abstract

Immiscible liquid systems are ubiquitous in industry, medicine and nature. Understanding

the relationship between the phase morphologies and fluid dynamics within the phases for

the conditions they experience is critical to understanding in these many occurrences. In

this paper, we detail a Smoothed Particle Hydrodynamics (SPH) model that facilitates

building of this understanding. The model includes surface tension, enforces strict

incompressibility, and allows for arbitrary fluid constitutive models. The validity of the

model is demonstrated by applying it to a range of model problems with known solutions,

including a liquid droplet falling under gravity through another quiescent liquid, Young-

Laplace equation, and droplet deformation and rupture under shear whilst confined.

Keywords: Incompressible Smoothed Particle Hydrodynamics; Newtonian liquids;

Immiscible liquid-liquid microfluidics; Droplet deformation; Drag coefficient

31

3.1 Introduction

Processing of immiscible liquid-liquid dispersions occurs widely in the manufacture of

foods [210], pharmaceuticals [211], cosmetics [212], paints [213], and oil [214]. Examples

of processes include emulsification [213, 215], encapsulation [216, 217], multiphase

reaction systems [218, 219], electrochemical processes [220], bioprocesses [221],

separation processes [222] such as liquid-liquid extraction [223], polymer blending [224],

and oil recovery [225]. Processes involving immiscible liquids are also found beyond

industry including in environmental clean-up [226], artificial oxygen carriers [227], oil

spills [228], and pyroclastic flows [229].

There is an increasing need to understand and exploit the link between liquid-liquid phase

morphology and fluid dynamics within the phases on the one hand and the conditions that

lead to them on the other. For example, the droplet size distribution of an emulsion

produced through agitation is a function of the balance between droplet breakup and

coalescence, which can be controlled by the surfactant and stabilizer concentration and

relative velocities of the phases [118-120]. Microfluidic production of encapsulates

provides a further example: in this case, encapsulate morphology can be varied through the

nature of the flow-focusing in the microchannels and, amongst other things, the

continuous-to-dispersed phase flow rate ratio [121, 122]. The challenges faced in

experimentally elucidating these types of relationships are significant, however. For

example, the visualization of the morphologies of the phases and flow fields therein are

still very much in their infancy [123-126]. Models that treat the phases and interfaces

between them explicitly have, therefore, an important role to play in building

understanding of liquid-liquid systems and exploiting this understanding in a systematic

way.

Models of liquid-liquid systems in which the individual phases and interfaces between

them are treated explicitly are long-standing [127-130]. The earliest models, which focus

on droplets in a continuous phase, include those of Taylor [131, 230], Mason and co-

workers [132, 133], Cox and co-workers [134, 135], and Acrivos and co-workers [136-

138] amongst others. Whilst these models were important in building understanding, they

are limited by a good number of simplifying assumptions [139], including negligible

32

inertial effects (i.e. small Reynolds numbers), small viscosity ratio ranges, and regular

droplet shapes (e.g. spheres; ellipsoids). Phenomenological approaches have been adopted

to overcome some limitations associated with wholly analytical models. For example,

Maffettone and Minale [87] used such an approach, determining the model parameters by

ensuring it matches analytical results in appropriate limits. Such models are, however, also

limited by underpinning assumptions such as, for example, specific droplet shapes and the

absence of any connection between the model parameters and underlying fundamentals.

Adoption of wholly numerical approaches can, in principle, overcome limitations faced by

analytical and related models. Some of the earliest numerical models include those of

Acrivos and co-workers [139-141], who studied the deformation and rupture of a viscous

droplet suspended in another liquid within extensional and shear flow fields. More recent

examples include the works of Loewenberg, Hinch and Davis [142-144, 146-148, 231],

which are based on a boundary-integral approach. Interface tracking techniques such as

Volume of Fluid methods (VOF) [150, 232] and level set methods [152, 153] have been

used even more recently. Whilst these works represent major advances in the field, they

have two significant disadvantages that lead to algorithmic complexity, computational

expense and numerical stability challenges [130, 165]: (1) the need to adjust the mesh as

the phases deform; and (2) the need to track the interfaces between the different phases.

The former can be eliminated by selecting a single set of mesh [233] with very high

resolution enough to correctly capture the interface without adjusting the mesh as the

phases deform but this again would be computationally very expensive. Additionally, the

treatment of coalescence and rupture of the interfaces between the phases is extremely

challenging if not impossible in many of these techniques [234, 235].

Particle-based approaches have more recently been used as a remedy to the issues faced by

mesh-based approaches. The most widely-adopted is the lattice Boltzmann method (LBM)

[32, 236]. This method as well as its less widely-used antecedent, lattice-gas automata

(LGA) [162, 163], are also not without their problems and limitations, however. For

example, Galilean invariance is lost in LGA, leading to a spurious density-dependent

factor multiplying the inertial term in the momentum conservation equation whose effect

cannot be removed [164]. LBM-based simulation of immiscible multiphase fluid flows, on

the other hand, tend to suffer from stability problems [40]. An alternative approach is

33

Dissipative Particle Dynamics (DPD) [164-166], which has been used to study the

rheology of complex fluids [167]. DPD is an off-lattice mesoscopic simulation technique

which involves a set of particles moving in continuous space and each particle represents a

cluster of atoms or molecules, and, as a result of the internal degrees of freedom associated

with each particle, the particle-particle interactions include random and dissipative

contributions[168]. However, the disadvantage of DPD is determination of liquid

properties (e.g. viscosity) and transport coefficients of the system such as self-diffusion

coefficient. Since the validity of the equilibrium properties and analytical expressions is

restricted to the limit of small time steps, DPD is not computationally as efficient as LBM

[169, 170]. A related but computationally far cheaper approach [209] of much longer

standing is Smoothed Particle Hydrodynamics (SPH) [171]. In addition to avoiding all the

issues associated with mesh-based methods, SPH has the advantages that it is derived

directly from the Navier-Stokes equations and, hence, can be easily extended to any non-

Newtonian matter.

In addition to being used to model multiphase fluid systems via the two-fluid model [177,

184], SPH has been used extensively to model such systems in which the interfaces are

explicitly resolved. Many of these ignore surface tension (e.g. [185-190]), making them of

limited value when interest lies in liquid-liquid systems where it plays an important role in

dictating behaviour. Some have, however, sought to include surface tension effects. The

first broad approach is that initially proposed by Morris [191], which embeds the

continuum surface force (CSF) approach [192] within SPH. This approach is

disadvantaged by the need to evaluate the curvature of interfaces, a numerically

challenging task that also leads to numerical instabilities. Adams and co-workers [29]

addressed this issue, although computational complexity and expense is still an issue. The

CSF-based SPH approach has since been extended to systems involving solid surfaces of

desired wettability [29, 193] and strictly incompressible systems [182, 183, 194, 195, 237-

240]. A second broad approach sees SPH combined with the Cahn-Hilliard model [196,

197]; this diffuse interface approach is yet to see wide use, however. The final broad

approach to including surface tension within SPH is via incorporation of inter-particle

forces that lead to phase separation. The earliest example of this approach was due to

Nugent and Posch [198], whose inter-particle force was inspired by the intermolecular

34

interaction term that arises in the van der Waals equation of state used to close their

compressible SPH model. This particular model, which is far simpler and cheaper than the

two alternative approaches mentioned above, has seen some use since (e.g. [199, 241])

along with similar models based on other inter-particle force models, including some

inspired by DPD [170, 201] along with various other more arbitrary forms [202], including

trigonometric [203] and inverse square law [204]. In all these cases, however, strict

incompressibility has not been imposed; In this paper, we report incorporation of surface

tension within strictly incompressible SPH framework by including a Lennard-Jones (LJ)

interaction between particles [242] in immiscibility model. The adoption of the LJ

interaction to bring about surface tension has the advantage that it mirrors the almost

universally used approach to incorporating solid boundaries within SPH, thus opening the

way to unifying the treatment of interfaces in SPH. The paper first provides details of the

SPH model and how it is parametrized to yield the desired surface tension. The new

method is then validated by comparing its predictions to analytical results and experiments

for a number of model problems, including a liquid droplet in a linear shear field and the

same falling freely in a quiescent continuous phase.

3.2 The Method

Governing Equations 3.2.1

Smoothed-particle hydrodynamics builds on the Navier–Stokes equations expressed in the

Lagrangian frame, which for an incompressible, isothermal system are of the form

∇. 𝐯 = 0 (1)

𝜌 𝑑𝐯

𝑑𝑡= ∇. 𝛔 + 𝜌𝐠 + 𝐅𝐼 (2)

where 𝜌, 𝐯 , and 𝛔 are the fluid density, velocity and stress tensor, respectively, and 𝐠 is

the acceleration due to gravity or other body forces. The term 𝐅𝐼 is introduced into the

momentum equation here so as to enable the imposition of immiscibility between the

different liquids as described in more detail below. The form of this force is selected in a

35

way that its sum across the entire multiphase system disappears, ensuring overall

conservation of momentum.

The stress tensor is described here by

𝛔 = −𝑃𝐈 + 𝛕 (3)

where 𝑃 is the pressure, 𝐈 the unit tensor, and 𝛕 the shear stress tensor. The latter for a

Newtonian fluid may be expressed as

𝛕 = −𝜇 [ ∇𝐯 + (∇𝐯)𝑇 ] (4)

where 𝜇 is the dynamic viscosity.

SPH Discretization 3.2.2

In SPH, the fluid is represented by a discrete set of particles of mass, 𝑚𝑖, and viscosity, 𝜇𝑖,

that move with the local fluid velocity, 𝐯𝑖. Here, each particle also carries a colour, 𝑐𝑖, to

indicate which of the immiscible fluids it belongs to. The velocity and other quantities

associated with any particle are interpolated at a position, 𝑟, through a weighted

summation of contributions from all neighbouring particles within a compact support of

size 𝑂(ℎ), as illustrated in Figure 3-1.

Figure 3-1 An illustration of an SPH weighting function with compact support sited on

particle-i (shown in red) that leads to the particle interacting with all other particles-j within

the support of size O(h).

kh

Kernel W(r)

i

rij

j

36

For example, the density of a particle-i is given by [172]

𝜌𝑖 =∑𝑚𝑗

𝑗

𝑊(𝑟𝑖𝑗, ℎ) (5)

where 𝑟𝑖𝑗 is the distance between particles i and j.

The pressure gradient associated with particle-i is given by [172, 207]

𝛻𝑃𝑖 = 𝜌𝑖∑𝑚𝑗 (𝑃𝑗

𝜌𝑗2+𝑃𝑖𝜌𝑖2)

𝑗

𝛻𝑖𝑊𝑖𝑗 (6)

where 𝑃𝑖 is the pressure associated with particle-i.

