VIETNAM NATIONAL UNIVERSITY HO CHI MINH CITY
HO CHI MINH CITY UNIVERSITY OF TECHNOLOGY
--------------------

MAI NGOC LUAN

NUMERICAL STUDY OF ELECTROHYDRODYNAMIC
ATOMIZATION BY OPENFOAM
Major: AEROSPACE ENGINEERING
Major code : 8520120

MASTER’S THESIS

HO CHI MINH CITY, July 2023


THIS THESIS IS COMPLETED AT
HO CHI MINH CITY UNIVERSITY OF TECHNOLOGY – VNU-HCM
Academic Supervisor: Assoc. Prof. Dr. Ngo Khanh Hieu

Examiner 1: Assoc. Prof. Dr. Le Tuan Phuong Nam

Examiner 2: Dr. Pham Minh Vuong

This thesis is defended at Ho Chi Minh City University of Technology,
VNUHCM on July 15th, 2023
Master's Thesis Committee:
1. Assoc. Prof. Dr. Vu Ngoc Anh

Chairman

2. Dr. Vuong Thi Hong Nhi

Secretary

3. Assoc. Prof. Dr. Le Tuan Phuong Nam

Examiner 1

4. Dr. Pham Minh Vuong

Examiner 2

5. Dr. Nguyen Song Thanh Thao

Member

Approval of the Chairman of Master's Thesis Committee and the Dean of
Faculty of Transportation Engineering after the thesis being corrected
CHAIRMAN OF THESIS
COMMITTEE

DEAN OF FACULTY OF
TRANSPORTATION ENGINEERING


VIETNAM NATIONAL UNIVERSITY HO CHI MINH CITY
HO CHI MINH CITY UNIVERSITY OF TECHNOLOGY

SOCIALIST REPUBLIC OF VIETNAM
Independence – Freedom – Happiness


MASTER THESIS ASSIGNMENT
Student name: MAI NGOC LUAN

Student ID: 2170729

Date of birth: 08/11/1997

Place of birth: Ho Chi Minh City

Major: Aerospace Engineering

Major code: 8520120

I.

THESIS’S TITLE:

Numerical study of electrohydrodynamic atomization by OpenFOAM.
(Phân tích hiện tượng phun tĩnh điện bằng phương pháp số sử dụng phần mềm
OpenFOAM)
II. THESIS ASSIGNMENT:
This thesis aims to develop a electrohydrodynamic solver based on the opensource software OpenFOAM to investigate the single cone-jet mode of
electrospray. The solver is physically verified and validated with preceding
literature and with experiment data under the consideration of liquid’s contact
angle condition. Additionally, the solver is enhanced to investigate the Taylor
cone formulation process under the effects of corona discharge. The outcome of
this thesis will serve as the basis for future numerical analyses of electrospray.
III. DATE OF ASSIGNMENT: 14/02/2023
IV. DATE OF COMPLETION: 09/06/2023

V. SUPERVISOR’S FULL NAME: Associate Professor Dr. Ngo Khanh Hieu

Ho Chi Minh City,……/……./2023
SUPERVISOR
(Sign and write full name)

HEAD OF DEPARTMENT
(Sign and write full name)

DEAN OF FACULTY OF TRANSPORTATION ENGINEERING
(Sign and write full name)

i


ACKNOWLEDGEMENT
This work is accomplished under a collaboration between VNU-HCM Key Lab. for
Internal Combustion Engine, Ho Chi Minh City University of Technology, Ho Chi
Minh City, Vietnam and School of Engineering and Built Environment, Griffith
University, Queensland, Australia.
I gratefully acknowledge Associate Professor Dr. Ngo Khanh Hieu, Dr. Dau Thanh
Van, Dr. Tran Canh Dung, Dr. Dinh Xuan Thien, Mr. Vu Trung Hieu and Mr. Vu
Hoai Duc for helping achieve this state of my thesis.
I would like to thank Dr. Nguyen Tan Hoi and CFD team at TechnoStar Vietnam
for supporting me during my Master course.
Additional thanks to Ms. Truong Van Ngoc for proofreading and grammatical
corrections.