Finally, the divergence of the shear stress tensor attached to a particle-i is given by [173]

(𝛻. 𝝉)𝑖 = 𝜌𝑖∑𝑚𝑗 (𝛕𝑗

𝜌𝑗2+𝛕𝑖𝜌𝑖2

)

𝑗

. 𝛻𝑖𝑊𝑖𝑗 (7)

The components of the shear stress tensor in this expression, which are derived from

Equation (4), are given by

𝜏𝑖𝛼𝛽 = −(∑

𝑚𝑗𝜇𝑗

𝜌𝑗 𝑣𝑖𝑗𝛽

𝑗

𝜕𝑊𝑖𝑗

𝜕𝑥𝑖𝛼 +∑

𝑚𝑗𝜇𝑗

𝜌𝑗 𝑣𝑖𝑗𝛼

𝑗

𝜕𝑊𝑖𝑗

𝜕𝑥𝑖𝛽) (8)

where 𝐯𝑖𝑗 = 𝐯𝑖 − 𝐯𝑗

We have employed the following quintic spline kernel to ensure the second derivatives of

the smoothing kernel that arise in the viscous SPH model are smooth, thus avoiding

instabilities that are known to arise with lower order kernels [33, 179]

37

𝑊(𝑟∗, ℎ)

=7

478𝜋ℎ2×

{

(3 − 𝑟∗)5 − 6(2 − 𝑟∗)5 + 15(1 − 𝑟∗)5 0 ≤ 𝑟∗ < 1

(3 − 𝑟∗)5 − 6(2 − 𝑟∗)5 1 ≤ 𝑟∗ < 2

(3 − 𝑟∗)5 2 ≤ 𝑟∗ < 3 0 𝑟∗ > 3

(9)

where 𝑟∗ = 𝑟 ℎ⁄ is a dimensionless radius.

Immiscibility Model 3.2.3

The immiscibility between the different fluids is enforced through the force, 𝐹𝐼, in

Equation (2), which is dependent on the colour of the particles. The force is based on the

following functional form between particle pairs-ij that was arrived at through

experimentation

𝜙𝑖𝑗 =

{

𝜀𝑖𝑗 (

𝐿0𝑟)12

𝑖𝑓 𝑐𝑖 ≠ 𝑐𝑗

−𝜀𝑖𝑗 (𝐿0𝑟)6

𝑖𝑓 𝑐𝑖 = 𝑐𝑗

(10)

where, 𝐿0 is a reference length, and 𝜀𝑖𝑗 is a parameter that is linked to the relevant

interfacial tension as outlined further below. The first part of this expression ultimately

leads to a short range repulsive force of equal magnitude but opposite direction between

pairs of different coloured SPH particles (i.e. different liquids), whilst the second part leads

to an somewhat longer ranged attractive force of equal magnitude but opposite direction

operating between pairs of particles of the same colour (i.e. same liquid). Although the first

is clearly necessary to cause phase separation, the less intuitive second force was found to

lead to improved behaviour from the model when included.

The force between particles is obtained from the gradient of the expression

(𝐅𝐼)𝑖𝑗 = −∇ϕ𝑖𝑗 (11)

Bringing together equations (10) and (11), the SPH formulation for the force acting on

particle-𝑖 that brings about phase separation is

38

(𝑭𝐼)𝑖 =∑𝑚𝑗

𝜌𝑗𝜙𝑖𝑗 𝛻𝑖𝑊𝑖𝑗

𝑗 (12)

Solution Algorithm 3.2.4

Combining equations (2), (6), (7), and (12) leads to the following SPH formulation for the

momentum equation:

𝑑𝒗𝑖𝑑𝑡

= −∑𝑚𝑗 (𝑃𝑗

𝜌𝑗2+𝑃𝑖𝜌𝑖2

)

𝑗

𝛻𝑖𝑊𝑖𝑗 +∑𝑚𝑗 (𝝉𝑗

𝜌𝑗2+𝝉𝑖𝜌𝑖2

)

𝑗

. 𝛻𝑖𝑊𝑖𝑗

−1

𝜌𝑖∑

𝑚𝑗

𝜌𝑗(𝑭𝐼)𝑗𝛻𝑖𝑊𝑖𝑗

𝑗

(13)

This is solved using a two-step predictor-corrector scheme based on that proposed by

Cummins and Rudman [175] for single phase strictly incompressible flows and later

extended to multiphase flows without surface tension by Shao and Lo [180]. Firstly, an

initial estimate of the particle velocities, 𝐯∗, is made using only the shear stress,

gravitational and surface force terms in Equation (13) at time step-t (indicated by the

subscript-t; particle indices have been dropped for convenience) [175, 180].

𝑣∗ = 𝑣𝑡 + (1

𝜌𝛻. 𝜏 + 𝑔 +

1

𝜌𝐹𝐼)

𝑡

∆𝑡 (14)

where ∆𝑡 is the time step size. The corresponding initial estimates for the particle positions

are then evaluated using [180]

𝒙∗ = 𝒙𝑡 + 𝒗∗∆𝑡 (15)

Use of these initial particle position estimates in Equation (5) would reveal that the

density𝜌∗, is not fixed as desired. Incompressibility is, thus, enforced by correcting the

initial estimates of the particle velocities using [180]

39

𝒗𝑡+1 = 𝒗∗ + ∆𝒗∗∗ (16)

where the velocity correction is evaluated using the pressure gradient term of the

momentum equation only

∆𝒗∗∗ = −1

𝜌∗𝛻𝑃𝑡+1 ∆𝑡 (17)

To obtain the pressure at the new time, 𝑃𝑡+1 , Equation (16) and (17) are combined and the

divergence is taken to give [175]

𝛻. (𝒗𝑡+1 − 𝒗∗

∆𝑡) = −𝛻. (

1

𝜌∗𝛻𝑃𝑡+1 ) (18)

Imposing the incompressibility condition at the new time step, ∇. 𝐯t+1 = 0, leads to the

Pressure Poisson Equation (PPE) [175]

∇. (1

𝜌∗∇𝑃𝑡+1 ) =

∇. 𝐯∗∆t

(19)

The left hand side is discretized based on Shao’s approximation for the Laplacian in SPH

[180], which is a hybrid of a standard SPH first derivative with a finite difference

computation [175]

∇. (1

𝜌∇𝑃)

𝑖

=∑𝑚𝑗 8

(𝜌𝑖 + 𝜌𝑗)2

(𝑃𝑖 − 𝑃𝑗). 𝐫ij. ∇𝑖𝑊𝑖𝑗

|𝐫𝑖𝑗|2+ 𝜔2

𝑗

(20)

where 𝜔 is a small value (e.g. 0.1ℎ ) to ensure the denominator is always non-zero.

The right hand side of equation (19) is discretised in SPH using

(∇. 𝐯∗)𝑖 = 𝜌𝑖∑𝑚𝑗 (𝐯∗𝑗

𝜌𝑗2 +

𝐯∗𝑖𝜌𝑖2) . ∇𝑖𝑊𝑖𝑗

𝑗

(21)

40

Discretization of the PPE equation leads to a system of linear equations, 𝐀𝐲 = 𝐛, in which

𝐲 is the vector of unknown pressure gradients to be determined, and the matrix 𝐀 is not

necessarily positive definite or symmetric. In the present work, the biconjugate gradient

algorithm [243] was used to solve this set of equations.

Once the velocity at the next time step is determined via use of Equation (16), the new

particle positions are finally obtained using

𝐱𝑡+1 = 𝐱𝑡 +1

2(𝐯𝑡 + 𝐯𝑡+1)∆𝑡 (22)

3.3 Results and Discussion

Young-Laplace and Parameterization of Immiscibility Model 3.3.1

The validity of the new method was first assessed by ensuring that a circular droplet of one

liquid is recovered when sitting in a second, quiescent immiscible liquid of the same

density (𝜂 = 𝜌𝑑 𝜌𝑐⁄ = 1) and viscosity (𝜆 = 𝜇𝑑 𝜇𝑐⁄ = 1), in line with physics. This was

done by observing the change in the shape of a stationary droplet over time using a time

step size of ∆𝑡∗ = ∆𝑡√𝛾 𝜌ℎ3⁄ = 0.1 when the drop is initially a square of size 6ℎ × 6ℎ

(ℎ = 1.25𝐿0, where 𝐿0 = 5 × 10−5𝑚) within a periodic domain of 24ℎ × 24ℎ; the total

number of SPH particles was 𝑁𝑝 = 961. As Figure 3-2 shows for a typical simulation, as

expected, the perimeter of the droplet gradually decreases as it changes from its initial

square shape to a circle. This figure also shows that the pressure difference across the

interface between the two phases, shown in dimensionless form (∆𝑃∗ = ∆𝑃ℎ 𝛾⁄ ), increases

as the droplet transforms in shape, once again in line with the physics where the pressure

difference is counter-balanced by the interfacial tension (in this case, 𝛾 = 0.045 N/m,

which corresponds to 𝜀 = 56.25 N/m2).

41

Figure 3-2 Typical variation through time, 𝒕∗ = 𝒕√𝜸 𝝆𝒉𝟑⁄ , of the circumference of an initially

square neutrally buoyant droplet suspended in a second continuous phase and the associated

change in pressure difference, ∆𝑷∗ = ∆𝑷𝒉 𝜸⁄ , across the interface between the two liquids.

Snapshots of the droplet along the transformation pathway are shown at various points; the

continuous phase is not shown for simplicity.

Figure 3-3 shows the variation of the pressure drop across the interface between the drop

and continuous phase as a function of the inverse of the drop size. The linear behaviour

seen in this figure clearly conforms to the Young-Laplace equation

∆𝑃 =

𝑅 (23)

The relationship between the interaction model parameter in equation (10), , and the

interfacial tension, , can be determined by repeating the simulations that lead to Figure 3-

3 for various values of the former. Doing this leads to Figure 3-4, this shows that the

interaction model parameter varies in a linear fashion with the interfacial tension.

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.5

0.6

0.7

0.8

0.9

1

0 200 400 600 800

Axi

s Ti

tle

Cd

rop

let

Axis Title𝑡 𝛾 𝜌ℎ3⁄

𝐶𝑑 𝑝 𝑡𝐶 𝑎

⁄ ∆𝑃∗

𝑡∗

∆𝑃∗

𝐶𝑑 𝑝 𝑡𝐶 𝑎

⁄

42

Figure 3-3 Variation of dimensionless pressure drop across the droplet interface with the

inverse of its dimensionless radius, 𝑹∗ = 𝑹 𝒉⁄ , for 𝜸 = 𝟎. 𝟎𝟏𝟒𝟐 N/m (obtained using 𝜺 = 𝟏𝟔

N/m2).

Figure 3-4 Variation of the interfacial tension, 𝜸, with the interaction model parameter that

induces immiscibility here, 𝜺.

Deformation of a Confined Droplet in a Linear Shear Field 3.3.2

The new method was further validated by considering through time the deformation under

linear shear of an initially circular droplet in a second immiscible liquid of the same

density (𝜂 = 𝜌𝑑 𝜌𝑐⁄ = 1) and viscosity (𝜆 = 𝜇𝑑 𝜇𝑐⁄ = 1), Figure 3-5. The initial droplet of

diameter 𝐷 = 17ℎ (ℎ = 1.2𝐿0, where 𝐿0 = 1 × 10−5 m) was located centrally within a

domain of height 𝐻 = 34ℎ and length 𝐿 = 100ℎ; the total number of SPH particles

was 𝑁𝑝 = 4961. To compensate truncation of the support domain of the kernel at the wall

0

5

10

15

20

25

30

35

40

0 0.1 0.2 0.3 0.4

P

* ×1

04

1/R*

0

0.1

0.2

0.3

0.4

0.5

0.6

0 100 200 300 400 500 600 700

[N

.m-1

]

[ N.m-2 ]𝜀 2⁄

𝛾

⁄

43

boundary, at initial state 3h layers of dummy particles are regularly arranged at the other

side of the wall boundary [110, 111]. Equal but opposite velocities of magnitude 𝑈 were

applied to the top and bottom of the domain with no slip boundary conditions (i.e. the top

and bottom were treated as moving solid surfaces) to yield a linear shear field with shear

rate 𝐺 = 2𝑈 𝐻⁄ . The state of the system was evolved through time using timesteps of size

∆𝑡∗ = 𝐺 ∆𝑡 = 0.001.