Ho Chi Minh City, July 2023
Mai Ngoc Luan


ii


ABSTRACT
Electrospray, or Electrohydrodynamic Atomization (EHDA) operates on the
principles of electrohydrodynamics which deal with the motion of fluids placed
inside an electrical field. When a fluid is subjected to an adequately strong electrical
field, its surface can be deformed, creating a meniscus from whose apex thin jets is
induced. Eventually, these jets are destabilized and disintegrated into microscale or
nanoscale charged droplets. Among the known operating regimes of electrospray,
the stable single cone-jet mode is the most desired and applicable because of its
stability, controllability, and high yield rate in comparison to other regimes.
In this thesis, we program an electrohydrodynamic solver to simulate the conejet mode based on the Taylor-Melcher leaky-dielectric model. The solution for the
electrostatic governing equations is additionally developed, coupling with OpenFOAM’s interFOAM to model incompressible time-dependent multiphase fluid
flow. The solver is physically verified and validated with preceding literature as
well as with experiment data under the further consideration of liquid’s contact
angle condition, followed by analyses on the effects of electrical conductivity,
voltage, surface tension, flow rate, and fluid viscosity on spray current and jet
diameter. Numerical results are in reasonable agreement with experiments and
consistent with preceding literature. Additional studies on different contact angles
are performed, suggesting potentially major impacts of this factor on the cone-jet
mode in high voltage and low flow rate circumstances. Furthermore, the
electrohydrodynamic solver is enhanced to investigate the Taylor cone formulation
process under the effects of corona discharge. Electrospray-corona simulation and
contrasting experiment with high-speed camera show significant improvement of
the numerical prediction for Taylor cone formation, implying the crucial role of
liquid wetting to the Taylor cone formation in numerical electrospray-corona
discharge studies.
Keywords: capillary nozzle, CFD, cone-jet, corona discharge, electrospray, liquid

wetting, OpenFOAM, Taylor cone, …
iii


TĨM TẮT LUẬN VĂN
Cơng nghệ phun sương bằng lực tĩnh điện được sử dụng để tạo ra sơn khí từ điện
áp lớn. Công nghệ này hoạt động dựa trên các nguyên tắc của điện thủy động lực
học dùng để giải quyết các vấn đề liên quan đến chuyển động của lưu chất trong
điện trường. Khi chất lỏng được đặt trong một trường điện trường đủ lớn, bề mặt
của nó bị biến dạng tạo ra một cấu trúc có dạng hình nón và từ đó tạo ra những tia
chất lỏng bắn ra từ đỉnh của hình nón này. Sau đó những tia chất lỏng bị phân tách
thành những vi hạt mang điện tích. Trong những chế độ hoạt động của cơng nghệ
này, chế độ đơn tia có khả năng ứng dụng cao nhất bởi vì tính ổn định, khả năng
điều chỉnh cao và phun hiệu quả của nó.
Trong luận văn này, tác giả phát triển một công cụ mô phỏng kết hợp tĩnh điện
và cơ lưu chất dựa trên mơ hình Taylor-Melcher leaky-dielectric để nghiên cứu chế
độ đơn tia. Bộ giải các phương trình tĩnh điện được phát triển thêm, kết hợp với bộ
giải interFoam có sẵn của OpenFOAM để mơ phỏng dịng chuyển động đa pha,
khơng nén, và phụ thuộc vào thời gian. Độ tin cây của công cụ mô phỏng này được
minh chứng bằng cách tái tạo các hiện tượng vật lý cơ bản và so sánh với các nghiên
cứu trước, cũng như là với thí nghiệm có tính tới sự ảnh hưởng của góc tiếp xúc của
lưu chất. Tiếp đó, sự ảnh hưởng của các yếu như điện dẫn, điện áp, sức căng bề mặt,
lưu lượng, độ nhớt lưu chất lên dịng điện phun và đường kính tia phun cũng được
phân tích. Kết quả mơ phỏng cho thấy sự đồng nhất hợp lý với các dữ liệu so sánh.
Thêm vào đó, phân tích các góc tiếp xúc lưu chất khác nhau thể hiện sự ảnh hưởng
lớn của yếu tố này trong trường hợp điện áp cao và lưu lượng thấp. Cuối cùng, công
cụ mô phỏng được cải tiến để xem xét q trình tạo thành của nón Taylor dưới sự
ảnh hưởng của hiện tượng phóng điện. Các kết quả mô phỏng cùng với dữ liệu thực
nghiệm cho thấy sự cải tiến rõ rệt trong việc dự đoán quá trình tạo thành của nón
Taylor, từ đó thể hiện vai trị tiềm năng của của tính dính ướt của lưu chất trong mơ

phỏng sự hình thành của nón Taylor dưới ảnh hưởng của hiện tượng phóng điện.
Từ khóa: ống mao dẫn, phương pháp số động lực học lưu chất, đơn tia, hiện tượng
phóng điện, sự dính ướt lưu chất, OpenFOAM, nón Taylor, …
iv