Under suitable conditions, the droplet can deform to take on an ellipsoidal shape as

illustrated in Figure 3-5. The character of this droplet may be described in part by the

deformation parameter [244]

𝐷𝑓 =𝑎 − 𝑏

𝑎 + 𝑏 (24)

where 𝑎 and 𝑏 are the lengths of the major and minor axes of the ellipsoid, respectively.

The other key parameter characterizing the droplet behaviour is the angle the major axis of

the ellipsoid subtends to the streamlines of the shear field, 𝜑.

Figure 3-5 Droplet shape and velocity field obtained from a typical SPH simulation

with the key characteristics shown.

Figures 3-6 illustrates the deformation and breakup of a droplet under a confined shear

flow through dimensionless times, 𝑡∗ = 𝐺 𝑡 , at Re = 𝜌𝑅2𝐺 𝜇⁄ = 0.1, λ = 1, D H⁄ = 0.5

and Ca = 𝜇𝑅𝐺 𝛾⁄ = 0.5 , a capillary number greater than the critical value where droplet

breakup is ensured. In comparison, Figure 3-7 indicates the velocity field and shape change

of the same droplet for a Ca value smaller than the critical value.

𝑈

𝑈

𝐻

𝐿

44

Figure 3-6 Variation of velocity field and droplet morphology with time for 𝐂𝐚 = 𝟎. 𝟓,

𝐑𝐞 = 𝟎. 𝟏, 𝝀 = 𝟏,and 𝑫 𝑯⁄ = 𝟎. 𝟓.

0 2 4 6 8 10 120

0.5

1

1.5

2

2.5

3

3.5

4

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 2 4 6 8 10 120

0.5

1

1.5

2

2.5

3

3.5

4

1 2 3 4 5 6 7 8 9 10 110

0.5

1

1.5

2

2.5

3

3.5

4

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 1 2 3 4 5 6 7 8 9 100

0.5

1

1.5

2

2.5

3

3.5

4

1 2 3 4 5 6 7 8 9 10 110

0.5

1

1.5

2

2.5

3

3.5

4

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 2 4 6 8 10 120

0.5

1

1.5

2

2.5

3

3.5

4

1 2 3 4 5 6 7 8 9 10 110

0.5

1

1.5

2

2.5

3

3.5

4

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 2 4 6 8 10 120

0.5

1

1.5

2

2.5

3

3.5

4

1 2 3 4 5 6 7 8 9 10 110

0.5

1

1.5

2

2.5

3

3.5

4

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 2 4 6 8 10 120

0.5

1

1.5

2

2.5

3

3.5

4

1 2 3 4 5 6 7 8 9 10 110

0.5

1

1.5

2

2.5

3

3.5

4

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 2 4 6 8 10 120

0.5

1

1.5

2

2.5

3

3.5

4

𝑡∗=0

𝑡∗=1

𝑡∗=3

𝑡∗=5

𝑡∗=9

𝑡∗=12

45

Figure 3-7 Variation of velocity field and droplet morphology with time for 𝐂𝐚 = 𝟎. 𝟐,

𝐑𝐞 = 𝟎. 𝟏, 𝝀 = 𝟏,and 𝑫 𝑯⁄ = 𝟎. 𝟓.

0 2 4 6 8 10 120

0.5

1

1.5

2

2.5

3

3.5

4

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 2 4 6 8 10 120

0.5

1

1.5

2

2.5

3

3.5

4

1 2 3 4 5 6 7 8 9 10 110

0.5

1

1.5

2

2.5

3

3.5

4

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 2 4 6 8 10 120

0.5

1

1.5

2

2.5

3

3.5

4

1 2 3 4 5 6 7 8 9 10 110

0.5

1

1.5

2

2.5

3

3.5

4

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 2 4 6 8 10 120

0.5

1

1.5

2

2.5

3

3.5

4

1 2 3 4 5 6 7 8 9 10 110

0.5

1

1.5

2

2.5

3

3.5

4

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 2 4 6 8 10 120

0.5

1

1.5

2

2.5

3

3.5

4

1 2 3 4 5 6 7 8 9 10 110

0.5

1

1.5

2

2.5

3

3.5

4

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 2 4 6 8 10 120

0.5

1

1.5

2

2.5

3

3.5

4

1 2 3 4 5 6 7 8 9 10 110

0.5

1

1.5

2

2.5

3

3.5

4

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 2 4 6 8 10 120

0.5

1

1.5

2

2.5

3

3.5

4

𝑡∗=0

𝑡∗=1

𝑡∗=3

𝑡∗=5

𝑡∗=7

𝑡∗=10

46

The velocity fields show changes in x-velocity component along the height of the domain.

This induces a velocity field inside the droplet which reduces the shear stress on the

droplet. As a result, the droplet starts elongation and a rotational motion so that a

circulating flow is set up inside the droplet. At higher Ca, viscous forces are more

important than interfacial forces and therefore, the droplet undergoes a larger deformation

and experiences a breakup, as indicated by figure 3-6.

Figure 3-8 compares the deformation parameter as obtained from the SPH simulations at a

confinement ratio of 𝐷 𝐻⁄ = 0.5 with the experimental results of Sibillo et al. [245] and

Taylor’s model for an unconfined droplet in a shear field [244]

𝐷𝑓 =19𝜆 + 16

16𝜆 + 16 𝐶𝑎 (25)

This figure shows that the deformations predicted by SPH match very well the

experimental results up to the Ca = 0.3 limit probed by Sibillo et al. [245]. The results also

match well those predicted by Taylor’s model up to this Capillary Number. However,

beyond this one can see a significant deviation from Taylor’s model, indicating that

confinement has an effect at higher Capillary Numbers where shear is starting to dominate

over surface tension.

Figure 3-8 Variation of the deformation parameter as defined in Equation (24) with Capillary

Number as predicted by SPH with Np=7701 (solid circles with linear fit shown as a line;

R^2=0.99), SPH with Np=4961 (open circles) Taylor theory as defined by Equation (25) (solid

line) [244], and experimental data (open squares) [245].

0

0.1

0.2

0.3

0.4

0.5

0.6

0 0.1 0.2 0.3 0.4 0.5

Df

Ca

47

Figure 3-9 compares the tilt angle of the ellipsoid (see Figure 3-5) as obtained from the

SPH simulations at a confinement ratio of 𝐷 𝐻⁄ = 0.5 with experimental data and the

value predicted from the Taylor model [244]

𝜑 =𝜋

4−(19𝜆 + 16)(2𝜆 + 3)

80(𝜆 + 1) 𝐶𝑎 (26)

Once again, comparison with experiment is very good. The deviation of the SPH values

from Taylor’s theory is modest but grows with Ca, indicating that the tilt angle is more

sensitive to confinement.

Figure 3-9 Variation of the droplet tilt angle with Capillary Number as predicted by SPH

with Np=4961(open circles), SPH with Np=7701 (solid circles), Taylor theory as defined by

Equation (25) (solid line) [244], and the experimental values of Sibillo [246] (open squares).

Deformation of Free-Falling Droplet 3.3.3

The new method was finally validated by considering the descent of an initially stationary

circular droplet under gravity through a second stationary liquid whose density, 𝜌𝑑 =

1150 𝑘𝑔 𝑚3⁄ differed from that of the continuous phase, 𝜌𝑐, by ∆𝜌 = 𝜌𝑑 − 𝜌𝑐 = 150

kg/m3. The droplet was initially of diameter 𝐷 = 14ℎ (ℎ = 1.2𝐿0, where 𝐿0 = 1 ×

10−4𝑚), and was located within a periodic domain of size 34ℎ × 34ℎ; the total number of

SPH particles was 𝑁𝑝 = 1681. The droplet motion and deformation was followed through

time (∆𝑡∗ = ∆𝑡√𝑔 𝐷⁄ = 0.004) until the terminal velocity was reached and the droplet

0

5

10

15

20

25

30

35

40

45

0 0.1 0.2 0.3 0.4 0.5

φ

Ca

48

shape remained unchanged (𝑡∗ = 45). A range of combinations of Morton Number, Mo =

𝑔∆𝜌𝜇𝑐4 𝜌𝑐

2γ3⁄ , Ohnesorge Number, Oh𝑑 = 𝜇𝑑 √ 𝜌𝑑𝛾𝐷⁄ , and Oh𝑐 = 𝜇𝑐 √ 𝜌𝑐𝛾𝐷⁄ and

Eötvös Number, 𝐸𝑜 = 𝑔 ∆𝜌 𝐷2 γ⁄ were considered.

Figure 3-10 compares the variation with time of the droplet descent speed, 𝑢𝑑∗ = 𝑢𝑑 √𝑔𝐷⁄ ,

obtained here from SPH with that predicted by Han and Tryggvason [247] using a finite

difference front tracking method. This figure shows that the two predictions are

qualitatively very similar. Whilst there are some quantitative differences between the two

predictions, these differences are small about 1.36%. Snapshots of the deforming droplet at

various times indicate how the deformation affects the droplet descent speed. After the

initial rise the droplet descent slows down as it starts deforming, until reaches to a steady

state condition.

Figure 3-10 Variation of the droplet descent speed with time as predicted by SPH (dash line)

and Han and Tryggvason [247] (solid line) for 𝛈 = 𝟏. 𝟏𝟓, 𝛌 = 𝟏, 𝐄𝐨 = 𝟏𝟎, 𝐎𝐡𝒅 = 𝟎. 𝟐𝟒 and

𝐌𝐨 = 𝟎. 𝟎𝟒. Snapshots of the deforming droplet are shown at various times

Figure 3-11 shows the evolution of the droplet descent speed with time for different Eötvös

Numbers at 𝜂 = 1.15, 𝑂ℎ𝑑 = 0.23, 𝑂ℎ𝑜 = 1.25. Although the falling velocity ends to a

steady state, the simulation results reveal some dissimilarity in the droplet velocity trends

for the different Eo. According to the figure at the fixed Ohnesorge numbers, the smaller

Eo reaches to the steady state faster, while for the larger Eo, the droplet experiences an

0

0.05

0.1

0.15

0.2

0.25

0.3

0 10 20 30 40 50

Axi

s Ti

tle

Axis Title𝑡∗

𝑢𝑑∗

49

overshoot in the descent speed that implies larger deformation for the falling droplet.

Figures 3-11 (b) and (c) show the droplet shape changes at the different Eötvös Numbers.

Figure 3-11 (a) variation of droplet descent speed with time as predicted by SPH for 𝜼 =𝟏. 𝟏𝟓, 𝑶𝒉𝒅 = 𝟎.𝟐𝟑, 𝑶𝒉𝒐 = 𝟏. 𝟐𝟓, 𝑬𝒐 = 𝟐𝟒 (solid line), and 𝑬𝒐 = 𝟏𝟒𝟒 (dash line). Droplet

deformation for (a) 𝑶𝒉𝒅 = 𝟎. 𝟐𝟑, 𝑶𝒉𝒐 = 𝟏. 𝟐𝟓, 𝑬𝒐 = 𝟐𝟒, and (c) 𝑶𝒉𝒅 = 𝟎. 𝟐𝟑, 𝑶𝒉𝒐 = 𝟏. 𝟐𝟓,

𝑬𝒐 = 𝟏𝟒𝟒 .