THE COMMITMENT
I hereby commit that:
-

This master thesis outline is done by me with guidance from Assoc. Prof.
Ngo Khanh Hieu, Dr. Dau Thanh Van, Dr. Dinh Xuan Thien, and with the
support of Dr. Canh-Dung Tran, Mr. Vu Trung Hieu and Mr. Vu Hoai Duc.

-

Design of the experiment apparatus and supporting experiments are carried
out by the research team at the School of Engineering and Built
Environment, Griffith University, Australia led by Dr. Dau Thanh Van. The
author’s contributions include program development, all disclosed numerical
simulations, data curation and visualization, and scientific discussions
presented in this thesis.

-

The data, numbers, results in this work except for specialized experiments
are done by me at Ho Chi Minh City University of Technology. Any
publication or article reusing the content of this work is dominantly authored
by me and explicitly declared in the “Publications” section of this thesis.


-

All of the references used in this work are cited fully and clearly in
information: name of the author(s), title, date of publication, place of
publication with highest precision in my knowledge.
Author,

Mai Ngoc Luan

v


TABLE OF CONTENTS
THESIS ASSIGNMENT .......................................................................................... i
ACKNOWLEDGEMENT....................................................................................... ii
ABSTRACT ........................................................................................................... iii
TÓM TẮT LUẬN VĂN......................................................................................... iv
THE COMMITMENT..............................................................................................v
TABLE OF CONTENTS ....................................................................................... vi
List of Tables .......................................................................................................... ix
List of Figures...........................................................................................................x
Nomenclature ....................................................................................................... xiv
Chapter 1.

Thesis introduction ...........................................................................1

1.1.

Motivation ................................................................................................1


1.2.

Objective(s) of the study ..........................................................................3

1.3.

Investigation subject and scope of study ..................................................4

1.4.

Literature review ......................................................................................6

Chapter 2.
2.1.

Background theories .......................................................................14

Electrostatics...........................................................................................14

2.1.1.

Electric charge and electric field ....................................................14

2.1.2.

Coulomb’s law ................................................................................15

2.1.3.

Gauss’s law .....................................................................................16


2.1.4.

Conservation of charge ...................................................................17

2.1.5.

Electrostatic force density ...............................................................18

2.2.

Computational Fluid Dynamics..............................................................18

2.2.1.

Navier-Stokes equations .................................................................18
vi


2.2.2.

OpenFOAM ....................................................................................19

2.2.3.

InterFOAM solver...........................................................................20

2.2.3.1.

Pressure-velocity coupling .........................................................20


2.2.3.2.

Volume of Fluid interface tracking method ...............................25

2.2.3.3.

Contact angle correction .............................................................28

Chapter 3.
3.1.

Electrohydrodynamic coupling procedure .....................................30

The Taylor-Melcher leaky-dielectric model...........................................30

3.1.1.

Fluidic field .....................................................................................32

3.1.2.

Electrostatic field ............................................................................32

3.1.3.

Corona discharge ............................................................................33

3.2.


Structure and solving process of interElectroFoam ...............................34

Chapter 4.
4.1.

Results and discussion ....................................................................36

Code validation .......................................................................................36

4.1.1.

Physical verification of interElectroFoam ......................................36

4.1.2.

Validation with previous literature .................................................41

4.1.3.

Validation with experiment results .................................................47

4.2.

Dimensionless analyses ..........................................................................53

4.3.

Contact angle effects on Taylor cone .....................................................60

4.4.


Simulation of corona discharge effects in electrospray .........................63

4.4.1.

Corona discharge condition assumptions .......................................64

4.4.2.

Simulation results on Taylor cone formulation ..............................64

Chapter 5.

Conclusion and prospective future research...................................71

Publications ............................................................................................................73
Reference ................................................................................................................74
vii


Appendix A. Experimental apparatus ....................................................................83
Appendix B. Additional experiment results ...........................................................85
Appendix C. Contact angle correction formulation ...............................................87
Appendix D. Additional simulation results ............................................................90
VITA.......................................................................................................................92

viii


List of Tables

Table 4-1. Air and Ethanol properties used in simulations. ...................................37
Table 4-2. Boundary conditions for the computational domain in physical parameters. .........................................................................................................................37
Table 4-3. Convergence criteria of the physical parameters. .................................38
Table 4-4. Description of validation test cases used in Singh [28]. .......................41
Table 4-5. Description of validation test cases used in Huh [38]. .........................44
Table 4-6. Physical properties of PEG-200. ...........................................................49

ix


List of Figures
Figure 1-1. Different forms of the Taylor cone [15]. ...............................................5
Figure 1-2. Captured developing stages of the Taylor cone [16]. ............................5
Figure 1-3. Corona discharge captured in electrospray [18]. ...................................6
Figure 2-1. Illustration of charged particles in space and their electric field [51]. 14
Figure 2-2. Directory structure of an OpenFOAM simulation...............................20
Figure 2-3. Solving procedure of the PISO algorithm. ..........................................23
Figure 2-4. Solving procedure of the SIMPLE algorithm. .....................................24
Figure 3-1. Flowchart of the present interElectroFoam. ........................................34
Figure 4-1. (a) The dimensions of the computational domain and; (b) The mesh description of physical verification simulation. .........................................................36
Figure 4-2. Phase fraction results of Ethanol (a) t = 300 ms,
ms

  4000 V

 0V

and (b) t = 15

. .......................................................................................................38


Figure 4-3. Electric field intensity variation with time of Ethanol electrospray. ...39
Figure 4-4. Additional simulation results of Ethanol electrospray: (a) Charge density accumulation of liquid interface (contour: charge density; black line: liquid
surface), (b) Backflow near the apex of the cone, (c) Vectors of total electrostatic
force acting on the interface at t = 0.3 ms. .............................................................40
Figure 4-5. Comparison of jet diameter variation with flow rate between Singh [28],
our interElectroFoam and scaling law for 2-μm nozzle. ........................................43
Figure 4-6. Comparison of jet diameter variation with flow rate between Singh [28],
our interElectroFoam and scaling law for 10-μm nozzle. ......................................43
Figure 4-7. Phase fraction simulation results in different applied voltage: (a) Huh’s
results [38], (b) interElectroFoam results. ..............................................................44
Figure 4-8. Electric field plotted along axial coordinate y = 8 μm of Huh [38] and
interElectroFoam. ...................................................................................................46
Figure 4-9. Electric field contour surrounding the cone, jet and droplets..............46

x


Figure 4-10. The dimensions of the computational domains for two nozzle configurations (a) Nozzle 1 and (b) Nozzle 2...................................................................47
Figure 4-11. The axisymmetric hybrid mesh model with (i) Feeding nozzle and (ii)
Ring electrode. The inset figure shows a close-up view of the mesh resolution
nearby the nozzle. The nozzle’s sharp edges are filleted at rfil  1.5%o.d. ...............48
Figure 4-12. The Taylor cone’s shape produced by different grid resolutions. .....49
Figure 4-13. (a) Experimental measurement of contact angle of PEG-200 on stainless steel plate and simulation of static droplet on wall boundary with different
contact angle conditions; (iii) is overlapped with (i) to demonstrate appearance
conformity. (b) Simulation shows different fluid propagation schemes due to
different contact angles on nozzle; (vi) No angle represents zero-gradient condition
 / n  0. In both (a) and (b), the gravitational acceleration vector is downward. 50

Figure 4-14. (a) Comparison between Taylor cone’s shape of experiment and simulation of N1 nozzle configuration; (b) Images of the (i) Taylor cone in experiment;

and (ii) 3-D rendered phase fraction from simulation. Cone length lc is the distance
from nozzle tip (L = 0 mm) to the intersection point of the black lines; cone angle
c is approximated by the angle formed by the black lines. ..................................51