Figure 3-12 illustrates the effect of 𝑂ℎ𝑐 and the corresponding Morton numbers on the

droplet shape and descent speed at the fixed 𝐸𝑜 = 24, and 𝑂ℎ𝑑 = 1.16. According to the

figure, the shape change occurs from spherical to oblate and indented oblate as 𝑂ℎ𝑐 and

Mo decrease, implying the less viscous droplet experiences the more deformation. These

results are in agreement with the shape regime maps developed by Han and Tryggvason.

Figure 3-12 (a) variation of droplet descent speed with time as predicted by SPH for 𝜼 =𝟏. 𝟏𝟓, 𝑶𝒉𝒅 = 𝟏.𝟏𝟔, 𝑬𝒐 = 𝟐𝟒, 𝑶𝒉𝒄 = 𝟎. 𝟎𝟓 (solid line), and 𝑶𝒉𝒄 = 𝟏. 𝟐𝟓 (dash line). Droplet

deformation for 𝑶𝒉𝒅 = 𝟏. 𝟏𝟔, 𝑬𝒐 = 𝟐𝟒, (b) 𝑶𝒉𝒄 = 𝟎. 𝟎𝟓 and 𝑴𝒐 = 𝟎. 𝟎𝟎𝟎𝟏𝟓, and (c)

𝑶𝒉𝒄 = 𝟏. 𝟐𝟓 and 𝑴𝒐 = 𝟓𝟖

0

1

2

3

4

5

0 1 2 3 4 5

Axi

s Ti

tle

Axis Title𝑥 − 𝑑𝑖𝑟𝑒𝑐𝑡𝑖𝑜 (× 𝑟𝑑)

−𝑑𝑖𝑟𝑒𝑐𝑡𝑖𝑜 (×𝑟 𝑑)

0

1

2

3

4

5

0 1 2 3 4 5

Axi

s Ti

tle

Axis Title𝑥 − 𝑑𝑖𝑟𝑒𝑐𝑡𝑖𝑜 (× 𝑟𝑑)

−𝑑𝑖𝑟𝑒𝑐𝑡𝑖𝑜 (×𝑟 𝑑)

(a) (b) (c)

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0 10 20 30 40 50 60

Axi

s T

itle

Axis Title

𝑢𝑑∗

𝑡∗

0

1

2

3

4

5

0 1 2 3 4 5

Axi

s Ti

tle

Axis Title𝑥 − 𝑑𝑖𝑟𝑒𝑐𝑡𝑖𝑜 (×𝑟𝑑)

−𝑑𝑖𝑟𝑒𝑐𝑡𝑖𝑜 (×𝑟 𝑑)

0

1

2

3

4

5

0 1 2 3 4 5

Axi

s Ti

tle

Axis Title𝑥 −𝑑𝑖𝑟𝑒𝑐𝑡𝑖𝑜 (×𝑟𝑑)

−𝑑𝑖𝑟𝑒𝑐𝑡𝑖𝑜 (×𝑟 𝑑)

(a) (b) (c)

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0 10 20 30 40 50 60

Axi

s T

itle

Axis Title𝑡∗

𝑢𝑑∗

50

3.4 Conclusions

In this work, a SPH-based model was developed to study immiscible liquid-liquid systems.

Strict incompressibility was enforced through solving the Pressure Poisson Equation. A

surface tension model was proposed to enforce immiscibility between the different fluids.

The model was validated against different model problems including Young-Laplace

equation, droplet deformation and breakup under confined shear field, and a liquid droplet

falling under gravity through another quiescent liquid. The results suggest that the model is

capable to predict phase morphologies and fluid dynamics of immiscible liquid-liquid

systems.

51

Smoothed Particle Hydrodynamics-based Chapter 4:

Simulations of Immiscible Liquid-Liquid Microfluidics

in a Hydrodynamic Focusing Configuration

H. Elekaei Behjati1, M. Navvab Kashani

1, H. Zhang

1, M.J. Biggs

1,2,*

1. School of Chemical Engineering, the University of Adelaide, SA 5005,

Australia.

2. School of Science, Loughborough University, Loughborough, LE11 3TU,

UK.

* m.biggs@lboro.ac.uk

52

Statement of Authorship

Title of Paper

Smoothed Particle Hydrodynamics-based Simulations of Immiscible

Liquid-Liquid Microfluidics in a Hydrodynamic Focusing

Configuration

Publication Status ☐Published ☐Accepted for Publication

☐Submitted for Publication ☒Publication Style

Publication Details Journal of Physics of Fluids

Author Contributions

By signing the Statement of Authorship, each author certifies that their stated contribution to the

publication is accurate and that permission is granted for the publication to be included in the

candidate’s thesis.

Name of Principal

Author (Candidate) Hamideh Elekaei Behjati

Contribution

Performed modelling and coding for simulation. Processed and

undertook bulk of data interpretation. Prepared manuscript as primary

author.

Signature

Date 23/08/2015

Name of Co-author Moein Navvab Kashani

Contribution Assisted with results analysis, data interpretation and manuscript

preparation.

Signature

Date 23/08/2015

Name of Co-author Hu Zhang

Contribution Provided advices on simulation.

Signature

Date 23/08/2015

Name of Co-author Mark J. Biggs

Contribution

Supervisor. Advised on experimental design, modelling, and data

interpretation. Advised on manuscript preparation, revised manuscript,

and corresponding author.

Signature Date 23/08/2015

53

Abstract

Smoothed Particle Hydrodynamics (SPH) was applied to the study of hydrodynamic

focusing of a liquid flowing in a deep microchannel by a second, immiscible less-viscous

liquid introduced at right angles through identical but symmetrically opposed

microchannels. To investigate the flow dynamics and topological changes of the two

phases, the phases and the interfaces between them were explicitly modelled. The fluids

were truly incompressible, viscous effects were treated rigorously, and surface tension was

included. The model is able to reproduce the experimental results of Cubaud and Mason

(Phys. Fluids. 20, 053302, 2008), including the flow morphology mapping with the

Capillary Numbers of the two liquids, and the characteristics dimensions of the focused

liquid phase as a function of the liquid flow rate ratio. Going beyond the experimental

results, the model was used to build understanding of the velocity fields.

Keywords: Microfluidics; Hydrodynamic focusing; Smoothed Particle Hydrodynamics;

Immiscible liquids; Surface tension.

54

4.1 Introduction

The potential for microfluidics to revolutionize the energy, chemical processing and

medical fields has attracted significant interest and research effort over the last few

decades [1, 4, 81, 248-253]. Whilst many microfluidic systems involve single phase fluids

only (e.g. lab-on-a-chip analysis of biological fluids [254, 255]), microfluidic processing of

multiphase fluids also offers a wealth of opportunities. For example, adding a second

immiscible fluid phase to single-phase microfluidics facilitates high specific interfacial

area, mixing and mass transfer, leading to increased reaction yields for mass transfer-

limited reactions [81, 248, 256, 257]. Multiphase microfluidics also provides possibilities

to precisely control fluid-fluid interfaces and flow patterns for a wide range of applications

ranging from production of novel materials [81, 258, 259] through to medicine [51, 260,

261]. The production of monodispersed droplets of controlled size in micro-emulsions for

pharmaceutical applications [104, 262] is just one example.

In the absence of electric or magnetic fields, the morphology of the phases in isothermal

two-fluid microfluidic flows is dependent on the balance between the two dominant forces

at play – viscous and surface tension – which can be quantified by the Capillary Number,

the relative flow rates of the two phases, the ratio of their viscosities, and the geometry of

the microfluidic pathways. For example, when one liquid (L1) is ‘focused’ by a second

immiscible liquid (L2) introduced through two side channels as illustrated in Figure 1(a),

the phase morphology can take on one of five characters depending on the Capillary

Numbers of the two streams, 𝐶𝑎𝑖, as indicated in Figure 1(b) [14]: (a) threading in which

L1 is focused into a thin thread by L2; (b) jetting where the thread breaks up due to

instabilities to produce droplets of L1 in L2 down-stream of the flow focusing junction; (c)

dripping in which the droplets of L1 are formed at constant frequency at the junction; (d)

tubing in which L1 occupies much of the outlet cross-section; and (e) viscous displacement

where fingers of L1 penetrate into the side channels. The thickness of the thread in the

threading regime, 𝜀, is related to the ratio of the flow rates [14]

𝜀

𝐻≈ (

𝑄12𝑄2

)1 2⁄

(1)

55

In the dripping regime, the droplet length, 𝑙𝑑, is given by [14]

𝑙𝑑𝐻≈1

2(

𝑄2𝑄1 + 𝑄2

𝐶𝑎2 )−0.17

(2)

In the jetting regime, the diameter of the droplets, 𝑑𝑗, and the thread length, 𝑙𝑗 are given by

[14]

𝑑𝑗

𝐻≈ 3.1 (

𝑄12 𝑄2

)1 2⁄

(3)

𝑙𝑗

𝐻≈

8 𝐶𝑗 𝜇1

𝜋𝐻2𝐶𝑎𝑐𝛾( 𝑄1𝑄22

)1 2⁄

(4)

where 𝐶𝑗 is a constant close to unity and 𝐶𝑎𝑐 is the critical capillary number which was

found to be approximately 0.1 [14]. A range of other configurations beyond that of Figure

4-1 have also been studied over the last two decades, including T-junctions [15, 35, 105],

Y-junctions [18], and concentric injection [110] to reveal a rich phase morphology

behaviour that is dependent on the nature of the flow focusing junction. For example, a

cross-junction like that shown in Figure 4-1 allows formation of symmetrical droplets

whilst only asymmetrical drops are possible with T-junctions [263].

.

(a)

𝐻

𝐻𝐶𝑎1 = 𝜇1𝑈1 𝛾⁄

𝑄1

𝑄2 2⁄

𝑄2 2⁄ , 𝐶𝑎2= 𝜇2𝑈2 𝛾⁄

𝑄1+ 𝑄2

56

(b)

Figure 4-1 (a) schematic of cross-junction flow focusing geometry considered in detail in [14]

and here, where 𝑸𝒊, 𝑪𝒂𝒊, 𝝁𝒊 and 𝑼𝒊 = 𝑸𝒊 𝑯𝟐⁄ are the flow rate, Capillary Number, viscosity

and superficial velocity of liquid Li, respectively, and 𝜸𝟏𝟐 is the interfacial tension for the

liquid pair; (b) phase morphology map based on the Capillary Numbers of the liquid L1

(𝑪𝒂𝟏) and the focusing liquid L2 (𝑪𝒂𝟐): (a) threading; (b) jetting; (c) dripping; (d) tubing;

and (e) viscous displacement. The numbers on the map refer to the conditions that were

considered in the study reported on here.

Though there has been extensive study of the effect of fluid properties and flow geometries

on phase morphology, there are still many gaps in understanding due to the sporadic

sampling of the range of materials, fluids, flow rates, channel dimensions and geometries

[264]. Additionally, understanding of the flow fields in microfluidic devices is even less

well explored because of the challenges faced in measuring these at such small scales.

Numerical studies can assist in filling this gap. For instance, De Menech et al [35] studied

transition regimes in a microfluidic T-junction using a phase field model to show the

importance of viscosity ratio for the droplet formation as well as the effect of confinement

b

0.01

0.1

1

0.0001 0.001 0.01 0.1

Ca

1

Ca2

b

c

e

a

d

be

1

2

3

5

7

8

9

10

11

12

13

141516b

a

cd

e

6

4

57

on the shape of droplet and slugs when the size of the side-channel is less than size of the

continuous phase channel. Similarly, the different flow patterns obtained through droplet

formation process in a T-junction microchannel have been numerically investigated by the

same group [250]. Liu and Zhang [265] studied the effects of flowrates, viscosity and

wettability on the process of droplet formation at a microfluidic junction using an

improved phase field lattice Boltzmann method. Also, Gupta and Kumar [161] applied

lattice Boltzmann method (LBM) to show how the dimensions of the channels in a

microfluidic T-junction influence the droplet size. Recently, Ngo et al. [266] investigated

numerically the droplet formation process in a double T-shaped microchannels using a

Volume of Fluid (VOF) model. Liu et al. [267] employed Dissipative Particle Dynamics

method to study multiphase flow behaviour in an inverted Y-shaped microchannel and a

microchannel network. Wu et al. [268] developed a multiphase LBM for droplet formation

in a microfluidic cross-junction flow geometry and demonstrated the dependence of

dripping flow pattern on both Capillary and Weber Numbers. Likewise, Liu et al. [20]

reported three different flow regimes for varied flowrate ratios at a fixed Capillary number

using LBM. Their study was, however, limited to small Capillary Numbers (Ca<0.01) and

viscosity ratios (𝛌1/50). Shi et al. [269] performed a comparative study on the dynamics

of the droplet formation in a microfluidic T-junction and a flow-focusing configuration

using LBM. They indicated that a spherical droplet can be formed through a flow-focusing

geometry, whereas T-junction is likely to produce an elliptical droplet.

Numerical modelling of immiscible liquid-liquid microfluidics requires the explicit

modelling of the two phases and the interfaces between them. This is necessary for two

reasons at least. Firstly, as the characteristic dimensions of the individual phases are

typically comparable to that of the confining geometry, homogenisation approaches are

inappropriate. The second reason is the fact that interest often lays in understanding the

detailed morphology of the phases, something that is not accessible in more mean field

models such as the two-fluid model. Although explicit modelling of the individual phases

and the interfaces between them through space and time is necessary, it is particularly

challenging. If mesh-based numerical methods are used, for example, the evolution of the

mesh in space and time requires complex and expensive algorithms that can bring error

propagation and numerical instabilities [130, 165]. These methods are also often unable to

58

handle the coalescence or break-up of phases (e.g. break-up of the thread in jetting to form

the droplets) [241, 270]. Meshless methods such as Smoothed Particle Hydrodynamics

(SPH) [27, 171] and the more expensive [209] DPD [271, 272] avoid these issues, on the

other hand.

In the work reported in this paper, an incompressible SPH-based model [273] is shown to

be able to reproduce the experimental results obtained by Cubaud and Mason [14] for the

flow-focusing geometry shown in Figure 4-1(a). The emphasis of the study was on the

effect of the flowrate ratio and Capillary Numbers. The paper first presents the details of

the SPH model in brief along with the details of the study. Results are then presented for

the range of conditions shown on the Capillary Number map in Figure 4-1(b) and

compared with the experimental data of Cubaud and Mason [14], thereby demonstrating

that the SPH method is suitable for building detailed understanding of the influence of

fluid properties and process conditions on the phase morphology. The flow fields are also

explored, adding understanding to the experimental results.

4.2 Model and Study Details

Model 4.2.1

A microfluidic right angled cross-junction consisting of three inlet microchannels and one

outlet microchannel as shown in Figure 4-1(a) was considered; the depth of the

microchannels was assumed to be much greater than their width, 𝐻, and study was

restricted to a plane away from the top and bottom of the channels (i.e. the simulations

were limited to two spatial dimensions). A Newtonian liquid L1 of viscosity 𝜇1 entered the

main inlet channel at a volumetric flow rate 𝑄1 = 𝑈1𝐻2. Likewise, a second Newtonian

liquid L2 of viscosity 𝜇2 < 𝜇1 was injected symmetrically into the two lateral inlet

channels each at a volumetric flow rate (𝑄2 2⁄ ) = 𝑈2𝐻2. This configuration leads to the

second liquid hydrodynamically focusing the first as outlined in the Introduction to the

report here.

The SPH-based model proposed in our earlier work [273] for simulation of immiscible

liquid-liquid flows was applied here to study the flow configuration outlined immediately

above; full details of the SPH model can be found in Elekaei et al. [273]. A total of 6562

59

SPH particles were initially arranged uniformly within the microchannels with a spacing of

𝐿0 = 1 × 10−5 m and the fluid particles were assigned an initial speed of 𝑈𝑖 = 𝑄𝑖 𝐻2⁄ . The

particles located in the main microchannel were initially designated as liquid L1, and those

in the two transverse channels as liquid L2. No-slip boundary conditions and confinement

of the SPH particles within the microchannel walls was implemented as described by

[273]. Dirichlet velocity boundary conditions were applied at the inlets and outlet of the

microchannels using a similar approach without, of course, the particle confinement.

Whenever a particle of a liquid leaves the outlet, it is fed back into the simulation through

the appropriate inlet, being sure to maintain symmetry of the lateral flows, thus conserving

the number of particles (and mass) in the simulations. The timestep size used was 1 ×

10−7 s, and the kernel size was 1.4 𝐿0.

Study Details 4.2.2

The study reported here focused on the two immiscible Newtonian liquids summarized in

Table 4-1. The channel cross sectional size was also set to 𝐻 = 100 m for all

simulations. The 16 Capillary Number pairs (𝐶𝑎1, 𝐶𝑎2) shown in Figure 4-1(b) were

considered by varying independently the superficial velocity of the two liquids, 𝑈𝑖 . These

conditions were selected to widely probe the capacity of the model to reproduce the map

determined experimentally by Cubaud and Mason [14], including the boundaries between

the different phase morphologies.

Table 4-1: Properties of liquid pair considered

Property/Fluid Glycerol (L1) PDMS Oil

(L2)

Viscosity (𝜇𝑖; Pa·s) 1.214 0.00459

Surface tension (𝛾12; N/m) 0.027

4.3 Results and Discussion

Figure 4-2 shows the simulation result for the Capillary Number doublet of (0.5, 0.5),

which corresponds to the point 1 in Figure 4-1(b). It is clear that the model is able to

reproduce the threading flow expected at this point. As shown in Figure 4-10, the same can

be said for the three other simulations undertaken in this part of the phase morphology

60

map. More quantitatively, as Figure 4-3 shows, the SPH model is able to predict very well

the variation of the thread thickness, 𝜀, obtained experimentally, which is defined by

equation (1); the relative error between the correlation and the model is on average 8%,

which is anticipated the deviations are smaller than the experimental uncertainties. The last

point in this figure at 𝑄1 𝑄2 = 0.24⁄ shows the upper limit for 𝑄1 𝑄2⁄ in the

aforementioned conditions for which a threading flow regime is valid.

Figure 4-2 also shows the velocity field at steady state, something that is difficult to

measure experimentally. As a result of a flowrate ratio of 𝑄1 𝑄2⁄ = 0.1 , the two branches

of the continuous phase suppress the dispersed phase at the cross-junction and makes a

thinning process which leads to the formation of a long thread of 𝐿1.

Figure 4-2 Representative outcome for threading regime at 𝐂𝐚𝟏 = 𝟎. 𝟓 and 𝐂𝐚𝟐 = 𝟎. 𝟎𝟐,

which corresponds to point 1 on Figure 4-1(b).

200 300 400 500 600 700 800200

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500

600

𝑥 (𝜇𝑚)

(𝜇𝑚)

61

Figure 4-3 Dimensionless thickness of threads as a function of flowrate ratio as evaluated

using the SPH model (open circles) and equation (1) (solid line).

Figure 4-4 shows the SPH prediction at a Capillary Number doublet of (0.1, 0.003), which

corresponds to point 2 on Figure 4-1(b). Once again, it is clear that the model is able to

reproduce the dripping flow expected at this point. As shown in Figure 4-11, the same can

be said for the three other simulations undertaken in this part of the phase morphology

map. Figure 4-5 shows, the SPH model is able to predict very well the variation of the

droplet length, 𝑙𝑑, obtained experimentally, which is defined by equation (2); the relative

error between the correlation and the model is on average less than 8%, which is

anticipated that the error is smaller than the experimental uncertainties.

Figure 4-4 also indicates the evolution of the droplet formation process. At higher flowrate

ratios that allows a larger momentum for the dispersed phase, 𝐿1 is broken into droplets.

According to the figure, initially the velocity of the dispersed phase reaches to its

maximum where occupies the width of the main channel until collapses under the effect of

the continuous phase momentum that acts against interfacial tension, and leads to a

thinning pinch-off process. The built-up pressure in the side channels makes acceleration

and drives L1 to squeeze until reaches to a constant superficial velocity of U1+U2. At the

end of pinching process, breakup occurs and interfaces retract as a result of surface

tension.

0

0.1

0.2

0.3

0.4

0.5

0 0.05 0.1 0.15 0.2 0.25 0.3

epsi

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H

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𝜀𝐻⁄

62

(a)

(b)

(c)

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600

𝑥 (𝜇𝑚)

(𝜇𝑚)

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𝑥 (𝜇𝑚)

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63

(d)

Figure 4-4 Representative outcome of the SPH model for dynamics of the droplet formation

in dripping regime at 𝐂𝐚𝟏 = 𝟎. 𝟏 and 𝐂𝐚𝟐 = 𝟎. 𝟎𝟎𝟑, which corresponds to point 2 on Figure 4-

1(b). The snapshots belong to (a) 𝐭∗ = 𝟎. 𝟎𝟕 , (b) 𝐭∗ = 𝟎. 𝟐 , (c) 𝐭∗ = 𝟎. 𝟑𝟖 , and (d) 𝐭∗ = 𝟎. 𝟒 ,

where 𝐭∗ = 𝐔𝟏𝐭 𝐇⁄

Figure 4-5 Dimensionless length of droplet as estimated by the SPH model (open circles) and

equation (2) (solid line)

Figure 4-6 shows the SPH prediction at a Capillary Number doublet of (0.3, 0.005), which

corresponds to point 3 on Figure 1(b). As with the two other cases considered thus far, it is

clear that the SPH model is able to reproduce the jetting flow expected at this point. As

shown in Figure S3, the same can be said for the three other simulations undertaken in this

part of the phase morphology map. Figure 4-7(a) shows, the SPH model is able to predict

very well the variation of the droplet length, 𝑑𝑗, obtained experimentally, which is defined

by equation (3); the relative error between the correlation and the model is on average 6%,

𝑥 (𝜇𝑚)

(𝜇𝑚)

200 300 400 500 600 700200

300

400

500

600

0.5

1

1.5

2

1 1.5 2 2.5 3 3.5 4

Axi

s Ti

tle

Axis Title

𝑙 𝑑𝐻⁄

𝑄2 (𝑄1+𝑄2⁄ 𝐶𝑎2 )−0.17

64

which is anticipated that the deviation is smaller than the experimental uncertainties.