Figure 4-15. Comparison between Taylor cone’s shape of experiment and simulation of N2 nozzle configuration; (b) Images of the Taylor cone in experiment
using e  5400 V ; and (b) 3-D rendered phase fraction from simulation using
s  6500 V. Comparison only demonstrates matching Taylor cone’s shape under

further consideration of liquid wetting. ..................................................................51
Figure 4-16. Depiction of parameters used in jet diameter interpolation scheme. 53
Figure 4-17. The variation of spray current and jet diameter with electrical conductivity. ........................................................................................................................54
Figure 4-18. Characteristics of electrospray: Spray current and jet diameter versus
applied voltage........................................................................................................55

xi


Figure 4-19. Characteristics of electrospray: Spray current and jet diameter versus
surface tension. .......................................................................................................55
Figure 4-20. Characteristics of electrospray: Spray current and jet diameter versus
electric capillary number estimated from applied voltage. ....................................56
Figure 4-21. Characteristics of electrospray: Spray current and jet diameter versus
electric capillary number estimated from the surface tension. ...............................57
Figure 4-22. Characteristics of electrospray: Spray current and jet diameter versus
flow rate. .................................................................................................................58
Figure 4-23. Characteristics of electrospray: Spray current and jet diameter versus
fluid viscosity. ........................................................................................................58
Figure 4-24. Fluidic velocity field and surficial charge density for different viscosities (a) μ=0.01 Pa.s; (b) μ=0.02 Pa.s; (c) μ=0.06 Pa.s. Vector fields show the
recirculating motion (generalized by red arrows); background contours show
surficial charge density. ..........................................................................................59

Figure 4-25. Taylor cone shape under different contact angles for (a) N1 configuration; and (b) N2 configuration with the attachment positions and the actual angles
between the liquid’s surface and the wall of the nozzle’s tip  annotated............60
a

Figure 4-26. Spray current and jet diameter versus contact angle for N1 and N2 configuration under original   3400V; Q  0.75 ml/h. ...................................................61
Figure 4-27. Spray current and jet diameter versus contact angle for N1 configuration with   3600V; Q  0.6 ml/h, respectively. .........................................................62
Figure 4-28. Taylor cone’s shape variation with contact angle with   3600 V for
N1 configuration. ....................................................................................................62
Figure 4-29. Taylor cone’s shape variation with contact angle with Q  0.6 ml/h for
N1 configuration. ....................................................................................................63
Figure 4-30. Annotation of corona discharge origin in the computational mesh domain. Inset illustrates the behavior of electric field and resulted corona discharge
under presented assumptions. The boundary value of corona discharge condition is
calculated by Eq. (3.19). .........................................................................................64
xii


Figure 4-31. High-frequency captured Taylor cone’s shape at six consecutive timesteps [t1 - t6] from (a) Simulation neglecting corona discharge; and (b) Simulation
involving corona discharge (   6700 V ), and (c) experiment. Electric field intensity
is rendered as background contour. Three stages of liquid progression annotated (i)
Propagating liquid: liquid is advancing from the inner edge to the outer edge of the
nozzle; (ii) Edges covered: liquid reached the outer edge and obstructs the corona
discharge; (iii) Jet forming: increased electric field is inducing jet at the tip of the
Taylor cone. ............................................................................................................65
Figure 4-32. Taylor cone’s shape, ionic wind velocity contour and charge cloud
from corona at t2 and t5. Background contours illustrate (a) Ionic wind velocity; and
(b) Charge density from corona, vector field represents ionic wind field, streamline
denotes electric field. Electrodes annotated (i) Nozzle, (ii) Ring electrode. Inset
shows a glowing region from nearby the outer edge of dry nozzle in high voltage,
which is an indicator of strong electric field. .........................................................67
Figure 4-33. Ionic wind velocity and maximum electric field intensity at the nozzle’s tip variation with discharge current with time to first jet induction tj annotated.

................................................................................................................................68
Figure 4-34. The illustration of charge cloud from corona for different discharge
currents. The boundary of the charge cloud is determined by a selected value of
 c  0.75 mC/m 3 in all cases for comparative purpose. ...........................................68

xiii


Nomenclature
Regular letters:
q
charge
electric flux density
D
Qencl
enclosed charge

C
C/m2
C

viscous stress per unit volume
body force per unit volume

N/m3
N/m3

face flux
discretized coefficients of u
discretized coefficients matrix


m3/s

S
A
pd

face area normal vector
area of discharge surface
new static pressure, p  g  x

m2
m2
Pa

x

position vector of a control volume
dimensionless electric capillary number

T

f fp , fb
F
a P, a N

H(u)

Ca e
dj


jet diameter

m

E
E

electric field,   
electric field intensity

V/m
V/m

Eon

onset electric field

V/m

Ez

V/m

fe

electric field component in the perpendicular
direction to the outlet
electrostatic force,