Similar to the dripping regime, a droplet formation and pinching process occurs in the

jetting regime but away from the junction after forming a very thin thread of the liquid in

the main channel, 𝐿1. Similarly, Figure 4-7(b) indicates the SPH results for variation of the

thread length of the jet, 𝑙𝑗, as defined by equation (4); the relative error between the

correlation and the model is on average 10%.

(a)

(b)

Figure 4-6 Representative outcome of the SPH model for dynamics of the jetting regime at

𝑪𝒂𝟏 = 𝟎.𝟑 and 𝑪𝒂𝟐 = 𝟎. 𝟎𝟎𝟓, which corresponds to point 3 on Figure 4-1(b)). The snapshots

belong to (a) 𝒕∗ = 𝟎. 𝟒𝟐 , and (b) 𝒕∗ = 𝟎. 𝟒𝟒 , where 𝒕∗ = 𝑼𝟏𝒕 𝑯⁄

400 600 800 1000 1200 1400200

300

400

500

600

𝑥 (𝜇𝑚)

(𝜇𝑚)

𝑥 (𝜇𝑚)

(𝜇𝑚)

400 600 800 1000 1200 1400200

300

400

500

600

65

(a) (b)

Figure 4-7 Dimensionless (a) droplet diameter (equation (3)), and (b) thread length (equation

(4)) in dripping regime as evaluated by the SPH model (open circles) and the related

correlations (solid line).

Figure 4-8 shows the SPH prediction at a Capillary Number doublet of (1.0, 0.006), which

corresponds to point 4 on Figure 4-1(b). As with the all the other cases considered thus far,

it is clear that the SPH model is able to reproduce the expected flow pattern (i.e. tubing).

The controlling agent in tubing regime is the viscous stress by which a long viscous core

occupies the most cross section of the main channel as can be observed by the figure.

Figure 4-8 Representative outcome of the SPH model for the tubing regime at 𝑪𝒂𝟏 = 𝟏. 𝟎

and 𝑪𝒂𝟐 = 𝟎.𝟎𝟎𝟔, which corresponds to point 4 on Figure 4-1(b).

0

0.2

0.4

0.6

0.8

1

1.2

0 0.05 0.1 0.15 0.2 0.25

Axi

s Ti

tle

Axis Title𝑄1 𝑄2⁄

𝑑 𝑗𝐻⁄

0

2

4

6

8

10

12

14

7E-11 8E-11 9E-11 1E-10 1.1E-10

Ax

is T

itle

Axis Title 𝑄1𝑄2 2⁄ 0.5

𝑙 𝑗𝐻⁄

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400

500

600

𝑥 (𝜇𝑚)

(𝜇𝑚)

66

And, finally, Figure 4-9 shows the SPH prediction at a Capillary Number doublet of (0.19,

0.0002), which corresponds to point 16 on Figure 4-1(b). Viscous displacement regime,

also known as fingering, happens at a large q, when the low momentum of L2 is not

enough to drive L1, and so L1 starts to occupy the side channels.

Figure 4-9 Representative outcome of the SPH model for the tubing regime at 𝑪𝒂𝟏 = 𝟎. 𝟏𝟗

and 𝑪𝒂𝟐 = 𝟎.𝟎𝟎𝟎𝟐, which corresponds to point 16 on Figure 4-1(b).

4.4 Conclusion

An incompressible Smoothed Particle Hydrodynamics-based model was applied to study

immiscible-liquid flows through microfluidic hydrodynamic focusing configuration for a

wide range of capillary numbers (0.001 < 𝐶𝑎 ≤ 1.0). For a Newtonian liquid pair with a

large viscosity ratio, the different flow patterns known as threading, dripping, jetting, and

tubing regime were achieved through our simulations under the effects of the capillary

number of each liquid phase, as expected by the experimental results of Cubaud and

Mason (Phys. Fluids. 20, 053302, 2008). The characteristics dimensions of the liquid phase

in the main channel such as the thickness of the thread in the threading regime, the droplet

length in the dripping regime, and the droplet diameter in the jetting regime were validated

by the experimental results. Moreover, dynamics of the focused liquid flow and the droplet

formation evolution were discussed for different flow patterns.

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4.5 Supplementary Information

(a)

(b)

(c)

Figure 4-10 Representative outcome for threading regime at (a) 𝐂𝐚𝟏 = 𝟎. 𝟓 and 𝐂𝐚𝟐 = 𝟎. 𝟎𝟎𝟖,

which corresponds to point 12 on Figure 4-1(b), (a) 𝐂𝐚𝟏 = 𝟎. 𝟒 and 𝐂𝐚𝟐 = 𝟎. 𝟎𝟏, which

corresponds to point 11 on Figure 4-1(b), and (a) 𝐂𝐚𝟏 = 𝟎. 𝟑 and 𝐂𝐚𝟐 = 𝟎. 𝟎𝟐𝟓, which

corresponds to point 10 on Figure 4-1(b).

200 300 400 500 600 700200

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400

500

600

𝑥 (𝜇𝑚)

(𝜇𝑚)

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300

400

500

600

𝑥 (𝜇𝑚)

(𝜇𝑚)

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68

(a)

(b)

(c)

Figure 4-11 Representative outcome for dripping regime at (a) 𝐂𝐚𝟏 = 𝟎. 𝟎𝟖𝟓 and 𝐂𝐚𝟐 =𝟎. 𝟎𝟎𝟐 which corresponds to point 15 on Figure 4-1(b), (a) 𝐂𝐚𝟏 = 𝟎. 𝟏 and 𝐂𝐚𝟐 = 𝟎. 𝟎𝟎𝟏,

which corresponds to point 8 on Figure 4-1(b), and (a) 𝐂𝐚𝟏 = 𝟎. 𝟎𝟓 and 𝐂𝐚𝟐 = 𝟎. 𝟎𝟎𝟎𝟔 which

corresponds to point 7 on Figure 4-1(b).

𝑥 (𝜇𝑚)

(𝜇𝑚)

200 300 400 500 600 700200

300

400

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600

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69

(a)

(b)

(c)

Figure 4-12 Representative outcome for jetting regime at (a) 𝐂𝐚𝟏 = 𝟎. 𝟏𝟗 and 𝐂𝐚𝟐 = 𝟎. 𝟎𝟎𝟓

which corresponds to point 15 on Figure 1(b), (a) 𝐂𝐚𝟏 = 𝟎.𝟏𝟕 and 𝐂𝐚𝟐 = 𝟎. 𝟎𝟎𝟖, which

corresponds to point 14 on Figure 1(b), and (a) 𝐂𝐚𝟏 = 𝟎. 𝟐 and 𝐂𝐚𝟐 = 𝟎. 𝟎𝟎𝟖 which

corresponds to point 13 on Figure 1(b).

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400

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600

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(𝜇𝑚)

𝑥 (𝜇𝑚)

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70

Effects of Flow Parameters and Geometrical Chapter 5:

Characteristics on the Dynamics of Two Immiscible

Liquids in a Hydrodynamic Focusing Microgeometry

H. Elekaei Behjati1, M. Navvab Kashani

1, H. Zhang

1, M.J. Biggs

1,2,*

1. School of Chemical Engineering, the University of Adelaide, SA 5005,

Australia.

2. School of Science, Loughborough University, Loughborough, LE11 3TU,

UK.

* m.biggs@lboro.ac.uk

71

Statement of Authorship

Title of Paper

Effects of Flow Parameters and Geometrical Characteristics on the

Dynamics of Two Immiscible Liquids in a Hydrodynamic Focusing

Microgeometry

Publication Status ☐Published ☐Accepted for Publication

☐Submitted for Publication ☒Publication Style

Publication Details Journal of Physics of Fluids

Author Contributions

By signing the Statement of Authorship, each author certifies that their stated contribution to the

publication is accurate and that permission is granted for the publication to be included in the

candidate’s thesis.

Name of Principal

Author (Candidate) Hamideh Elekaei Behjati

Contribution

Performed modelling and coding for simulation. Processed and

undertook bulk of data interpretation. Prepared manuscript as primary

author.

Signature

Date 23/08/2015

Name of Co-author Moein Navvab Kashani

Contribution Assisted with results analysis, data interpretation and manuscript

preparation.

Signature

Date 23/08/2015

Name of Co-author Hu Zhang

Contribution Provided advices on simulation.

Signature

Date 23/08/2015

Name of Co-author Mark J. Biggs

Contribution

Supervisor. Advised on experimental design, modelling, and data

interpretation. Advised on manuscript preparation, revised manuscript,

and corresponding author.

Signature Date 23/08/2015

72

Abstract

Recently we reported a numerical study on immiscible-liquid flows in hydrodynamic

focusing microchannels using Smoothed Particle Hydrodynamics (SPH) technique. The

optimal design of the system, however, needs a detailed understanding of the dynamics of

flows and the capability to predict such behaviours. To respond to this need, we utilized

our SPH-based simulation to predict the behaviour of the multiphase flow system for a

wider range of flow parameters including the flowrate ratio, viscosity ratio and capillary

number of each liquid phase. The flowrate ratio of the continuous phase to the dispersed

phase affects the flow behaviour and causes different flow patterns depending on 𝐶𝑎1

and 𝐶𝑎2 . The droplet length also appears to be influenced by the both flowrate quantity

and flowrate ratio in dripping regimes. In addition, we discuss the viscosity ratio effects on

the dynamics of the two-phase flow. Subsequently, we explore the effects of geometry

characteristics such as channel size, width ratio, and the angle between the main and side

inlet channels, on the flow dynamics in a hydrodynamic focusing configuration. According

to our findings, increasing the side channel width causes longer droplets, and the right-

angled design effects the most efficient focusing behaviour.

Keywords: Microfluidics; Hydrodynamic focusing; Smoothed Particle Hydrodynamics;

Immiscible liquids; Surface tension.

73

5.1 Introduction

The flow structures formed by two-phase flows in microchannels have attracted significant

scientific and industrial attention in recent years. The importance of the micro-systems lies

in the dominance of interfacial forces such as viscosity and surface tension over bulk

forces, which enable controlled flow patterns, such as droplet formation, with precise

control over the size of the droplets. Such controlled behaviour of two immiscible liquids

interacting in an emulsion inside a microchannel is highly needed for a variety of industrial

applications such as pharmaceuticals, food processing, chemical processing, enhanced oil

recovery, and biotechnology [274-277].

Nakajima et al [17, 96] and Knight et al[97] were pioneers in generating monodisperse

micro-droplets, the former through an array of cross-flow microchannels and the latter

using a hydrodynamic focusing microchannel. Over the past decade, a number of research

studies have reported developments being made in the field of droplet formation and

breakup using different microchannel configurations, including T-junctions [15, 35, 105,

278], Y-junctions [18], Cross-junctions [20, 38], Hydrodynamic focusing and flow

focusing [14, 16, 100, 106-109], Concentric injection [110] and Membrane Emulsification

[111-113]. As is addressed by these studies, a potential strategy to increase the degree of

monodispersity during droplet production in the micro-geometries is to draw from the

characteristic features of multi-stream laminar flow systems, using continuous flows. In

laminar flows, the multi-liquid flows through microchannels without any turbulence;

interfacial forces dominate inertia and make it possible to control the flow structure, size of

the formed droplets and threads by altering the relative flowrates of the multi-stream.

Hydrodynamic focusing is a successful example of monodispersed droplet formation

micro-configuration, wherein a centrally located flow is narrowed or hydrodynamically

focused by two neighbouring flows through cross-like microchannels. Xu et al[16]set up a

three-stream laminar flow configuration to hydrodynamically focus a more viscous liquid

surrounded by two inviscid liquids in order to facilitate a final breakup through control of

the flow rate ratio. Cubaud and Mason [14] performed a comprehensive experimental

study on Newtonian immiscible liquids in a hydrodynamic focusing microfluidic device.

They investigated the competition between viscosity and interfacial tension effects on the

74

creation of capillary threads and droplets using hydrodynamic focusing over a large range

of viscosities and interfacial tensions at a fixed geometry, which provides a useful range of

flowrate ratios, viscosity ratios and Ca. In addition, they continued their experiments on

miscible fluids in order to investigate miscible interfacial morphologies and flow patterns

[109]. Recently, Kunstmann-Olsen et al[108] studied experimentally and theoretically a

two-dimensional microfluidic cross-flow and showed that the local geometry of the

microchannel-junction affects the hydrodynamic focusing behaviour.

As demonstrated by Baroud et al. [277], the parameters by which the flow pattern of

liquids interacting in a microchannel can be influenced, are very extensive and variables

include the nature of the liquids, the details of the flow injections, the geometry of the

channels, the surface characteristics and so forth. Despite various studies on droplet

formation through microfluidic devices reported so far, a full understanding of the flow

physics and the effective process parameters is still missing. On the other side,

experimental studies for microchannel flows are quite challenging and expensive, so

analytical solutions at such a small scale are limited to a few simple cases. Therefore,

numerical studies can be complementary to these experiments. To date, there are only a

limited number of numerical investigations in the literature. For instance, Wu et al. [38]

presented an improved immiscible lattice Boltzmann model for simulation of immiscible

two-phase flows in a cross-junction microchannel and also studied the effect of Ca on

droplet formation in such a cross-flow systems. Furthermore, Liu et al. [20] studied the

effect of flowrate ratios and capillary numbers on droplet formation in a microfluidic

cross-flow configuration, using a multiphase lattice Boltzmann model. They used a limited

range of capillary numbers (Ca<0.01) and viscosity ratios (≤1/50), but a variety of

flowrate ratios in their simulation and recognized three different flow regimes for their

different flowrate ratios at a fixed capillary number. In spite of these numerical

simulations, much effort is still required to accurately control interfaces and capture all the

key parameters for the design development of microfluidic devices.

Nevertheless, modelling of multiphase micro-flows is not easy because of some of their

special properties like free surfaces, moving interfaces, largescale differences in each

dimension, and microscale phenomena [279]. Numerical studies involving microchannel

flow simulations using mesh-based methods face a variety of obstacles. The most

75

important problem lies in the difficulties associated with generating the meshes, which

would be a problematic task, time consuming and an expensive process for problems with

complex geometry. Another challenge in mesh-based simulations is to track the movement

of the interfacial boundaries of multiphase fluid systems. This is an expensive operation

which affords another source for modelling error and numerical instability [130, 165].

Meshless methods have been the focus of research over the past few decades, in order to

develop the next generation of more efficient computational schemes designed for more

complex problems. One preferred meshless method is Smoothed Particle Hydrodynamics

(SPH) [171, 182].

Recently, we presented a novel attempt to model the flow behaviour of two immiscible

liquids in a hydrodynamic focusing micro-geometry using an incompressible Smoothed

Particle Hydrodynamics (SPH) method. The dynamics of the two-phase flows at various

capillary numbers were studied through the SPH-based simulation. Moreover, the

simulation results were compared and validated against the existing experimental data and

the comparison showed good agreement between the model prediction and experimental

outcomes [280]. The efficiency of droplet formation in microchannels is mainly affected

by the nature of the liquids and flow parameters such as viscosity, interfacial tension,

flowrate ratio, capillary number and geometrical characteristics. Therefore, in the present

work, we develop our SPH-based simulation to numerically study the effects of flow

parameters and geometry on immiscible-liquid flow behaviour in hydrodynamic focusing

microfluidics. The results are discussed in three sections covering the effects of the

flowrate and flowrate ratio, the effects of the viscosity ratio, and finally the effects of the

inlet channels’ geometry.

5.2 Model Details

A microfluidic right angled cross-junction consisting of three inlet microchannels and one

outlet microchannel, as shown in Figure 5-1, was considered; the depth of the

microchannels was assumed to be much greater than their width, H, and the study was

restricted to a plane away from the top and bottom of the channels (i.e. the simulations

were limited to two spatial dimensions). A Newtonian liquid, L1, of viscosity 𝜇1 entered

the main inlet channel at a volumetric flow rate of 𝑄1 = 𝑈1𝐻2. Likewise, a second

76

Newtonian liquid L2 of viscosity 𝜇2 < 𝜇1 was injected symmetrically into the two lateral

inlet channels each at a volumetric flow rate (𝑄2 2⁄ ) = 𝑈2𝐻2. This configuration leads to

the second liquid hydrodynamically focusing the first as outlined in the Introduction to the

report here.

Figure 5-1 Schematic of cross-junction flow focusing geometry

The SPH-based model proposed in our earlier work [273] for simulation of immiscible

liquid-liquid flows was applied here to investigate the effects of various flow parameters

and geometry characteristics; full details of the SPH model, simulation parameters and

boundary conditions can be found in Elekaei et al. [273, 280].

5.3 Results and Discussion

Effects of flowrate and flowrate ratio 5.3.1

To investigate the influence of flowrate ratio, we keep the size of the microchannels

constant at 𝐻 = 100 𝜇𝑚, and consider a reference liquid pair and constant flowrate of the

main channel phase 𝑄1, whilst changing the flow rate of the continuous phase 𝑄2 and

thus 𝐶𝑎2. Firstly, we study the effect of flowrate ratio 𝑞 =𝑄2

𝑄1 on the flow behaviour of each

liquid phase in a pair, including an 80 volume percentage of Glycerol in water as L1 with

𝐻

𝐻𝐶𝑎1 = 𝜇1𝑈1 𝛾⁄

𝑄1

𝑄2 2⁄

𝑄2 2⁄ , 𝐶𝑎2= 𝜇2𝑈2 𝛾⁄

𝑄1+ 𝑄2

77

𝜇1 = 0.077 Pa. s and PDMS oil as 𝐿2 with 𝜇2 = 8.2 × 10−4 Pa. s plus the interfacial

tension of 𝛾12 = 0.0304 −1 [14] at constant 𝐶𝑎1.

The simulation results show various flow regimes at different flowrate ratios. Figure 5-2

represents the influence of the flowrate ratio on the flow pattern of the aforementioned

liquid pair with a fixed viscosity ratio of µ1

µ2= 94, for which 𝑄1 = 25 𝜇𝑙 𝑚𝑖 ⁄ and

𝐶𝑎1 = 0.1 are fixed, while 𝑄2 and 𝐶𝑎2 are variable with regard to a range of flowrate

ratios at 0.1, 0.5, 1, 2, 3, 5, 10, and 20.

(a) (b)

(c) (d)

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Po

siti

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(𝜇𝑚)

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(e) (f)

(g) (h)

Figure 5-2 Effect of flowrate ratio on the flow pattern at (a) q=0.1, (b) q=0.5, (c) q=1, (d) q=2,

(e) q=3, (f) q=5, (g) q=10, and (h) q=20, with a fixed viscosity ratio of 𝛍𝟏

𝛍𝟐= 𝟗𝟒

The flow structure of the liquid pair changes from the displacement pattern in Figure 5-

2(a) and the tubing regime in Figures 5-2(b) and (c), to dripping and jetting regimes as

shown in Figures 5-2(d) to 5-2(f) respectively, and finally the threading regime in Figures

5-2(g) and (h), as the volume fraction of the continuous phase 𝛼 =𝑄2

𝑄1+ 𝑄2 increases.

Moreover, it can be understood from the figures that the flowrate ratio affects the size of

the formed droplets and the thickness of the threads, as well as the thickness of the core

plug flow in the tubing regime. Comparing Figure 5-1(d) and (e) indicates that the droplet

length in the dipping regime decreases when the flowrate ratio 𝑞 =𝑄2

𝑄1 or the continuous

phase volume fraction 𝛼 increases.

In addition to the flowrate ratio, the flow rate quantity itself can influence the flow regime

and associated features at a constant flowrate ratio. To study this effect, we have

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79

performed a number of simulations in several groups for which various flowrates of the

dispersed phase at a fixed flowrate ratio were considered. Figure 5-3 indicates a flow

pattern map for the above liquid pair, based on the capillary numbers of both liquids, 𝐶𝑎1

and 𝐶𝑎2, and the flowrate ratio q. Note that changes in 𝐶𝑎1 and 𝐶𝑎2, here, were caused

simply by the changes in the flowrates, 𝑄1 and 𝑄2.

Figure 5-3 Flow regime map in terms of 𝐂𝐚𝟏, 𝐂𝐚𝟐 and flowrate ratio. Dripping (solid square),

displacement (solid circles), tubing (solid triangle), threading (solid diamond), and jetting

flow regime (solid pentagon).

According to the Figure 5-3, variations of the flowrate quantities even at the same flowrate

ratio can change the flow regime. In addition, the figure shows that at smaller values

of Q1 for the liquid pair, the dripping regime is achievable for a wider range of q and Ca2.

Furthermore, Figure 5-4 shows how the flowrate quantity can affect the length of the

droplets of the aforementioned liquid pair in the dripping flow regime at a constant

flowrate ratio of 2. Increasing the flowrate of the dispersed phase results in a smaller

droplet and more stabilized tail, as is obvious in the Figures 5-4(a) to (c) and measured by

Figure 5-4(d).

80

(a) (b)

(c) (d)

Figure 5-4 Effect of the flowrate quantities on the droplet size for the constant flowrate ratio

of q=2 and at (a) 𝐐𝟏 = 𝟏𝟎 µ𝐥

𝐦𝐢𝐧 , 𝐂𝐚𝟏 = 𝟎. 𝟎𝟒, 𝐂𝐚𝟐 = 𝟎. 𝟎𝟎𝟎𝟖𝟔 , (b) 𝐐𝟏 = 𝟐𝟓

µ𝐥

𝐦𝐢𝐧 , 𝐂𝐚𝟏 =

𝟎. 𝟏, 𝐂𝐚𝟐 = 𝟎. 𝟎𝟎𝟐𝟐 and (c) 𝐐𝟏 = 𝟒𝟎 µ𝐥

𝐦𝐢𝐧 , 𝐂𝐚𝟏 = 𝟎. 𝟏𝟔𝟖, 𝐂𝐚𝟐 = 𝟎. 𝟎𝟎𝟑𝟔 , (d) dimensionless

droplet diameter vs. flowrate of 𝐐𝟏

The effect of the viscosity ratio 5.3.2

Structure of two-phase flows in microchannels is controlled by the domination of one of

the competing forces such as inertia, viscous forces and interfacial tension. In addition, the

relative importance of the viscous stress of one phase acting on the other is a factor in

determining the flow pattern. Viscous forces tend to deform droplets and keep the interface

of the two-liquid phase smooth by dissipating the energy of the perturbations at the

interface. To find out the effects, we have implemented three groups of simulations for

Glycerol as the dispersed phase and PDMS oil as the continuous phase. During each group

of simulations 𝜇1 = 1.214 Pa. s and 𝛾12 = 0.027 −1and other parameters such as the

flowrates and flowrate ratio have been kept constant whilst a range of viscosity ratios,

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1

1.2

1.4

1.6

1.8

2

0 10 20 30 40 50 60

Axi

s Ti

tle

Axis Title𝑄1 [µl ]⁄

dH⁄

81

denoted by 𝜆 =𝜇1

𝜇2 have been considered. Figure 5-5 shows the simulation results for the

fixed flowrate ratio of q=10 and the different viscosity ratios as shown in Figure 5-5(a)

λ=1484, Figure 5-5(b) λ=264, and Figure 5-5(c) λ=24. By decreasing the viscosity ratio

and increasing viscosity of the continuous phase, the viscous stress of 𝐿2 on the interface

increases, which results in a change in the flow pattern from a tubing flow regime to a

threading regime.