N/m3

f

surface tension force

N/m3

g
lref

gravitational acceleration
characteristic length

m/s2
m

Id

discharge current

A

i.d
I cond

inner diameter
conductive current

mm

A

I conv

convective current

A

i.dr
J
Je

ring inner diameter
current density
corona current density

n0

uncorrected interface normal vector

mm
A/m2
A/m2

xiv


nc

corrected interface normal vector


nw

boundary wall normal vector

o.d

outer diameter
pressure
liquid flow rate
Reynold number
interelectrode distance

p

Q
Re

re
d fil

S
t
u
ur

mm
Pa
ml/h; m3/s
mm


nozzle sharp edges fillet diameter

mm

outlet area
time
fluid velocity
artificial compression term

m2
s
m/s

lc

liquid velocity component in the perpendicular m/s
direction to the outlet
cone length
mm

vi

ionic wind velocity

tj

time to first jet induction

Uz


m/s
s

Greek letters:


p





0
r


pressure relaxation factor
phase fraction of liquid
permittivity
permittivity of free space

F/m
F/m

dielectric constant
m2/s
Pa.s



e

fluid kinematic viscosity
fluid viscosity
contact angle
angle between uncorrected interface normal vector
and wall’s normal vector
mean curvature of free surface
electrical conductivity of fluid

e

mobility of charge of gas

m2/Vs



c

fluid density
corona volumetric charge

kg/m3
C/m3

e

volumetric charge density


C/m3





0

xv

m-1
S/m




surface tension
electric potential

c

cone angle

a
e

actual contact angle between liquid’s interface and
nozzle’s wall
magnetic permittivity


e

electric characteristic time, o / e

m

magnetic characteristic time, e 0e elref2



N/m
V

M

electrostatic Maxwell stress tensor
Abbreviations:
AC
alternating current
CF
carbon fibre
CSF
continuum surface force
EHDA
electrohydrodynamic atomization
N1
nozzle 1
N2
nozzle 2
VOF

volume of fluid
Subscripts:
e
experiment
s
simulation
l
liquid
g
gas
σ

xvi

H/m

N/m2


Chapter 1. Thesis introduction
This chapter briefly presents the rationale, aim(s), object and range of study of
this thesis and the review of literature related to the problem of interest.

1.1. Motivation
Electrohydrodynamic Atomization (EHDA), or electrospray operates on the
principles of electrohydrodynamic which deals with the motion of fluids placed
inside an electrical field. Fluid surface can be controllably deformed, elongated,
finally broken up and dispersed into much smaller droplets compared to the size
feeding tube (nozzle’s diameter in the case of electrospray) which is useful for a
wide range of applications. First recorded by William Gilbert [1], the phenomenon

together with its underlying science have been studied intensively and developed
into technologies that enrich people’s life in different ways. Along with the
advances of micro-/nanotechnologies, electrospray, due to its great potential, has
been found to be useful in chemical, biological, pharmaceutical, internal
combustion, propulsion, and manufacturing applications. In particular
 For chemical/biological/pharmaceutical applications:
o Mass spectrometry: technique to measure mass-to-charge ratio of
ions, producing results called mass spectra which are used to
determine the elemental or isotopic signature of a sample, the masses
of particles and of molecules and to illuminate the chemical identity
or structure of molecules and other chemical compounds [2].
o Food industry: electrospray is used to encapsulate bioactive
compounds, enhance aromatic properties, manufacture smart
packaging, edible films, and coatings [3-5].
o Inhalable drug/antigen delivery: micro-/nanoparticles produced by
electrospray provide better efficacy-to-safety ratio compared to other
methods. EHDA has been used in delivering treatments for bronchial

1


asthma, lung sicknesses and cancer, influenza virus, etc. [6].
Electrospray can produce drug-loaded particles with a core-shell
structure to improve drug protection and drug release accuracy [7].
Additionally, ionic wind cooperated with electrospray device is
employed to reduce the cumulative charge in the particles and
transport them to a target in front of the nozzle [8, 9].
 Internal combustion applications: electrospray atomizes fuels to generate
sprays, monodisperse droplets with micro/nanoscale diameters to enhance
combustion stability and emissions [10, 11].