(a) (b)

(c)

Figure 5-5 Effect of viscosity ratio of (a) λ=1484 and Ca2=0.0036, (b) λ=264 and Ca2=0.02, and

(c) λ=24 and Ca2=0.2, on flow pattern at q=10 and Ca1=0.5

Figure 5-6 indicates how the droplet formation can be influenced by the viscosity ratio at

the flowrate ratio of q=8 and 𝐶𝑎1 = 0.1. As can be seen from the figure, at high values of λ

(when 𝜇2 is very small), there must be an increase in the flowrate of the continuous phase

at the same capillary number of the dispersed phase in order to achieve a dripping flow

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82

regime in the outlet channel. A greater disparity in viscosities of the two phases allows for

a larger deformation and deviation from droplet generation. Furthermore, the figure

indicates that longer droplets are formed at higher viscosity ratios.

(a) (b)

(c)

Figure 5-6 Effect of viscosity ratio of (a) λ=1484 and Ca2=0.00054, (b) λ=264 and Ca2=0.003,

and (c) λ=24 and Ca2=0.033 on flow pattern at q=8 and Ca1=0.1

Figure 5-7 displays the results of another group of simulations when q=4.4 and 𝐶𝑎1 = 0.3.

By increasing 𝜇2 , the viscous shear of the continuous phase acting on the dispersed phase

increases, resulting in a change in the two-phase flow regime from displacement as shown

in figure 5-7(a) to a jetting regime in figure 5-7(b) and then to a threading flow regime in

figure 5-7(c).

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(a) (b)

(c)

Figure 5-7 Effect of viscosity ratio of (a) λ=1484 and Ca2=0.0009, (b) λ=264 and Ca2=0.005

and (c) λ=24 and Ca2=0.054, on flow pattern at q=4.4 and Ca1=0.3

Effect of geometry 5.3.3

In this section, we investigate the effects of the width ratio and inlet angle on the two-

phase flow behaviour in the crossed microchannels.

5.3.3.1 Width ratio effect

The width ratio of the main channel 𝑊2 with the side channel 𝑊1 at a constant channel

height H can affect the length of the droplets in a dripping flow regime. In an attempt to

study this effect, we have implemented the simulations at a fixed width 𝑊1 =

100 𝜇𝑚 and various width ratios, denoted as 𝜁 =𝑊2

𝑊1 . Figures 5-8 represent the simulation

results for effects of 𝜁 on the droplet size when 𝑞 = 2 and 𝐶𝑎1 = 0.04. According to the

figure, the wider the side channel width, the longer the droplet length.

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(a) (b)

(c)

Figure 5-8 Effect of width ratio of (a) 𝛇 = 𝟎. 𝟓 and Ca2=0.0036, (b) 𝛇 =1 and Ca2=0.0009, and

(c) 𝛇 =2 and Ca2=0.00026 on droplet length at 𝐪 = 𝟐 and 𝐂𝐚𝟏 = 𝟎. 𝟎𝟒

Figure 5-9 summarises the simulation results related to the width ratio effect on the droplet

length as a function of the flowrate ratio at 𝐶𝑎1 = 0.04. The figure confirms that there is a

good agreement between the predicted results and the values calculated from the following

correlation that was obtained by Cubaud and Mason [14] for droplet length measurement

in a dripping regime:

𝑙𝑑𝐻≈1

2(

𝑄2𝑄1 + 𝑄2

𝐶𝑎2 )−0.17

(1)

By increasing the width ratio in a dripping flow regime the size of the droplets increases at

every flowrate ratio in this range. In addition, the changing trends at each 𝜁 are quite

similar with approximately equal difference which shows the identical effects of width

ratio on the droplet size against the different flowrate ratios.

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Figure 5-9 Normalized droplet length in terms of the width ratio 𝛇 and the flowrate ratio q,

SPH simulation (solid diamond) and Equation (1) (open diamond) for𝛇 = 𝟎. 𝟓, SPH

simulation (solid circle) and Equation (1) (open circle) for 𝛇 = 𝟏, and SPH simulation (solid

square) and Equation (1) (open square) for 𝛇 = 𝟐

5.3.3.2 Inlet angle effect

In order to study the effect of the inlet angle, which is made by the main channel and the

side channel in the cross-like configuration, three different angles were investigated and

compared at the fixed flowrate ratio q=2, viscosity ratio λ=94, and width ratio 𝜁 = 1 with a

width of 100 μm. The results obtained by the simulations at 𝜃 = 45°, 90°, 135° are shown

in Figures 5-10 (a), (b) and (c) respectively. As is observed in the figures, increasing the

inlet angle leads to an increase in the droplet length. In addition, the breakup process is

affected by changing the inlet angle from 90⁰. According to the results, the sharp-angled

design delays the droplet breakup, whereas the obtuse-angled design hastens the

separation. In the right-angled design, the maximum flow focusing is obtained, however,

for a 45°or 135°-angled design, the inclined flow direction of the continuous phase

influences the focusing behaviour unfavourably. This effect arises from the decomposition

of the inertial force into a smaller vertical component plus a horizontal component which is

co-current with the dispersed phase in the 45° design and counter-current with that in the

135°design. Nonetheless, according to Figures 5-8 and 5-10, at a constant flowrate ratio,

0.8

1

1.2

1.4

1.6

1.8

2

2.2

2.4

2.6

2.8

0 2 4 6 8 10 12

Axi

s Ti

tle

Axis Title

𝑙 𝑑

𝑞

86

the influence of the inlet angle is less important as compared with the effect of the width

ratio.

(a) (b)

(c)

Figure 5-10 Effect of an inlet angle of (a) θ=45⁰, (b) θ=90⁰ and (c) θ=135⁰ on the droplet

length at q=2, Ca1=0.1 and Ca2=0.002

5.4 Conclusion

In this study, we investigated numerically the behaviour of immiscible-liquid pairs

interacting through hydrodynamic focusing microchannels, based on the Smoothed Particle

Hydrodynamics (SPH) method. We discussed the various aspects of the system, including

the effects of the capillary numbers of both dispersed and continuous phases, the influence

of important flow parameters and geometrical characteristics. Different flow patterns

including viscous displacement, tubing, jetting, dripping and threading regimes were

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obtained through variations of the flowrate ratio. The results show that the dripping regime

only happens for q≥1, meaning that L2 must be at least equal to or greater than L1 to

achieve a dripping flow regime. In addition, we indicated that the flowrate quantity and

flowrate ratio affect the droplet length in dripping regimes. We studied the effects of the

viscosity ratio on the flow structures and the dynamics of the droplets. As a consequence, a

large value of λ imposes an increase in the flowrate of the continuous phase with the same

capillary number of the dispersed phase to attain a dripping regime in the outlet channel.

Subsequently, we investigated how the geometry of the channel junction influences the

flow pattern and the hydrodynamic focusing. More specifically, we discussed the effects of

the different width ratios as well as the inlet angle of the main and side channels in cross-

like microfluidic configurations. According to our findings, increasing the side channel

width causes longer droplets, and the right-angled design gives the most efficient focusing

behaviour.

88

Conclusion Chapter 6:

This thesis is focused on outlining work that was focused on developing the application of

a meshless explicit numerical simulation method for the study of immiscible liquid-liquid

flows in microfluidic systems, so as to improve understanding of the fundamentals of

flows in such devices and develop a design capacity for immiscible-liquid microfluidics.

This work was undertaken through two related developments. The first one involved

developing a Smoothed Particle Hydrodynamics (SPH) model for explicit numerical

simulation of immiscible liquid systems. The model includes the surface tension, enforces

strict incompressibility, and allows for arbitrary fluid constitutive models. The validity of

the model was demonstrated by applying it to a range of interfacial problems with known

solutions, including a liquid droplet falling under gravity through another quiescent liquid,

Young-Laplace equation, and the deformation and breakup of a droplet under a shear field,

whilst confined.

The second development in this research has been divided into two parts: the first involved

the application of the proposed model for the study of hydrodynamic focusing of a liquid

flowing in a deep microchannel by a second immiscible liquid introduced at right angles

through identical but symmetrically opposed microchannels. The model was shown to be

able to reproduce the experimental results of Cubaud and Mason (Phys. Fluids. 20,

053302, 2008), including the flow morphology mapping with the Capillary Numbers of the

two liquids, and the characteristic dimensions of the focused liquid phase as a function of

the liquid flow rate ratio. Going beyond the experimental results, the model was used to

build understanding of the velocity fields.

The second part involved building a basis for the optimal design of the microfluidic system

by determining the effects of various flow parameters and geometry characteristics on the

flow dynamics and topological changes of the multiphase flow system. The different flow

patterns, including displacement, tubing, jetting, dripping and threading regimes were

identified as a result of changes in the capillary numbers of the interacting liquids. The

effects of flowrate quantity and flowrate ratio on the droplet length in dripping regimes

were also shown. The viscosity ratio effect was studied and it was demonstrated that a high

value of the viscosity ratio imposes an increase in the flowrate of the continuous phase,

89

with the same capillary number of the dispersed phase, to attain a dripping regime in the

outlet channel. Moreover, it was indicated how the geometry of the channel junction

influences the flow pattern and hydrodynamic focusing. In this regard, the effects of the

different width ratios, as well as the inlet angles of the main and side channels, were

investigated in the cross-like microfluidic configuration. We concluded that increasing the

side channel width causes longer droplets, and the right-angled design gives the most

efficient focusing behaviour. The two-dimensional model developed in this thesis is

expected to predict the parameters and variables with less than 10% deviation from the

experimental results: that is reasonably useful for the prediction of the hydrodynamics of a

3D system. These developments will provide great insights for designing microfluidic

devices involving immiscible liquid-liquid flows.

In terms of future work, there is clearly an opportunity to expand on the work concerned

with different microfluidic configurations such as T-shaped junctions. It would also be of

interest to extend the method to study double-emulsion microfluidic systems. In the case of

the numerical simulation work, this could also be extended to a three-dimensional model.

In addition, the model can be developed for fluids beyond those considered here, including

non-isothermal conditions and non-Newtonian fluids. The standard SPH algorithm

explicitly solves the conservation equations that is not efficient for the prediction of the

rheological properties of non-Newtonian fluids with high viscosity and therefore an

implicit SPH method is also required [281]. Before this happens, however, parallelisation

of the code and adoption of more efficient methods for solving the Pressure Poisson

Equation will be necessary.

90

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