 Propulsion applications: colloid electrospray thrusters produce thrust
through electrostatic acceleration of charged liquid droplets. First successful
orbit mission of electrospray thruster was the LISA Pathfinder spacecraft
[12].
 Manufacturing applications:
o Electrohydrodynamic jet printing (e-jet printing): the employment of
e-jet printing can produce sub-micrometer resolution for patterning or
to fabricate devices in electronics or other areas of technology by use
of functional or sacrificial inks [13].
o Powder technology: multiple electrosprays system enabled more
energy efficient controlled production of particles. Produced powder
particles can be used as components for the fabrication of other
materials, for example, paints, emulsions, as ingredients used in food,
cosmetics or surface coating [14].
All of the mentioned applications are based on the single cone-jet spraying
regime of electrospray due to its ability to generate smaller particles or structures,
higher controllability, stability and yield rate in comparison to different methods.
Commonly recorded spraying modes of electrospray include dripping, cone-jet
(single cone-jet and multiple cone-jet), microdripping, simple jet and ramified jet
and spindle [15]. As single cone-jet is the most desirable spraying regime of
2


electrospray, hereinafter it will be the sole research subject and briefly referred to
as cone-jet or electrospray or EHDA.
The popularity of EHDA, or specifically the single cone-jet mode, arises the
need for developing numerical modelling to reproduce, predict, characterize
prominent properties as well as optimize the operational characteristics of this
electrohydrodynamic phenomenon. The ultimate goal of this development is to
produce a numerical method that is able to satisfy industrial demands and replace

intricate experiments which often involve complex apparatus construction and
consume a large amount of time. Furthermore, from an academic perspective, if
computational simulations are proven physically accurate, they can also provide
vivid explanations for empirical manifestations. This could potentially help
researchers to achieve a deeper insight into the discovered phenomenon, which
would play as the catalyst for new ideas as well as the precursor for novel
advancements.

1.2. Objective(s) of the study
Understanding the aforementioned challenges, this study introduces the
development of a numerical method coupling electrostatic system and fluid
dynamic system to simulate the electrospray’s single cone-jet mode. This newly
programmed solver will then be verified and validated by contrasting with
established scaling laws, preceding literatures and experiments provided by the
research team from Griffith University, Australia (GU). The detailed procedure of
this study is listed as follow.
 Firstly, we develop a solver based on the open-source software OpenFOAM,
in which the newly implemented electrostatic solution is combined with the
solver interFoam already integrated inside the OpenFOAM package. The
resulting solver is referred to as interElectroFoam.
 Secondly, series of test cases are conducted to verify the physical conformity
of the electrohydrodynamic solver by comparing the simulated phenomena
with some well-known observations reported in existing literatures. In this
3


step, we will also quantitatively contrast the achieved results with some
selected publications to consolidate the reliability of the interElectroFoam
solver.
 Thirdly, we run simulations whose settings (injecting fluids’ properties,

device’s sizes, etc.) are consistent with GU’s results and then visually
compares the two sets of data to further validate the solution.
Simultaneously, we carry out numerical investigations on the impacts of
different conditions such as voltage, flow rate, fluid properties to the
operation of electrospray and contrast the outcome with similarly reported
results from literature. Moreover, brief analyses on the impacts of contact
angle on the characteristics of the cone-jet mode is presented.
 Fourthly, as the preceding steps are completed, we continue to enhance the
solver to analyze the corona discharge and ionic wind generation and their
qualitative correlation with electrospray. The definition of corona discharge
and ionic will be presented briefly in the next section.
Overall, this thesis aims to develop a simplified OpenFOAM-based solver to
simulate the electrospray’s single cone-jet mode with reasonably acceptable
accuracy. This solver does not consider intensive break-up model for ejecting
droplets, thermophysical effects and any evaporation model. Validating test cases
and further analyses including the impact of corona discharge are performed in
simplified axisymmetric 2-D simulations, which will be discussed in detail later in
this thesis.

1.3. Investigation subject and scope of study
Electrospray ionization’s cone-jet mode is one of the most fascinating
functioning modes in which the meniscus takes the shape of a cone whose apex
ejects a jet giving rise to droplets when broken up [15]. Taylor [16] thoroughly
described the formation of a liquid meniscus under the influence of applied electric
potential stage by stage. Firstly, the liquid surface is warped into a cone-like shape
owing to the change in equilibrium of surficial forces, then a jet is thrown out of the
4


apex’s center point and disintegrated into extremely small droplets. Finally, the

conical shape and the jet collapse. He concluded in this study that the interfacial
forces are in equilibrium condition if the semi-vertical angle of the conical shape
stays at approximately 49.3̊. The conical shape and its behaviors described by
Taylor is later commonly known as the Taylor cone. Different forms of the Taylor
cone and the captured development of the Taylor cone reported by Taylor are
presented in Fig. 1-1 and Fig. 1-2, respectively.

Figure 1-1. Different forms of the Taylor cone [15].

Figure 1-2. Captured developing stages of the Taylor cone [16].

Stable cone-jet mode requires a specific range of applied voltage and input flow
rate which vary with electrode’s configurations, interelectrode distance and the
liquid’s properties. If the mentioned conditions are not satisfied, the cone-jet mode
would be unstable or replaced by other modes, such as spindle or multi-jet, whose
characteristics might be thoroughly different. In this thesis, we only concentrate on
investigating the stabilized cone-jet mode because of its popularity and practicality.
In this thesis, we also aim to further develop a numerical simulation that takes
into account the impact of the corona discharge effects in electrospray. In
5


electrospray, when high electric potential is applied, the air in the vicinity of the
nozzle can be ionized, resulting in various forms of electrical discharge from either
the nozzle tip or the liquid surface. Under the influence of electric field in the
interelectrode region and surrounding ionized gas, a stream of charge, corona
discharge, can be created flowing down the potential gradient. This movement of
ions transfers momentum into surrounding neutral molecules, giving rise to ionic
wind, around the nozzle which can affect electrospray processes both positively and
negatively [17]. Corona discharge or ionic wind can be considered as an entirely

different research field due to its complexity and broad scope of applications. In this
work, we only aim to qualitatively investigate the ionic wind phenomenon in
electrospray by simplified numerical method. We expect the outcome of our work
would adequately represent characteristics of ionic wind in electrospray as observed
in experiments, and therefore, provide a reliable tool for future quantitative
research. An illustration of corona discharge in electrospray is provided in Fig. 1-3.
This figure is taken from Pongrác et al. [18] where the effect of water’s conductivity
and DC corona discharge on some electrospray mode has been investigated. In this
case, the intense discharge indicated by purple glow can be seen originating from
the sharp edge of the nozzle.

Figure 1-3. Corona discharge captured in electrospray [18].

1.4. Literature review
Electrospray’s scope of applications is large, leading to an accordingly large
collection of research related to this problem. Prominent early works are the report
of Rayleigh [19], Zeleny [20], Taylor and Melcher [16, 21] describing very

6


fundamental aspects of this electrohydrodynamic phenomenon both experimentally
and theoretically. Taylor’s work [16] was so pioneering that the electrospray’s
common conical shape was named after him. Shortly after this publication, Taylor
and Melcher [21] introduced a mathematic model, named leaky-dielectric, to
explain the interrelationship between the physical aspects involved and approximate
the force exerted on the surface of the liquid. In its most elementary form, the leakydielectric model comprises Stokes equations to account for fluid motion, Gauss’s
Law, an expression of current conservation considering Ohmic conductivity with
magnetic effects neglected, and expressions for calculating the Maxwell’s stress
tensor on the fluid-fluid interface from the Coulombic force and the polarization

force [1, 21]. This approach has been utilized in various succeeding analyses to
numerically represent the Taylor cone and its behaviors. Researchers may use
different methods to solve the related electrohydrodynamic equations, yet the basic
principles remain unchanged.
Among early researches, Lastow and Balachandran superseded the heat transfer
equation by electrostatic system in the commercial Ansys’ CFX module to develop
a numerical tool to simulate the liquid cone formation and atomization [22] and
later used the program to investigate a double-layer spray nozzle used for atomizing
water and weak saline solutions in the low-voltage stable cone jet mode [23]. Both
works compared droplets’ diameter variation with flow rate of CFD result,
experiment data and also scaling law and reported good agreement. The numerical
model in these works did not include a breakup model, so the authors can only
estimate the droplets’ diameter from jets’ size. Similarly, Sen et al. [24] integrated
the leaky-dielectric model into Flow3D package to investigate the characteristics of
a novel carbon fiber emitter (CF) electrospray for mass spectrometric (MS) studies.
In this research, the authors calculated jets’ diameter and electric current and
contrasted their results with experiments and existing literatures. Excellent
agreement with both empirical data and published works has not only proven their
program’s reliability but also the potential of their CF electrospray. The same idea

7